Composite reinforcements for optimum performance
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Related titles: Non-crimp fabric composites (ISBN 978-1-84569-762-4) Non-crimp fabric (NCF) composites are composites that are reinforced with woven mats of straight (non-crimped) fibres. Straight fibres deform much less under tension. NCF composites are being used in applications in the aerospace, automotive, civil engineering and wind turbine sector where strength is important. Non-crimp fabric composites reviews production, properties and applications of this important class of composites. Interface engineering of natural fibre composites for maximum performance (ISBN 978-1-84569-742-6) There is a growing trend towards the use of natural (sustainable) fibres as reinforcements in composites. One of the major mechanisms of failure in such composites is the breakdown of the bond or interface between the reinforcement fibres and the matrix. When this happens, the composite loses strength and fails. By engineering the interface between fibres and the matrix, the properties of the composite can be manipulated to give maximum performance. This book reviews both processing and surface treatments to improve interfacial adhesion in natural fibre composites as well as ways of testing their resulting properties. Polymer carbon nanotube composites (ISBN 978-1-84569-761-7) Polymer carbon nanotube composites reviews the use of carbon nanotubes as reinforcements in a polymer matrix, creating a new class of nanocomposites with useful properties and wide potential applications. Part I discusses preparation and processing techniques. Part II analyses key properties and ways of characterising polymer carbon nanotube composites. The final part of the book covers some of the important applications of this new group of materials. Details of these and other Woodhead Publishing materials books can be obtained by: ∑ visiting our web site at www.woodheadpublishing.com ∑ contacting Customer Services (e-mail:
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Composite reinforcements for optimum performance Edited by Philippe Boisse
Oxford
Cambridge
Philadelphia
New Delhi
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Published by Woodhead Publishing Limited, 80 High Street, Sawston, Cambridge CB22 3HJ, UK www.woodheadpublishing.com Woodhead Publishing, 1518 Walnut Street, Suite 1100, Philadelphia, PA 19102-3406, USA Woodhead Publishing India Private Limited, G-2, Vardaan House, 7/28 Ansari Road, Daryaganj, New Delhi – 110002, India www.woodheadpublishingindia.com First published 2011, Woodhead Publishing Limited © Woodhead Publishing Limited, 2011 The authors have asserted their moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials. Neither the authors nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Control Number: 2011935449 ISBN 978-1-84569-965-9 (print) ISBN 978-0-85709-371-4 (online) The publisher’s policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp which is processed using acid-free and elemental chlorine-free practices. Furthermore, the publisher ensures that the text paper and cover board used have met acceptable environmental accreditation standards. Typeset by Replika Press Pvt Ltd, India Printed by TJI Digital, Padstow, Cornwall, UK
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Contents
Contributor contact details
Part I Materials for reinforcements in composites
xiii 1
1
Fibres for composite reinforcement: properties and microstructures
A. R. Bunsell, Mines ParisTech, France
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10
Introduction Fineness, units, flexibility and strength Comparison of materials Organic fibres Glass fibres Chemical vapour deposition (CVD) monofilaments Carbon fibres Small-diameter ceramic fibres Conclusions References
3 4 6 8 17 19 22 27 30 30
2
Carbon nanotube reinforcements for composites
32
A. W. K. Ma, University of Connecticut, USA and F. Chinesta, Ecole Centrale de Nantes, France
2.1 2.2 2.3 2.4
Carbon nanotubes (CNTs) Carbon nanotube (CNT) polymer composites Performance and applications References
32 36 46 47
3
Ceramic reinforcements for composites
51
J. Lamon, CNRS/National Institute of Applied Science (INSA Lyon), France
3.1 3.2
Introduction Ceramic fibers: general features
3
51 53
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Contents
3.3 3.4 3.5
Fracture strength: statistical features Mechanical behavior at high temperatures Fiber–matrix interfaces: influence on mechanical behavior Mechanical behavior of composites: influence of fibers and interfaces Conclusion References
3.6 3.7 3.8
57 64 73 74 81 82
Part II Structures for reinforcements in composites
87
4
Woven reinforcements for composites
89
D. Coupé, Snecma Propulsion Solide, France
4.1 4.2 4.3 4.4 4.5 4.6 4.7
Introduction: from the beginning of wearing to technical applications Technology description Woven fabric definitions Applications for composite reinforcements Conclusion and future trends Acknowledgment Sources of further information and advice
89 90 100 113 114 115 115
5
Braided reinforcements for composites
116
A. Gessler, EADS Deutschland GmbH, Germany
5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12
Introduction Fundamentals of braiding Braiding technologies for preforming Key parameters for using braiding machines Characteristics and properties of braided textiles Mandrel technologies Further processing Typical applications Limitations and drawbacks Future trends Sources of further information and advice Reference
116 116 120 134 136 144 149 151 154 155 155 156
6
Three-dimensional (3D) fibre reinforcements for composites
157
A. P. Mouritz, RMIT University, Australia
6.1 6.2
Introduction Manufacture of three-dimensional (3D) fibre composites
157 160
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Contents
6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11
Microstructure of three-dimensional (3D) fibre composites Delamination fracture of three-dimensional (3D) fibre composites Impact damage resistance and tolerance of three-dimensional (3D) fibre composites Through-thickness stiffness and strength of three-dimensional (3D) fibre composites Through-thickness thermal properties of three-dimensional (3D) fibre composites In-plane mechanical properties of three-dimensional (3D) fibre composites Joint properties of three-dimensional (3D) fibre composites Conclusions References
7
Modelling the geometry of textile reinforcements for composites: WiseTex
S. V. Lomov, Katholieke Universiteit Leuven, Belgium
7.1 7.2
Introduction Generic data structure for description of internal geometry of textile reinforcement Geometrical description of specific types of reinforcements Geometrical model as a pre-processor for prediction of mechanical properties of the reinforcement Conclusion References
7.3 7.4 7.5 7.6 8
Modelling the geometry of textile reinforcements for composites: TexGen
A. C. Long and L. P. Brown, University of Nottingham, UK
8.1 8.2 8.3 8.4 8.5 8.6 8.7
Introduction: rationale and background to TexGen Implementation Modelling theory Rendering and export of model Applications Future trends References
vii
168 175 183 184 186 187 195 196 197 200 200 202 204 225 232 235 239 239 240 246 254 257 262 263
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viii
Contents
Part III Properties of composite reinforcements
265
9
In-plane shear properties of woven fabric reinforced composites
J. Cao, Northwestern University, USA, J. Chen, University of Massachusetts at Lowell, USA and X. Q. Peng, Shanghai Jiao Tong University, P. R. China
9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9
Introduction Fabric properties Experimental setups of the trellis-frame test Experimental results of the trellis-frame test Experimental setups of the bias extension test Experimental results of the bias extension test Conclusions Acknowledgments References
267 270 270 280 286 294 298 301 301
10
Biaxial tensile properties of reinforcements in composites
306
V. Carvelli, Politecnico di Milano, Italy
10.1 10.2 10.3 10.4 10.5 10.6
Introduction Experimental analysis Analytical model Numerical modelling Conclusions References
306 308 318 324 328 329
11
Transverse compression properties of composite reinforcements
333
P. A. Kelly, The University of Auckland, New Zealand
11.1 11.2 11.3 11.4 11.5 11.6
Introduction Transverse compression of composite reinforcements Inelastic response of fibrous materials Inelastic models of reinforcement compression Future trends References and further reading
333 334 344 351 359 360 367
12
Bending properties of reinforcements in composites
E. de Bilbao, Institut Universitaire de Technologie d’Orléans, France
12.1 12.2 12.3
Context Improved cantilever test Results and discussion
267
367 375 383
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Contents
12.4 12.5 12.6
Conclusions Acknowledgement References
392 393 393 397
13
Friction properties of reinforcements in composites
J. L. Gorczyca, K. A. Fetfatsidis and J. A. Sherwood, University of Massachusetts at Lowell, USA
13.1 13.2 13.3
Introduction Theory Testing methodologies (static and dynamic friction coefficients) Experimental data Modeling of thermostamping Conclusion References
13.4 13.5 13.6 13.7 14
Permeability properties of reinforcements in composites
V. Michaud, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland
14.1 14.2 14.3 14.4 14.5 14.6 14.7
Introduction The permeability tensor Saturated permeability modelling for fibre preforms Unsaturated permeability modelling Permeability measurement methods Conclusion and future trends References and further reading
Part IV Characterising and modelling reinforcements in composites 15
Microscopic approaches for understanding the mechanical behaviour of reinforcements in composites
D. Durville, Ecole Centrale Paris/CNRS UMR 8579, France
15.1 15.2 15.3
Introduction Interests and goals of the approach at microscopic scale Modelling approach to textile composites at microscopic scale Application examples Conclusions References
15.4 15.5 15.6
ix
397 401 411 418 422 428 428 431
431 432 436 444 445 451 451
459
461 461 463 465 477 482 485
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x
Contents
16
Mesoscopic approaches for understanding the mechanical behaviour of reinforcements in composites
E. Vidal-Sallé, INSA Lyon, France and G. Hivet, Polytech Orléans, France
16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8
Introduction Mechanical behaviour of the reinforcement Mechanical behaviour of the yarn Geometric modelling Behaviour identification and finite element modelling Finite element simulations, use and results Conclusions and future trends References
486 488 498 503 511 519 522 523
17
Continuous models for analyzing the mechanical behavior of reinforcements in composites
529
X. Q. Peng, Shanghai Jiao Tong University, P. R.China and J. Cao, Northwestern University, USA
17.1 17.2 17.3 17.4 17.5 17.6 17.7
Introduction Continuum mechanics-based non-orthogonal model Non-orthogonal constitutive model for woven fabrics Specific application for a plain weave composite fabric Validation of the non-orthogonal model General fiber-reinforced hyperelastic model Specific fiber-reinforced hyperelastic model for woven composite fabrics 17.8 Conclusions 17.9 Acknowledgment 17.10 References 18
X-ray tomography analysis of the mechanical behaviour of reinforcements in composites
P. Badel, Ecole des Mines de Saint-Etienne, France and E. Maire, INSA-Lyon, France
18.1 18.2 18.3 18.4
Introduction X-ray tomography of composite reinforcements Analyses of the structure of a textile reinforcement Application of the mechanical behaviour of woven reinforcements to finite element simulations Conclusion References
18.5 18.6
486
529 533 535 537 552 556 558 562 563 563 565
565 566 571 578 585 586
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xi
19
Flow modeling in composite reinforcements
E. Ruiz and F. Trochu, École Polytechnique de l’Université de Montréal, Canada
588
19.1 19.2 19.3 19.4 19.5 19.6 19.7
Introduction Governing flow equations Analytical solution Numerical solution Application examples Conclusions References
588 589 591 594 603 613 613
20
Modelling short fibre polymer reinforcements for composites
616
P. Laure, Université de Nice-Sophia Antipolis, France and L. Silva and M. Vincent, Mines ParisTech, France
20.1 20.2 20.3 20.4 20.5 20.6
Introduction Observations Models Computation of fibre orientation in injection moulding Conclusions References and further reading
616 617 622 631 645 647
21
Modelling composite reinforcement forming processes
651
P. Boisse and N. Hamila, Université de Lyon, France
21.1 21.2 21.3 21.4 21.5 21.6 21.7
Introduction A mesoscopic approach Continuous approaches The semi-discrete approach Discussion and conclusion Acknowledgements References
651 653 656 661 668 668 669
Index
672
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Contributor contact details
(* = main contact)
Editor
Chapter 2
P. Boisse Université de Lyon LaMCoS Insa Lyon France
A. W. K. Ma* Institute of Materials Science/ Chemicals Materials and Biomolecular Engineering University of Connecticut 97 North Eagleville Road Storrs, CT 06269–3136 USA
E-mail:
[email protected]
Chapter 1
E-mail:
[email protected]
A. R. Bunsell Mines ParisTech (formerly Ecole des Mines de Paris) Centre des Matériaux BP 87 91003 Evry Cedex France E-mail:
[email protected]
F. Chinesta UMR CNRS Ecole Centrale de Nantes 1 rue de la Noe BP 92101 44321 Nantes Cedex 3 France
Chapter 3 J. Lamon CNRS/National Institute of Applied Science (INSA Lyon) MATEIS Laboratory 21 Avenue Jean Capelle 69621 Villeurbanne Cedex France E-mail:
[email protected]
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xiv
Contributor contact details
Chapter 4
Chapter 7
D. Coupé Safran Group Snecma Propulsion Solide France
S. V. Lomov Department MTM Katholieke Universiteit Leuven Kasteelpark Arenberg, 44 B-3001 Leuven Belgium
AEC 112 Airport Drive Rochester, NH 03867 USA E-mail:
[email protected]
Chapter 5 A. Gessler EADS Deutschland GmbH Innovation Works 81663 München Germany E-mail:
[email protected]
Chapter 6 A. P. Mouritz School of Aerospace, Mechanical and Manufacturing Engineering RMIT University GPO Box 2476 Melbourne Victoria Australia 3000 E-mail:
[email protected]
E-mail:
[email protected]
Chapter 8 A. C. Long* and L. P. Brown Faculty of Engineering University of Nottingham University Park Nottingham NG7 2RD UK E-mail: Andrew.Long@nottingham. ac.uk Louise.Brown@nottingham. ac.uk
Chapter 9 J. Cao* Department of Civil and Environmental Engineering Northwestern University 2145 Sheridan Rd Evanston, IL 60208-3111 USA E-mail:
[email protected]
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Contributor contact details
J. Chen Advanced Composite Materials and Textile Research Laboratory Department of Mechanical Engineering University of Massachusetts at Lowell One University Avenue Lowell, MA 01854 USA
Chapter 11
E-mail:
[email protected]
E-mail:
[email protected]
X. Q. Peng Department of Plasticity Shanghai Jiao Tong University 1954 Huashan Road Shanghai P. R. China 200030
Chapter 12
E-mail:
[email protected]
Chapter 10 V. Carvelli Department of Structural Engineering Politecnico di Milano Piazza Leonardo da Vinci 32 20133 Milano Italy E-mail:
[email protected]
xv
P. A. Kelly Department of Engineering Science Faculty of Engineering University of Auckland Private Bag 92019 Auckland Mail Centre Auckland 1142 New Zealand
E. de Bilbao Institut Universitaire de Technologie d’Orléans Département Génie Mécanique et Productique 16 rue d’Issoudun BP 16729 45067 Orléans Cedex 2 France E-mail:
[email protected]
Chapter 13 J. L. Gorczyca, K. A. Fetfatsidis and J. A. Sherwood* Department of Mechanical Engineering University of Massachusetts at Lowell One University Avenue Lowell, MA 01854 USA E-mail:
[email protected] [email protected] Konstantine_Fetfatsidis@ student.uml.edu
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xvi
Contributor contact details
Chapter 14
Chapter 17
V. Michaud Laboratoire de Technologie des Composites et Polymères (LTC) Ecole Polytechnique Fédérale de Lausanne (EPFL) CH 1015 Lausanne Switzerland,
X. Q. Peng* Xiongqi Peng Department of Plasticity Shanghai Jiao Tong University 1954 Huashan Road Shanghai P. R. China 200030
E-mail:
[email protected]
E-mail:
[email protected]
Chapter 15 D. Durville MSSMat Laboratory Ecole Centrale Paris/ CNRS UMR 8579 Grande Voie des Vignes 92290 Châtenay-Malabry France E-mail:
[email protected]
Chapter 16 E. Vidal-Sallé* INSA Lyon LAMCOS – Bâtiment Joseph Jacquard 20 avenue Albert Einstein 69621 Villeurbanne Cedex France E-mail:
[email protected]
G. Hivet Polytech Orléans 8 rue Léonard de Vinci 45072 Orléans Cedex 2 France
J. Cao Department of Civil and Environmental Engineering Northwestern University 2145 Sheridan Rd Evanston, IL 60208-3111 USA E-mail:
[email protected]
Chapter 18 P. Badel * Center for Health Engineering LCG CNRS UMR5146 Ecole des Mines de Saint-Etienne 158, cours Fauriel 42023 Saint-Etienne Cedex 2 France E-mail:
[email protected]
E. Maire Université de Lyon, CNRS INSA-Lyon MATEIS UMR5510 69621 Villeurbanne Cedex France E-mail:
[email protected]
E-mail:
[email protected]
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Contributor contact details
Chapter 19 E. Ruiz* and F. Trochu Research Center on High Performance Composites (CCHP) École Polytechnique de l’Université de Montréal C.P. 6079, succ. Centre-Ville Montréal Québec, H3C 3A7 Canada E-mail:
[email protected]
Chapter 20 P. Laure* Laboratoire J.-A. Dieudonné CNRS UMR 6621 Université de Nice-Sophia Antipolis Parc Valrose 06108 Nice Cedex 02 France
xvii
L. Silva and M. Vincent Cemef CNRS UMR 7635 Mines ParisTech BP 207 06904 Sophia Antipolis Cedex France E-mail:
[email protected] [email protected]
Chapter 21 P. Boisse* and N. Hamila Université de Lyon LaMCoS Insa Lyon France E-mail:
[email protected]
E-mail:
[email protected]
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1
Fibres for composite reinforcement: properties and microstructures
A. R. B u n s e l l, Mines ParisTech, France
Abstract: There are many types of fibres that can be used as reinforcements. Their remarkable properties are exploited in many composite materials and structures, enabling applications which otherwise would be almost impossible. Their long fine forms allow original shapes to be made and their often extraordinary mechanical properties push back the limits of structural materials. Some fibres are familiar, as they are used in textiles, but fibres also find technical applications, such as in reinforcing tyres. Other fibres combine great stiffness and strength with light weight which traditional structural materials cannot rival. These properties depend on the atomic structures of the fibres and these are induced by the way they are made. This chapter relates how the manufacture of the fibres controls their properties and how these are determined by their microstructures. Key words: synthetic fibre reinforcements, manufacture, microstructure, mechanical characteristics.
1.1
Introduction
The form of a fibre explains much of the interest in them as reinforcements. They are fine filaments of matter with diameters often of the order of 10 microns and lengths that can range from centimetres to virtually continuous. Before they were used for composite materials it was their shape, long and fine, that made them ideal for use in textiles (Morton and Hearle, 2008). Fine filaments are extremely flexible and can be woven even if in tension they are very stiff. Many features of fibre-reinforced materials can be seen to be related to their textile uses. Like cloth, most fibre-reinforced composites are two-dimensional, although a woven structure can be seen to have an out-of-plane thickness on the fine scale. Of course there are thick cloths and three-dimensional weaving, but the basic nature of fibre-reinforced composites is two-dimensional. Other materials, such as metals, are made in bulk form and have to be either machined or deformed to be made into sheets, which are expensive operations. A woven cloth can be draped around a complex shape with curvatures in two orthogonal planes because of the way the fibres can move slightly with respect to one another but a metal plate has to be deformed plastically. For example, the bulkhead in an aircraft like the A380 makes use of the drapability of carbon fibre cloth 3 © Woodhead Publishing Limited, 2011
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Composite reinforcements for optimum performance
in its manufacture, so this aspect of fibre characteristics is important in the manufacture of composites. Drapability is only one advantage, however, as matter in the form of a filament often possesses mechanical properties which cannot be achieved with the same material in bulk form. The strength of a body is related to the probability of finding a defect in it, and because of the very small volume per unit length of fibres they are often much stronger than the same material when in bulk form. If the structure of the fibre can be aligned parallel to the fibre axis, the stiffness of the material can also be increased, and some carbon fibres possess Young’s moduli which approach the theoretical maximum that nature can give. Alignment does mean that the structure becomes anisotropic, which can be either a drawback or a useful characteristic, depending on the application. Nature has evolved along the lines of fibre reinforcement because, unlike other, often fully crystalline materials, composite structures can possess a wide range of characteristics which can be optimized to best meet the needs of different parts of a structure. Wood, with its obvious grain, is a clear example of a natural composite, but all of nature, in both the animal and vegetable worlds, is based on structures which have inspired the development of man-made fibre-reinforced composites. Again, the form of the fibre is optimal for reinforcement. The aspect ratio, that is the ratio of the length to the diameter of a fibre, can be very great, and reinforcement theory shows that this long, fine form is ideal for effecting a transfer of loads from the composite matrix to the usually mechanically more robust fibres.
1.2
Fineness, units, flexibility and strength
The small diameters of most fibres present a difficulty not encountered with bulkier materials. Most materials are characterized by their failure stress, which is the force to produce failure divided by the cross-sectional area or stiffness, calculated as the ratio of stress to strain at the beginning of the deformation curve. This was not usually possible with fibres until the 1960s when the scanning electron microscope was developed and fibre cross-sections could be measured accurately. However, measuring the cross-section still remains a difficulty for characterizing fibres. As fibres have diameters of the order of 10 mm and many fibres, particularly natural fibres, do not have regular cross-sections, even the best optical microscopes are of little help. The resolution of a microscope is limited by the wavelength of light, which is around 0.6 mm, so that accurate measurement of the diameter of even a cylindrical fibre presents serious difficulties. For this reason the fibre industry developed units based on weight per unit length. The traditional unit was the ‘denier’, defined as the weight in grams of 9000 metres of the fibre. A typical textile fibre might be 15 denier. More recently the international unit
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Fibres for composite reinforcement
5
for fibres has been given the name ‘tex’ and is defined as one gram per 1000 metres. This means that the tex is less fine as a unit than the denier so that the decitex (dtx) is often used, as being one gram per 10,000 metres, very close to the denier. The strength of a fibre is given as the force to produce failure (gram force, for example) per textile unit (denier or tex). This can be seen to be related to traditional engineering units of strength, as it is equal to the force multiplied by the length and divided by the weight:
Force ¥ length/weight = force ¥ length/(volume ¥ density)
= force ¥ length/(length ¥ cross-section ¥ density) = force/(cross-section ¥ density) As force/cross-section is the engineering definition of stress, it can be seen that strengths given in textile units are related to engineering units through the density of the fibres. To understand the importance of the diameter in determining the flexibility of a fibre, it should be noted that the relationship governing its bending can be shown to be a function of Fl3/ED4, in which F is the applied force to produce bending, l is the fibre length, E is its initial modulus or stiffness and D its diameter (Bunsell, 2009). This relationship shows that the bending of a fibre is controlled by the reciprocal of the diameter to the fourth power. Reducing the diameter to half of its original value makes the fibre 16 times more flexible. This means that materials which are very stiff, that is, have a very high Young’s modulus, can still be flexible if they are made into fibres of small diameter. Strength is not an intrinsic property of a material as it depends on the presence of defects. Usually, the bigger the defect the weaker the material. This observation is described by Weibull statistics which reasons that a material is analogous to a chain, the strength of which depends on the weakest link amongst the n links making up the chain. If the probability of failure of a link, under an applied stress s, is P0, the probability of its survival is 1 – P0. The probability of survival of the chain is therefore
1 – Pn = (1 – P0)n
1.1
Pn = 1 – exp[n ln(1 – P0)]
1.2
or
Let’s consider a body of volume V to be divided into small volumes V0 analogous to links in a chain. In this case n is analogous to V/V0. If we consider that V0 is constant, which is like saying that all the links in a chain would have the same length, the value of ln(1 – P0) depends only on the stress, so that for the whole specimen the risk of failure is (from Eq. 1.1):
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Composite reinforcements for optimum performance
È PV = 1 – exp Í– Î
ÚV
Ê ˆ˘ f Á s , 1 ˜ ˙ dV Ë V0 ¯ ˚
1.3
Weibull put f (s, 1/V0) equal to [(s – su)/s0]m, where s is the applied stress, su is the stress below which there is no probability of failure, and the two material parameters s0 and m are measures of the density and variability of the flaws in the material. It should be noted that s0 does not have the units of stress. In a straight tensile test on fibres the stress s is constant throughout, so that we can write: È Ês – s u ˆ m ˘ PV = 1 – exp Í– Á V˙ Ë s 0 ˜¯ ÎÍ ˚˙
1.4
The parameter m is known as the Weibull shape parameter or simply the Weibull modulus and it quantifies the scatter of the strengths in a population of fibres. The value of m is often low, around 6, similar to that of bulk ceramics, because there is usually a large scatter in strengths in a population of fibres. A material with less scatter would have a higher value for its Weibull modulus. The scatter in fibre strengths is an important characteristic and determines the kinetics of the failure of a fibre bundle and ultimately how many fibrereinforced composites fail. For a more detailed exposition of Weibull statistics see Bunsell and Renard (2005). A consequence of the form of a fibre, which is explained by Weibull statistics, is that fibres are generally much stronger than the same material in bulk form. Imagine a plate of glass with dimensions 100 mm ¥ 100 mm: it could be tested to failure and its failure stress determined. Now if the glass is converted into glass fibres, each fibre would be found to be much stronger than the original plate. Weibull explains that if 100 mm of fibre were tested, it would have a much smaller volume than the plate and therefore there would be a much lower probability of finding a significant defect in it. If all the fibres were assembled next to one another so as to reconstruct a plate and they were embedded in a matrix to make a composite, the new structure would be much stronger than the original glass plate even if the matrix were not very strong. This is because of the intrinsically higher strength of glass in fibre form and also because individual fibre breaks would be isolated by stress transfer to other fibres by shear of the matrix.
1.3
Comparison of materials
Composite materials are structural materials and as such compete with others which are more traditional. The densities and Young’s moduli of some of these materials are compared in Table 1.1. The values have been rounded
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Table 1.1 Typical properties of some bulk materials compared to those of fibres and fibre-reinforced composites Material Specific gravity
Young’s modulus (GPa)
Specific modulus (GPa)
Bulk materials Steel Aluminium Titanium Beryllium Oak Copper Nickel Tungsten Concrete
7.9 2.7 4.5 1.8 0.6–0.9 8.9 8.8 19.6 2.4
200 76 116 289 11 110 21 400 30
25.3 28 25.7 161 12–18 12.4 2.4 20.4 12.5
1.38 1.15 1.4 1.53 1.45 1.56 0.96 2.5 1.8 2.16
15 5 12 65 135 280 117 72 295 830
10.8 3.3 8.5 43 93 180 122 27.6 164 384
11 45 83 143–179 500
12–18 16 58 87–109 270
Fibres Polyester PET fibre Nylon (PA66) fibre Spider silk fibre Flax fibre Kevlar fibre Zsylon fibre Dyneema polyethylene fibre Glass fibre Carbon (high strength – H.S.) fibre Carbon (ultra high modulus – H.M.) fibre Fibre composites Oak wood Glass fibre composites Kevlar fibre composite H.S. carbon fibre composite H.M. carbon fibre composite
0.6–0.9 2.08 1.43 1.64 1.85
off for ease of comparison, and where the materials are not perfectly elastic the initial slope of their stress–strain curves has been taken as their Young’s modulus. Stiffness is often a critical characteristic of a structure and so is its weight, so that the specific modulus, which is Young’s modulus divided by specific gravity, is an important characteristic. The driving force for the development of many composite materials has been the combination of high mechanical properties, particularly stiffness and low density. Table 1.1 shows that the common structural metals steel, aluminium and titanium have very similar specific moduli so that to maintain the same stiffness there is little to be gained by substituting one for another, if weight is important. Beryllium is a metal and the fourth element in the Periodic Table. It is used for some rather exotic applications such as for some structures in satellites but it is toxic, particularly its oxide, which makes it unsuitable for
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most uses. It can be seen that some synthetic fibres possess specific moduli much higher than those of the traditional structural materials. Glass fibres do not have remarkable specific moduli but nevertheless find very wide use in composite materials. This is usually not to gain weight but for low cost, as glass fibres are much cheaper than many other reinforcements, and for ease of manufacture of complex parts, which is another characteristic of composite materials that makes them attractive. Fibres usually possess much higher strengths than the same material in bulk form, for the reasons explained above, and some have remarkable stiffness, usually due to the alignment of their atomic structures. These high-performance fibres can be seen to possess remarkable specific moduli, far exceeding the values of traditional materials. Carbon fibres can be made with a very wide range of Young’s moduli and the values given in Table 1.1 are some typical values. The properties given for the composites are those of unidirectional composites containing a volume fraction of fibres of 60% loaded in the fibre direction. The technical, synthetic fibres, which have been developed since the middle of the twentieth century and which have permitted fibre-reinforced composites to be produced, are the subject of this chapter.
1.4
Organic fibres
Several people unsuccessfully tried to produce artificial fibres before regenerated cellulose fibres were patented in 1884, in France by Count Hilaire de Chardonnay (Moncrieff, 1982). These first artificial fibres were made of cellulose nitrate and were very inflammable. Ten years later in 1894, Charles Frederick Cross and his collaborators Edward John Bevan and Clayton Beadle, in England, patented their process, which gave rise to viscose fibres. This was a fibre which could be used for textile purposes, and commercial production began in 1905 by the Courtauld’s company, in England. The fibres were made by extrusion of the cellulose solution into a bath of sulphuric acid. The goal of producing artificial fibres was to produce a cheaper and therefore more accessible silk. The viscose fibres did find uses in textile applications but also in reinforcing rubber, typically for the emerging tyre industry. These fibres represent the link between natural and synthetic fibres, as rayon fibres make use of the structure of cellulose molecules which occur naturally in all plants. The cellulose for making rayon fibres is usually obtained from wood.
1.4.1
Thermoplastic fibres
It was not until around 1938 that the first truly synthetic organic fibres were produced. These were polyamide 6.6 fibres, developed by the work of
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Carothers at DuPont in the USA, and almost simultaneously, polyamide 6 was produced by the German company I.G. Farben. Both types of polyamide fibres contain the amide group, consisting of a carbon atom linked to an oxygen atom and both linked to a nitrogen atom. This group lies within the backbone of the polyamide macromolecule. Polyamides are more generally known as nylons. The work that gave rise to polyamide 6.6 fibres came from a widerranging study of both polyamides and polyesters, the latter being based on the ester group, consisting of a carbon atom linked to an oxygen atom and both linked to another oxygen atom. This ester group lies in the backbone of any polyester polymer. Although the work on polyesters was initially abandoned, because of the instability of the molecular structure, it inspired work in England by J. R. Whinfield and J. T. Dickson of the Calico Printers Association. These two chemists found that by associating the ester group with an aromatic group, consisting of a cycle of six carbon atoms, poly(ester terephthalate) (PET) could be produced. This polymer overcame many of the limitations of other polyesters that had been produced as it resisted hydrolysis and possessed a melting point of 260°C. The polyester fibres that were developed from PET and commercially launched by ICI in the UK around 1947 have become the most widely produced fibres of all types. Both nylon and polyester (PET) fibres find wide technical uses, such as in reinforcing rubber in tyres, walkways, fan belts, hoses, flexible drives, moving walkways and other applications. The aromatic ring in the PET molecule can be seen in Table 1.2. It confers on the polymer improved thermal and chemical stability as well as interesting mechanical properties. The PET fibres have the same melting point as PA66 fibres but around three times the stiffness. This stiffness, however, is still low compared to that of most structural materials. Both nylon and polyester fibres are semi-crystalline thermoplastics and are drawn from the melt; however, stiffer organic fibres were later developed by using the technology of liquid crystals. The addition of the aromatic ring, seen in the PET molecule, defined the routes which chemists would take to produce ever more complex chemical structures, to produce organic fibres with higher tensile stiffness that are used in high-performance composites.
1.4.2
Aramid fibres
The aramid family of fibres is made up of aromatic polyamides, some of which possess remarkably high Young’s moduli more than 20 times that of conventional polyamide fibres. The term aramid comes from an abbreviation of aromatic polyamide and covers the range of wholly aromatic polyamides so as to distinguish them from the linear aliphatic polyamides. The aramids
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Maximum elastic modulus (GPa)
Poly(p-phenylene benzobisoxazole), PBO [Zylon], Toyobo
Poly(p-phenylene terephthalamide) [Kevlar], DuPont
Poly(m- phenylenediamine- isophthalamide) [Nomex], DuPont
HN
O N
N
CO
O
NH
CO
O
H NH
C
O
N
CO
CH2
CO
CH2 n
280
135
17
15
650
550
360
260
CO
Polyethylene terephthalate [polyester], ICI
O
230
260
Melting or decomposition temperature °C
Polyamide 6 4 [Nylon 6], —NH—CH [ ]nˆ 2—CH2—CH2—CH2—CH2—CO— I.G. Farben
Polyamide 6/6 5 [Nylon 6.6], —NH—CH [ ]nˆ 2—CH2—CH2—CH2—CH2—CH2—NH—CO—CH2—CH2—CH2—CH2—CO— DuPont
Fibre type Repeat unit in the macromolecule
Table 1.2 Some molecules used to produce organic fibres
Fibres for composite reinforcement
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consist of stiff molecules due to the aromatic rings which are not flexible. The result is a family of polymers which are more chemically resistant and thermally stable than aliphatic polyamides due to the greater degree of alignment of the molecular structure that is possible. Some of the aramids possess much higher strengths and tensile stiffness, the latter being due to the para- linkage of the aromatic rings, which means that they are joined parallel to the main molecular chain. This limits the compliance of the molecule. The properties of the most common classes of organic fibres are shown in Table 1.3, and it should be clear how those fibres based on cyclic molecular structures are much stiffer than the first thermoplastic fibres that were produced. An early type of aramid fibre which has found wide use is poly(metaphenylene isophthalamide). This fibre is sold by its makers, DuPont, under the name of Nomex. This polymer contains meta-oriented phenylene radicals, which means that stresses through the molecule are not supported by straight covalent bonds, resulting in a tensile stiffness which is still only about that of highly drawn PET. The fibre is, however, flame resistant and thermally stable to around 100°C over the melting point of the aliphatic polyamides and has good dielectric properties. As a consequence the fibre is used in apparel for which resistance to heat is required such as fire officers’ uniforms and racing drivers’ overalls. It is also used in chopped form to make a paper material that is widely used as a honeycomb in advanced composite structures. The low specific gravity of Nomex (1.38) compared to aluminium (2.7), which is also used for honeycomb structures, gives it an obvious advantage for applications for which weight saving is important, such as in aircraft. In order to obtain high strength and above all high stiffness, the aramid polymers should contain predominantly para-oriented aromatic units. This is the case in poly(para-phenylene terephthalamide) (PPTA) which is the polymer used to make the Kevlar fibre that has been available from DuPont since 1972. A similar fibre is the Twaron fibre, originally produced by Akzo, in the Netherlands and now produced there by the Japanese company Teijin (Pergoretti and Traina, 2009). Table 1.3 Typical properties of most commonly commercially available organic fibres Fibre Polymer Diameter Density (µm) (g/cm3)
Young’s modulus (GPa)
Strength (GPa)
Strain to failure (%)
Polyamide 66 Polyester Kevlar 49 Zylon Polyethylene (Dyneema)
<5 <18 135 280 117
1 0.8 3 5.8 3
20 15 4.5 2.5 3.5
PA66 PET PPTA PBO PE
20 15 12 12 38
1.2 1.38 1.45 1.56 0.96
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PPTA is insoluble in most solvents, but a 10% solution of the polymer in concentrated (>98 wt%) sulphuric acid results in the production of a mesophase or liquid crystal solution. The rod-like PPTA molecules are randomly oriented in the solution but, beyond a critical concentration, locally the molecules are attracted together and adopt an ordered arrangement in small domains to achieve better packing. The domains are still randomly arranged with respect to one another and the solution retains the flow properties of a liquid, but in polarized light it can be seen to be locally oriented. When the solution is passed through the holes of a spinneret the induced shear forces orient the molecular structure so that a highly oriented fibre can be produced without drawing, as shown schematically in Fig. 1.1. The result is a highly oriented and well-organized molecular structure for these types of fibres. The molecules are oriented parallel to the fibre axis so that the covalent bonds in the structure support any applied load. However, the presence of the amide group, which is not straight, leads to less than
1.1 A schematic representation of a liquid crystal solution being extruded through a spinneret to form a highly oriented fibre.
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theoretical maximum stiffness for an organic fibre. Table 1.3 shows how the development of aramid fibres produced filaments which were around 30 times stiffer than nylon 66 fibres and had remarkable specific properties in tension. However, inevitably, in the plane normal to the fibre axis the secondary bonds, which are hydrogen and van der Waals’ bonds in PPTA fibres, make the fibres very anisotropic. The presence of hydrogen bonds means that these fibres absorb significant amounts of water. On a molecular scale, the fibres behave like a bundle of filaments, with each filament representing a PPTA molecule. A bundle is strong in tension but in all other directions, particularly compression, it is weak. For this reason these highly oriented organic fibres are not suitable for structures which are subjected to compressive loads. Figure 1.2 shows that Kevlar fibres deform plastically in compression, rather than break as brittle fibres do (Lafitte and Bunsell, 1985). When an aramid fibre is bent, the convex surface goes into compression and this relieves the tensile stress which is developed in the convex surface. It means that these fibres can be bent to a zero radius of curvature without breaking and this makes them both difficult to cut and useful for absorbing energy. A major application for composite materials reinforced with these highly oriented organic fibres is in impact absorption, whether in the case of explosions, for
1.2 This Kevlar fibre is bent around another fibre. Both have diameters of 12 mm and it can be seen that the concave side of the fibre has developed slip bands by the plastic deformation in shear of the polymer. In this way the fibre accommodates the compression rather than breaking (Lafitte and Bunsell, 1985).
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flak jackets or used with other elastic fibres so as to make a hybrid composite capable of resisting shocks. Another mechanism which absorbs energy is fibrillation, and these highly oriented organic fibres made by liquid crystal technology usually fail like a blade of grass, clearly revealing their fibrillar microstructure, as can be seen from Fig. 1.3 showing a broken Zylon fibre. The Zylon fibre, made by Toyobo in Japan, has been commercially available since 1998 and has the highest tensile stiffness of all commercially available organic fibres. This fibre is 40% stiffer than steel for a fifth of the density. The fibre is made from poly-p-phenylenebenzobisoxazole (PBO) (Pergoretti and Traina, 2009). This is one of the polybenzazoles containing aromatic heterocyclic rings, which were developed by the US Air Force as polymers that resist heat better than Kevlar. As the molecule is straight, the Zylon fibre, as shown in Tables 1.2 and 1.3, has a tensile stiffness twice as great as that of Kevlar. When Zylon fibres are first produced they absorb some water, around 2 wt%, because of porosity in the fibre structure due to solvent evaporation. Treatment of the fibre reduces this to around 0.2 wt%. Such a low water uptake is due to the transverse bonds being purely van der Waals but must increase anisotropy. Other organic fibres are in development, such as the M5 fibre made from polypyridobisimidazole (PIPD) which was produced by the researchers from the Dutch company Akzo Nobel and is now being developed by Magellan Systems International with suggestions that it will be commercially produced by DuPont. The PIPD molecule is related to PBO, and the Young’s modulus claimed for the fibre of 330 GPa is even higher. The producers claim that
1.3 A broken Zylon fibre, revealing its highly oriented fibrillar microstructure.
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there is a highly significant difference in the behaviour of the PIPD fibre as lateral cohesion is determined by hydrogen bonds. As mentioned above, the PBO fibre relies on van der Waals’ bonds for lateral cohesion which, however, are very weak. Hydrogen bonds are 10 times stronger, which means that the M5 fibre should have superior radial and compressive strengths. The presence of hydrogen bonds will probably mean that the fibre will absorb water, however.
1.4.3
High-modulus polyethylene fibres
The stiffness of fibres such as Kevlar or Zylon is a consequence of the alignment of the main molecular chains parallel to the fibre axis. This is achieved by the liquid crystal technology discussed above, but aligning the main molecular chains of even the simplest polymer parallel to the fibre axis would, theoretically, produce a tensile Young’s modulus greater than that of steel. The difficulty is how to do it. One of the simplest polymers is polyethylene, consisting of a backbone of carbon atoms to which are attached hydrogen atoms. The simplicity of the molecule has allowed the DSM company, in the Netherlands, to produce polyethylene fibres with an aligned molecular structure and so consequently produce a high-modulus fibre (Vlasblom and van Dingenen, 2009). High-modulus polyethylene fibres are produced using a dilute (<5%) sol-gel, in which the polymer is dissolved in a solvent and then spun. The low concentration of the polyethylene allows the macromolecules sufficient space to become disentangled and, by optimization of the process, to produce fibres with aligned molecular structures. The gel is extruded and the solvent evaporated to make the filaments. High-modulus polyethylene fibres produced in the Netherlands by DSM in collaboration with Toyobo in Japan under the name Dyneema and in the USA by Honeywell under the name Spectra have properties rivalling those of the aramid fibres with a lower specific gravity of 0.97. Unlike the fibres made by liquid crystal technology, these fibres do not contain thermally stable cycles of atoms, so that the fibres are more sensitive to temperature effects. These fibres are limited in temperature to a maximum of 120°C and suffer from creep. The melting point is around 146°C. The Dyneema fibres are therefore composed of aligned polyethylene molecules, which results in a high-modulus fibre when loaded in tension but also a highly anisotropic structure. The fibrillar nature of the fibre can be seen in Fig. 1.4. The anisotropy of the fibre means that, like the liquid crystal fibres, it is weak in compression. Figure 1.5 shows a kink band produced by a compression wave generated by the failure of the Dyneema fibre in tension. High-modulus polyethylene fibres are finding markets as high-strength
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1.4 The fibrillar nature of Dyneema high modulus polyethylene fibres is clearly seen in this micrograph.
1.5 A compression kink band in a Dyneema fibre following a tensile test.
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mooring ropes for large ships and oil platforms as well as for flak jackets and similar anti-ballistic structures.
1.5
Glass fibres
Glass filaments were probably made in antiquity. The beginning of an industrial process for producing fine glass filaments was demonstrated in Great Britain in the nineteenth century, and they were used as a substitute for asbestos in Germany during the First World War. However, the first commercial production dates from 1931 when two American firms, Owen Illinois Glass Co. and Corning Glass Works, developed a method of spinning glass filaments from the melt through spinnerets. The two firms combined in 1938 to form Owens Corning Fiberglas Corporation. Initially the glass fibres were destined for filters and textile uses, but the development of thermosetting resins opened up the possibility of fibre-reinforced composites, and in the years following the Second World War the fibre took a dominant role in this type of material. Today, glass fibres represent about 99% by weight of all fibre reinforcements used in composites, if rubber-based composites are not considered. Fibres of glass are produced by extruding molten glass, at a temperature around 1250°C, through holes with diameters of 1–2 mm in a spinneret and then drawing the filaments to produce fibres having diameters usually between 5 and 15 mm (Jones, 2009). The spinnerets usually contain several hundred holes so that a strand of glass fibres is produced. Several types of glass exist but all are based on silica (SiO2) which is combined with other elements to create speciality glasses. The compositions and properties of the most common types of glass fibres are shown in Table 1.4. The most widely used glass for fibre-reinforced composites is called E-glass, while glass fibres with superior mechanical properties are known as S-glass. E-glass is the most widely used glass in fibre production, type S is a glass with enhanced mechanical properties, and type C glass resists corrosion in an acid environment. Glass fibres are easily damaged by abrasion, either with other fibres or by coming into contact with machinery in the manufacturing process, so they are coated with a protective coating known as a size. The purpose of the size is both to protect the fibre and to hold the strand together. The size may be temporary, usually a starch–oil emulsion, to aid handling of the fibre, which is then removed and replaced with a finish to help fibre matrix adhesion in the composite. There are two main types of coupling agent used in composites. These are organometallic compounds, typically of cobalt, nickel, lead, titanium or chromium, or, more commonly, organosilanes. They are applied as an aqueous solution. An example of an organosilane is shown in Fig. 1.6. It reacts with the glass surface through hydrogen bonds. The
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Table 1.4 Compositions (wt%), density and mechanical properties of glasses used in fibre reinforcement production Glass type
E
S
C
SiO2 54 65 Al2O3 15 25 CaO 18 MgO 4 10 B2O3 8 F 0.3 Fe2O3 0.3 TiO2 Na2O K2O 0.4
65 4 14 3 5.5
Density Strength at 20°C (GPa) Elastic modulus at 20°C (GPa) Failure strain at 20°C (%)
2.49 2.8 70 4.0
2.54 3.5 73.5 4.5
R
R
Si
Si
O
O H
2.49 4.65 86.5 5.3
8 0.5
O p
O H
H
O
O
Si
Si
H
Glass surface
1.6 Glass fibres are always coated with a size, the most common of which is based on silane.
size usually has several additional functions which are to act as a coupling agent and lubricant and to eliminate electrostatic charges. Continuous glass fibres may be woven, as are textile fibres, made into a non-woven mat in which the fibres are arranged in a random fashion, used in filament winding or chopped into short fibres. In this latter case the fibres are chopped into lengths of up to 5 cm and lightly bonded together to form a mat, or chopped into shorter lengths of a few millimetres for inclusion in moulding resins. The structure is vitreous with no definite compounds being formed and no crystallization taking place. An open network results from the rapid cooling that takes place during fibre production with the glass cooling from about 1500°C to 200°C in 0.1–0.3 seconds. Despite this rapid rate of cooling
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there appear to be no appreciable residual stresses within the fibre and the structure is isotropic. The result is that the Young’s moduli of glass fibres are very similar to that of bulk glass of the same type and the fibres are not anisotropic, like most other reinforcements. Glass is set to remain the most widely used reinforcement for general composites. Glass fibres are cheaper than most other relatively high-modulus fibres and because of their flexibility do not require very specialized machines or techniques to handle them. Their elastic modulus is, however, low when compared to many other fibres, and the specific gravity of glass, which for E-type glass is 2.54, is relatively high. The poor specific value of the mechanical properties of glass fibres means that they are not ideal for structures requiring light weight as well as high properties, although considerations such as their relatively low cost compared to other high-performance fibres and the ease of composite manufacture means that they are used in some such structures.
1.6
Chemical vapour deposition (CVD) monofilaments
The first fibre produced with a much increased Young’s modulus compared to glass fibres was the boron fibre made by a CVD technique onto a substrate which was a tungsten wire. Boron, the fifth element in the Periodic Table, is the lightest element with which it was found practical to make fibres. The first boron fibres were produced in the USA at the beginning of the 1960s. They had a Young’s modulus exceeding 400 GPa and quite extraordinary strength in compression. Silicon carbide fibres, produced by a similar technique to that used to make boron fibres, can be used as a reinforcement for metal matrix composites, including titanium and intermetallic matrix composites. Silicon carbide fibres, are based on materials which are readily available, and the substrate can be either tungsten wire or a carbon filament, which is cheaper (Wawner, 1988). The production of both boron and silicon carbide fibres by CVD takes place in a reactor which consists of a glass tube, which for commercial production is usually 1–2 metres in length and most often vertical. Schematic representations of the arrangements for the production of both types of fibres are shown in Fig. 1.7. The substrate can run continuously through the reactor by the use of mercury seals at each end. These ensure that the reactive gases are contained in the reactor and also allow the substrate to be heated by electric current over the deposition length. The choice of the substrate is governed by the need to use a material which maintains its strength to around 1100°C and is an electrical conductor. Heating the substrate, usually with a dc current, is a necessary part of the process which determines the deposition rate, which must be constant or irregularities in the fibre diameter
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Composite reinforcements for optimum performance Supply reel
Mercury contact
Supply reel
W
W or C
BCl3 + H1
Boron deposition reactor
Mercury contact
Silane Hydrogen propane Argon Silane Hydrogen
HCl Silane Propane Argon Bw SiCw or SiCc Take-up reel
Take-up reel
1.7 Schematic views of the reactors for producing boron and silicon carbide fibres by CVD.
occur with a subsequent fall in strength. Tungsten wire was chosen for the production of boron fibres and can be used also for SiC fibres, although carbon substrates are an alternative. The tungsten wires have a diameter of around 12 mm but the carbon cores are usually around 30 mm. It is therefore evident that the fibres produced have much bigger diameters than most other reinforcements used for composite materials. As explained above, the large diameter leads to the production of fibres which are not very flexible and cannot be woven, which limits their use. Boron fibres are made by mixing boron trichloride with hydrogen, which results in boron being deposited according to the following reaction:
2BCl3(g) + 3H2(g) Æ 2B(s) + 6HCl(g)
Passage through the reactor takes 1–2 minutes and results in a fibre with a diameter of 140 mm. During the deposition process the small boron atoms diffuse into the tungsten core to produce complete boridization and the production of WB4 and W2B5. This leads to an increase in diameter to 16 mm
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which induces considerable residual stresses, putting the core into compression and the neighbouring boron mantle into tension. A final annealing process puts the fibre surface into compression. The fibres were initially of interest for all types of composites destined for mainly aerospace structures, but their use in resin composites was curtailed by the development of cheaper, highperformance carbon fibres that became available in the late 1960s. Boron fibres were successfully used to reinforce aluminium, and the American space shuttle made extensive use of this type of composite in its structure. Silicon carbide fibres made by CVD are produced on both tungsten and carbon cores in similar reactors to those use for the production of boron fibres, except that multiple injection points are used for the introduction of the reactant gases. Various carbon-containing silanes have been used as reactants. In a typical process, with CH3SiCl3 as the reactant, SiC is deposited on the core as follows:
CH3SiCl3(g) Æ SiC(s) + 3HCl(g)
Figure 1.8 shows the cross-section of a SiC fibre made by CVD that has been used to reinforce titanium. Like the boron fibres, these SiC fibres usually have a diameter of 140 mm. The tungsten core can be seen in the centre and a protective layer can also be seen on the fibre surface. This layer is to protect the SiC from the highly reactive molten titanium during manufacture of the composite. Several types of protection are used; the best known is
1.8 A cross-section of a SiC fibre made by CVD, used to reinforce a metal matrix.
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produced during the later stages of fibre production and consists of a CVD layer, the composition of which varies from being rich in silicon to being rich in carbon before becoming rich again in silicon on the fibre surface. This SCS coating is typically 3 mm in thickness. The greatest interest for SiC fibres made by CVD is for the reinforcement of titanium for use in jet engines so as to reduce weight.
1.7
Carbon fibres
Carbon is the sixth lightest element and the carbon–carbon covalent bond is the strongest in nature (4000 kJ/mole); however, the arrangements of the bonds and the distances between the carbon atoms can vary, so giving many types of carbon, including graphite, diamond and amorphous forms.
1.7.1
Carbon fibres from cellulose
Many fibres can be converted into carbon fibres, the basic requirement being that the precursor fibre carbonizes rather than melts when heated. In 1901 Edison used bamboo fibres, which like all plants are made up of cellulose, and converted them into carbon filaments for his electric light bulbs. He therefore became the first person to patent a carbon fibre but they proved very brittle and after three years were replaced by tungsten. Interestingly, however, the carbon fibres developed in the USA in the 1950s and early 1960s used viscose fibres regenerated from cellulose (Bacon, 1973). This proved a slow process as the carbon yield from cellulose is only 24% and mechanical properties were not high; however, such fibres have low thermal conduction properties and are still used in carbon–carbon heat shields and brake pads.
1.7.2
Carbon fibres from polyacrylonitrile (PAN)
Most carbon fibres nowadays are made by a process developed and commercially produced first in Great Britain in the mid-1960s and then in Japan, where production began in 1970, based on the conversion of a modified form of polyacrylonitrile (PAN) fibre. PAN has a carbon yield of 67% by weight (Matsuhisa, and Bunsell, 2009). The PAN molecule has a backbone of carbon atoms and consists of long linear molecules. Heating in air around 275°C causes the fibre to turn black due to oxygen interacting with the PAN. This causes crosslinking, producing a three-dimensional infusible atomic network. The crosslinked fibre is then heated in nitrogen above 1000°C to produce a carbon fibre. This process is conducted with the fibre in tension, as the matter that is lost causes a reduction in volume and without the tensile load on the fibre would result in a fibre with poor
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structural organization at the atomic level. The properties of carbon fibres made from PAN precursors depend on the temperature of pyrolysis. The different stages in carbon fibre production are shown in Fig. 1.9. Highest strength is achieved between 1500°C and 1600°C but the Young’s modulus continues to increase with increasing temperature. The reasons for this change in strength and modulus with pyrolysis temperature lie at the level of the atomic structure of this type of carbon fibre, which was closely studied by Oberlin and Guigon (1988). The structure which results from the pyrolysis of PAN is highly anisotropic, with the basic structural units, formed by small plates of the carbon atom groups, arranged in hexagons, aligned parallel to the fibre axis. They form imperfectly stacked layers called turbostatic layers with the interlayer spacing being greater than that of graphite. There is complete rotational disorder in the radial direction and the relatively poor stacking of the carbon atoms means that a graphite structure is never achieved. The structure contains pores which account for the density of the fibres being less than that of fully dense carbon. Table 1.5 shows some typical properties of carbon fibres; however, there exists a considerable range of properties for these fibres. Pyrolysis temperatures above 1600°C produce an increase in size of the basic structural units which evolve, becoming larger plates and more continuous ribbon-like structures. The structure tends towards that of graphite but never attains the close packing of this crystalline form of carbon.
Polymerization
AN Acrylonitrile
Wet or dry spinning
PAN
PAN fibre
Polyacrylonitrile
Oxidation 200–300°C 2–3 hours in air
Oxidized fibre
Precursor fibre
Carbonization
High temperature treatment
1000–1500°C 5 min in N2
2500–3000°C 1 min in N2 Surface treatment
Sizing
High strength
High modulus
1.9 Carbon fibres made from PAN can be processed at different temperatures in order to change their properties.
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Table 1.5 Typical properties of PAN- and pitch-based carbon fibres Fibre type Diameter Density Strength Failure Young’s (µm) (g/cm3) (GPa) strain modulus (%) (GPa) Ex-PAN High strength (first generation) High strength (second generation) High modulus (first generation) High modulus (second generation) Ex-pitch Oil-derived pitch Oil-derived pitch (high modulus) Coal tar pitch Coal tar pitch (high modulus)
7 5 7 5
1.80 1.82 1.84 1.94
4.4 7.1 4.2 3.92
1.8 2.4 1.0 0.7
250 294 436 588
11 11 10 10
2.10 2.16 2.12 2.16
3.7 3.5 3.6 3.9
0.9 0.5 0.58 0.48
390 780 620 830
The carbon fibre surface consists of planar BSUs aligned parallel to the fibre surface and along the axis direction. There are no pendant bonds available so that carbon fibres are surface treated using an oxidation process or other methods to make the fibre surface more reactive, so as to improve adhesion between the fibre surface and matrix resin. After surface treatment, the carbon fibres are coated with a sizing agent to facilitate composite manufacture. PAN-based carbon fibres were originally produced with diameters of 7 mm and this first generation of fibres is still the most widely used; however, a second generation of carbon fibres is available with diameters of 5 mm. This change in diameter reduces the cross-section by a half and the result is that the strengths of these fibres are considerably improved when compared to the earlier fibres. The above discussion on Weibull statistics gives the explanation: that the smaller volume per unit length means that there are fewer defects to weaken the fibres. In addition the smaller diameter allows for greater regularity in the structure so that the modulus is increased. Figure 1.10 compares the appearances of the first-generation fibres (left) with the latest on the right. In addition to the smaller diameter, it can be seen that the later fibres have been produced with improved control over the spinning process so as to produce smoother fibres. This also improves the strengths of the fibres. Figure 1.11 shows the range of fibre properties available from PAN-based carbon fibres. Together with the improvement in tensile strength, the compressive strength of carbon fibre has also improved (Norita et al., 1988). Therefore the presence of defects is also an important factor for the compressive strength. Another important factor for the compressive strength of single filaments is crystallite size. Carbon fibres can be categorized into low modulus carbon fibres (<200 GPa), standard modulus carbon fibres (~230 GPa), intermediate modulus carbon fibres (~300 GPa) and high modulus carbon fibres (>350 GPa). Fibres
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1.0 µm
25
1.0 µm
Tensile strength (GPa)
1.10 A comparison of a first-generation PAN-based carbon fibre (left) and a fibre of the most recent generation (right).
8
Ultra high strength
6
High modulus
4 Ultra high modulus 2 200
400 600 Young’s modulus (GPa)
800
1.11 The range of properties from commercially available PAN-based carbon fibres.
with higher moduli than 600 GPa are sometimes called ultra high modulus carbon fibres and are sometimes known as graphite fibres. The highest tensile modulus of PAN-based carbon fibres is 700 GPa. PAN-based carbon fibres are produced in a range of tows, the finest of which originally contained 1000 fibres. Such tows were used for the highest quality aerospace structures but with improved control, experience in manipulating the fibres and a drive to reduce costs, tows of 3000, 6000 and 12,000 filaments became standard. This trend has continued and tows of
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24,000 fibres are increasingly used, with tows of even 48,000 carbon fibres becoming to be seen as standard. Tows containing hundreds of thousands of carbon fibres seem to be the future for at least non-aerospace applications as fibre production cost is reduced in this way. The large tows then have to be split into tows with smaller numbers of fibres so that they can be used in composite manufacture.
1.7.3
Carbon fibres from pitch
Carbon fibres made from pitch, which is the residue of petroleum refining or coal tar distillation in the steel industry, were developed in the USA in the early 1970s and in Japan in the 1980s. The high carbon yield of pitch, which approaches 90%, makes it an attractive and potentially cheap source for making carbon fibre precursors (Lavin, 2000). Cost, however, is increased by the purification processes which are necessary for this naturally occurring material. Some carbon fibres are made from pitch with no attempt to align their structures, and the result is fibres with a low Young’s modulus of around 40 GPa. These fibres are used for their chemical inertness and low cost for the reinforcement of cement and concrete. The biggest producer of this type of isotropic pitch-based carbon fibre is Kureha in Japan. In the early 1970s, Union Carbide in the USA developed high performance carbon fibres from mesophase pitch (Volk, 1975). In this method the pitch is treated so that the molecular structure is ordered to give a nematic (onedimensional) liquid crystal structure. Mesophase pitch melts around 300°C and is then processed around 400°C. The polymerized pitch molecules are made up of chains of aromatic hexagonal units each consisting of six atoms of carbon. When the polymerization is sufficiently advanced the molecules form spheres which grow until a phase inversion occurs, at which point the previous situation with the discrete spheres being embedded in the isotropic medium reverses and the molecules become the medium consisting of a continuous nematic liquid crystalline phase, called a mesophase. The precursor pitch fibres are then spun from the melt. After production of the precursor fibres the production processes of pitch-based carbon fibres are fundamentally the same as those of PAN-based carbon fibres. The arrangement of the atomic structures of the carbon fibres produced from pitch can be more regular than that obtained from PAN precursors, with a graphite structure being achieved in some types of these fibres. As a consequence the results are higher moduli and lower production temperatures, with the result that ultra high modulus carbon fibres made from pitch are cheaper than those made from PAN. The maximum tensile modulus of a carbon fibre is limited to that of a graphite single crystal, which is 1050 GPa. Carbon fibres made from coal tar pitch are commercially available with moduli of 935 GPa. The regularity of the arrangements of the carbon atoms
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means that the basic structural units of the carbon are larger than in PANbased fibres and this results in lower strengths, particularly in compression. As Table 1.5 shows, the tensile strength of pitch-based carbon fibre does not vary with pyrolysis temperature as is the case in producing carbon fibres from PAN.
1.8
Small-diameter ceramic fibres
There are two families of small-diameter ceramic fibres (Bunsell and Berger, 1999). Glass fibres are based on oxides, and shortly after their development discontinuous oxide fibres were produced as refractory insulation for furnaces and similar applications (Bansal, 2005). Fibres based on silicon carbide were produced much later, inspired by the production of carbon fibres from PAN.
1.8.1
Oxide fibres
Oxide fibres for reinforcement were first produced in the 1970s, primarily in the USA and Great Britain. Saffil fibres developed by ICI are short fibres with small diameters of around 3 mm. They are based on alumina (Al2O3) and around 3% silica (SiO2). They find use as reinforcement for aluminium. DuPont produced the first continuous a-alumina fibre but did not proceed to commercial exploitation. This was due to its brittle nature. Alpha-alumina is the most crystalline form of alumina and has a Young’s modulus of around 400 GPa. Other producers, notably 3M, have produced other fibres based on alumina, often combined with other oxides to limit phase transformation of the alumina into the most brittle alpha phase (Wilson, 2009). 3M produces a range of oxide fibres under the name of Nextel and they are successfully used at temperatures up to around 1000°C. However, the ionic bonds in oxides make them susceptible to creep at higher temperatures. The Nextel 720 fibre has a complex structure consisting of a-alumina and mullite phases. This means that the fibre has a complex crystallographic structure with no easy slip planes. It is the most advanced oxide fibre but is limited to temperatures below 1100°C because it is metastable in the presence of alkaline impurities (Deleglise et al., 2002). Table 1.6 shows some examples of oxide fibres together with some of their properties.
1.8.2
Silicon carbide fibres
Silicon carbide filaments can be used at temperatures above 1000°C. This is due to the structure of SiC containing covalent bonds, as in carbon. It is this similarity with carbon that encouraged the use of these fibres to replace
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99% Al2O3, 0.2–0.3% SiO2, 0.4–0.7% Fe2O3
99.9% Al2O3 10–12
10
Alumina–silica Saffil Ltd Saffil 95% Al2O3, 5% SiO2 1–5 based fibres 15 Sumitomo Altex 85% Al2O3, 15% SiO2 Chemicals 3M Nextel 312 62% Al2O3, 24% SiO2, 14% B2O3 10–12 and 8–9 3M Nextel 440 70% Al2O3, 28% SiO2, 2% B2O3 10–12 3M Nextel 720 85% Al2O3, 15% SiO2 12
a-Al2O3 based Mitsui Almax fibres Mining 3M Nextel 610
2 1.8 1.7 2.1 2.1
3.2 2.7 3.05 3.4
1.9
1.02
3.2
3.75
3.6
1.12 1.11 0.81
0.8
0.67
0.5
0.3
152 190 260
210
300
370
344
Fibre type Manufacturer Trade Composition Diameter Density Strength Strain to Young’s mark (wt%) (mm) (g/cm3) (GPa) failure modulus (%) (GPa)
Table 1.6 Examples of oxide-based fibres
Fibres for composite reinforcement
29
carbon fibres in composites used for very high-temperature applications, such as rocket nozzles. Silicon carbide in bulk form can be used in air up to 1600°C whereas carbon oxidizes from 400°C. The commercial development of silicon carbide fibres in the early 1980s was the direct consequence of the work of Yajima and his colleagues in Japan (Yajima et al., 1976). Nippon Carbon made these fibres commercially available under the name of Nicalon. Later, Ube Industries began producing Tyranno fibres with similar properties but slightly different compositions. The early fibres were produced in an analogous fashion to that used to make carbon fibres from PAN. A polycarbosilane organic precursor was used. It not only contained the elements necessary for SiC but was structured in a way that was reminiscent of the b-crystalline form of the ceramic. The precursor fibres were made infusible by heating in air, then heating to a higher temperature in a controlled atmosphere produced the ceramic fibre. Unfortunately, unlike in carbon fibre manufacture, the oxygen that was used to crosslink the polymer remained in the structure. The early fibres did not have the properties of bulk SiC due to the small (<2 nm) SiC grains being embedded in an amorphous matrix phase created by the presence of oxygen. A second generation of fibre was produced by Nippon Carbon, called Hi-Nicalon, for which oxygen was replaced by electron irradiation of the precursor so as to induce crosslinking. This fibre had enhanced properties but still not those of fully crystalline SiC (Berger et al., 1995). A third generation of SiC fibres has emerged with almost fully sintered structures (Bunsell and Piant, 2006) and these fibres retain their properties up to 1400°C. The manufacturers of these latest SiC fibres are the Japanese companies Nippon Carbon and Ube Industries as well as the American company COI Ceramics. Table 1.7 gives some of the properties of the three generations of SiC-based fibres.
Table 1.7 Examples of the three generations of SiC-based fibres Trade mark Manufacturer Average Room diameter temperature (µm) strength (GPa)
Room temperature Young’s modulus (GPa)
First gen. Second gen. Third gen.
Nicalon 200 Hi-Nicalon
Nippon Carbon 14 Nippon Carbon 12
3 2.8
200 270
Tyranno SA3 Sylramic Sylramic iBN Hi-Nicalon Type-S
Ube Industries 7.5 COI Ceramics 10 COI Ceramics 10 Nippon Carbon 12
2.9 3.2 3.5 2.5
375 400 400 400
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1.9
Conclusions
There exist fibres to reinforce all classes of materials – elastomers, polymers, metals and ceramics – so that composite materials cover the whole spectrum of structural materials. They are also found in some structures which experience extremes in temperature. The reinforcements often bring to the composites their remarkable properties which can be indispensible for certain types of applications. The properties of the fibres depend on the elements from which they are made and these are generally found at the beginning of the Periodic Table, which means that they are light in weight. Strength and stiffness are not linked to density so that the fibres allow lightweight composite structures to be made and this has been the most important characteristic sought for many applications. However, the fineness of the fibres also allows structures to be envisaged that otherwise would be difficult to produce, so that manufacturing considerations can be a reason for using fibre-reinforced materials. Fibres are extraordinary forms of reinforcement and their properties have allowed composites to be developed and have enabled a remarkable revolution in structural materials to occur.
1.10
References
Bacon R (1973) Chemistry and Physics of Carbon, Vol. 2, Marcel Dekker, New York. Bansal N P (2005) Oxide Fibers – Handbook of Ceramic Composites, Kluwer Academic Publishers, Boston, MA. Berger M-H, Hochet N and Bunsell A R (1995) J. Microscopy, 177(3), 230–241. Bunsell A R, ed. (2009) Handbook of Tensile Properties of Textile and Technical Fibres, Woodhead Publishing, Cambridge, UK, and CRC Press, Washington, DC. Bunsell A R and Berger M-H (1999) Fine Ceramic Fibers, Marcel Dekker, New York. Bunsell A R and Piant A (2006) J. Mater. Sci. 41, 823–839. Bunsell A R and Renard J (2005) Fundamentals of Fibre Reinforced Composite Materials, CRC Press, Taylor & Francis Group, London and Boca Raton, FL. Deleglise F, Berger M-H and Bunsell A R (2002) J. Eur. Ceram. Soc. 22, 1501–1512. Jones F R (2009) ‘Structure and properties of glass fibres’ in Bunsell A R (ed.), Handbook of Tensile Properties of textile and technical fibres, Woodhead Publishing, Cambridge, UK, 529–573. Lafitte M H and Bunsell A R (1985) Polym. Eng. Sci., 25(3) 182–186. Lavin J G (2000) ‘Carbon fibres’ in Hearle J W S (ed.), High Performance Fibres, Woodhead Publishing, Cambridge, UK, 156–190. Matsuhisa Y and Bunsell A R (2009) ‘Tensile failure of carbon fibers’ in Bunsell A R (ed.), Handbook of Tensile Properties of textile and technical fibres, Woodhead Publishing, Cambridge, UK, 574–602. Moncrieff R W (1982) Man-Made Fibres, 6th ed., Butterworth Scientific, London. Morton W E and Hearle J W S (2008) Physical Properties of textile fibres, 4th ed., Woodhead Publishing, Cambridge, UK.
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Norita T, Kitano A and Noguchi K (1988) ‘Compressive strength of fiber reinforced composite materials – Effect of fiber properties’, 4th Japan–US Conference on Composite Materials, 548. Oberlin A and Guigon M (1988) ‘The structure of carbon fibres’ in Bunsell A R (ed.), Fibre Reinforcements for Composite Materials, Elsevier, Amsterdam, 149–210. Pergoretti A and Traina M (2009) ‘Liquid crystalline organic fibres and their mechanical behaviour’ in Bunsell A R (ed.), Handbook of tensile properties of textile and technical fibres, Woodhead Publishing, Cambridge, UK, 354–436. Vlasblom M P and van Dingenen J L J (2009) ‘The manufacture, properties and applications of high strength, high modulus polyethylene fibers’ in Bunsell A R (ed.), Handbook of tensile properties of textile and technical fibres, Woodhead Publishing, Cambridge, UK, 437–485. Volk H F (1975) ‘High performance carbon fibers and cloth from pitch’ in Scala E et al. (eds), Proc. 1975 Conf. on Composite Materials, Vol. 1, 64–69. Wawner F E (1988) ‘Boron and silicon carbide fibers’ in Bunsell A R (ed.), Fibre Reinforcements for Composite Materials, Elsevier, Amsterdam, 371–426. Wilson D (2009) ‘The structure and tensile properties of continuous oxide fibers’ in Bunsell A R (ed.), Handbook of Tensile Properties of Textile and Technical Fibres, Woodhead Publishing, Cambridge, UK, 626–650. Yajima S, Okamura K, Hayashi J and Omori M (1976) J. Am. Ceram. Soc. 59, 324– 327.
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2
Carbon nanotube reinforcements for composites
A. W. K. M a, University of Connecticut, USA and F. C h i n e s t a, Ecole Centrale de Nantes, France
Abstract: Carbon nanotubes (CNTs) belong to a novel class of nanoscale fibers. Their high aspect ratio and superior physical properties make them an appealing candidate to be incorporated into polymeric materials for mechanical reinforcement and conductivity enhancement. This chapter provides an overview of the development of CNT polymer composites. We first discuss the structure and physical properties of CNTs as individual nanoscale entities. We then review different techniques available for processing them into composites and the challenges involved. Two different flow models are introduced, and we conclude this chapter by discussing the performance and potential applications of some CNT polymer composites. Key words: carbon nanotube, flow modeling, polymer composite, processing.
2.1
Carbon nanotubes (CNTs)
2.1.1
Structure and properties
Carbon nanotubes (CNTs) are rolled cylinders of graphene sheets having one or multiple layers.1 Single-walled CNTs (SWNTs), with one layer of graphene, can be metallic or semi-conducting, depending on how the graphene sheet is rolled up into a cylinder2,3 (Fig. 2.1). All existing synthesis methods produce a mixture of SWNTs with different diameters and electronic properties. This poses one of the biggest hurdles in using CNTs for electronic applications.4–6 Producing large quantities, say grams, of a single electronic type of SWNTs, through growth7,8 or subsequent separation,9,10 remains an active research area. There is always a certain length and diameter distribution in a CNT sample, meaning that CNTs are ‘polydispersed’ in terms of both diameter and length. Diameter of SWNTs typically varies from 0.4 nm to 5 nm, whereas that of multi-walled CNTs (MWNTs) varies from 1.4 nm to >100 nm. As-produced SWNTs are found as bundles (Fig. 2.2), where the SWNTs are held together by van der Waals forces,11 making it difficult to obtain a uniform dispersion with ordinary solvents and mixing methods (see Section 2.2.2). Lengths of CNTs typically vary from hundreds of nanometers 32 © Woodhead Publishing Limited, 2011
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2.1 Different types (zig-zag, chiral, and armchair) of single-walled CNTs (SWNTs) can be conceptualized as graphene sheets rolled up into a cylinder at different angles. Image produced by Michael Ströck.
to hundreds of microns, but millimeter12 and centimeter13 long CNTs have also been reported. Table 2.1 summarizes some of the physical properties of CNTs. Individual CNTs have high tensile strength (~37 GPa), high modulus (~1 TPa), and yet low density (~1.4 g/cm3). They are ballistic conductors, where no heat is dissipated during conduction. As a result, they can carry a current density of 109 A/cm2 or even higher, orders of magnitude larger than normal metals (~105 A/cm2). CNTs are semi-flexible filaments with a persistence length on the order of tens of microns,14 larger than that of rigid-rod polymers (p-phenylene terephthalamides (PPTA): 30 nm and poly(p-phenylene-2,6benzobisoxazole) (PBO): 60–120 nm), but can undergo reptation – snakelike slithering motion – in a confined environment.15 Recently, SWNT has been used as a model system to confirm the ‘reptation theory’ proposed by P. G. de Gennes for polymer. In many ways, CNTs behave like a rigid-rod
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2.2 TEM of a bundle of SWNTs produced using the high-pressure carbon monoxide disproportionation (HiPco) method. Reproduced with permission from Chemical Physics Letters, Elsevier. Table 2.1 Physical properties of CNTs
CNT
Density Tensile strength Young’s modulus Thermal conductivity Electrical resistivity Current-carrying capacity Persistence length
Refs 3
1.4 g/cm (for SWNT) 37 GPa 6, 25 1 TPa 25 1750–5800 W/m K 26 <10–4 W cm 27 >109 A/cm2 6 26–138 μm (depending on CNT diameter and number of walls) 14
polymer and have been referred to as the ‘ultimate polymer’ given their superior properties.16,17 Similar to rigid-rod polymers like PPTA and PBO or nanorods like tobacco mosaic virus, CNTs can phase transition to liquid crystals when dispersed at high concentration in liquids.18–22 From the liquid crystalline state with orientational order, CNTs can be further processed into ordered films23 and aligned fibers.21,24
2.1.2
Synthesis and costs
There are various ways of producing CNTs. CNTs can be produced by applying high voltage (known as the ‘electric arc discharge’ method25) or
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striking a high-purity graphitic source with one or two laser beams (known as the ‘laser ablation method’26). Alternatively, CNTs can be produced from catalytic reactions of a wide range of hydrocarbons.11,13,27–29 Catalyst particles can be pre-patterned onto a substrate or produced in situ during the synthesis by introducing catalyst precursor as an aerosol. The latter is also known as the ‘floating catalyst’ method.29 Different methods give different yields and different levels of purity; the diameter and length distributions of CNTs also vary accordingly. Typical impurities are fullerenes, metal catalyst residues, amorphous carbon, and graphitic particles (in the case of laser ablation-produced CNTs). Combination of techniques such as filtration, centrifugation, thermal oxidation and acid digestion has been developed to remove impurities.30–33 Caution, however, should be taken because some of these purification techniques create defects, introduce chemical functional groups, and/or shorten CNTs.34–36 Current cost data for as-produced SWNTs and MWNTs from different sources are provided in Table 2.2. Although MWNTs are usually more defective than SWNTs, MWNTs have received more attention for composite applications, mainly because of their lower material costs.
2.2
Carbon nanotube (CNT) polymer composites
2.2.1
Early development
P. M. Ajayan (formerly Université Paris-Sud, now Rice University), H. D. Wagner (Weizmann Institute of Science, Israel), and their co-workers were among the first to study CNT composites. Ajayan et al.37 embedded CNTs in epoxy and realized that some of the CNTs can be aligned by cutting the CNT epoxy composite into thin slices (50–200 nm). A few years later, there were some studies on stress transfer in CNT composite.38–41 Like conventional carbon fiber composites, Raman spectroscopy has proven to be a powerful tool in understanding the stress transfer in CNT polymer composites. Wagner et al.38 measured the compression of CNT with Raman spectroscopy during the cooling of a CNT epoxy composite, and deduced the Young’s moduli for the embedded SWNT and MWNT. Schadler et al.39 noticed a difference in Young’s modulus under tension and under compression, and hypothesized Table 2.2 Current cost data for CNTs Manufacturer
Synthesis method
Price (US$/g)
Purity
Carbon Solutions SouthWest Nano Technologies (SweNT) Nanocyl
Electric arc discharge method CO disproportionation (with Co and Mo as catalyst) Catalytic carbon vapor deposition
$50/g (SWNT) 40–60% $750/g (SWNT); >90% $5/g (MWNT) $6–$485/g Varies (SWNT, MWNT)
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that only the outer layer of an MWNT is in tension and that the difference is due to poor stress transfer into inner layers. In 1999, Calvert published a review on early development work of CNT composite.42 As succinctly pointed out in the review, several pioneers soon realized the challenges involved in the development of CNT polymer composite, namely (1) to disperse CNT bundles and (2) to improve interfacial interaction between the matrix and the CNT. As shown in Fig. 2.3, the field of CNT polymer composite expanded rapidly after 2000 as CNTs become more readily available, thanks to advances in more efficient and larger-scale synthesis. To date, there is a wealth of literature data on blending CNTs with different types of thermosets and thermoplastics; a comprehensive list of various CNT-polymer systems can be found in a recent review paper by Bose et al.43
2.2.2
Processing techniques
The first step in processing CNT polymer composite usually involves mixing CNTs into a low-viscosity solvent or directly blending them into polymer melts using a twin-screw mixer44,45 (Fig. 2.4). Nevertheless, mechanical mixing does not always guarantee a uniform dispersion, given the tendency of CNTs to aggregate.46–48 There are some techniques to facilitate the 160 145 140
Number of publications
120
115
100
89
80
70
60
54 40
40
31
20 0
140
16 1
2
4
6
5
1994 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Year
2.3 Number of journal article publications per year in CNT polymer composite from 1994 to 2010. A total of 723 article publications were obtained from Web of Knowledge using “Title= (nanotube or CNT) AND Topic= (composite and polymer)” as the search phrase.
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Carbon nanotube (Number of walls, diameter, length, polydispersity, chemistry)
Purification (Thermal oxidation, acid wash, etc.)
Polymer matrix (Chemical functionality, viscosity)
Dispersion (Sonication, high shear mixing, twin screw mixer, etc.)
Material forming (Solvent evaporation, photo/thermal curing, in-situ polymerizatin, extrusion, etc.)
Composite fibers, films (Level of dispersion, CNT alignment, physical properties)
2.4 Typical steps involved in producing CNT polymer composites.
dispersion of CNTs, namely sonication in surfactant solutions, high shear mixing, and chemical functionalization.35,49–51 Unfortunately, many of these techniques are ineffective in debundling CNTs and are destructive – they shorten the length of CNTs, disrupt the pristine sp2 hybridization, and impair their intrinsic properties. One unusual and yet effective way to disperse CNTs is to use strong acids.21,52,53 CNTs are found to be soluble in oleum (sulfuric acid saturated with SO3), chlorosulfonic acid, and their mixtures, and can be further spun into neat solid fibers.21,44,45,54 Interestingly, notable high-performance PPTA fibers (DuPont’s Kevlar® and Teijin’s Twaron® fibers) have been produced using a similar method – by spinning from highconcentration solutions of PPTA in oleum. The effectiveness of dispersion depends largely on the choice of polymer matrix (chemistry and viscosity), the chemical functionality of the CNTs, and the mixing method.39,44 Several research groups have been active in establishing the understanding of the flow behavior of CNT in a wide range of polymeric liquids or melts: Dr Petra Pötschke (Leibniz Institute), Erik Hobbie (formerly NIST, now North Dakota State University), Professor Malcolm Mackley (University of Cambridge), and Professor Matteo Pasquali (Rice University). The current understanding is that low simple shear flow induces aggregation of CNTs that are attractive in nature.46,55,56 An extreme example is the experimental observation of a highly anisotropic type of aggregates in confinement.57 High shear flow disentangles CNT aggregates
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and subsequently aligns CNTs in the shear direction,47,58,59 but compared to simple shear flow with a rotational component, extensional flows are more effective in aligning CNTs.60 One important question is how to assess the uniformity of dispersion.44 Light microscopy is useful for assessing dispersion quality at the micron scale. Although individual CNTs are nanoscale objects, they absorb light strongly,61,62 and poor CNT dispersions usually contain large aggregates that are optically resolvable.57 For assessing dispersion quality on a nanoscale, electron microscopy is needed,63,64 but a solid specimen is preferred and the results can be partial. For liquid samples, one interesting idea is to use rheology. It should be noted that judging the degree of mixing based merely on the value of the storage modulus from visoelasticity measurements can be misleading.44 The proper way to do this is to measure the (complex) viscosity as a function of mixing time.50 The minimum mixing time required for a certain mixing method is determined as the mixing time beyond which the viscosity reading becomes steady. It is best to use more than one technique to get an accurate picture of dispersion quality, given the limitations of each of them. After dispersion, a mixture of CNT and polymer melts can be extruded directly into fibers or films. For solvent-based mixing, the solvent is usually removed by evaporation, followed by an in situ polymerization process.65,66 For thermoset systems, the monomers are usually crosslinked using photoor thermal curing.47,63 It is interesting to note that CNTs absorb UV light strongly, and can increase the curing time significantly as a result.47,67 Nevertheless, like carbon black, the added CNTs can protect the composite against photo-degradation.58 There are several research groups capable of producing neat CNT fibers and films,54,68,69 and one can further impregnate these structures with polymer resin.
2.2.3
Flow modeling of functionalized and nonaggregating CNTs
Chemically functionalizing CNTs can prevent aggregation, and the flow model of a dilute or semi-dilute suspension of non-aggregating CNTs in a Newtonian matrix is similar to the one developed for short fiber suspensions.58 The momentum balance equation, neglecting the mass and inertia terms, reads: div s = 0 2.1 where s is the total stress tensor. The mass balance equation for an incompressible fluid gives:
div v = 0
2.2
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The constitutive equation for a dilute suspension with high-aspect-ratio (>1000) rigid CNTs can be written as s = –PI + 2hD + f = –PI +
2.3
where P denotes the pressure, I is the unit tensor, h is the matrix viscosity, D is the strain rate tensor (symmetric component of the velocity gradient tensor) and sf is the anisotropic viscous component of the stress tensor due to the presence of fibers. The suspension microstructure is defined by CNT orientation. It is assumed that the CNTs are ellipsoidal and rigid and that they are immersed in a Newtonian matrix whose kinematics is defined by its velocity field. If p denotes the unit vector aligned in the CNT axis direction, then its evolution will be given by the Jeffery equation:70 dp = W · p + k D · p – k (D :(p ƒ p))p dt
2.4
l 2 – 1, l is the CNT aspect ratio (length to diameter ratio), and where k = 2 l +1 Ω denotes the vorticity tensor. The tensor product ƒ of vectors a and b is defined as (a ƒ b)ij = aibj, and ‘:’ denotes the tensor product contracted twice (i.e. (a : b) = ∑∑aijbji). Although the Jeffery equation was first proposed for an isolated fiber immersed in a Newtonian fluid, it can be applied to a population of fibers or CNTs if the suspension is dilute and the interaction between CNTs is negligible. However, defining the fluid microstructure using the orientation of each CNT in the suspension is not helpful for the formulation of a mesoscale model. Instead, a more useful way of describing the microstructure is to use a kinetic theory approach which introduces an orientation distribution function y(x, p, t) such that y(x, p, t)db represents the probability of finding, at point x and time t, a CNT whose orientation is within the interval defined by p and p + dp. In this expression, the physical coordinates (x, t) (space and time) can be distinguished from the conformation coordinate, p, which describes the orientation defined on the unit surface S(0,1). This distribution function satisfies the normality condition:
ÚS(0,1) y (x, p, t ) dp = 1, "x, "t
2.5
The consideration of an orientation distribution function allows certain moments to be defined, namely the second-order moment, also known as the second-order orientation tensor: a(x, t ) =
ÚS(0,1) p ƒ p y (x, p, t ) dp
2.6
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and the fourth-order moment, also known as the fourth-order orientation tensor: A(x, t ) =
ÚS(0,1) p ƒ p ƒ p ƒ p y (x, p, t ) dp
2.7
The evolution of the orientation distribution function is governed by the Fokker–Planck equation: dy Ê ∂p ˆ Ê ∂y ˆ + ∂ y = ∂ d dt dpÁË dt ˜¯ ∂p ÁË r ∂p ˜¯
2.8
where the advection field dp/dt is given by the Jeffery equation (Eq. 2.4), and dy/dt is the material derivative, i.e. dy ∂y = + v · grad xy dt ∂t
2.9
∂y ∂x
2.10
where grad xy =
dr given in Eq. 2.8 is the rotary diffusion coefficient of the CNTs and is a key parameter to describe the balance between flow-induced alignment and orientation randomizing events such as thermal Brownian motion. For CNTs suspended within epoxy (10 Pa.s), dr was determined to be on the order of 0.005 s–1, consistent with theoretical predictions.71 Deriving a constitutive equation for non-aggregating CNT suspension involves the use of spatial homogenization and statistical averaging. Together with other simplifying hypotheses for high-aspect-ratio fibers, the stress tensor f due to the presence of CNTs is defined: Iˆ Ê t f = 2hN p (A : D) + b dr Á a – ˜ 3¯ Ë
2.11
where Np is a scalar parameter that depends on CNT concentration and aspect ratio whereas in dilute suspensions b is a parameter that depends on the number concentration of CNTs and viscous drag coefficient.72 The second term on the right-hand side of Eq. 2.11 essentially accounts for the effect of Brownian motion, but in general, other randomizing events such as CNT–CNT hydrodynamic interaction can be accounted for by replacing dr with an ‘effective diffusion term’. Neglecting the Brownian contribution gives a shear stress: 12 = hg· + 2hNpA1212g· 2.12 where A1212 is the 1.2–1.2 component of the fourth-order tensor A and we can define the apparent viscosity:
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ha =
t 12 = h(1 + 2N p A1212 ) g
41
2.13
This simple model is able to describe experimental findings as shown in Fig. 2.5. 1000
8 Dr = 0.005 s–1 Np
6 4 2
100 ha [Pa.s
Np = 7
0 0
Np = 4
0.2 Conc. [%]
0.4
10 Epoxy 0.05% CNT 0.2% CNT 0.33% CNT
Np = 0.5
1 0.01
0.1
1 · g R [s–1] (a)
10
100
100
ha [Pa.s
Np = 7 Dr = 0.01 s–1 Dr = 0.005 s–1
10 Dr = 0.0025 s–1 0.01
0.1
1 · g R [s–1] (b)
10
100
2.5 (a) Orientation model fitting of 0.05%, 0.2% and 0.33% functionalized non-aggregating CNTs suspended in epoxy resin. Np and Dr are the fitting parameters. (b) Sensitivity of the model to Dr given the experimental data of 0.33% treated CNT suspension and Np = 7. Reprinted with permission from Journal of Rheology 53(3), 547 (2009). Copyright 2009, The Society of Rheology.
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2.2.4
Flow modeling of aggregating CNTs
The orientation model described in Section 2.2.3 fails to capture the experimentally observed shear-thinning characteristic for aggregating CNT suspensions. A different model named the Aggregation/Orientation (AO) model has been developed.48 The AO model considers a hierarchy of CNT aggregate structures within an untreated CNT suspension, where the shear viscosity is controlled by the state of aggregation and CNT orientation. The Fokker–Planck description is modified to incorporate aggregation/ disaggregation kinetics and a detailed derivation for the AO model is included in this section. The aggregation/orientation distribution function in the AO model is written as y(x, t, p, n), where n Œ [0,1] describes the state of aggregation (n = 0 corresponds to CNTs that are free from entanglement and n = 1 represents a CNT aggregate network). It is assumed that the flow is steady and homogeneous and that different populations (n) are present within the control volume considered in the balance equation. In more complex flow situations where spatial flow inhomogeneity is present, the material derivative in the Fokker–Planck equation contains the orientation distribution function gradient and it does not reduce to the temporal partial derivative. Such spatial dependence does not introduce great difficulties in numerical modeling and is commonly encountered in the simulation of forming processes involving flows in complex geometries. For an arbitrary population n (where n ≠ 0 nor 1), the population increases as a result of the aggregation of smaller aggregates (r < n) or the disaggregation of larger aggregates (r > n). On the other hand, the population n decreases because of the disaggregation of population n forming less entangled aggregates (r < n) or the aggregation of n forming more entangled aggregates (r > n). The aggregation kinetics involves an aggregation velocity (vc) as well as a disaggregation velocity (vd). The resulting balance equation related to the distribution function reads:
{
} {
}
Ï ∂ ∂y (p, n ) ¸ dp ∂ Ì– ∂p y (p, n ) dt + ∂p dr (n ) ∂p ˝ Ó ˛ Ïv r =n y (p, r )dr + v r =1 y (p, r )dr ¸ dÚ Ô c Úr =0 Ô r =n +Ì =0 r =n r =1 ˝ Ô– vdy (p, n ) Ú dr – vcy (p, n ) Ú d r Ô dr r =0 r =n ˛ Ó
2.14
The distribution function y(p, n) can be obtained by solving Eq. 2.14 and the fourth-order orientation tensor can be expressed as: A(n ) = Ú p ƒ p ƒ p ƒ p y (p, n ) dp
2.15
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The stress contribution due to the presence of CNTs for a multi-population system can be obtained by generalizing Eq. 2.11:
t f = 2h
1
Ú0
N p (n )(A(n ) : D) dn
2.16
The rheological model described above provides a first approximation for describing a suspension with aggregating CNTs and contains four parameters: the concentration and aspect-ratio parameter Np, the rotary diffusion coefficient (dr) and the aggregation and disaggregation velocities (vc and vd). To simplify the analysis and minimize the degrees of freedom, the following assumptions are made, but other more complex models could be proposed: 1. Np varies linearly with the population variable n. If we assume that N pmin << N pmax and that N pmin ª 0, then Np(n) ª N pmaxn (see Eq. 2.17a below). (note: N pmin = Np (n = 0) and N pmax = Np (n = 1).) 2. In the modeling of associative polymers, a number of authors have assumed that chains within an aggregate network can undergo affine deformation and therefore dr(n = 1) ª 0. In the case of CNT suspensions, there is, however, experimental evidence showing that the opposite is true.57,61 CNT aggregates tend to rotate in simple shear flow, and CNTs within aggregate are more isotropically oriented compared with disentangled CNTs. The rotary diffusion coefficient is therefore assumed to increase linearly with the population n as given in Eq. 2.17b. 3. vd increases linearly with the shear rate and takes the form as given in Eq. 2.17c below, where g·max is a characteristic shear rate above which the suspension viscosity coincides with the matrix viscosity. At high shear rates, vd remains constant and vd = vdmax (2.17d). 4. vc decreases linearly with the shear rate and takes the form as given in Eq. 2.17e. For a shear rate higher than g·max, the aggregation velocity becomes zero (Eq. 2.17f). ÏN p (n ) = N pmax n + N pmin (1 – n) n ª N pmax n Ô max Ôdr (n ) = dr n Ô Ï Ô max Ê g ˆ if g ≤ g max Ôv (g ) = ÔÔ vd ÁË g max ˜¯ Ì Ôd Ì Ô v max if g > g max Ô ÓÔ d Ô Ï Ô max – g ˆ max Ê g max Ôv (g ) = ÔÌ vc ÁË g max ˜¯ if g ≤ g max Ôc Ô 0 if g > g max ÔÓ Ó
2.17a 2.17b 2.17c 2.17d 2.17e 2.17f
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The AO model has been tested against the experimental evolution of apparent steady shear viscosity for four different concentrations of untreated CNT suspensions (0.025%, 0.05%, 0.1%, and 0.25%), and reasonable fitting results have been obtained (Fig. 2.6).
2.2.5
Challenges and trends in modeling CNT composite systems
The models described in the previous sections (2.2.3 and 2.2.4) can capture the flow-induced microstructure evolution for CNTs suspended within a Newtonian matrix. Strictly speaking, such models should not be applied to suspensions with a viscoelastic matrix, even though one might still be able to fit the models to experimental data. For instance, it is well known in the context of short fiber suspensions that the Jeffery equation is only valid for Newtonian fluid flows. However, the equation has been practically applied to simulate processes involving thermoplastic which clearly exhibits viscoelastic behavior, and in some cases the predictions agree reasonably well with the experimental findings. Modeling the flow of CNTs through a porous medium is of high relevance to industrial forming processes such as injecting CNT/resin into preformed structures within a mold. There are two major difficulties, and they constitute an active area of research: (1) The flow induced orientation of CNT occurring on microscopic scale, within micro-channels with complex topologies, and (2) CNT concentration varies as a function of flow-path length as some of the CNTs are retained (‘filtered out’) by the micro-channels in preformed structures. Microscopic flow analysis is necessary if one is interested in evaluating the induced orientation, because the orientation distribution depends on the local flow kinematics. It is important to recall that in many applications CNTs enhance electrical conductivity, and in the case of composite laminates an out-of-plane CNTs orientation is required for obtaining the conduction in the transverse direction. Concerning the filtration of CNTs, different types of filtration kinetics have been proposed. It is commonly assumed that both the retention and the resuspension rates are proportional to the amount of CNTs passing through and retained respectively. By further assuming that resuspension of retained CNTs is insignificant, the retention will depend only on the velocity and concentration of CNTs in the flow. In addition to resin transfer molding, it is also possible to use infusion techniques with shorter flow path-lines, or direct incorporation of CNTs into pre-impregnated thermostable or thermoplastic sheets. It is important to understand and model complex CNT flows in these different forming processes to ensure a good dispersion and precise control over orientation distribution in the conformed parts.
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1 0.1
10
100
1 Shear rate (s–1] (a) 0.25% CNT
1 Shear rate (s–1] (c) 0.05% CNT
N pmax = 208 Drmax = 1 ¥ 10–4 s–1 b = 350
10 0.1
100
10
10
N pmax = 1339 Drmax = 5 ¥ 10–5 s–1 b = 182
1 Shear rate (s–1] (b) 0.1% CNT
10
10
N pmax = 556 Drmax = 3.58 ¥ 10–5 s–1 b = 222
1 Shear rate (s–1] (d) 0.025% CNT
N pmax = 23 Drmax = 0.0541 s–1 b = 907 1 0.1
10
100
10 0.1
100
1000
2.6 Aggregation/Orientation (AO) model fitting to experimental ha data at different CNT concentration levels: (a) 0.25%, (b) 0.1%, (c) 0.05% and (d) 0.025%. Experimental data are represented by unfilled circles () and the best-fit values of max max x Npmax , Drmax and b = v dma /v cmax were obtained from least-square fitting. (Npmax a and gmax in Eq. 2.17 were set to 0 and 100 –1 s respectively.) Reprinted with permission from Journal of Rheology 52(6) 1311 (2008). Copyright 2008, The Society of Rheology.
ha [Pa.s]
ha [Pa.s]
1000
ha [Pa.s] ha [Pa.s]
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2.3
Performance and applications
One appealing attribute of CNTs is their high aspect ratio (>1000), and their capability of forming a percolating network at relatively low concentration (compared to carbon black and carbon fibers).6 Sandler et al.73 achieved a conductivity of 10–2 S/m at a loading of 0.1 vol%, whilst a conductivity of 10–6 S/m is sufficient for electrostatic dissipation applications. Another potential application is to introduce a small amount of CNTs into automotive composites to facilitate the electropainting process. The electrical conductivity of CNT composites depends on type of microstructure present, which in turns depends on the the chemical pre-treatment of CNTs43 and the application of external fields (flow, electric, and magnetic) during processing.44 Hightemperature annealing can improve electrical conductivity in processed CNT polymer composite by allowing CNTs to connect better.74 This observation is in line with an argument put forward by Zakri and Poulin75 that weakly attractive CNTs are more efficient in forming a percolating network. For mechanical reinforcement, in 1999 Andrews et al.76 created a composite from CNT and petroleum pitch and reported that incorporating 5 wt% CNT increased the tensile strength and Young’s modulus by 90% and 150% respectively. Their composite fibers had a tensile strength of ~830 MPa and a modulus of ~78 GPa. A year later, P. Poulin (Université Bordeaux, France) and co-workers reported the production of high-loading CNT composite fibers by first dispersing CNTs in a surfactant solution and then condensing the CNTs in a polyvinyl alcohol (PVA) solution.77 Their composite fibers had a tensile strength of 150 MPa and a Young’s modulus of 9–15 GPa. Attempts have also been made to combine CNT with rigid-rod polymers like PBO and PPTA. Kumar et al.78 published the result of PBO/SWNT composites, and reported a 50% increase in tensile strength for PBO fibers containing 10% of SWNT. More recently, Deng et al.79 observed that CNT only marginally improved the mechanical properties of the PPTA/SWNT (by ~15%) at a low fiber draw ratio. At higher draw ratios, the orientation of polymer decreased and the mechanical properties degraded. In general, CNTs can be used for applications where carbon fibers prevail. Additionally, CNT polymer composite can potentially be used for electromagnetic interference (EMI) shielding,80 photo-degradation protection,67 fire retardation 81, sensing,82 and solar cell applications.83 Nevertheless, there are only a few examples of CNT polymer composite being currently used in commercial products, such as in sporting equipment and high-end yachts. Market penetration of CNT polymer composite is slow, mainly because of the high costs of CNTs compared with other traditional materials such as carbon fibers and carbon black. Even though a smaller amount of CNT is needed to achieve similar performance, CNT polymer composite may not pass the cost/benefit test. In the short term, lowering costs of CNT will definitely
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47
expedite the commercial application of CNT polymer composites. For highend applications, the quality and precise control over CNT diameter, length, and electronic properties still remain a grand challenge. Like the development of carbon fiber composite, continuing efforts have been made to improve the interfacial interaction between CNT and the matrix through chemical functionalization.43,84 There is also a trend of combining CNT with other nanostructures (such as graphene and fullerene) to create novel, functional 3-D composite structures.85
2.4
1. 2. 3. 4. 5. 6.
7.
8.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
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41. C. Bower, R. Rosen, L. Jin, J. Han and O. Zhou, Applied Physics Letters 74 (22), 3317–3319 (1999). 42. P. Calvert, Nature 399 (6733), 210–211 (1999). 43. S. Bose, R. A. Khare and P. Moldenaers, Polymer 51 (5), 975–993 (2010). 44. A. W. K. Ma, K. M. Yearsley, F. Chinesta and M. R. Mackley, Proceedings of the Institute of Mechanical Engineers Part N: Journal of Nanoengineering and Nanosystems 222, 71–94 (2009). 45. P. Potschke, T. D. Fornes and D. R. Paul, Polymer 43 (11), 3247–3255 (2002). 46. S. Lin-Gibson, J. A. Pathak, E. A. Grulke, H. Wang and E. K. Hobbie, Physical Review Letters 92 (4), 048302 (2004). 47. A. W. K. Ma, M. R. Mackley and F. Chinesta, International Journal of Material Forming 1 (2), 75–81 (2008). 48. W. K. A. Ma, F. Chinesta, A. Ammar and M. R. Mackley, Journal of Rheology 52 (6), 1311–1330 (2008). 49. Y. Y. Huang and E. M. Terentjev, International Journal of Material Forming 2, 63–74 (2008). 50. Y. Y. Huang, S. V. Ahir and E. M. Terentjev, Physical Review B 73 (12), 125422 (2006). 51. D. Tasis, N. Tagmatarchis, V. Georgakilas and M. Prato, Chemistry – a European Journal 9 (17), 4001–4008 (2003). 52. S. Ramesh, L. M. Ericson, V. A. Davis, R. K. Saini, C. Kittrell, M. Pasquali, W. E. Billups, W. W. Adams, R. H. Hauge and R. E. Smalley, Journal of Physical Chemistry B 108 (26), 8794–8798 (2004). 53. W. Zhou, P. A. Heiney, H. Fan, R. E. Smalley and J. E. Fischer, Journal of the American Chemical Society 127 (6), 1640–1641 (2005). 54. V. A. Davis, PhD dissertation, Rice University, 2006. 55. E. K. Hobbie and D. J. Fry, Physical Review Letters 97 (3), 036101 (2006). 56. S. S. Rahatekar, K. K. K. Koziol, S. A. Butler, J. A. Elliott, M. S. P. Shaffer, M. R. Mackley and A. H. Windle, Journal of Rheology 50 (5), 599–610 (2006). 57. A. W. K. Ma, M. R. Mackley and S. S. Rahatekar, Rheologica Acta 46 (7), 979–987 (2007). 58. A. W. K. Ma, F. Chinesta and M. R. Mackley, Journal of Rheology 53 (3), 547–573 (2009). 59. A. W. K. Ma, F. Chinesta, M. R. Mackley and A. Ammar, International Journal of Material Forming 1 (2), 83–88 (2008). 60. A. W. K. Ma, F. Chinesta, T. Tuladhar and M. R. Mackley, Rheologica Acta 47 (4), 447–457 (2008). 61. S. Lin-Gibson, J. A. Pathak, E. A. Grulke, H. Wang and E. K. Hobbie, Physical Review Letters 92 (4) (2004). 62. Z. P. Yang, L. J. Ci, J. A. Bur, S. Y. Lin and P. M. Ajayan, Nano Letters 8 (2), 446–451 (2008). 63. Z. H. Fan and S. G. Advani, Polymer 46 (14), 5232–5240 (2005). 64. P. Pötschke, A. R. Bhattacharyya and A. Janke, European Polymer Journal 40 (1), 137–148 (2004). 65. Z. J. Jia, Z. Y. Wang, C. L. Xu, J. Liang, B. Q. Wei, D. H. Wu and S. W. Zhu, Materials Science and Engineering a – Structural Materials Properties Microstructure and Processing 271 (1–2), 395–400 (1999). 66. M. Moniruzzaman and K. I. Winey, Macromolecules 39 (16), 5194–5205 (2006). 67. A. W. K. Ma, PhD dissertation, University of Cambridge, 2009.
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68. Y. L. Li, I. A. Kinloch and A. H. Windle, Science 304 (5668), 276–278 (2004). 69. M. Zhang, K. R. Atkinson and R. H. Baughman, Science 306 (5700), 1358–1361 (2004). 70. G. B. Jeffery, Proceedings of the Royal Society of London Series A – Containing Papers of a Mathematical and Physical Character 102 (715), 161–179 (1922). 71. M. Doi and S. S. Edwards, The Theory of Polymer dynamics (Clarendon, Oxford, 1986). 72. R. G. Larson, The Structure and Rheology of Complex fluids (Oxford University Press, New York and Oxford, 1999). 73. J. Sandler, M. S. P. Shaffer, T. Prasse, W. Bauhofer, K. Schulte and A. H. Windle, Polymer 40 (21), 5967–5971 (1999). 74. S. Abbasi, P. J. Carreau, A. Derdouri and M. Moan, Rheologica Acta 48 (9), 943–959 (2009). 75. C. Zakri and P. Poulin, Journal of Materials Chemistry 16 (42), 4095–4098 (2006). 76. R. Andrews, D. Jacques, A. M. Rao, T. Rantell, F. Derbyshire, Y. Chen, J. Chen and R. C. Haddon, Applied Physics Letters 75 (9), 1329–1331 (1999). 77. B. Vigolo, A. Penicaud, C. Coulon, C. Sauder, R. Pailler, C. Journet, P. Bernier and P. Poulin, Science 290 (5495), 1331–1334 (2000). 78. S. Kumar, T. D. Dang, F. E. Arnold, A. R. Bhattacharyya, B. G. Min, X. F. Zhang, R. A. Vaia, C. Park, W. W. Adams, R. H. Hauge, R. E. Smalley, S. Ramesh and P. A. Willis, Macromolecules 35 (24), 9039–9043 (2002). 79. L. Deng, R. J. Young, S. van der Zwaag and S. Picken, Polymer 51 (9), 2033–2039 (2010). 80. H. M. Kim, K. Kim, C. Y. Lee, J. Joo, S. J. Cho, H. S. Yoon, D. A. Pejakovic, J. W. Yoo and A. J. Epstein, Applied Physics Letters 84 (4), 589–591 (2004). 81. B. Schartel, P. Pötschke, U. Knoll and M. Abdel-Goad, European Polymer Journal 41 (5), 1061–1070 (2005). 82. C. Wei, L. Dai, A. Roy and T. B. Tolle, Journal of the American Chemical Society 128 (5), 1412–1413 (2006). 83. E. Kymakis, E. Koudoumas, I. Franghiadakis and G. A. J. Amaratunga, Journal of Physics D – Applied Physics 39 (6), 1058–1062 (2006). 84. P. M. Ajayan and J. M. Tour, Nature 447 (7148), 1066–1068 (2007). 85. N. Halonen, A. Rautio, A. R. Leino, T. Kyllonen, G. Toth, J. Lappalainen, K. Kordas, M. Huuhtanen, R. L. Keiski, A. Sapi, M. Szabo, A. Kukovecz, Z. Konya, I. Kiricsi, P. M. Ajayan and R. Vajtai, Acs Nano 4 (4), 2003–2008 (2010).
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3
Ceramic reinforcements for composites
J. L a m o n, CNRS/National Institute of Applied Science (INSA Lyon) France
Abstract: Ceramics need to be reinforced because of their inherent brittleness and lack of reliability. Only continuous fibers are able to arrest the cracks through deflection at fiber–matrix interfaces. Only those fibers that can withstand the high temperatures required by composite processing (>1000°C) without significant damage can be used. Efficiency of reinforcement depends strongly on interface/interphase characteristics. Features of the mechanical behavior, as well as the influence of reinforcement on composite mechanical behavior and properties, are also discussed. Key words: ceramic matrix composites, fibers, interfaces, mechanical behavior, toughness, reliability, fatigue, strength, fracture statistics.
3.1
Introduction
Ceramics need to be reinforced because of their inherent brittleness and lack of reliability. Ceramic matrix composites are built primarily for toughness. Ceramic materials exhibit favorable mechanical properties that are retained at high temperature, which make them candidates for structural use at high temperatures and in severe environmental conditions. The interesting properties are high tensile strength, very high compressive strength, high Young’s modulus and low density at temperatures up to 1500°C. Because of their low fracture toughness, currently less than 6 MPa √m, ceramics are very sensitive to microstructural flaws, which initiate cracks that propagate catastrophically. Proper reinforcement of ceramics is aimed at increasing the resistance to crack propagation by introducing elements that arrest the cracks. Only continuous fibers are able to arrest the cracks through deflection at fiber–matrix interfaces. Damage tolerance requires strong fibers and weak interfaces. Composite strength requires strong fibers and strong multifilament tows. Discrete elements like particles, whiskers and short fibers have been introduced into monolithic ceramics. They are unable to arrest the cracks that zigzag between the obstacles. Whiskers have been the most attractive discrete reinforcement, in the past. Incorporating whiskers into a ceramic matrix may improve resistance to crack growth, making the composite less sensitive to flaws. However, fracture toughness is not increased significantly 51 © Woodhead Publishing Limited, 2011
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when compared to the ceramic matrix: once a crack initiates, its propagation is catastrophic. Although whisker strengthening and toughening are limited, they can be sufficient for certain applications like cutting tools: SiC-whiskerreinforced alumina or silicon nitride. At least one successful whisker-reinforced composite is already being marketed. An SiC-whisker-reinforced alumina cutting-tool material is being used to machine nickel-based superalloys. But, there is a major drawback as whiskers are dangerous for health. Continuous-fiber-reinforced ceramics do not fail catastrophically. After matrix failure, the fiber can still support a load. Reinforcement by continuous fibers provides fracture-toughness values that may be much larger than those for metals; they may approach infinity in certain composites like C/SiC or C/C. In CMCs, only those fibers that can withstand the high temperatures required by composite processing (above 1000°C) without significant damage can be used. Other high-temperature requirements to be met include long-term high-temperature stability, creep resistance and oxidation resistance. There is a wide spectrum of continuous-fiber-reinforced ceramic matrix composites (CMCs) depending on the chemical composition of the matrix and fibers. Non-oxide CMCs reinforced by non-oxide fibers have been the most studied, because carbon and silicon carbide fibers display the highest properties for use at high temperature. Second, for compatibility reasons, non-oxide fibers can be combined essentially to non-oxide matrices. However, carbon fibers degrade in oxidizing atmosphere at temperatures as low as 450°C, and they must be protected. SiC-based fibers are much more resistant to oxidation. Oxide fibers are inherently resistant to oxidation, but they have limited creep resistance and undergo grain growth at high temperatures, which causes strength degradation. Further, they display much higher densities than carbon fibers. Despite these drawbacks, alumina-based CMCs have been extensively studied. This chapter focuses on continuous reinforcement by ceramic fibers because they are more applicable to high-temperature use: SiC, carbon or oxide fibers are capable of withstanding high temperatures (to about 2500°C for carbon fibers). Diameter is about 7–20 mm. High-stiffness and large diameter SiC fibers that consist of a thick layer of silicon carbide which has been deposited on a thin fiber substrate of tungsten or carbon were developed initially to reinforce aluminum and titanium matrices. Although they have been used as reinforcement in silicon nitride, their use in CMCs is difficult since they cannot be woven, because of their high stiffness and large diameter. Diameter of the final product is about 140 mm. The basic requirement for a tough ceramic composite, whatever the chemistry, is that cracks that initiate in the matrix do not propagate into fibers, but bypass them by deflecting into fiber–matrix debonding cracks. This requires a tailored interface in the region between fiber and matrix. Fiber–matrix interfaces or interphases (interphase is a coating on fiber aimed
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at controlling crack deflection) play an important role in fiber reinforcement. Thus, efficiency of reinforcement of fiber-reinforced ceramics depends strongly on interface/interphase characteristics. One cannot address reinforcement without discussing interface/interphase issues. In addition to developments in fibers, advances in controlling the interfacial bond between matrix and fibers have led to further mechanical property improvements of ceramic–ceramic composites. The interfacial bond must be optimized to promote favorable toughening mechanisms such as crack deflection and crack bridging. The interphase allows crack deflection to be tailored and mechanical behavior to be controlled. Without proper interface control, a brittle polyphase material results, rather than a toughened composite. Features of the mechanical behavior of CMCs, as well as the influence of reinforcement on mechanical behavior and properties, are also discussed in this chapter.
3.2
Ceramic fibers: general features
3.2.1
Oxide fibers
Oxide fibers that are commercially available are mostly based on alumina ceramics [1–4]. Table 3.1 lists the main properties of available oxide fibers. They display high values of tensile strength and Young’s modulus, and diameters as fine as 10–12 mm. As oxides they are resistant to oxidation at high temperature. However, they creep under load at temperatures above 1100°C. When subjected to high temperatures over long times, they are sensitive to grain growth. Nextel oxide fibers are the most widely used reinforcements for continuous fiber oxide–oxide composites [1–3]. Nextel 610 fiber has the highest strength and elastic modulus (3.1 GPa and 380 GPa respectively), but it is limited by creep to <1000°C. Nextel 720 fiber has lower room-temperature strength (2.1 GPa) but higher creep resistance which allows use at higher temperatures (up to 1200°C). Sapphire (single crystal Al2O3) fibers are also available. Their cost and diameter (>50 mm) limit their use in composites. Other oxide fiber types have been lightly investigated [1]. Current research activities are focused on the development of oxide fibers with enhanced creep properties and reduced grain growth rates. Oxide fibers are typically used in woven form. A limited amount of work has been completed on 3-D woven oxide–oxide composites.
3.2.2
Non-oxide fibers
Non-oxide fibers exhibit superior tensile strength and creep resistance to oxide fibers. They possess comparable Young’s modulus and diameter. Table 3.1 lists the main properties of SiC-based ceramic fibers. Non-oxide fibers are used in various 2-D or 3-D forms. © Woodhead Publishing Limited, 2011
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SiC fibers Nicalon Hi-Nicalon Tyrano-S Tyranno-SA Sylramic Sylramic-iBN
C
25 28 27 <0.3 – 15 – 5 – Other
B 2O 3 B 2O 3 – Fe3O3 SrO2, Y2O3 – – – –
Other
Nippon Carbon 58 31 11%O Nippon Carbon 63.7 35.8 <1%O UBE Industries 50.4 29.7 17.9%O, 2%Ti UBE Industries 67.8 31.3 <1%O, <2%Al Dow Corning/COI 66.6 28.5 O, B, Ti, N COI/NASA
S
63 70 73 >99 89 85 100 95 100
3M 3M 3M 3M 3M 3M Mitsui Mining Sumitomo Chemical Saphikon Inc.
Oxide fibers Nextel 312 Nextel 440 Nextel 550 Nextel 610 Nextel 650 Nextel 720 Almax Altex Sapphire
SiO2
Composition Al2O3
Manufacturer
Fiber
Table 3.1 Fiber properties
3.0 2.8 3.3 2.8 3.2 3.2
1.7 2.0 2.0 3.1 2.6 2.1 1.0 1.8 3.0
192 269 170 380 400 400
2.55 2.74 2.35 3.10 3.0 3.0
150 2.7 190 3.1 193 3.0 380 3.9 358 4.1 260 3.4 320 210 435 3.8
Strength Modulus Density (GPa) (GPa) (g/cm3)
10–20 10–20 11 7.5–10 10 10
10–12 10–12 10–12 10–12 10–12 10–12 15 10–15 70–250
Diameter (µm)
3.5 3.1 4.5 5.4 5.4
8.8 –
3.0 5.3 5.3 8.0 8.0 6.0
CTE (ppm/°C)
Ceramic reinforcements for composites
55
Lower creep rates are observed at temperatures above 1200°C, even under high stresses, whereas oxide fibers can barely exceed 1000°C [5–12]. For example, Sylramic fibers show less than 1% creep strain after 1000 hours at 1350°C and 100 MPa stress, and Hi-Nicalon type S fibers show less than 0.5% after 60 hours at 1350°C and 850 MPa stress. Creep strain of 10–8/s is obtained at 1000°C and 100 MPa stress on the most creep-resistant oxide fiber (Nextel 720), at 1400°C and 300 MPa stress on Tyranno SA3, and at 1350°C and 850 MPa on Hi-Nicalon type S. Tyranno SA3 and Hi-Nicalon type S exhibit higher resistance to creep than Hi-Nicalon. The creep resistance is commensurate with low oxygen content, SiC grain size and the small amount of amorphous phase. It has been shown to be improved after high-temperature treatment under various atmospheres. Creep behavior of Hi-Nicalon can be improved by using high-temperature treatment that eliminates the amorphous phase and organizes a better carbon structure [10].
3.2.3
Carbon fibers
Carbon fibers degrade in an oxidizing atmosphere at temperatures as low as 450°C. They are stable in non-oxidizing environments up to 2800°C. There is a wide variety of carbon fibers. They can be classified in several categories, based on Young’s modulus and strength, precursor fiber material and final heat treatment [13]. Categories based on precursor fiber material are as follows: ∑ ∑ ∑ ∑ ∑ ∑
PAN-based carbon fibers Pitch-based carbon fibers Mesophase pitch-based carbon fibers Isotropic pitch-based carbon fibers Rayon-based carbon fibers Gas-phase-grown carbon fibers.
In general, it is seen that the higher the tensile strength of the precursor the higher is the tenacity of the carbon fiber. The characterization of carbon fiber microstructure by X-ray scattering and electron microscopy techniques has shown that most carbon fibers consist of crystallites that are arranged around the longitudinal axis of the fiber [14–18]. Depending on the orientation of the layer planes with respect to the axis, Young’s modulus values span a wide range from a few GPa to more than 1000 GPa, and thermal conductivity from a few to several hundreds of Wm–1K–1 [14–19]. In high-modulus fibers, planes are highly oriented parallel to the axis. Young’s modulus decreases with increasing orientation angle. Based on Young’s modulus, carbon fibers are classified into five groups [13]:
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∑ Ultra-high-modulus, type UHM (modulus > 450 GPa) ∑ High-modulus, type HM (modulus 350–450 GPa) ∑ Intermediate-modulus, type IM (modulus 200–350 GPa) ∑ Low modulus and high-tensile, type HT (modulus < 100 GPa, tensile strength > 3.0 GPa) ∑ Super high-tensile, type SHT (tensile strength > 4.5 GPa). In PAN-based fibers, the linear chain structure is transformed to a planar structure during oxidative stabilization and subsequent carbonization. Basal planes oriented along the fiber axis are formed during the carbonization stage. Wide-angle X-ray data suggest an increase in stack height and orientation of basal planes with an increase in heat treatment temperature. A difference in structure between the sheath and the core was noticed in a fully stabilized fiber. The skin has a high axial preferred orientation and thick crystallite stacking. However, the core shows a lower preferred orientation and a lower crystallite height. Based on final treatment temperature, carbon fibers are classified into: ∑
Type I, high-heat-treatment carbon fibers (HTT), where final heat treatment temperature should be above 2000°C and can be associated with high-modulus type fiber. ∑ Type II, intermediate-heat-treatment carbon fibers (IHT), where final heat treatment temperature should be around or above 1500°C and can be associated with high-strength type fiber. ∑ Type III, low-heat-treatment carbon fibers, where final heat treatment temperatures are not greater than 1000°C. These are low-modulus and low-strength materials. Tensile strength and modulus are significantly improved by carbonization under strain when moderate stabilization is used. Overall, the strength of a carbon fiber depends on the type of precursor, the processing conditions, heat treatment temperature and the presence of flaws and defects. Carbon fibers exhibit elastic behavior at room temperature. Stiffening as the load increases has been observed on fibers with intermediate Young’s modulus (around 200 GPa), i.e. with an amount of crystallites than can align in the fiber axis direction [20]. PAN-based fibers typically buckle on compression and form kink bands at the innermost surface of the fiber. However, similar high-modulus type pitch-based fibers deform by a shear mechanism with kink bands formed at 45° to the fiber axis. Carbon fibers are very brittle. The layers in the fibers are formed by strong covalent bonds. The sheet-like aggregations allow easy crack propagation. On bending, the fiber fails at very low strain.
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3.3
Fracture strength: statistical features
3.3.1
Single fibers
57
As usual with brittle ceramics, fracture data of single filaments exhibit a significant scatter, since fracture is induced by flaws that have a random distribution. An important consequence is that the fracture stress is not an intrinsic characteristic. It is instead a statistical variable which depends on several factors, including the test method, the size of test specimens and the number of test specimens [21]. Therefore, a universal reference value of fiber fracture strength cannot be recommended. It is widely accepted that the Weibull model satisfactorily describes the statistical distribution of failure strengths of single filaments under tensile loads: P = 1 – exp{– ∫(s/s0)m dV/V0}
3.1
where P is the probability of failure, s is the stress, s0 is the scale factor, m is the Weibull modulus, V is the volume of the specimen and V0 is the reference volume (1 m3 is generally used); m reflects the scatter in data, and s0 is related to the mean value of strength. The strength for a given geometry and stress state can be determined using equation (3.1). However, m, s0 and V0 must be available. It is important to note that the estimate of s0 depends on V0 [21]. It is substantially different if V0 = 1 m3 or 1 mm3. This dependence is ignored in most papers. When V0 is not given, the estimate of s0 is meaningless. The strength cannot be determined safely. Unfortunately, reliable s0 values (characteristic strength in a few papers) cannot be recommended here, until the authors have completed their papers. The values of the Weibull modulus of most commercial carbon and ceramic fibers at room temperature are rather small (Table 3.2). They span a range from 4 to 12. Scatter in m is generally observed, inherent to the use of an estimator to construct the Weibull plot, and of limited sample size. For a single gauge length and uniform tensile stress, equation (3.1) reduces to m ÏÔ Ê ˆ ¸Ô P = 1 – exp Ì– V Á s ˜ ˝ ÔÓ V0 Ë s 0 ¯ Ô˛
3.2
The Weibull modulus, m, is determined as the slope of the ‘Weibull plot’ of ln(ln(1/1 – P)) vs. lns: ln(ln(1/1 – P)) = m ln s + k
3.3
where k is a constant. This can be done either graphically or using methods of curve fitting like linear regression analysis, the least-squares method or
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Table 3.2 Tensile properties at room temperature (gauge length = 50 mm): sr is the tow failure stress, ac is the critical fraction of individual fiber failures, mf and s0 are the statistical parameters of single filaments derived from tow tensile stress–strain curves
Mechanical characteristics
Fibers
sr(MPa)
Ef (GPa)
ac
Nicalon [49] 180 0.17 Hi-Nicalon [49] 1839 270 0.13 Hi-Nicalon S [49] 2776 420 0.14 Alumina Nextel 720 [45] Alumina Nextel 650 [45] Alumina Nextel 610 [45] Carbon PAN-based T300 [59] 2000 205 0.18 Carbon PAN-based HTA [59] 2900 220 0.05 Pitch-based carbon [59] 320 43 0.03 a
mf 4.6 6.8 7.1 7.6 6.8 10.1 4 9 12
s(MPa)a 8.4 61 99
85 120
V 0 = 1 m 3.
the maximum likelihood estimator [21]. For Weibull plots, P = (i – 0.5)/n is recommended to estimate fracture probability, where n is the number of fibers tested and i is the rank of strength for each fiber. Another method of determining the Weibull modulus is to measure fiber strength as a function of gauge length. With increasing gauge length, the chance of finding a large flaw increases, so fiber strength decreases. The effect of gauge length is given by the equation
ln s = (–1/m) lnL + k¢
3.4
where L is gauge length and k¢ is a constant. Thus, the slope of a log–log plot of strength vs. gauge length is –1/m.
3.3.2
Multifilament tows
Multifilament tows represent a fundamental entity in textile composites. They comprise several hundreds or thousands of single filaments. They progressively carry the load as the matrix is damaged, and they control ultimate failure [22]. Multifilament tows are elastic and damage tolerant, whereas single fibers are brittle. Two different types of bundle tensile behavior are observed depending upon loading conditions and fiber [23]: ∑ ∑
Either a non-linear force–strain relation associated with individual fiber breaks for carbon or certain SiC-based fiber tows under controlled deformation Or a two step non-linear force–strain relation (Fig. 3.1) for ceramic fibers under controlled deformation or load. The first step of stable failure
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160 Hi-Nicalon
140
Simulation
Force (N)
120 100 NLM 202
80 60 40 20 0
0
0.5
1 Strain (%)
1.5
2
3.1 Tensile load–strain curves for SiC-based multifilament tows showing controlled behavior (simulation) and two-step behavior (experimental results) (Hi-Nicalon and NLM 202 Nicalon fibers).
involves individual failures of a fraction of the fibers. The second step of unstable failure results from the failure of those fibers that survived during the first step. The fraction of fiber fractures at maximum force in both types of tow response controls the failure of composites. It takes a critical value ac = Nc/N0, where N0 is the initial number of fibers, Nc is the number of broken fibers at maximum load, and ac is a constant (Table 3.2). The particular fiber that fails at maximum load is referred to as the ‘critical fiber’. It has rank Nc when strength data are ordered from smallest to largest. Relationships between tow tensile behavior and fiber properties have been modeled by several authors. Daniels [24] considered tows containing parallel and non-contacting fibers. He demonstrated that tow strengths are described by a normal distribution for large numbers of fibers. Coleman [25] proposed a relationship between tow and fiber strengths and evidenced a significant drop in tow strength when compared to mean fiber strength. Phoenix and Taylor [26, 27] introduced effects of non-uniform loading resulting from fiber misalignment and scatter in fiber lengths within tows. Calard and Lamon [23] introduced random load sharing. They predicted tow strength drops when compared with the Daniels model. The bundle model Bundle models are based upon the following hypotheses [24, 25]: the bundle contains Nt identical and parallel fibers (radius Rf, length l), and the © Woodhead Publishing Limited, 2011
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statistical distribution of fiber strengths is described by the Weibull model (equation (3.2)). When a fiber fails, equal load sharing is assumed. This means that the load is carried equally by all the surviving fibers, whereas the broken fiber no longer carries any load. Under load-controlled conditions, the load that was carried by the broken fiber is shared equally by the surviving fibers which experience overloading by an increment Dsi: Ds i =
si N t – Ni
3.5
where i designates the fiber that failed, si is the stress that was operating on this fiber before failure, and Ni is the number of broken fibers. Ultimate failure occurs when Dsi > si+1 – si where si+1 is the strength of the fiber having rank i + 1, the strengths being in ascending order. at this stage, the surviving fibers are generally unable to withstand the load increment Dsi. Most of them fail catastrophically. Failure becomes unstable at maximum load. Under strain-controlled conditions, there is no overloading of surviving fibers when a fiber fails: Dsi = 0. As a consequence, failure is a stable phenomenon. The ratio a (s) of the number of broken fibers N to the total number of fibers Nt is approximately equal to the failure probability when Nt is large and when fiber failures are equally probable events:
a (s ) = N = P(s ) Nt
3.6
Consequently, the total force F(s) applied to the bundle is: m ÏÔ Ê ˆ ¸Ô F (s ) = N i(1 (1 – a (s )) ))sf s = N i sf exp Ì– V Á s ˜ ˝ ÔÓ V0 Ë s 0 ¯ Ô˛
3.7
with sf the fiber cross-sectional area. The maximum force Fmax is given by one of the following conditions, depending on the loading mode: dF = 0 (stable failure under deformation-controlled conditions) ds 3.8 a = ac (unstable failure under load-controlled conditions) 3.9 The first condition of stable failure (equation (3.8)) yields
s max
Ê ˆ = s0Ám V ˜ Ë V0 ¯
–1 m
3.10
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Ê 1ˆ a (s max ) = 1 – exp Á – ˜ Ë m¯ Fmax
Ê Vˆ = F (s max max ) = N t sf s 0 Á m ˜ Ë V0 ¯
61
3.11 –1 m
Ê 1ˆ expÁ – ˜ Ë m¯
3.12
The second condition of unstable failure (equation (3.9) yields Fmax = F(ac) = NtsfsF(1 – ac)
3.13
where sF is the strength of the tow at instability. Fmax corresponds now to tow ultimate strength. It can be shown that ac is given by equation (3.11), which implies that sF = smax. As a consequence, Fmax is also given by equation (3.12) in the presence of unstable failure. The only difference between both tow behaviors under load- or strain-controlled conditions lies in the domain beyond Fmax. To estimate scattering of the maximum force Fmax, the binomial function b(Nt, a(smax)) (or ac respectively) can be considered [28], which is naturally equal to the critical number of broken fibers Nc (statistical definition): Nc = b(Nt, a(smax))
3.14 2
Thus, the expectation E( ) and the variance s ( ) of the critical number of broken fibers Nc are deduced from the expectation and the variance of the binomial function, which finally yields the expectation and the variance of the maximum force Fmax: E(Nc) = Nt(smax) fi E(Fmax) = Nt(1 – a(smax)) sfsmax
3.15
s2(Nc) = Nt(1 – a(smax)) a(smax) fi s2(Fmax) = (sfsmax)2 Nt(1 – a(smax)) a(smax)
3.16
This allows calculation of the coefficient of variation Cv, given by Daniels [24], McCartney and Smith [29] and Gurvich and Pipes [30]: Cv (Fmax ) =
s (Fma max x) = E (Fma max x)
a (s max max ) N t (1 – a (s m max ))
3.17
An important consequence of equation (3.17) is that the coefficient of variation is small when the total number of fibers Nt is high: Nt = 500 in this paper. Therefore, according to theory the maximum forces (tow strengths) should not be scattered at all. Experimental results are at variance with theory: an unpredicted scatter in tow strength is observed (Fig. 3.2). This scatter results essentially from imperfect local load sharing attributed to fiber interactions (such as friction) or dynamic effects. Local load sharing induces a drop in tow strengths (Fig. 3.3) and an enhanced scatter in data when
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Probability of failure
0.8
0.6
0.4
0.2 ac
Tows Single filaments
0 0.3
0.4
0.5
0.6
0.7 0.8 Strain (%)
0.9
1
1.1
1.2
3.2 Statistical distributions of strains-to-failure for SiC-based single filaments and multifilament tows. 120 100 0.00
Force (N)
80 0.10 60 0.25
40 20 0
0.40
0
0.5
1 Strain (%)
1.5
2
3.3 Influence of fiber interactions by random load sharing on the tensile load strain behavior of multifilament tows (modeling) [23].
compared to equal load-sharing conditions. Imperfect local load sharing in tows involves fewer fibers than local load sharing in composites does. Fiber stress redistribution depends on the degree of interaction of the broken fiber with its neighbors: random load sharing [23]. Although tow strength data are influenced by extrinsic factors, testing of tows is an interesting technique to determine fiber properties. Taking into
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account random local load sharing allows sound predictions of tow strengths and associated scatter [23].
3.3.3
Filament–tow relations: tow-based testing methods for determination of single filament properties
It is worth pointing out that tow strength is determined by the critical number of fibers broken individually at maximum force:
s(Fmax) = s(a = ac)
3.18
whereas the average filament strength corresponds to the particular value a = 0.5: s– = s(a = 0.5) 3.19 Since a < a, s(F ) < s–. Both strengths can be determined using equation c
max
(3.20) provided that m, s0 and V0 are available; ac is derived from m using equation (3.11). When the theoretical ac is identical to the experimental value at Fmax, this indicates that there was no fiber interaction during the tests (equal load sharing). The failure characteristics of single filaments can be extracted from the tensile stress–strain curves determined on tows [31–33], or by using the standardized method described in [34] (Table 3.2). For this purpose, fiber strengths are derived from the force and the effective bundle section, taking into account the number of fibers broken individually. The number (j) of fibers broken at load Fj is derived from the compliance Cj:
Nj = N0(1 – C0/Cj)
3.20
where N0 is the initial number of intact fibers in the tow and C0 is the corresponding compliance. For the Weibull plot, P = Nj/N0 = 1 – C0/Cj can be used since the sample size is generally large, owing to the number of filaments present in a tow. Deformation instead of stress can be used (Fig. 3.2), provided an appropriate method of strain measurement is employed [33]. Using deformation presents an important advantage, since measurement of fiber sections is not required. Then, deriving stresses on fibers from strains is straightforward when Young’s modulus is available. The tow testing technique is interesting because of the significant sample size which can be obtained using a single test. However, artifacts may result from fiber misalignement. Specimen preparation and testing require much care. The degree of fiber interaction can be checked from the comparison of theoretical and experimental values of ac. Data for ac and statistical parameters are reported in Table 3.2. The
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database is not complete because authors are sometimes interested only in the Weibull modulus. Anyway, as pointed out above, available data may not be reliable. Therefore, it is recommended to determine the statistical parameters instead of using data from sources whose reliability has not been checked or established.
3.4
Mechanical behavior at high temperatures
At intermediate temperatures SiC-based fibers are sensitive to subcritical crack growth. Strength degradation of SiC single filaments starts at temperatures exceeding 1000°C [35, 36], and creep at temperatures over 1100°C [37]. It was also found that growth of a silica layer at the fiber surface and oxygen diffusion are enhanced under load [38]. Temperatures below 1000°C are referred to as intermediate. This temperature range has not been investigated on oxide fibers. Carbon fibers exhibit viscoelastic behavior at intermediate temperatures, and a viscoplastic behavior at high temperatures in inert environments [20]. The transition temperature between both regimes is commensurate with the degree of organization of carbon structure. It increases from 1200–1400°C for isotropic fibers to 2000–2200°C for highly organized fibers.
3.4.1
Strength degradation and oxidation at high temperature
Non-oxide fibers experience oxidation-related effects in the high-temperature range (above 900°C). However, this temperature may be lower for fibers that contain larger amounts of oxygen. Almost all the results found in the literature have been determined on single filaments. The authors were essentially interested in creep and strength retention after heat treatment or oxidation at high temperatures. The strength retention at room temperature generally decreases with increasing the heat treatment temperature [8, 39, 40] and the load [40]. At temperatures above 1000°C, the oxidation was found to cause the growth of SiC grains, a drop of resistivity and degradation of strength for Nicalon and Hi-Nicalon fibers [39]. At all temperatures, the formation of a uniform silicon oxide layer at the surface of fibers has been observed (Fig. 3.4). A very thin oxide film was formed at temperatures between 650 and 730°C [41]. Microstructure observations after heat treatment at 1300°C revealed that oxidation and loading accelerated new flaw nucleation and growth, resulting in stress corrosion cracks in Hi-Nicalon fibers [40]. Oxidation in air at 800–1000°C only slightly decreased the strength of Nicalon fibers, whereas a decrease of strength and no noticeable change of the Young’s modulus were reported for Tyranno fibers at lower temperatures (650–730°C) [41]. This strength decrease was related to flaw size increase after oxidation [41].
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3.4 SEM micrograph showing evidence of slow crack growth and an oxide layer on the surface of a SiC-based Hi-Nicalon fiber, after static fatigue under 250 MPa at 1000°C during 13 days.
Strength retention (MPa)
2500 2000
1500
1000
500
Monofilament Hi-Nicalon Single filament Hi-Nicalon
0 0
500
1000 Temperature (°C)
1500
3.5 Strength retention for SiC-based Hi-Nicalon single filaments and multifilament tows at elevated temperatures.
Figure 3.5 shows that Hi-Nicalon single filaments retain more than 90% of their room temperature strength at 1300°C. By contrast, the strength data measured on multifilament strands show a steep decrease from 400°C. Yun and DiCarlo also reported strength data measured on Hi-Nicalon tows at high temperatures in air [42]. They found that tensile strength degradation started at 300°C. This phenomenon results from fiber bonding by a SiO2 oxide layer that grows at the surface of fibers. E. Lara-Curzio modeled the effect of oxidation on the stress-rupture time behavior of fiber bundles and composites [43, 44]. The delayed failure of
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fiber bundles was attributed to formation of a silica layer on the surface of fibers; the thickness of this layer was introduced in place of the critical flaw size in the linear fracture mechanics equation of strength, leading to a fiber bundle strength decrease with time as t–1/4 [43, 44], which corresponds to a fiber-independent stress exponent n = 4. Figure 3.6 compares the relative high-temperature multifilament strand or roving strength of all available oxide Nextel fibers. Since the number and diameter of the fibers is different for each type of fiber, the data are presented as a percentage of room temperature strength. All fibers retain more than 90% of their room-temperature strength at 800°C. Nextel 720 showed no degradation in strand up to 1100°C. Strength degradation is related to creep resistance. Nextel 720 comprises a majority of large 0.5 mm creep-resistant mullite grains; as a result, Nextel 720 fiber has 150–200°C higher temperature capability than other Nextel fibers [45]. With PAN-based carbon fibers, the strength increases up to a maximum at 1300°C and then gradually decreases in inert atmosphere, whereas the elastic modulus has been shown to increase with increasing temperature [20].
3.4.2
Static fatigue under constant load at intermediate temperatures: subcritical crack growth
Several researchers have demonstrated since the 1960s the sensitivity of refractory materials to slow (subcritical) crack growth [46, 47]. Slow crack
Strength retention (%)
100
80
60
40 610 650 720
20
Strain rate = 0.68 mm/min
0 0
200
400
600 800 1000 Test temperature (°C)
1200
1400
3.6 Comparison of relative tensile strength retention of multifilament strands of Nextel 312, 440, 550, 610, 650 and 720 fibers at elevated temperatures [45].
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growth leads to failure, which implies that the strength is time dependent at high temperatures. Delayed failure is observed for SiC-based fibers under constant load at intermediate temperatures (Fig. 3.7) [48, 49]. It is indicated by long rupture times under stresses much lower than the strength at room temperature (Fig. 3.7). The data on single filaments exhibit a significant scatter, the rupture times seeming to decrease when the stresses are increased (Fig. 3.8). But the 10000000 Tows Single filaments
1000 100
100000 1 day
10
Hours
Rupture time (s)
1 month 1000000 1 week
10000 1
1h 1000
0.1 100 1 min
0.01
10 100
600
1100 Applied stress (MPa)
1600
2100
3.7 Stress–rupture time data determined on single SiC-based filaments (Hi-Nicalon) at 800°C. 10000000
1000
1000000
g = 1.05E+30x–0.45
10
10000
1
1000 100 10 1 100
100
0.1 Tow 500°C Two 800°C Single fiber 500°C Single fiber 800°C 300
500
Lifetime (h)
Lifetime (s)
100000
0.01 g = 3.36E+26x–0.34
0.001
700 900 1100 1300 1500 1700 1900 Applied stress (MPa)
3.8 Stress–rupture time diagrams for Hi-Nicalon single fibers and multifilament tows at 500°C (under 1500 MPa) and 800°C (under 1100 MPa).
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trend is not as clear as with the tows. This statistical variability in rupture times is similar to that observed with bulk ceramic specimens, which results from the presence of different critical flaws within the fibers. The stress–rupture time data obtained for tows align on a curve that the following power law fits satisfactorily:
tsn = A
3.21
where t is the rupture time, s is the applied stress, and n and A are constants depending respectively on material and environment. The above power law often describes non-linear time-dependent responses such as that dictated by slow crack growth activated by the environment under low stresses in ceramics [50]. However, the response of tows must be regarded as a remarkable feature, since it is at variance with that observed for bulk ceramics and that of single fibers for which the experimental stress–rupture time data do not range in a single line, so that no monotonous trend is shown. The constant A depends on the initial flaw size. It takes the same value for those specimens which possess flaws with the same initial size [50]. The response of tows can be related to the critical fiber-dictated failure mode, as discussed above. When ac is a constant, the critical fibers correspond to identical initial flaw sizes. A and n are estimated from the stress–rupture time diagram using a regression technique (Table 3.3). The stress exponent n does not display temperature dependence. It seems to depend on fiber. A depends on temperature. Close values of n were estimated: 8.4 for the Hi-Nicalon tows and 7.2 for the HiNicalon S ones (Table 3.3); n was smaller (n = 2.8) and A was larger (A = 2 ¥ 1029 MPan s at 600°C) for Nicalon tows [51]. This indicates that the Hi-Nicalon and Hi-Nicalon S tows are less sensitive to slow crack growth than the Nicalon ones. The delayed failure of SiC-based fiber bundles at temperatures below 800°C results from the subcritical crack growth of the surface defects by the oxidation of the grain boundaries (free carbon) and the SiC nanograins or silicon oxycarbide at the crack tip [38]. Both phenomena may contribute simultaneously (Nicalon fibers) or sequentially (Hi-Nicalon S and SA3 fibers, Table 3.3 Subcritical crack growth constants for low oxygen content SiC-based Hi-Nicalon and Hi-Nicalon S fibers in ambient air at elevated temperatures T (°C)
Hi-Nicalon tows n
Hi-Nicalon S tows n
A (MPa s)
500 8.45 1.05 ¥ 1030 600 800 8.34 3.36 ¥ 1026
n
A (MPan s)
7.25 7.24
3.15 ¥1026 3.33 ¥ 1024
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which contain a little free carbon not connected). Growth of a silica layer at the fiber surface is not responsible for fracture. In the presence of free carbon, the crack length increase is attributed to the consumption of free carbon by oxidation. In the presence of silicon carbide or silicon oxycarbide, the silicon oxide formed at the defect/crack tip causes a local volume increase (by a factor of 2.1 for SiC and by less for SiOxCy) which induces a tensile stress Ds. Considering the atomic concentrations of free carbon, those fibers with the largest fractions of free carbon are the less resistant to static fatigue:
Tyranno SA3 ª Hi-Nicalon S < Hi-Nicalon ª Nicalon
(ª2%)
(ª3%)
(ª17%)
(ª15%)
∑
Nicalon fiber contains large amounts of very reactive free carbon (≈15 at%), and very small b-SiC nanograins (~5 nm) embedded in silicon oxycarbide. These elements are very sensitive to oxidation due to their structure (free carbon) or their size (Valhas and Laanani demonstrated that the smaller the grains, the faster they oxidize [52]). In consequence both mechanisms can operate simultaneously and this is consistent with the small resistance to fatigue of this fiber. ∑ Hi-Nicalon fiber is essentially made of bigger b-SiC nanograins (~10 nm) and a free carbon network (the electrical conductivity is high: about 1 S/cm, compared to the Hi-Nicalon S and Nicalon fibers: 10–5 S/cm). The oxidation of this network from the surface defects should be the predominant mechanism of slow crack growth. As a consequence, longer rupture times would be expected when compared to Nicalon fibers. ∑ Hi-Nicalon S fiber contains much bigger b-SiC nanograins (~20 nm) without the silicon oxycarbide, and a little free carbon located at grain boundaries and not connected. The oxidation of the b-SiC nanograins would contribute to slow crack growth. The resistance to static fatigue is greater than that of Hi-Nicalon fiber because free carbon is not connected and the SiC grains are bigger. ∑ Tyranno SA3 fiber possesses the same amount of free carbon in the surface as the Hi Nicalon S fiber, but the difference lies in the SiC grain size: 50–100 nm versus 20 nm for the Hi-Nicalon S fiber. The longer rupture times exhibited by the SA3 fiber can be attributed to the presence of much bigger grains, which oxidize more slowly than the smaller ones present in the other SiC-based fibers [52]. In this case the oxidation rate of the SiC grains would determine slow crack growth. Both phenomena of oxidation of SiC grains and free carbon may contribute sequentially to slow crack growth in the Hi-Nicalon S and Tyranno SA3 fibers. Finally, the lifetime–microstructure relationships can be highlighted by comparing rankings of resistances to static fatigue, and microscopic features which control the reactivity of constitutive elements:
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Lifetime: Tyranno SA3 > Hi-Nicalon S > Hi-Nicalon > Nicalon
Free carbon: (ª2%)
Grain size: (ª50–100 nm) (ª20 nm)
(ª3%)
(ª17%)
(ª15%)
(ª10 nm)
(ª5 nm)
In order to improve the fatigue resistance of these SiC-based fibers in air at intermediate temperatures between 400 and 800°C, the manufacturers will have to reduce drastically the amounts of oxygen and free carbon near the fiber surface and also to eliminate the surface defects, in order to limit the oxidation.
3.4.3
Creep at high temperatures
Figure 3.9 compares the creep rate of oxide Nextel fibers at 1100°C [45]. Creep differed by a factor of 10–100 times between each type of fiber. The stress exponent for Nextel 720 and 610 is 1.8 less than for the other fibers. Nextel 720 fiber has the best creep resistance than any commercially available polycrystalline oxide fiber. This effect was attributed to the presence of creep-resistant mullite grains. However, further reductions in creep of oxide fibers are possible, as shown on YAG fibers. Typical creep curves that were obtained for near-stoichiometric SiCbased fibers are shown in Fig. 3.10 [12]. Steady-state creep was observed after a more or less long primary creep stage, depending on the fiber: after 0.00001
Test temperatures 1100°C
Strain rate (s–1)
0.000001
Nextel 610
1E-07
Nextel 650
1E-08
Nextel 720 (1200C)
1E-09
Nextel 720
1E-10 10
100 Load (MPa)
1000
3.9 Comparative creep rate of Nextel fibers at 1100°C as a function of stress [45].
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3.5 3
Strain (%)
2.5 2 1.5 1 0.5
Experiment Prediction
0 0
48
96
144 192 Time (h) (a)
240
288
336
0.7 0.6
Strain (%)
0.5 0.4 0.3 0.2 0.1
Experiment Prediction
0 0
12
24
Time (h) (b)
36
48
60
3.10 Creep curves obtained on (a) Hi-Nicalon and (b) Hi-Nicalon S SiC-based fibers at 1350°C, and under (a) 850 and (b) 500 MPa: comparison with prediction.
about 140 h for Hi-Nicalon fibers at 1200°C, about 72 hours for SA3 fibers at 1200°C, about 8 hours for SA3 fibers at 1250°C, and about 8 hours for Hi-Nicalon S at 1350°C. The creep results reported by most authors were obtained during much shorter tests (<48 hours). Thus, it may be anticipated
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that certain tests were not sufficiently long, so that a true secondary creep stage was probably not reached. Creep curves obtained under incremental temperature steps show evidence of tertiary creep at temperatures above 1600°C [12] (Fig. 3.11). Creep curves are described by the following well-accepted equations of deformations in the primary and secondary stages (Fig. 3.10):
ee = s E0
3.22
ep = sA[1 – exp(–pt)]
3.23
es = bsnt
3.24
e = ee + ep + es
3.25
where subscripts e, p and s refer respectively to elastic regime, primary and secondary creep, s is the stress on the fiber, E0 is the initial fiber Young’s modulus, t is time, and A, b, n and p are constants. Based on microstructure analysis, the fibers can be considered to be a mixture of wrinkled carbon-layer packets and SiC grains. Possible controlling creep mechanisms may involve grain boundary sliding, carbon diffusion, dewrinkling, deformation and sliding of carbon crystallites [37]. Primary creep can be attributed to viscoelastic deformation of carbon at grain boundaries. Secondary creep of low oxygen content fibers (n ≈ 2.5) may be attributed to grain boundary sliding, without grain elongation and a glassy phase (Rachinger mechanism), and diffusion of al, C or Si at grain 8 7
1700°C
6
Strain (%)
5 1650°C
4 1600°C 1550°C
3 1500°C 2
1450°C
1400°C
Test stop for SEM observation
1350°C
1 0 0
12
24
36
48 Time (h)
60
72
84
96
3.11 Creep of SiC-based SA3 fiber under a stress of 150 MPa and in the 1350–1700°C temperature range.
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boundaries in SA3 fiber and diffusion of carbon or silicon within grain in Hi-Nicalon S fiber. In polycrystalline ceramics, accommodation results from diffusion and fold formation at triple junctions [53]. In SiC fibers, it probably involves carbon deformation. Tertiary creep was shown to be due to an increase in stress as the load-bearing fiber area is reduced by volatilization of Si.
3.5
Fiber–matrix interfaces: influence on mechanical behavior
The fiber–matrix interfacial domain is a critical part of composites, because load transfers from the matrix to the fiber and vice versa occur through the interface. Interface characteristics depend on the fiber’s ability to be more or less strongly bonded to the matrix or to a coating. The interface must not be too strong in order to obtain crack deflection by fibers, although excessively weak interfaces are detrimental to composite strength. High strength requires efficient load transfers from fibers to the matrix. This is obtained with rather strong interfaces, provided deflection can occur (Fig. 3.12). Efforts have been directed towards optimization of interface characteristics with respect to composite behavior, through control of the fiber–matrix bond. In CVI SiC/SiC composites, fiber matrix interfaces consist of a thin Matrix crack
Fiber/matrix interphase Strong interface cohesive failure mode
Weak interface adhesive failure mode
Load transfers Limited debonding
Energy absorption Important debonding Fiber pullout Sliding friction
Multiple matrix cracking
Strengthening
Toughening
3.12 Schematic diagram summarizing relationships between degree of fiber–matrix bonding and composite strength and toughness.
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coating layer (less than 1 mm thick) of one or several materials deposited on the fiber (interphase). Enhanced interfaces have been obtained using fibers that have been treated in order to increase the fiber–coating bond [54, 55]. The concept of enhanced interfaces has been established on CVI SiC/SiC composites with SiC-based fibers, and PyC or multilayered (PyC/SiC)n fiber coatings. Less interesting results have been achieved with BN interphases [56] or carbon fibers. In CVI SiC/SiC composites with PyC-based fiber coatings, the interfacial shear stress (which reflects the strength of the fiber–matrix bond) ranges between 10 and 20 MPa for the weak interfaces obtained with as-received fibers, and between 100 and 300 MPa for the strong interfaces obtained with treated fibers. In the presence of weak fiber–coating bonds, the matrix cracks generate a single long debond at the surface of fibers (adhesive failure type, Fig. 3.13). The associated interface shear stress is low, and load transfers through the debonded interfaces are poor. The matrix is subjected to low stresses and the volume of matrix that may experience further cracking is reduced by the presence of long debonds. Matrix cracking is not favored. The crack spacing distance at saturation as well as the pullout length tends to be rather long (>100 mm). Toughening results essentially from fiber–matrix sliding friction over the interface crack. However, as a result of matrix unloading, the fibers carry most of the load, which reduces the composite strength. The corresponding tensile stress–strain curve exhibits a narrow curved domain limited by a stress at matrix saturation which is smaller than ultimate strength (Fig. 3.14). In the presence of stronger fiber–coating bonds, the matrix cracks are deflected within the coating (cohesive failure type, Fig. 3.13) into short and branched multiple cracks. Short debonds as well as improved load transfers allow further cracking of the matrix. Sliding friction within the coating as well as multiple cracking of the matrix increases energy absorption, leading to toughening. Limited debonding and improved load transfers reduce the load carried by the fibers, leading to strengthening. The associated tensile stress–strain curve exhibits a wide non-linear domain and the stress at matrix cracking saturation is close to ultimate failure (Fig. 3.14).
3.6
Mechanical behavior of composites: influence of fibers and interfaces
Fibers deeply influence the mechanical behavior of ceramic matrix composites depending on various parameters: ∑
Elastic properties with respect to matrix properties. It is worth pointing out that there are a wide variety of fibers and matrices with elastic modulus ranging from low values (roughly, a few GPa) to very large ones (≈400
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I
I
Microcrack
M
(SiC/PyC)n interphase
F
I
Microcrack
Microcrack
M
PyC interphase
F
I
Microcrack
M
(SiC/PyC)n interphase
F
(b) Strong F/I bond
3.13 Schematic diagrams showing crack deflection (a) at the fiber surface with as-received fibers which generate rather weak interfaces, and (b) within the coating with treated fibers which are more strongly bonded to the coating.
(a) Weak F/I bond
Microcrack
M
PyC interphase
F
76
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Stress (MPa)
300 (b) IV 200
III
II
100
I 0 0
0.2
0.4
0.6 0.8 Deformation (%)
1
1.2
3.14 Tensile stress–strain curves obtained with (a) treated fibers (enhanced interface) and (b) as-received fibes (weak interfaces) in SiC/SiC composites with a pyrocarbon coating on fibers.
GPa for SiC-based fibers or matrices, and ≈700 GPa for carbon fibers). This situation is quite different from that prevailing in organic matrix composites for which the matrix elastic modulus is always ≈5 GPa, whatever the matrix is. ∑ Ability to be more or less strongly bonded to the matrix as discussed in the previous section. ∑ Resistance to fracture, and statistical distribution of strength data.
3.6.1
Tensile stress–strain behavior
Damage-sensitive stress–strain behavior is obtained when the contribution of the matrix to load carrying is initially significant, as observed when the elastic modulus of fibers (Ef) is comparable to the matrix one (Em). Then, Em degrades as matrix damage proceeds. Damage consists of transverse cracks with respect to fibers oriented in the loading direction. Damage-sensitive behavior exhibits non-linear stress–strain relationships. Saturation of matrix damage is indicated by a point of inflection, at transition between domains II and III in Fig. 3.14. Then the ultimate portion of the curve reflects the deformation of fibers (domain IV in Fig. 3.14). Fiber failures may initiate prior to ultimate fracture. The influence of matrix elastic modulus is illustrated by the mixtures law, which provides satisfactory trends for continuous fiber-reinforced composites:
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E c = E mV m + E fV f
77
3.26
where Ec is the composite elastic modulus, Vm is the volume fraction of the matrix, and Vf is the volume fraction of fibers oriented in the loading direction in a two-dimensional woven composite. In two-dimensional (2D) SiC/SiC and C/SiC composites, the matrix is stiffer than the fibers: Em (~410 GPa) > Ef (200–400 GPa), Vm ~ Vf. It carries a significant part of the applied load. In C/C composites, the fibers are much stiffer than the matrix: Em (~30 GPa) < Ef (200–400 GPa), Vm ~ Vf. The fibers carry most of the load. The matrix is subject to low stresses. As shown by equation (3.27) derived from (3.26) for small Em, the behavior is dictated by fibers. The stress–strain curve is essentially linear, as it reflects a damage-insensitive behavior.
E c ~ E f V f
3.27
In those damage-sensitive unidirectional composites under on-axis tensile loads, the basic damage phenomena involve multiple microcracks or cracks that form in the matrix, perpendicular to the fiber direction, and that are arrested by the fibers at fiber–matrix interfaces. In composites reinforced with fabrics of fiber bundles, matrix damage is influenced by the multilength scale structure [22]. In 2D SiC/SiC and C/SiC composites, damage consists essentially of the formation of cracks between the longitudinal tows perpendicular to the fiber axis and their deflection by the tows and, in a second step, of matrix cracks within the longitudinal tows and their deflection by the filaments. In 2D C/SiC composites, the first step of damage initiates during cooling down from the processing temperature as a result of the large coefficient of thermal expansion of the carbon fiber compared to that of the SiC matrix. The major modulus loss (70% in 2D SiC/SiC) is caused by the first families of cracks located on the outside of the longitudinal tows (deformations < 0.2%). By contrast, the microcracks within the longitudinal tows are responsible for only a 10% loss. The substantial modulus drop reflects important changes in load sharing: the load becomes carried essentially by the matrix-coated longitudinal tows (tow reloading). During microcracking in the longitudinal tows, load sharing is affected further, and the load becomes carried essentially by the filaments (fiber reloading). The elastic modulus reaches a minimum described by the following equation:
1
Emin = 2 EfVf
3.28
where Vf is the volume fraction of fibers oriented in the loading direction. Equation (3.28) reflects the fact that the mechanical behavior is fully controlled by the fiber tows oriented in the direction of loading (Fig. 3.15).
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Composite reinforcements for optimum performance Longitudinal tow Macropore
Transverse tow
Layer
0.5 mm
3.15 Micrograph showing the multiple length scales structure of 2D CVI SiC/SiC composite.
3.6.2
Damage tolerance and fracture toughness
Damage tolerance and resistance to crack propagation depend strongly on fibers, first through interfaces, and second through fiber strength. The density of matrix cracks is enhanced by rather strong interfaces: the crack spacing distance may be as small as 10–20 mm whereas the crack spacing distance is at least 10 times larger in the presence of rather weak interfaces. In the presence of a notch or a pre-existing main macroscopic crack, a process zone of matrix microcracks is generated at the tip of the notch or the crack. As indicated in a previous section, crack density is commensurate with interface strength. Extension of the main crack results from the random failures of the fiber bundles located within the process zone [54]. Matrix cracking is an alternative mechanism of energy dissipation. Strain energy release rate values ranging from 3 to 8 kJ/m2 have been determined on CVI SiC/SiC composites respectively with weak or strong interfaces [54]. The corresponding values of the J-integral ranged from 11 kJ/m2 (weak interfaces) to 29 kJ/m2 (strong interfaces). These values are quite high by at least two orders of magnitude when compared to the SiC matrix.
3.6.3
Fatigue behavior
During cyclic fatigue at room temperature, matrix damage appears during the first cycles. Fatigue resistance is governed by the damage of fibers and
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fiber–matrix bonds. Owing to fiber resistance, composites like 2D SiC/SiC are insensitive to fatigue at room temperature, and under low loads. At high temperatures or in aggressive environments, the fibers may be sensitive to strength degradation by corrosion or by subcritical crack growth. As a consequence, composite resistance to fatigue decreases.
3.6.4
Ultimate failure
Ultimate failure generally occurs after saturation of matrix cracking, when the load is carried by the longitudinal tows. The fibers break when the applied load is close to maximum. Matrix damage and ultimate failure are successive phenomena. Ultimate failure is controlled by the tows. The ultimate failure of a tow of parallel fibers involves two steps, as discussed above: ∑ ∑
a first step of stable failure a second step of unstable failure.
During the first step, the fibers fail individually if the load is increased. In the absence of fiber interactions, the load is carried by the surviving fibers only (equal load sharing). Fiber interactions cause tow weakening leading to premature failures. The ultimate failure of a tow (second step) occurs when the surviving fibers cannot tolerate the load increment resulting from a fiber failure. The ultimate failure of a longitudinal tow coated with matrix results from the same two-step mechanism and involves global load sharing when a fiber fails. In the presence of multiple cracks across the matrix and associated interface cracks, the load-carrying capacity of the matrix is tremendously reduced or annihilated. The matrix-coated tows behave like dry tows subject to the stress state generated by the presence of matrix cracks. The ultimate failure of a matrix-coated tow occurs when a critical number of fibers have failed. This mechanism operates in the tows within textile CVI SiC/SiC composites. The ultimate failure of a composite is caused by the failure of a critical number of broken tows (≥1), depending on the stress state: ~1 under uniaxial tension, >1 in bending. It is worth pointing out that such a failure mechanism differs from that observed in polymer matrix-impregnated tows, where the fibers fail first and then local load sharing prevails when a fiber fails.
3.6.5
Reliability
Fiber stochastic features influence ultimate failure of ceramic matrix composites. Figure 3.16 shows that strength magnitude and scatter decrease from single fibers to tows, then to infiltrated tows and finally to woven composites.
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Density (1/MPa)
0.01 0.008 0.006
Composites (1D)
0.004 Tows
0.002 0 0
Fibers 500
1000 1500 Stress (MPa)
2000
2500
3.16 Strength density functions for SiC fibers (NLM 202), SiC multifilament tows, SiC/SiC (1D) minicomposites and 2D SiC/SiC composites.
As a result of the previously mentioned two-step failure mechanism, the ultimate failure of an entity is dictated by the lowest extreme of the strength distribution pertinent to its constituent, i.e. tows versus filaments, infiltrated tows versus fibers, and 2D composites versus infiltrated tows. The lowest strength extremes correspond respectively to the critical number of individual fiber breaks (ac ~ 17% for SiC-based NicalonTM and Hi-NicalonTM fibers) and to the critical number of tow failures (≥1 depending on stress state). The gap between tows and SiC-infiltrated tows results from strength determination: tow strength corresponds to the number of fibers carrying the load (effective cross-sectional area) whereas the strengths of infiltrated tows and composites correspond to the total cross-sectional area of specimens. During the successive damage steps, the flaw populations are truncated, which leads to a homogeneous ultimate population of flaws in the longitudinal tows at ultimate failure [57]. This progressive elimination of flaws strongly influences ultimate failure characteristics. Composite strength data exhibit a limited scatter and an insignificant dependence on the stressed volume and loading conditions (Figs 3.17 and 3.18). Thus, the flexural strength is 1.15 times as large as the tensile strength [57, 58] when measured on specimens having comparable sizes. The Weibull model is not appropriate to describe the volume dependence of strength data [57], since the weakest link concept is violated due to the presence of the cumulative damage process. However, the Weibull modulus (m) can be extracted from the statistical distribution of strength data: m is in the range 20–29 for 2D woven SiC/SiC composite. This value provides an index of the scatter in strength data. It reflects a small scatter.
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1
Probability
0.8
0.6
0.4
0.2
0 200
250 300 Ultimate stress (MPa)
350
3.17 Statistical distribution of strength data for 2D woven SiC/SiC composite. Influence of specimen dimensions on ultimate failure in tension: d 8 ¥ 30 mm2 s 160 ¥ 120 mm2 [57]. 1 Tensile tests
Failure probability
0.8
Three-point bending tests
0.6
Four-point bending tests
0.4
0.2
0 260
280
300 320 340 Maximum stress (MPa)
360
380
3.18 Strength distributions for 2D woven SiC/SiC composites tested under various loading conditions: tension, three-point bending and four-point bending [57].
3.7
Conclusion
Using continuous refractory fibers is the most efficient way to obtain strong and tough ceramic matrix composites. Owing to a wide variety of potential
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matrix materials, the fibers must meet several requirements. They must withstand the high temperatures required by composite processing above 1000°C. Their thermoelastic and fracture characteristics must compare favorably with those of the matrix. The influence of fibers on the mechanical behavior and on the properties of a composite depend on the matrix properties. The thermoelastic properties influence the stress-strain behavior. When the matrix is stiffer than the fibers (as typified by SiC/SiC and C/SiC composites) the stress–strain behavior is non-linear and damage sensitive. The matrix is able to share the applied load with the fibers, so that load transfers from the fibers to the matrix have to be sought. Toughening and strengthening depend on load transfers through fiber–matrix interfaces, i.e. the degree of bonding of fiber to the matrix. When the matrix is less stiff than the fibers (certain C/C composites) the behavior is damage insensitive. Most of the load is carried by the fibers, as in organic matrix composites. Improvement of damage tolerance is not the primary requirement to be met with such damage-insensitive composites. The thermal expansion of the fiber must also be matched to that of the matrix, since ceramic matrix composites are made at high temperatures. Fiber resistance to fast or delayed fracture determines composite ultimate strength under monotonous or fatigue loading. Fiber strength may be affected by temperature and environment, as a result of various phenomena like subcritical crack growth, creep and oxidation. Fast fracture data exhibit a statistical distribution which follows satisfactorily the Weibull model. The scatter in strength data is tremendously reduced through a truncation process when using multifilament tows. The multifilament tows are a fundamental entity in textile composites. They control ultimate failure and determine scatter in composite strength. They exhibit damagetolerant behavior which depends on the statistical distribution of filament strengths. Tow characteristics can be tailored with respect to composite strength. In damage-sensitive composites, the bonding of fibers to the matrix is of primary importance for the deflection of matrix cracks and for subsequent load sharing. Fiber–matrix interface characteristics depend on the fiber used. By applying appropriate treatment to the fiber surface, fiber–matrix bonding can be enhanced, leading to stronger and tougher composites.
3.8
References
1. K. A. Keller, G. Jefferson, R. J. Kerans, ‘Progress in oxide composites’, Annales de Chimie Science des Matériaux, 30, 659–671 (2005). 2. B. Clauss, ‘Fibers for ceramic matrix composites’, in: Krenkel W. (ed.) Ceramic Matrix Composites: Fiber-reinforced Ceramics and their Applications, Chapter 1, pp. 1–19, Wiley-VCH, Weinheim, Germany, (2008).
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3. K. A. Keller, G. Jefferson, R. J. Kerans, ‘Oxide–oxide composites’, in: Bansal N. P. (ed.) Handbook of Ceramics and Glasses, Kluwer Academic Publishers, New York, pp. 377–421 (2005). 4. A. Bunsell, ‘Oxide fibers’, in: Bansal N. P. (ed.) Handbook of Ceramics and Glasses, Kluwer Academic Publishers, New York, pp. 3–31 (2005). 5. J. A. DiCarlo, H.-M. Yun, ‘Non-oxide (silicon carbide) fibers’, Bansal N. P. (ed.) Handbook of Ceramics and Glasses, Kluwer Academic Publishers, New York, pp. 32–53, 2005. 6. H.-M. Yun, J. C. Goldsby, J. A. DiCarlo, ‘Tensile creep and stress-rupture behavior of polymer derived SiC fibers’, in: Singh J. P. and Bansal N. P. (eds) Advances in Ceramic-Matrix Composites II, pp. 17–28, American Ceramic Society, Westerville, OH, (1995). 7. H.-M. Yun, J. A. DiCarlo. ‘Comparison of the tensile, creep, and rupture strength properties of stoichiometric SiC fibers’, report NASA/TM-1999–209284 (1999). 8. A. R. Bunsell, A. Piant, ‘A review of the development of three generations of small diameter silicon carbide fibers’, J. Mater. Sci., 41, 823–839 (2006). 9. R. Bodet, X. Bourrat, J. Lamon, R. Naslain, ‘Tensile creep behavior of a silicon carbide-based fiber with a low oxygen content’, J. Mater. Sci., 30, 661–677 (1995). 10. G. Chollon, R. Pailler, R. Naslain, P. Olry, ‘Correlation between microstructure and mechanical behaviour at high temperatures of a SiC fiber with a low oxygen content (Hi-Nicalon)’, J. Mater. Sci., 32, 1133–1147 (1997). 11. J. A. DiCarlo, H.-M. Yun, ‘Microstructural factors affecting creep-rupture failure of ceramic fibers and composites’, in: Sheppard L. (ed.) Ceramic Material systems with Composite structures, pp. 119–134, American Ceramic Society, Westerville, OH (1998). 12. C. Sauder, J. Lamon, ‘The tensile creep behavior of SiC-based fibers with low oxygen content’, J. Am. Ceram. Soc., 90(4), pp. 1146–1156 (2007). 13. R. R. Hegde, A. Dahiya, M. G. Kamath, ‘Carbon fibers’, MSE 554, Nonwovens Science and Technology II, Spring 2004, http://web.utk.edu/~mse/pages/textiles/ carbon%20fibers.htm. 14. A. Oberlin, J. Goma, J. N. Rouzaud, Chimie Physique, 81, 701–710 (1984). 15. D. J. Johnson, J. Phys. D: Appl. Phys., 20(3), 287–291 (1987). 16. X. Bourrat, E. J. Roche, J. G. Lanvin, Carbon, 28(2/3), 435–446 (1990). 17. R. Bacon, W. A. Schalamon, J. Appl. Polym. Sci., Appl. Polym. Symp., 9, 285 (1969). 18. B. T. Kelly, Physics of graphite, Applied Science Publishers, London, pp. 196–222 (1981). 19. C. Sauder, J. Lamon, ‘Prediction of elastic properties of carbon fibers and CVI matrices’, Carbon, 43, 2044–2053 (2005). 20. C. Sauder, J. Lamon, R. Pailler, ‘The tensile behavior of carbon fibers at high temperatures up to 2400°C’, Carbon, 42, 715–725 (2004). 21. J. Lamon, Mechanics of brittle fracture and damage: Statistical probabilistic Approaches (in French, Mécanique de la rupture fragile et de l’endommagement: Approches Statistiques-Probabilistes), Editions Hermès-Lavoisier, Paris, 2007. 22. J. Lamon, ‘A micromechanics-based approach to the mechanical behavior of brittlematrix composites’, Comp. Sci. Technol., 61, 2259–2272 (2001). 23. V. Calard, J. Lamon, ‘Failure of fiber bundles’, Comp. Sci. Technol., 64, 701–710 (2004).
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24. H. E. Daniels, ‘The statistical theory of the strength of bundles of threads I’, Proc. R. Soc., A183, 405–435 (1945). 25. B. D. Coleman, ‘On the strength of classical fibers and fiber bundle’, J. Mech. Phys. Sol., 7, 60–70 (1958). 26. S. L. Phoenix, H. M. Taylor, ‘The asymptotic strength distribution of a general fiber bundle’, Adv. Appl. Prob., 5, 200–216 (1973). 27. S. L. Phoenix, ‘Probabilistic strength analysis of fiber bundle structures’, Fiber Sci. Technol., 7, 15–31 (1974). 28. D. Breysse, ‘A probabilistic model for damage of concrete structure’, Proc. 2nd Int. Symp. on Brittle Matrix Composite (BMC 2), Cedzyna, Poland, 20–22 September, pp. 237–247 (1988). 29. L. N. McCartney, R. L. Smith, ‘Statistical theory of the strength of fiber bundles’, ASME J. Appl. Mech., 105, 601–608 (1983). 30. M. Gurvich, R. Pipes, ‘Strength size effect of laminated composites’, Comp. Sci. Technol., 55, 93–105 (1995). 31. Z. Chi, T. Wei Chou, G. Shen, ‘Determination of single fiber strength distribution from fiber bundle testing’, J. Mater. Sci., 19, 3319–3324 (1984). 32. N. Lissart, J. Lamon, ‘Evaluation des propriétés de monofilaments à partir d’essais de traction sur mèches’, Comptes-rendus des 9° Journées Nationales sur les Composites (JNC9), edited by J.P. Favre and A. Vautrin, AMAC, 2, 589–598 (1994). 33. M. R’Mili, T. Bouchaour, P. Merle, ‘Estimation of Weibull parameters from loose bundle tests’, Comp. Sci. Technol., 56, 831–834 (1996). 34. ENV1007-5, ‘Advanced technical ceramic–ceramic composites ñ Methods of test for reinforcements – Part 5: Determination of distribution of tensile strength and tensile strain to failure of filaments within a multifilament tow at ambient temperature’, European Committee for Standardization, CEN TC 184 SC1, Brussels (1997). 35. M. Berger, N. Hochet, A. R. Bunsell, ‘Small diameter SiC-based fibers’, in: Bunsell A. R. and Berger M. H. (eds) Fine Ceramic Fibers, p. 265, Marcel Dekker, New York (1999). 36. H. M. Yun, J. A. DiCarlo, ‘Time/temperature-dependent tensile strength of SiC and Al2O3-based fibers’, in: Bansal N. P. and Singh J. P. (eds) Ceramic Transactions, Vol. 74, Advances in Ceramic-matrix Composites II, pp. 17–26, American Ceramic Society, Westerville, OH, 1996. 37. R. Bodet, X. Bourrat, J. Lamon, R. Naslain, ‘Tensile creep behaviour of a silicon carbide fiber with a low oxygen content’, J. Mater. Sci., 30, 661–677 (1995). 38. W. Gauthier, F. Pailler, J. Lamon, R. Pailler, ‘Oxidation of silicon carbide fibers during static fatigue in air at intermediate temperatures’, J. Am. Ceram. Soc., 92(9), 2067–2073 (2009). 39. T. Shimoo, K. Okamura, W. Mutoh, ‘Oxidation behavior and mechanical properties of low-oxygen SiC fibers prepared by vacuum heat-treatment of electron-beam-cured poly(carbosilane) precursor’, J. Mater. Sci., 38, 1653–1660 (2003). 40. J. J. Sha, J. S. Park, T. Hinoki, A. Kohyama, ‘Tensile properties and microstructure characterization of Hi-Nicalon SiC fibers after loading at high temperature’, Int. J. Fracture, 142, 1–8 (2006). 41. Y. Gogoti, M. Yoshima, ‘Oxidation and properties degradation of SiC fibers below 850°C’, J. Mater. Sci. Lett., 13, 680–683 (1994). 42. H. M. Yun, J. A. DiCarlo, Ceram. Eng. Sci. Proc., 4, 61–67 (1996). 43. E. Lara-Curzio, ‘Stress-rupture of Nicalon/SiC continuous fiber ceramic composites in air at 950°C’, J. Am. Ceram. Soc., 80(12), 3268–3272 (1997).
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44. E. Lara-Curzio, ‘Oxidation induced stress-rupture of fiber bundles’, J. Eng. Mater. Technol., 120, 105–109 (1998). 45. D. M. Wilson, L. R. Visser, ‘High performance oxide fibers for metal and ceramic composites’, Proc. Fibers and Composites Conf., Barga, Italy, 22, May (2000). 46. R. J. Charles, W. S. Hillig, Symposium on Mechanical Strength of Glass and Ways of Improving It, Florence, Italy, 1962. Union Scientifique Continentale du Verre, Charleroi, Belgium (1962). 47. S. M. Wiederhorn, ‘Sub-critical crack growth in ceramics’, in: Bradt R. C., Hasselman D. P. and Lange F. F. (eds) Fracture Mechanics of Ceramics, Vol. 2, pp. 613–646, Plenum Press, New York, 1974. 48. S. Bertrand, R. Pailler, J. Lamon, ‘Influence of strong fiber/coating interfaces on the mechanical behaviour and lifetime of Hi-Nicalon/(PyC/SiC)n/SiC minicomposites’, J. Am. Ceram. Soc., 84(4), 787–794 (2001). 49. W. Gauthier, J. Lamon, ‘Delayed failure of Hi-Nicalon and Hi-Nicalon S multifilament tows and single filaments at intermediate temperatures (500–800°C)’, J. Am. Ceram. Soc., 92(3), 702–709 (2009). 50. R. W. Davidge, J. R. McLaren, G. Tappin, ‘Strength–probability–time (SPT) relationships in ceramics’, J. Mater. Sci., 8, 1699–1705 (1973). 51. Ph. Forio, F. Lavaire, J. Lamon, ‘Delayed failure at intermediate temperatures (600°C–700°C) in air in silicon carbide multifilament tows’, J. Am. Ceram. Soc., 87(5), 888–893 (2004). 52. C. Valhas, F. Laanani, ‘Thermodynamic study of the thermal degradation of SiC- based fibers: influence of grain size’, J. Mater. Sci. Lett., 14(22), 1558–1561 (1995). 53. T. G. Langdon, ‘Grain boundary deformation process’, in: Bradt R. C. and Tressler R. E. (eds) Deformation of Ceramic Materials, pp. 101–125, Plenum Press, New York, (1974). 54. C. Droillard, J. Lamon, ‘Fracture toughness of 2D woven SiC/SiC CVI composites with multilayered interphases’, J. Am. Ceram. Soc., 79(4), 849–858 (1996). 55. R. Naslain, ‘Fiber–matrix interphases and interfaces in ceramic matrix composites processed by CVI’, Composite Interfaces, 1, 253–258 (1993). 56. F. Rebillat, J. Lamon, A. Guette, ‘The concept of a strong interface applied to SiC/ SiC composites with a BN interphase’, Acta Mater., 48, 4609–4618 (2000). 57. V. Calard, J. Lamon, ‘A probabilistic–statistical approach to the ultimate failure of ceramic-matrix composites – Part I: experimental investigation of 2D woven SiC/ SiC composites’, Compo. Sci. Technol., 62, 385–393 (2002). 58. J. C. McNulty, F. W. Zok, ‘Application of weakest-link fracture statistics to fiber-reinforced ceramic-matrix composites’, J. Am. Ceram. Soc., 80, 1535–1543 (1997). 59. T. Laguionie, ‘Determination of carbon fiber strengths using multifilament tows’, unpublished work (2002).
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4
Woven reinforcements for composites
D. C o u p é, Snecma Propulsion Solide, France
Abstract: Weaving technology, well known for several millennia, is fundamental to composite reinforcements. This chapter gives an overview of processes for the manufacture of woven fabrics, from basic ones used in textile applications to recent developments in the field. Definitions and applications of bi-dimensional (2D) and tri-dimensional (3D) fabrics, and of special woven shapes, are discussed. A projection of the future of weaving technologies in composite applications is made. Key words: composite woven fabrics, 2D fabrics, 3D fabrics, composite reinforcements, woven preform.
4.1
Introduction: from the beginning of weaving to technical applications
A woven fabric is characterized by two distinct thread directions (usually at right angles), interlaced with a repetitive pattern. Threads that run lengthways make up the ‘warp’, while threads that run from side to side represent the ‘weft’ or ‘fill’. The origins of weaving are not precisely known, but it seems to have been first used in the prehistoric period. Some specialists think our ancestors could have been inspired by spiders’ webs or birds’ nests. Weaving was initially used for making the tools of everyday life, like wooden fishing nets, or shelter components such as floors and roofs. The first hand-woven clothes seem to have appeared in Mesopotamia and Turkey (8000 bc), followed by Egypt (5000 bc), where linen, cotton and wool fibers were used (perhaps most famously in their linen wrapping tapes for mummies). During the same period, cotton and wool were being woven on the South American continent. Following the remarkable invention of silk spinning technology around 2700 bc, China exported silk fabrics all over the world. Woven fabrics were made on hand looms for millennia. Mechanization only appeared in the eighteenth century, and was improved during the nineteenth century by the introduction of Jacquard technology. Further major developments occurred in the twentieth century, with perhaps the most significant being the replacement of shuttles with rapiers for the insertion of fill-direction fibers. Other fill-insertion technologies have since 89 © Woodhead Publishing Limited, 2011
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been developed (projectile, air jet, etc.), but these are for very specific fibers and fabrics. Technical fibers like glass, ceramic and carbon fiber were also developed and popularized during the last century. These fibers typically have a high modulus and a very short elongation factor (less than 2%), and are transformed into fabrics in a radically different way from natural or polymeric fibers. Weaving is the most popular method of fabric manufacture, but many developments were needed before it could be used successfully on structural composites. Fiber manufacturers and technical weavers faced significant challenges when first attempting to produce fabrics from these fibers. Each step of the technology had to be optimized: spinning, twisting, warp setup, shed opening, fill insertion, etc. The quality of the sizing is particularly important, because it must be compatible with the composite matrix, and able to protect the fiber during the transformation process. These problems were overcome, and weaving technology has been used on glass fibers for more than 50 years, and carbon fibers for 30 to 40 years.
4.2
Technology description
The main principle of weaving is based on the interlocking of warp and fill fibers. As shown in Fig. 4.1, each warp end (a single yarn or tow in the warp direction) goes through a heddle eye and through a reed. Some eyes are moved up and others are moved down to form a shed, the internal triangle between two warp webs where the fill is inserted. The sequence is as follows: 1. 2. 3. 4.
The The The The
shed is open. loom moves the warp forward one space. fill is inserted. reed beats up to place the fill in the right position.
Then a new shed is created, with a different warp end at the top and bottom, and the sequence is repeated. A schematic weaving machine or loom, as presented in Figs 4.1 and 4.2, is composed of different parts: ∑ The back of the machine is the warp presentation side. ∑ The central area is where interlacing between warp and weft occurs. ∑ The front of the loom is the weaver side, where the fabric is wound.
4.2.1
Warp presentation
There are two main ways to present the warp side on a loom: a creel or a beam. Depending on fiber presentation, fiber type, and the desired properties of the woven fabric, one of these two processed is used.
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Heddle warp
A Cloth
Warp
Harnesses
Cloth roll
B
Warp beam
4.1 Basic weaving loom (courtesy of Staübli). Reed
Frames
Shed
Warp ends
Fabric
4.2 Weaving loom schema (courtesy of Staübli).
Creel technology A creel is a bobbin support that allows each yarn to be pulled independently, as shown in Fig. 4.3. A rudimentary creel can be made by simply placing bobbins on a rack (like those used for glass roving), while very sophisticated creels have tension control on each bobbin.
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4.3 Creel example.
The creel process has two main advantages: ∑ ∑
Bobbins can be used directly from the fiber manufacturer. Bobbins can run independently, which allows the loom to pull different lengths depending on the fabric pattern or shape.
Yarn tension may be less homogeneous than with beam technology, due to the positioning of the bobbins in the creel, and the yarn angle between creel and harness. An additional device between creel and harness, called an ‘eye board’, can be introduced in order to steer the warp straight through the harness from the back of the loom to the front. Beam technology A beam is a large cylinder onto which all warp ends can be wound, as shown in Fig. 4.4. Beam technology is very convenient for providing equal tension on each yarn, which is important in producing fabric with regular patterns. For technical fibers, like carbon or glass, the quality of the beam winding must be perfect. If the tension of some yarns is different from that of others, individual tows will have different lengths, and the fabric quality will be poor. For production, looms with warp beams are much more compact than looms with creels. However, an intermediary warping process is necessary, for winding yarn from the bobbins to the beam.
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Section beam warping
4.4 Beam example (courtesy of Staübli).
4.2.2
Crossing warp head
The central part of the loom consists of a head, which controls warp lift, shed, the reed, and the fill-insertion device. Two main technologies have been developed for the head on an automated loom: ∑ ∑
The earliest and simplest is the harness frame or dobby process, which uses several frames holding heddles that are lifted by different cams. The second is the Jacquard process, in which each warp end can be lifted individually by the head.
Harness frame process (cam or dobby) Harness frame technology, created in the eighteenth century, is a natural development of the hand loom, where the warp is separated into two yarn webs to form the shed and lifted by hand or by pedals. After fill insertion, each web is crossed to interlace the fill. Two webs are sufficient for plain weave, the simplest and most popular pattern (Fig. 4.1). With harness frames, each warp web passes through the heddles of the same frame, as shown in Fig. 4.5. To make more complex patterns, the number of frames can be increased. The maximum number of frames on the dobby of a regular loom is of the order of 28. For some very specific applications, customized looms have been developed that have a double dobby with 56 frames. However, increasing the number of frames requires a drastic increase in the lift of the rear frames, in order to keep a good shed for the fill insertion. This can be a problem because technical fibers are very sensitive to large variations in shed opening.
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Warp
4.5 Harness heddles (courtesy of Staübli).
Harness frame is the most convenient method for creating simple, regular fabrics like plain weave, twill and satin. Fabrics woven from composites are almost exclusively manufactured with this technology. Jacquard process Jacquard technology, invented by Joseph Marie Jacquard at the beginning of the nineteenth century, enables the manufacture of more complicated patterns. The Jacquard process is a mechanical technology that independently lifts each heddle of the harness (see Fig. 4.6). Originally, it was controlled by a device that read holes punched in pasteboard cards, where each hole corresponded to a warp lift. Although these mechanical heads are still used, it is increasingly common for Jacquard heads to be controlled by electronic systems. This technology is necessary for complex pattern definitions such as 3D weaving patterns, or complex net shape preforms for composite parts, which require a large degree of freedom for the interlacement of warp yarns.
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y
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4.6 Jacquard process (courtesy of Staübli).
Although some technical weavers have used this technology for years, it is not yet a popular method for producing composite preforms.
4.2.3
Take-up systems
The ‘take-up system’ is the mechanism that pulls woven fabric out of the central part of the loom, and so controls weft count. For regular fabrics it is composed of a set of different mandrels (two or three), as shown in Figs 4.1 and 4.2. The first two are dedicated to advancing the fabric and controlling the weft count, while the last one is for winding the woven fabric with a uniform tension. For regular production, almost all woven fabrics are manufactured using this technology.
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In the manufacture of thick fabrics (more than ~3 to 5 mm), particularly 3D woven fabrics, the textile structure can be damaged by winding on a regular mandrel even if the diameter of the mandrel is increased. A special device with clamps to pull the preform straight out of the loom is therefore used. These systems are usually custom built.
4.2.4
Fill insertion
Different processes have been used to insert the weft across the width of the fabric: ∑
The earliest is the shuttle process, which was used for handmade fabrics and is still used today for applications like narrow fabrics for ribbons. ∑ The rapier process was born in the twentieth century and is now the most popular production approach. ∑ Other processes, such as projectile, needle, air jet and water jet, are used only for very specific purposes. Shuttle process This process uses a device called a shuttle to move yarn across the fabric width. A small bobbin of yarn is incorporated into the shuttle. The yarn is held at one side of the fabric, and unwinds from the bobbin as the shuttle is thrown through the shed opening to the other side. Originally, the shuttle was moved by hand as shown in Fig. 4.7. During the eighteenth century, John Key invented the flying shuttle. This shuttle, running on wheels, moved over the lower side of the warp web as shown in Fig. 4.8. Two wooden tenders connected by a small rope, which was pulled by hand, propelled the shuttle. A century later, it was improved by the addition of a multi-shuttle device, able to change the type of yarn in the weft direction. For technical applications, like weaving composites, this technology is used rarely and only for specific purposes such as narrow tapes (carbon or glass UD, for example), or to achieve a continuous fill at the edge without selvedge (to weave a cylindrical fabric, for example). The winding and unwinding inherent to this process can introduce some non-negligible additional twisting to the fill yarn. This may be unacceptable for some fragile yarns and some composite applications. Rapier process This process consists of inserting the fill with a clamping device, called a rapier, which takes the yarn from a fixed bobbin on one side of the loom and
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4.7 Hand loom.
4.8 Flying shuttle loom.
transfers it to the other side. The end of the fill is cut on the bobbin side after insertion. The technology was developed at the beginning of the twentieth century, and was popularized during the second half of that century. Different types of rapier are used (Fig. 4.9): ∑ ∑
Rigid ones that stay straight during weaving Flexible rapiers that are wound and unwound during weaving, allowing the use of more compact looms ∑ Double rapiers, with one placed on each side, and an exchange of the fill yarn from one to the other in the middle of the shed.
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Solid
Shuttle
Rapier
Cripper
Air jet
Fluid
Air
Water
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4.9 Fill insertion technologies.
A rapier can be much slimmer than a shuttle, because it does not have to carry the bobbin. The shed opening can then be reduced, which means a smaller lift of the harness. This is important, because a smaller lift reduces the tension variation on the yarn, which is desirable for technical fibers with low elongation factors. This technology is extensively used for the production of woven fabrics in composite applications. Other processes Other fill insertion technologies exist, but are seldom used for technical woven fabrics. These technologies include: ∑
The projectile process, in which the fill yarn is transferred by clamping it to the back of a projectile, which is launched from one side of the loom to the other (see Fig. 4.9). ∑ Air jets. The fill yarn is pushed through the whole width of the fabric by an air nozzle or a series of air nozzles, without the use of any other device (i.e. without a shuttle or projectile) (see Fig. 4.9). ∑ Water jets. The same as air jets but the fluid power is provided by water instead of air (see Fig. 4.9). ∑ Needles. This technology replaces the shuttle or rapier in the production of narrow fabrics. Needle and yarn go through the shed to the other side of the loom. The yarn is maintained by a hook at the opposite side of the
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loom and the needle comes back with the yarn, which inserts a double fill into the same shed opening. Use of these technologies is mainly driven by the desire to increase loom productivity. They are applicable only for specific fibers, such as low-weight synthetic or cotton fibers.
4.2.5
Fabric width
Weaving looms have a large range of widths: ∑
Small looms are able to weave anything from a few millimeters to several centimeters. Mostly used for ribbons or tapes, they can use frame or Jacquard technologies for the harness and shuttle or rapier (or sometimes needle) for the fill insertion. For small widths, shuttle looms can have several heads in parallel, each weaving fabric with the same pass. ∑ Regular looms with a fabric width between 1 and 2.5 meters are able to use all the technologies explained previously. ∑ Large looms, which are more than 4 to 5 meters and sometimes up to 15 meters or more, are used for specific applications like carpets or paper machine clothing, and employ shuttle or projectile processes for fill insertion.
4.2.6
Other weaving technologies
Some marginal technologies have been developed that can be useful for specific applications. All are based on woven structure interlacements, but some have more directions and some have different fiber orientations. ∑
Circular weaving. A technology for weaving circular fabric exists, which produces very open fabrics for bags with synthetic fibers, but it is not used for technical fibers. With another technology, a tubular fabric can be woven on a shuttle loom by weaving two flat layers at the same time, and adjusting the weaving at the edge to keep the same pattern. The continuous fill provided by shuttle weaving eliminates discontinuities at the edges. Such fabric can be cut on a helix to obtain a ± 45° fabric. ∑ ±45° weaving technology. Some prototypes have been developed for manufacturing a woven ±45° fabric, but their use is limited. ∑ Tri-axial fabrics. There is a process for weaving in three directions, which produces a fabric with fiber with 0°/+60°/–60° orientation. This complex technology produces a very open fabric, but a single layer of this fabric is quasi-isotropic. ∑ Four-directional fabrics. Some prototypes have been developed to produce a 4 ¥ 45° woven fabric in 2D and even 3D. This technology is limited in width, and is not yet in production. © Woodhead Publishing Limited, 2011
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4.3
Woven fabric definitions
Having provided an overview of existing technologies, we next have to define all the possible fabric styles that weaving technologies can produce. The following description progresses from the simplest to the most complicated.
4.3.1
2D patterns
Patterns for woven fabrics are defined by the smallest repeating unit cell, which describes the interlacing of the warp and weft. The term ‘2D (two-dimensional)’ is used to identify standard fabric with one layer of warp and one layer of fill. There are three main types of 2D pattern: plain weave, twill and satin. Plain weave definition Plain weave (PW) is the basic and simplest pattern. The unit cell (or weave repeat) of this pattern is made up of two warps and two wefts. As shown in Fig. 4.10, the first warp yarn goes over the first fill and under the second fill, while the second warp goes under the first fill and up the second. Plain weave was the most popular pattern amongst early weavers. The harness requires a minimum of two frames, with odd warp ends running through the heddle of the first frame to create the first warp web, and even warp ends going through the second frame to create the second. These two webs alternate between the top and bottom position for each fill insertion.
4.10 Plain weave pattern (Wisetex image).
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The surface of the fabric looks like a checkerboard (as Fig. 4.11) on both sides. Basket weave pattern is derived from PW. Two or more warp ends alternately interlace two or more fill ends, as can be seen in Fig. 4.12. Twill definition The weave repeat of a twill weave (TW) pattern has a minimum of three warps and three weft columns. It is generated by shifting a base pattern by one warp and one weft column, relative to the adjacent warp column. This shifting is constant across the width of the fabric and is always in the same direction. The surface of the fabric is characterized by diagonal stripes, and the two sides typically differ in appearance. The most simple twill pattern is the 2 ¥ 1 TW (see Fig. 4.13), called 2 ¥ 1 because each warp end goes over two fills and under one. The bottom side of the fabric will show two fills running over two warps, then under one. The term ‘warp face’ is used for the top side (because warp predominates), and the term ‘weft face’ or ‘fill face’ is for the bottom side.
Top
Bottom
4.11 Each side of PW.
Top
Bottom
4.12 Basket pattern (Wisetex image).
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Top
Bottom
4.13 Twill pattern 2 ¥ 1 (Wisetex image).
TW 2 ¥ 2
TW 3 ¥ 1
4.14 Examples of TW (Wisetex image).
Looms need a minimum of three frames to create three different warp webs. These three webs alternate, sometimes with two webs at the top of the shed and one at the bottom, sometimes with one at the top and two at the bottom. See Fig. 4.14 for some examples of different variations of twill patterns. The 2 ¥ 2 TW is a balanced pattern because each side has the same number of warps interlacing up and down, which means the length of warp and fill will be the same on each side. Satin definition Satin is the third basic pattern family. It has a relatively small number of interlocking locations, which are scattered to avoid the diagonal appearance of twill patterns. The satin family is characterized by the same basic interlocking pattern for each warp, which is shifted several fill columns relative to the adjacent warp. In contrast to the twill pattern, the shift is not uniform for all warp columns in the repeats. The basic pattern of each warp is under one fill and over several. An ‘N’ harness satin (NHS) means that the warp goes under one and
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over (N – 1) fill ends. The weave repeat is N warp and N weft columns. This shifting is always more than 1 and less than (N – 1), and is constant overall. The amount of shift in the fill between two adjacent warps must not be a divisible factor of N. For example, for a 5HS the shift must be two or three, and for an 8HS it must be three or five (see fig. 4.15). The 4HS pattern is an exception to the usual rule for satin. It has an irregular shifting pattern because the rules defined above cannot be satisfied. For this special case, interlacing displacement between two adjacent warps alternates between two and one fill column shifts, as one can see in Fig. 4.16.
5HS
Top
Bottom
8HS
4.15 5HS and 8HS patterns (Wisetex image).
4.16 4HS pattern (Wisetex image).
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There is a significant difference between the appearance of the two sides of a satin fabric: one has a majority of visible warp and is called the ‘warp face’, and the other has a majority of visible weft and is called the ‘weft face’. Properties of the three basic 2D pattern families If the three types of 2D patterns are compared, several rules applicable to composite applications can be defined. ∑
Firmness and drapability. Because it has the highest level of interlacement, PW is the most stable pattern, which means it is the least formable. In contrast, HS patterns are definitively the most formable fabrics but can be more difficult to handle and to keep in their original geometry. For regular TW patterns, the formability is in between PW and HS. ∑ Weight and thickness. HS patterns are potentially the best choice for the heaviest and thickest fabrics. Greater thickness is made possible by the low level of interlocking. In contrast PW, the most interlaced pattern, is more appropriate for lightweight fabrics. ∑ Symmetry. The only patterns that create the same effect on both side are PW and N ¥ N TW. All the other patterns have a warp and fill face. ∑ Crimp (wave induce by yarn interlocking). Due to the interlacement at every fill, the PW pattern introduces more crimp into the fabric than TW or HS. For composite applications, mechanical properties in the x–y direction (i.e. in plane) can be significantly affected by a higher crimp level.
4.3.2
3D patterns
Compared with 2D patterns, which are far more common, 3D (threedimensional) fabrics only appeared in the composite field for R&D applications in the 1980s. However, such patterns have been used in the textile industry for more than 100 years. Three-dimensional woven fabrics are defined by an interlacement of several layers of warp and weft linked through the thickness. These patterns can be manufactured directly on a loom. Jacquard woven cloth, including furnishing fabrics, can be considered 3D woven fabrics but they do not have a regular pattern and have more crossing layers. These layers give particular color and design effects. This is also true for carpet fabrics, but when they are woven on a mechanical loom, two fabrics are woven at the same time. The woven pattern can be considered to be a 3D pattern, as can be seen in Fig. 4.17. It can be argued that the first technical application for 3D woven fabrics
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4.17 Carpet fabric pattern (patent nos EP0919652A2 and EP1375714A1).
a
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involved their use as straps for carrying heavy loads. These patterns are simple enough to be woven on frame looms, and are very efficient. A combination of the properties of the yarn, the pattern, and the friction between filaments makes such straps robust. Another technical application for 3D woven patterns is in paper making clothing. These fabrics, usually woven from polyamide and polyester monofilaments, have a specific texture on one side to obtain the proper paper surface, and a specific porosity to extract water from the paper through the fabric. A classification into orthogonal, angle interlock and multilayer patterns is proposed for the different patterns used for 3D woven fabrics in composite applications. Orthogonal 3D patterns Yarns in orthogonal 3D patterns are orientated in the three principal orthogonal directions of volume (x, y, z), as shown in Fig. 4.18. Some warp yarns are straight in the x direction (warp ends), fill yarns are straight in the y direction (fill ends), and some warp ends (called ‘warp weavers’) interlace the other yarns and are, at times, orientated in the z direction (corresponding to the thickness of the fabric). This is the simplest 3D pattern, and it can be manufactured on frame harness looms if the number of layers is not too high (maximum ~20 layers). The structure of this pattern permits multi-rapier insertions during the same pass, which can be an efficient way to increase productivity. Angle interlock patterns Angle interlock patterns do not have warp weavers in the z direction (across the thickness of the fabric). Interlock patterns are defined by two criteria:
4.18 Orthogonal 3D pattern (Wisetex image).
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Warps in the same warp column interlace several fill layers (to become a 3D fabric linked through the thickness). Every warp has the same path in the same warp column every fill layer.
We can define two families within angle interlock patterns: through-thethickness interlock and ply-to-ply interlock. Through-the-thickness interlock patterns, as shown in Fig. 4.19, have warp columns crossing all the weft layers of the fabric. In ply-to-ply interlock patterns, the warp is interlaced, and only goes through some weft layers as shown in Fig. 4.20. The EADS trademark 2.5D (Fig. 4.21) is a member of the ply-to-ply interlock family. Multilayer patterns Multilayer patterns look like several layers of regular 2D fabrics with some interlacement between the layers. For example, the pattern shown in Fig. 4.22 is a multi-PW pattern. The basic pattern of each layer is a PW, but
4.19 Through-the-thickness interlock pattern example (Wisetex image).
4.20 Ply-to-ply interlock pattern example (Wisetex image).
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4.21 2.5D pattern (Wisetex image).
4.22 Example of multilayer PW pattern (Wisetex image).
some warps are going from one layer into another to create links that make the 3D pattern. Multilayer patterns have certain benefits: the weave repeat can be quite simple and they can be made on a harness frame loom if only a few layers are needed (i.e. up to five or six). Another example of a multi-HS fabric is shown in Fig. 4.23. A loom with a Jacquard head is usually needed for this type of fabric if more than three layers are required. Properties of 3D patterns for composite applications Three-dimensional fabrics have some general properties, but these rules can be slightly modified, depending on the type of yarn and the warp/fill count for each layer.
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4.23 Example of multilayer 4HS pattern (Wisetex image).
Firmness Orthogonal 3D patterns produce very strong, hard preforms, and the geometry of these preforms is very difficult to modify. This characteristic can be explained by the yarn directions. The preform directly out of the loom can have a fiber content close to 50–55%, which is quite high. To increase it to 60–65%, the thickness has to be reduced by roughly 20%. It is practically impossible to do this with a thick preform without skewing the fiber alignment or damaging the fiber. Through-the-thickness interlocks are less firm than orthogonal 3D patterns, but they can be quite hard too, if there is a significant quantity of straight warp in the pattern. They are more compressible than orthogonal patterns because there is no fiber in the thickness direction. The most soft and compressible patterns that can be produced without damaging the yarn positioning are those without straight warp fibers. All the warps in the same column follow the same path, and are able to move together naturally without being fixed by another warp column with the same property. The multilayer patterns are not hard, and are easily compressed. More de-bulking is required for these patterns, especially when the interlacing goes through only one or two layers. Drapability The drapability of these 3D woven fabrics follows the same trend as firmness. An orthogonal 3D is not formable for complex shapes due to the structure
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of the interlacement. Interlock patterns without straight warp and multilayer patterns are the only ones that produce nice formable fabric. Other parameters like tow size, warp/weft count and warp/weft ratio can be optimized to improve the formability of a preform. Mechanical properties The best properties in the z direction are obtained with orthogonal 3D patterns, and this can be easily explained by fiber directions. The lowest through-the-thickness properties are found in multilayer patterns that are close to 2D fabrics. Interlock patterns are in between. In the in-plane (x–y) direction, properties depend on warp and fill crimp. Using patterns with straight tows like orthogonal 3D or through-the-thickness interlock can be an advantage, but this depends on the quantity of warp weavers. For interlock patterns, depending on the specific pattern, the tow size and the warp and fill count, the crimp can be reduced to give x–y properties close to those of the other types of 3D fabrics. For multilayer patterns, the x–y properties are not significantly different from those of the basic equivalent 2D fabric.
4.3.3
Specific shapes
Most fabrics are flat and can be wound onto a roll as they are manufactured, but there are ways of producing other shapes. Contour woven fabrics These fabrics are woven with a special shape, which is created by a particular device on the loom. One example is shown in Fig. 4.24. The most popular use for such fabrics is in fan case containment for aircraft engines. Spiral fabrics By using another special device on the loom, weaving technology permits the manufacture of helical fabrics, as shown in Fig. 4.25. The two yarn directions in the fabric are radial for the weft and circumferential for the warp. Tubular or conical fabrics One way to produce a tubular fabric is to weave a flat double layer on a shuttle loom, with a special interlacement at the edge to keep the regular pattern. With the same technology but using a more complex design, the edge junction of the double layer fabric can follow a triangular shape. After © Woodhead Publishing Limited, 2011
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4.24 Example of contour weaving (patent no. EP0302012A1).
4.25 Spiral woven fabric.
creating an opening between the two layers, the shape of the fabric is conical. This technology had been used for composite noses for military aircrafts. Double-wall fabric When manufacturing carpet or velvet, two woven fabrics are produced simultaneously (top and bottom) with interconnecting yarns. These yarns are cut in the middle to give the ‘fluffy’ texture on one side. If the interconnecting
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yarns are not cut, the same technology can be used to produce a double wall preform linked by vertical yarns. Complex shapes 3D patterns can be used to weave flat fabrics with several layers, and carefully positioned bifurcations can be made between layers to obtain preforms than can be unfolded to produce complex shapes. Figure 4.26 shows some examples of unfolded preforms for stiffeners, Pi preforms, and other complex shapes. A multitude of combinations are possible. Even if textile definition is more complicated than in regular fabric, creating complex woven shapes may ultimately simplify manufacturing of composite parts. For example, a complex part can be made with only one 3D woven preform, instead of using several plies, which sometimes require additional bonding technologies. 3D woven fabric can also produce a net shape preform directly on the loom by removing some layers and letting some warps end without interlacement. After weaving, extra floats can be trimmed to obtain the complex net shape preform. For example, an aircraft engine fan blade can be made from a 3D woven complex shape preform, as shown in Fig. 4.27
4.26 Examples of complex shapes.
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4.27 Snecma composite fan blade.
4.3.4
Woven fabric characterization
The different parameters of a woven fabric are given below: ∑
Definition of the yarn or tow, including type, linear weight, sizing and twist ∑ Warp and fill count, including the number of warp and fill ends per unit length ∑ Pattern ∑ Weight of the fabric ∑ Thickness under pressure ∑ Warp and fill crimp. This is the difference between the woven yarn length and the fabric length. It is expressed as the percentage of extra length that represents the additional length needed for the pattern interlacement. ∑ Selvedge. This is a specific pattern with a high level of special interlacement on each edge of the fabric. Its function is to fix the fill ends to avoid the risk of fraying. Usually selvedge is done with a different yarn type than is used in the rest of the fabric. ∑ Fabric width and piece length.
4.4
Applications for composite reinforcements
The following is an overview of applications that make significant use of woven fabrics for composite parts:
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∑
∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑
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Civil and military aircraft: for the new generation of civil aircraft, composite parts represent around 50% of total weight. Woven fabrics have a large proportion of composite reinforcement for use in the fuselage, wings, engine nacelles, landing gears, and brakes. Helicopters need composites for the same parts as planes but at higher percentages. Space and defense: for missiles and launch vehicles with polymeric and thermostructural matrix composites, and for satellite applications. Wind energy: for blades with carbon and glass fibers, and even for masts. Sport and leisure: for jet skis, snowboards, wakeboards and canoes, frames and components for racing cycles, motorcycle accessories, fuel tanks, wheels and helmets, tennis racquets and golf clubs. Marine: for hulls, decking, spars and mast. Glass fibers are used most, but carbon fibers are sometimes used in racing boats. Industry: for robotics and high-speed automation, printed circuit boards and composite tooling. Automotive industry, particularly sports cars: specific applications include bodies, chassis, and front and rear wings. Ballistics: glass or aramid fabrics are used for helmets, body armor, vehicle armor, and secure door structures for buildings and aircraft.
Woven fabrics are not the only reinforcements used for all these applications. They are challenged by other technologies like UD tapes, fiber placement or NCF (non crimp fabric) products.
4.5
Conclusion and future trends
What will be the future of weaving in the composites business? 2D woven fabric was the first technology used for composite structural reinforcements, and is still the most popular. More recent technologies like NCF and fiber placement are challenging this technology and have replaced it in some applications. These new technologies can potentially be cheaper or offer better fiber orientations, and will continue to replace woven parts. However, they are not such mature technologies as weaving, and for many applications introducing them will take time. For example, in the aerospace businesses it will take several years to introduce these technologies, because new products must be certified. Meanwhile, technical weavers are continuing to improve 2D technologies in order to increase loom speed, and are working on spreading heavier and cheaper tows to produce fabrics with the same weight. 3D weaving and complex shapes have high potential for expansion. Combined with RTM processes, they can limit the number of steps from © Woodhead Publishing Limited, 2011
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fiber to final part. There is only one preform to mold, and it is already close to the net shape, as opposed to several plies that must be cut, collated, and sometimes linked together. However, design offices will need to think differently if 3D weaving and complex shapes are to be introduced industrially on a large scale. There are important differences between them and the more familiar laminate composites. Perhaps most importantly, defining the right preform is more time consuming. 3D techniques are just beginning to find industrial applications, in the same way that 2D carbon woven fabrics were 40 years ago. Several years later these fabrics became industrial products – can we expect the same from complex shapes and 3D woven fabrics?
4.6
Acknowledgement
Many thanks to Jon Goering for correction of this chapter.
4.7
Sources of further information and advice
Some book references for weaving and textile technologies: ∑
Woven Textile Structure: Theory and Applications, B K Behera and P K Hari, Indian Institute of Technology, India Woodhead Publishing Series in Textiles, No. 115. ∑ 3-D Fibrous Assemblies: Properties, Applications and Modelling of Three-dimensional Textile Structures, J Hu, Hong Kong Polytechnic University, Hong Kong, Woodhead Publishing Series in Textiles, No. 74. ∑ Handbook of Technical Textiles, A. Richard Horrocks, Subhash Anand and S. Anand, Woodhead Publishing Series in Textiles, No. 12. Some website references for woven composite reinforcement manufacturers: http://ww3.albint.com/AEC http://www.ballyribbon.com/ http://www.cytec.com http://www.hexcel.com http://www.porcher-ind.com/ http://www.sigmatex.com/ http://www.3tex.com/ Some website references for weaving loom manufacturers: http://www.lindauerdornier.com http://www.mueller-frick.com/ http://www.staubli.com/en/textile/ © Woodhead Publishing Limited, 2011
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5
Braided reinforcements for composites
A. G e s s l e r, EADS Deutschland GmbH, Germany
Abstract: This chapter describes braiding technologies used for the production of net-shaped or near-net-shaped composite preforms. The advantages and limitations of the braiding process are explained with examples of practical applications. Because mandrels play an important role in braiding, various mandrel technologies and materials are described together with their different applications. Key words: braiding, unidirectional braiding, overbraiding of mandrels, robot-assisted braiding, 3D-braiding, braided textiles, braiding mandrels.
5.1
Introduction
For over 100 years, industrial braiding technology has been used in the manufacture of consumer goods as varied as shoe laces and power cords for flat-irons. Most braids, however, are highly specialized technical products with demanding requirements for durability, reliability and quality. Climbing ropes, marine ropes and metallic jackets for cables and high-pressure tubes are the best-known examples of these technical applications. Tubes made out of glass fibers were among the first braided products used for composites, mainly for sport and leisure applications. Fibers like carbon and aramid soon followed. In the early 1990s, braiding became more significant when preform-infusion technology developed in composite manufacturing. Braiding could be used in net-shaped fiber preforms for complex composite parts by overbraiding of contoured mandrels (the device around which the braid is formed). It soon it became clear that this technology was not only cost-effective and quick but also highly versatile. Many companies around the world started to research ways of mass-producing braided composites. Braided preforms are now well established in high-tech applications, e.g. crash absorbers in sports cars, aircraft propeller blades and even structural aircraft parts.
5.2
Fundamentals of braiding
Simple braids are textiles with interlocked fibers in two directions. At least three strands are needed to build a braid. Unlike in woven fabrics, where the warp runs along the textile and the weft is exactly perpendicular (90°), the 116 © Woodhead Publishing Limited, 2011
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basic characteristic of braiding is that the yarns are run at an angle from one rim of a braided ribbon to the other or like a helix around a tubular support. Figure 5.1 compares weave, flat braid and tubular braid. The yarn angles in a braid depend on the relative speed of the bobbins and the take-up. The faster the take-up, the smaller the angle. Angles between 10° and close to 90° can be achieved. It is important to achieve a closed braid with a 100% coverage to avoid resin-rich areas in the composite (see Fig. 5.2). The fiber angle, the total number of yarns and the circumference of the mandrel are in a direct relationship. This relationship is the key feature in the design of a braided preform and the most important limitation of braiding technology: with the exception of 3D braiding, the number of yarns is always fixed. No technical solutions to change the number of yarns in the braiding process have been found so far. In 1748 Thomas Wattford (Manchester, England) filed the first patent for a braiding machine. Industrial production of the so-called ‘Barmer Flechtmaschinen’ started in 1767 in Barmen, a part of Wuppertal (Germany). The basic principle (Fig. 5.3) has remained more or less unchanged. A braider consists of: ∑ ∑
The bobbins holding the yarns Bobbin carriers to keep the yarn under tension
Weave
Flat braid
Tubular braid
5.1 Comparison of weave, flat braid and tubular braid.
Open braid
Closed braid
5.2 Open braid (coverage about 80%) and closed braid (coverage 100%).
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A machine bed with an engraved groove to guide the bobbin carriers as they are pushed forward by the horn gears A take-up mechanism to pull the completed braid from the machine.
The variations of modern braiding machines are numerous. The machine size and the number of bobbin carriers depend on the type of braid required. Figure 5.4 shows some examples. The versatility of modern preforming machines depends on two basic features: ∑ ∑
Overbraiding of contoured mandrels The combination of the braider with an industrial robot to handle the mandrel.
The braider is fully controlled by the robot program. The machine speed can be continuously adapted to the mandrel diameter to produce a braid with the required yarn angle. The robot also allows smooth and rapid changes of
Clockwise running bobbins
Counter-clockwise running bobbins Take-up
Braid point
Braid
Carrier Horn gear Horn
Carrier foot track
5.3 Braiding machine principle and machine elements.
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5.4 Top to bottom: net yarn braiding machine, offshore rope braider, composite preform machine.
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the mandrel position and angle relative to the braiding machine, which is helpful in dealing with discontinuities in the preform. Additional machine adaptations have been made to deal with carbon fiber rovings. These started with special bobbin carriers and yarn guiding elements and ended up with the so-called ‘radial braider’ with bobbins running on the inside of the machine. Other fibers of technical interest, like glass or aramid rovings, have also benefited from these developments, though there is a drawback: the bobbin capacity on a radial braider is a lower (about 5–10% depending on the fiber type). As a result, production has to be stopped more often to load the braider with a new set of bobbins.
5.3
Braiding technologies for preforming
Although there is a huge number of different braider types, sizes and configurations for different technical applications, machines for production of preforms split into two basic types: the so-called ‘3D braider’ and – most important – the circular braider, as shown in Fig. 5.4. Machine size is defined by the yarn and preform size and is constrained by the need for 100% coverage. Braiders with 144 or 196 bobbin carriers are typical. Filament yarns like carbon rovings can be narrowed, but each yarn type has its specific maximum lay-up width. As a rough-and-ready rule, a 144 carrier braider is able to cover a mandrel with a circumference of about 300 mm at an angle of 45° with an 800 tex HT carbon roving. Machines need not be small, because it is possible to work with a reduced number of bobbins without serious effects on the braid quality, even when the bobbins are unevenly distributed on the machine.
5.3.1
Circular braiding
Conventional braiders This type of braider has its horn gears on the front of the machine. Due to the path of the bobbin, which follows a wavelike pattern (like a sine shape), the distance between the bobbin and machine center is never constant. To keep the yarns under tension, the bobbin carriers (Fig. 5.5) are equipped with a spring-loaded faller. This pulls a certain length of the yarn back into the bobbin carrier when it moves from the outside to the innermost position on the machine. These braiders are the standard machines for the production of tubes and ropes. For these applications, the machines are combined with a take-up mechanism and a braiding ring, located at a distance of about half of the machine diameter in front of or above the yarn outlet of the bobbins. Figure 5.6 shows a typical configuration. © Woodhead Publishing Limited, 2011
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5.5 Bobbin carrier of a conventional braider.
The braiding ring defines the distance known as the ‘braiding point’, the point at which the yarns meet each other. It influences the angle of the braid as well as the angle at which the yarns leave the bobbin carriers. The maximum angle at the bobbin is defined by the stroke of the tensioner. In the extreme position, when the braiding ring is located in the plane of the yarn outlet, the tensioner must cover the whole amplitude of the bobbin path. This configuration is of special importance for mandrel overbraiding (see Section 5.3.2). It requires an advanced type of bobbin carrier (Fig. 5.7), able to deal with not only the amplitude of about 120 mm but also the fragility of the carbon rovings as well. On the average, each segment of the yarn is pulled back and forth over the rollers and guiding elements about 30 times until it finally leaves the bobbin carrier. There is a delicate balance between the low yarn tension required and the force to pull the roving back into the reservoir. As a result, these bobbin carriers require careful maintenance. The advanced bobbin carrier design has substantially reduced yarn damage and broken filaments. New radial braiding machines promise to reduce the problem even further. Radial braiders First introduced by Muratec in early 1990, the so-called ‘radial braiders’ or ‘tunnel braiders’ were further developed to the late 1990s by the German
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5.6 Typical braiding machine configuration.
5.7 Advanced bobbin carrier used for overbraiding with carbon rovings.
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braiding machine manufacturer Herzog. The horn gears are no longer located on the front but inside the machine. The bobbin carriers point to the center of the machine, avoiding a 90° redirection of the yarns. Figure 5.8 illustrates a Herzog radial braider. The major benefit of a radial braider is the reduction of the relative movement between the yarn outlet and the braiding point to a few millimeters, because the bobbin path is now perpendicular to the machine base plane. Because yarn consumption now normally exceeds the change in length, there is no pullback of the yarns into the bobbin and no repetitive crinkling and straightening. As a result, the risk of yarn damage is drastically reduced compared to a conventional braider. The complexity of the bobbin carriers on a radial braider is similar to that on a conventional machine (see Fig. 5.5). The tensioner no longer needs to compensate for the amplitude of the bobbin path. The capacity of the faller now allows asymmetric positioning of the mandrel in the machine and reduces the risk of tangling the yarns when the direction of the mandrel is changed (Fig. 5.9). Another advance in reducing fiber damage is the use of a baffle, covering
5.8 Herzog radial braider.
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5.9 Radial braider bobbin carrier optimized for carbon roving processing.
about one-third of the spool. It reduces friction between the roving and the coil by reducing the take-up angle and the area affected when the yarn scrapes over the coil (Fig. 5.10). Radial braiders need a minimum machine size because of the angle between the horn gears. The first problem is to run the bobbin carriers smoothly around the machine. Noise and wear of machine parts increase in proportion with decreasing machine diameter. The second problem derives from the limited room for movement by the bobbin carriers. This reduces spool capacity, particularly for smaller machines. If there is a minimum size required, there is no theoretical limitation to the maximum size of machines. The design flexibility regarding the number and diameter of horn gears and bobbin capacity is constantly growing with machine size. One of the biggest radial braiders, with 288 bobbins and
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5.10 Effect of the baffle (realized as an open cage for demonstration).
a diameter of about 4.6 meters, is located at the University of Dresden (Germany): see Fig. 5.11. Independent of machine size, radial braiders are comparatively slim and both face planes can be kept completely flat. Motors and feeding systems for 0° yarns can be located on the outside. Hence two or more machines can be positioned side by side to produce a multilayer textile.
5.3.2
Overbraiding of mandrels
Overbraiding can be regarded as the most flexible technology for net-shaped or near-net-shaped textile preforming. At a first glance the preform shape is solely defined by the shape of the mandrel. But this is not the complete
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5.11 Radial braider at the University of Dresden.
picture. After removing the textile from the mandrel, other preforming methods like draping, folding or combination with other preforms can be used to create a huge variety of preforms. The technology of overbraiding needs a combination of a braider with a manipulation device for the mandrel. In some cases, a simple and lowcost linear track serves this purpose. However, as the preform becomes more complex, an industrial robot (see Fig. 5.4) can be regarded as the best option, particularly as the price is low enough to compete with most individual solutions. If the braider is connected to the robot control as an external axis, the complete production sequence can be automated by using the robot programming language. In 1991 Rosenbaum [1] described a machine configuration with a moving braider and a fixed mandrel. This is an optimized solution for very long mandrels for two reasons: first, the mandrel is supported on both sides, and, second, the required floor space is only about 10–20% longer than the mandrel, whereas it has to be more than twice the mandrel length if the braiding machine is fixed. Nowadays advanced braiding machines combine both a moving braider and an industrial robot. Two industrial robots, one on either side of the braider, can be coupled to one integrated unit if a doublesided support of the mandrel is required.
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The production of a preform is carried out by moving the mandrel in the braider back and forth. Each machine path creates a new layer. The stacking sequence depends on the type of braid (see below, Section 5.5.1). Wall thickness transitions (by two layers) can be achieved by changing the direction of motion, when only a part of the mandrel is covered. To create a neat edge on the inversion, the braid needs to be fixed. A ring of needles around the mandrel is a simple but unreliable solution. Clamping devices provide a better solution. They have sheet metal inside to form the edge and a stamp outside to ensure fixation when the sheet is removed. Figure 5.12 shows how wall thickness transitions can be achieved. The symmetrical construction of a radial braider is an optimized solution for overbraiding. This symmetry means that no deviations between forward and backward braiding limit the process and the flat shape of the yarn path in the machine maximizes the space for manipulating the mandrel (see Fig. 5.13). Compared to winding, overbraiding offers more freedom regarding part shape. Because of the self-supporting nature of braids, there is no limitation to fiber lay-up. Mandrels with more complex changes in curvature and circumference can be handled (Fig. 5.14 shows examples). Even undercuts can be handled by applying shaping devices. As shown in Fig. 5.15. The braid can be manufactured as a positive, and removing part of the mandrel before curing builds the undercut.
5.3.3
Three-dimensional (3D) braiding
The braiding technologies described above are limited to the production of textiles similar to woven fabrics. The preforms are built layer by layer
5.12 Realization of wall thickness transitions.
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5.13 Room for mandrel manipulation on different braider configurations.
with the mandrel as the shape-defining element. 3D braiding is completely different. The machine consists of a flat array of horn gears, connected by switches. The path of each individual bobbin can be freely chosen by altering the switches between two states. One state allows the bobbin to move from the actual horn gear to its neighbor; the other forces the bobbin to stay on the same horn gear and move in a new position 90° clockwise. On first-generation 3D braiders, all horn gears are turned simultaneously in one direction. Second-generation machines allow a horn gear to stay in position or to move either forwards or backwards. Figure 5.16 shows the design principle, and Fig. 5.17 shows a first-generation 3D braider with a 10 ¥ 10 horn gear array.
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5.14 Examples of overbraiding of extreme mandrel shapes.
This design means that the bobbins can move freely over the complete machine bed. The fiber architecture is no longer layer oriented. Defined by the programmable movement pattern, the yarns can now travel through the preform in three dimensions. On top of this the braid can be changed in shape, e.g. from a round to a rectangular cross-section, from solid to hollow, from a T to an L and H shape. The braid can even split up into two or more strands and then reconnect again later. Figure 5.18 shows an example for a 3D braid (T-profile), and Fig. 5.19 the machine setup. One basic method of operation is to reproduce the cross-section of the braid on the machine bed. A uniform 3D architecture can then be achieved by a standard moving pattern of the bobbins. More advanced 3D braids are much more complex. This was the reason for developing process simulation techniques at the same time as developing more advanced machines. Parameters to take into account include the moving pattern, the take-up speed and position, the dynamic forces influencing the yarns, yarn friction behavior, stiffness, etc. A basic problem of 3D braiding is the contradictory effects of increasing machine size. On the one hand, machines need to carry as many bobbins as possible. The total area of the product in cross-section is limited only
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B
A
A
5.15 Realization of an undercut by draping: A: curing tools; B: auxiliary braiding mandrel.
Take-up direction Standing threads input channel
3D-braid
Bobbin
Bobbin path Horn gear
Switch point (cycle)
Switch point (transfer)
5.16 Design principle of a 3D braider.
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5.17 First-generation 3D braider with a 10 ¥ 10 horn gear array.
5.18 Example of a 3D braid (T-profile).
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5.19 Machine setup for the T-profile shown in Fig. 5.18.
by the sum of braiding and standing yarns working simultaneously. On the other hand, a wide machine bed needs an increased pull-back capacity of the bobbins, which causes potential problems. The most serious is the early tangling of fibers away from the braiding point in the outer areas of the machine bed (Fig. 5.20). Compared to circular or radial braiders, where the horn gears rotate with a speed of up to 200 rpm, the productivity of 3D braiders is quite low, because the process is not continuous. Every production step has two parts: all switches have first to be set in their new position and, after that, the bobbins are pushed into the next position by turning the horn gears by 90°. Modern 3D braiders can execute around one production step per second, which is about 4–6 times faster than the speed of a first-generation machine. Second-generation 3D braiders have a modular design based on units with 4 ¥ 4 horn gears. The arrangement of the units is flexible and can be adapted to particular production needs. The basic configuration is nine units in a horizontal 3 ¥ 3 arrangement. By removing the center unit, the machine can be used for overbraiding of long straight mandrels. Optimizing the number and position of the units, e.g. in a T or L shape, helps to increase productivity, since the spare units can be used for other production requirements. An example of the selective use of individual units is shown in Fig. 5.21.
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5.20 Tangled fibers on a 3D braider instead of a compact braid.
5.21 Second-generation 3D braider.
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5.3.4
Other braiding technologies
Although not yet built, Georges Cahuzac has invented and patented a machine designed to produce an interlocked multilayer textile called ‘AEROTISS® 3.5 D fabric’ (Fig. 5.22). The textile is intended to overcome the impact damage sensitivity of CFRP materials with the help of rovings running between the layers to build multiple connection points. This type of textile requires a vertical braider consisting of multiple rows of bobbin tracks connected with self-shifting switches to manufacture a tube about 1 meter in circumference. It requires a machine with 5400 moving bobbins to build a 3 meter-wide 21-layer 3.5 textile with a thickness of about 3 mm. Production costs for the textile would be dominated by the cost of the machinery, including the bobbin carriers, and the time needed to change the bobbins.
5.4
Key parameters for using braiding machines
This section describes ways of obtaining an optimized setup of the braiding machine with respect to mandrel size and behavior of the braiding yarns. High-tech applications need smooth and flawless fiber architectures with a homogeneous fiber content of about 60%. Areas of neat resin in the laminate cannot be accepted. As a consequence, the coverage rate has to be as close to 100% as possible. Detailed investigations in recent years have shown that the so-called ‘UD-braids’ (see Section 5.5.1 below) offer superior material properties
5.22 Aerotiss 3.5D fabric.
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over standard biaxial and triaxial braids. Only UD-braids have been used for further development of braiding technology in EADS laboratories. For this reason, the formulas below are simplified for the use with UD-braids, where only one yarn angle and only a single layer thickness has to be considered. Formulas for biaxial and triaxial braids can be found in Rosenbaum [1], for example.
5.4.1
Braiding angle and coverage rate
The coverage rate depends on four determining factors: the mandrel circumference, the braiding angle, the number of braiding yarns and the width of the yarns after lay-up. Assuming a coverage rate of 100%, the number of yarns for the machine setup can be calculated like this:
N = (2pRM cos g)/wY
where RM = radius of mandrel, g = braiding angle, and wY = yarn lay-up width. Twist-free carbon rovings are the standard material used to produce highperformance composites. Most manufacturers deliver carbon rovings optimized for textile processes. These rovings also usually give the best results for braiding. The lay-up width can be easily reduced to produce a higher layer thickness. On braiding machines, where effective spreading devices cannot be used, the maximum width depends on the titer (linear density) of the yarn and on the type of sizing. Because the lay-up width is also influenced by the shape of the mandrel, experimental trials in the workshop are still the best option to find out the right roving width for the calculation of the braid. On mandrels with a changing circumference, the braid must be matched to the maximum circumference. If the braiding angle remains constant, the rovings are compressed in smaller mandrel areas and the layer thickness and areal weight increase. With a given fiber angle, the mandrel size is limited by the maximum number of bobbins the machine can carry. On the other hand, the machine setup can be adapted to smaller mandrels by leaving some bobbin positions empty or by choosing rovings of a lower titer for the complete setup or on just some of the bobbins. If possible, empty bobbins or those with different yarns should be distributed homogeneously on the machine.
5.4.2
Areal weight and layer thickness
The calculation of the areal weight of the braid is the second basic step in the preparation of a machine setup. Assuming a UD-braid, a fiber volume fraction of 60%, a coverage of 100% and HT fibers, layer areal weights in the
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range of 280 g/m2, corresponding to a layer thickness of about 0.25 mm, can be regarded as standard. A formula for the calculation of areal weight is:
mA = (Nf/1000cos (g))/2pRM
where N = number of yarns, f = yarn fineness in tex, g = braiding angle, and RM = radius of mandrel. Lower areal weights (from 230 to 200 g/m2 or lower) will need further processing. The braid is first produced with gaps between the yarns (coverage rate < 100%). The rovings must then be spread on the mandrel, for example with the help of a laminating roller. Online spreading mechanisms on the bobbin outlet and on or after the braiding ring are under development.
5.5
Characteristics and properties of braided textiles
While there is a huge number of books on the manufacturing of braided ribbons and strands for fashion and household goods, they are of very little help in producing technical textiles for high-performance fiber reinforcements.
5.5.1
Textile variations on circular braiders
With the exception of 3D braiding, all commonly known preforming techniques are based on circular braids. Biaxial braids A biaxial braid is the standard textile on a circular braider. The kind of weave (Fig. 5.23) differs depending on the position of the bobbins on the horn gears. No practical difference can be recognized between the two machine setups, neither in lay-up behavior nor in material properties. Both textiles show a wavy surface texture (Fig. 5.24) in combination with the typical fiber undulation known in woven fabrics. The layer thickness is twice the yarn height on the crossing points and effectively zero on the meeting point of four intertwining yarns. As a result, biaxial braids have some significant disadvantages for high-performance applications: ∑ higher preform compressibility in the thickness direction ∑ lower stiffness and strength in tension and compression ∑ inhomogeneous fiber content and risk of resin accumulations ∑ difficulty in reaching fiber volume fractions higher than 50%. Nevertheless some applications may benefit from other features deriving from the rough structure of the textile. First, there is a better interlock between
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(b)
5.23 Different weave types of biaxial braids: (a) standard setup leads to a 2 over 1 twill weave; (b) tandem setup leads to a 2 over 2 plain weave.
5.24 Typical surface quality of a biaxial braid.
the layers, leading to slightly higher interlaminar shear strength. Second, the gaps between the yarns and layers provide resin channels which allow faster infiltration or the use of resin systems with higher viscosity. The inherent symmetry of each biaxial braiding layer is a further advantage, especially for the production of flat preforms. After covering the mandrel with a series of braided layers, the mandrel can be removed and the preform flattened. The resulting layer sequence is always symmetrical.
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Triaxial braids Most circular braiders are equipped with feeding units for ‘standing ends’ through the hollow axles of the horn gears. These yarns are oriented along the axis of the preform, evenly distributed around the mandrel. Being fixed in position in the middle of the braided layer, they offer a cost-effective way to add some additional stiffness in a longitudinal direction. Triaxial braids also offer the highest productivity in terms of lay-up rate. However, triaxial braids have some disadvantages. Problems of fiber undulation, preform compressibility, resin content and distribution are even worse than in biaxial braids. In addition: ∑ ∑
The ‘standing end’ yarns are not completely straight. The total number of the standing ends is limited to the number of horn gears. ∑ The standing ends cause additional fiber undulation and waviness in the textile. Figure 5.25 shows the typical surface quality of a triaxial braid. The low number of standing ends and the undulation of each fiber reduce their contribution to increased stiffness and strength. The total length of the ‘straight yarns’ is about 5–15% longer than the preform, as a series of laboratory tests has shown.
5.25 Typical surface quality of a triaxial braid.
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UD-braids UD-braiding has been developed to overcome the drawbacks of bi- and triaxial braids deriving from fiber crimp. By replacing the carbon rovings on one set of bobbins (running clockwise or counterclockwise) with a very thin and pliant yarn, the residual undulation in the rovings can be reduced to a minimum. This is because the undulation is completely taken up by these so-called ‘support yarns’. Figures 5.26 and 5.27 show the principle and the effect of UD-braiding. Such a UD-braided layer more resembles a unidirectional fiber tape or a layer of a non-crimp fabric than a conventional braided layer. Various comparisons of material properties confirm this impression (see Fig. 5.28). The selection of the support yarns depends on the application and especially on the applied resin system. Thermoplastic yarns with a low melting point, for example, can be used to stabilize the preform for later handling and post-processing after a short heat treatment to glue the layers together. The support yarn can also carry special agents for resin modification or it could completely dissolve in the resin. Although not very robust, the support yarns (and that is the reason for their name) hold the rovings in position during the braiding process. This allows higher flexibility for fiber lay-up compared to winding. Support yarns have some drawbacks: ∑
Only one single + or – layer is produced in one machine path, with the complementary layer formed when the mandrel is pulled backward. ∑ When the mandrel is removed and the preform flattened, the layer stacking is no longer symmetrical. ∑ Yarns fed as standing ends cannot be used.
Auxiliary yarn Reinforcement fiber Braiding ring Mandrel
5.26 Principle of UD-braiding.
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Conventional braid
UD braid
5.27 Effect of UD-braiding. COM strength
800
COM modulus
80 70
600
60
500
50
400
40
300
30
200
20
100
10
Compression strength (GPa)
Compression strength (MPa)
700
0
0
NC-braid
Conventional braid
NCF
5.28 Comparison of compression properties (NCF, UD-braid, conventional braid).
As a result, the improvements in material quality are offset by a reduction in productivity of about 50%. Nevertheless UD-braiding is a serious candidate for weight-critical structural applications in aerospace engineering.
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5.5.2
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Textile variants on 3D braiders
The possible fiber architectures of 3D braids are in principle unlimited, which means there are many possible ‘textile variants’. As mentioned in Section 5.3.3, the production of a complex three-dimensional preform depends very much on factors such as the distance of the braiding point from the machine bed, the friction behavior of the yarns, yarn tension (e.g. for different bobbin positions on the machine bed) and, of course, the braiding program itself. Whereas the production process on a circular braider can still be optimized by experiments in the workshop, this is not possible for a 3D braided preform. For this reason the simulation of the 3D braiding process was developed in parallel with the development of the technology (see Fig. 5.29). Early versions of the software do not allow the simulation of mandrel overbraiding. The interfaces of the various programs are now user-friendly enough to consider using them for the simulation of circular braids. Indeed, current simulation development has focused more on conventional braiding technologies because the number of applications is many times more than for 3D braids.
5.5.3
Auxiliary technologies
Because braiding, especially UD-braiding, is limited to the production of layers with + or – orientation, it has to be combined with technologies for the other yarn orientations and for local thickness variations. Winding of 90° layers Winding devices (Fig. 5.30) similar to those used in the cable industry are used to produce a yarn orientation of nearly 90°. A complete orientation of 90° is impossible due to the spiral nature of the lay-up. With common carbon rovings, a winder can typically carry two bobbins without disturbing the layer orientation.
5.29 Simulation of a 3D braided H-profile.
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5.30 Winding device.
5.31 Problem of slopes with wound 90° layers.
Problems with wound layers can be observed when the mandrel shape changes. Slopes are difficult to handle, because the fiber tends to slip down (Fig. 5.31). A more serious problem is fiber undulation which is caused when the stair-like winding pattern is corrugated by the next braiding layer. This problem can only be solved with a higher number of smaller rovings, but this drives up the price of the material significantly.
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Feeding devices for 0° fibers As described above, the standing ends of triaxial braids do not work properly for high-quality preforms. Feeding devices for 0° fiber tapes (see Fig. 5.32) can be a suitable replacement as long as the tape is of constant width and is applied on a flat surface without being bent in a lateral direction. Such feeders can be located directly in front of the braiding machine. In this configuration, the tape does not need additional fixation because it is clamped by the braiding yarns. There is also no need for an active movement of the tape. Pick and place units for local reinforcements Structural elements like beams or struts often need local increases in wall thickness for load-bearing points. The braiding process itself is not able to deal with these small local increases in thickness. They often do not even cover the whole circumference of the mandrel. For an optimized CFRP design, they should be interleaved into the layer stacking of the preform. The application of pre-manufactured patches (Fig. 5.33) using a thermoplastic binder provides a solution. Industrial robots equipped with suitable end-effectors (e.g. heated stamps with a vacuum system) may be considered as a universal device for the handling of these patches. It is relatively simple to achieve a synchronized continuous movement of stamp and mandrel during binder activation.
5.32 Tape feeder for 0° fibers.
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5.33 Local fiber patch on a beam preform.
Displacers One problem with braids is that they are not able to run smoothly around holes or inserts, because there is no force to close the fibers around the hole. Instead fibres congregate around holes, producing very uneven fiber distribution and misalignment of the fibers, which reduces the strength of the component. The use of displacers (Figs 5.34 and 5.35) does not provide a sufficient solution to this problem.
5.5.4
Hybrid braids
A braiding machine can be equipped with a combination of different fibers at the same time. Although a mix of carbon rovings and a low-performance fiber like glass is not particularly useful in improving performance, a mix of carbon and aramid fibers might be useful in crash scenarios where the tough aramid can prevent the part from total disintegration.
5.6
Mandrel technologies
Preform complexity is often limited by the problem of demolding. The design and the material of the mandrel can be even more challenging than the design of the part itself, especially when a high degree of automation has to be achieved.
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5.34 Displacers for small and big openings.
5.35 Part manufactured with the aid of displacers for in situ integration of inserts.
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5.6.1
Mandrel materials
There is a wide range of suitable mandrel materials. The selection depends on three main questions: ∑
Is the mechanical performance of the mandrel material sufficient to bear the load during the braiding operation? ∑ Is the material able to withstand the thermal loads of the following processes (preforming, curing)? ∑ How can the demolding be achieved?
5.6.2
Persistent mandrels
It is acceptable for some applications that lightweight mandrels can be left in the final product. Usually these ‘persistent’ mandrels are made of structural or non-structural foam cores. A non-structural core does not contribute to the final behavior of the part. It simply carries the preform during production and, if expanding foam is used, it can supply inner pressure during the curing process in a closed mold. In the case of a structural core, the part can be considered as a sandwich construction where the mandrel transfers shear loads between the fiber-reinforced walls. In either case, the mandrel needs to be of a closed cell material and resistant to resin chemistry and process parameters (heat and temperature) to prevent the resin from filling up the inner cavities of the part. The mandrel adds significant weight to the part, not only through the mandrel material itself, but also through the resin which fills the outer cells of the foam at the interface to the skin material. As the volumetric weight of the foam reduces with bigger cells, the weight of the interface layer increases considerably. The optimum cell size with respect to the weight trade-off must be calculated individually for each part.
5.6.3
Removable mandrels
There are numerous options to select a mandrel material, as long as the geometry of the part makes it possible to remove a solid mandrel from the part. If this is not possible, a demountable mandrel should be considered as the second-best option. Figure 5.36 shows how to split the mandrel at the smallest diameter point or in a junction area of two straight mandrel parts. For more complex geometries only meltable or soluble materials can be used. Experiments have been performed with a number of materials. Wax with high melting point (60°C < tM < 120°C) Such waxes are a good choice in combination with room-temperature resins. They are easy to process and show low shrinkage. While their stability is © Woodhead Publishing Limited, 2011
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5.36 Examples for demountable mandrels for a nozzle and a JFprofile.
not very good, wax mandrels with a diameter of 50 mm and a length of 1 m have been used successfully with single-sided clamping by a robot. A thin wax film on the inner surface of the part after demolding is unavoidable. Pressed salt Pressed salt cores show good mechanical stability and can be used for all resin types and curing temperatures. In principle they are very attractive but a dilution time of days to a month (if it has to be done through small holes) is a major drawback. Low melting alloys (120°C < tM < 180°C) These alloys are very stable. A wide range of different melting points is available. Extensive tests have shown that alloys with a melting temperature of about 160°C fit the curing cycle of high-temperature resins perfectly. During infiltration and up to the gel point of the resin, the mandrel is still rigid. After that point, when the part has hardened enough for a free-standing final cure, the alloy starts to melt. There are two major drawbacks. The first is technical. Due to its weight, even a small residue of alloy can ruin the weight benefit of a part. The second problem derives from the alloy components. To realize the low melting points, poisonous metals like lead, cadmium and
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even mercury have been included in alloys. These materials are meanwhile unacceptable from a safety and environmental point of view. Bindered sand Like salt and low-melting alloys, bindered sand is well known from metal casting. By using the right binder, the material disintegrates when a certain temperature is exceeded. After that, the sand becomes pourable again and can be demolded even through small holes. The material is cheap, easy to process and very stable, but it is also porous and it can be penetrated completely by the low-viscosity RTM resins. These cores cannot be used without a sealing bag. Thin foils of polyethyleneterephthalate (also known from household oven bags) do a good job, though crinkles cannot be avoided for complex geometries. The foils are fixed to the part surface after curing. Problems may arise from resin residues in the wrinkles of the foil and if the sealing bag is damaged. This may happen during the braiding operation or during closing of the mold, when the preform is being compacted. In that case, the resin flows into the sand core and the part is irreparably lost. Aquacore™ Aquacore is very similar in the way it works to bindered sand. Consisting of lightweight micro-balloons and a water-soluble binder, it is an optimized system for all resin systems known for FRPs. Even after high-temperature curing cycles, it dissolves easily and quickly under flowing water. Like sand cores, Aquacore mandrels are porous. In combination with low-viscous RTM resins, a sealing bag is strongly recommended. Tests with the sealant by the manufacturer of Aquacore and with other paintable sealants have been only partly successful. The risk of cracks in the coating is very high. Evacuated sand-filled tubes Experiments with evacuated tubes filled with sand have been very successful. A silicon rubber tube matching the contours of the part and filled with finegrain sand becomes stiff after applying a vacuum of about 100 mbar. It is very important to ensure the sand is densely packed without cavities which could collapse. Filling the tube in an outer mold using a shaker device has yielded the best results. For long and slim mandrels, an additional inner metallic support structure can improve rigidity and handling. Figure 5.37 shows a ‘peanut bag’ mandrel After braiding, the preform and the mandrel are put into the female closed mold. The mandrel tube can then be pressurized. This densifies the preform and straightens the fibers before resin injection starts. After curing, the sand
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5.37 ‘Peanut bag’ mandrel.
can be poured out of the tube and the tube becomes loose. Both can be reused.
5.7
Further processing
5.7.1
Stabilization of the preform
There are different reasons for carrying out a stabilization step after braiding. This step, where a binder in the preform is activated, e.g. by application of heat, is sometimes also called ‘preforming’. When the braid must be removed from the mandrel, preforming is often used to hold the fibers in position for handling and transport. The reduction of the bulkiness of the braid is another reason for a preforming step, especially prior to RTM curing. The cavity of an RTM tool is matched to the end contour of the part. To reach the highfiber volume content in the part, the preform must be compacted without displacing the fibers when the mold is closed. This is very complicated and needs movable mold components in many cases. It may not be possible if the part geometry is too complex. Binders can be thermoplastics or thermosets, and can be supplied as powders, granules or yarns. The choice depends on the application. A reliable method is to combine the reinforcement fibers with thin thermoplastic yarns on some of the bobbins. Precise dosing and distribution of the binder is then very simple. For UD-braiding, the support yarns can take over the additional role of the binder. It is essential to consider the compatibility of binder and resin for each individual combination: ∑ ∑
The presence of a binder can reduce the permeability of the preform and may cause problems during infiltration. Thermoplastic binders can reduce the mechanical performance of the material, particularly its moisture properties.
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∑ ∑
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Washout effects on the resin front can cause a very uneven distribution of the material in the final part. Thermoset binders may cause a two-phase matrix of unpredictable quality.
The amount of binder should be kept as low as possible.
5.7.2
Cutting, folding and preform mounting
The possibility of producing a huge variation of net-shaped preforms by cutting and folding of braids and by the combination of sub-preforms can be illustrated with examples of different types of stiffener profiles. The so-called JF-profile is a material- and weight-optimized version of the H-profile where one half of one flange is removed. This profile type is used, for example, in aircraft fuselage frames. It features a non-constant curvature and often a variable height of the profile web. The foot of the profile is mounted to the skin. Figures 5.38 and 5.39 show the production steps for such profiles. Similar manufacturing methods have been tested for other types of profiles. A special production method has been developed for H-profiles. It is called
5.38 Manufacturing steps for a JF-profile.
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5.39 Braided C-, Z- and LCF-profiles. 1. Milling of foam mandrels (reusable metal tools for serial production) 2. Overbraiding of the mandrels (UD-braid) 3. Stacking of the braided mandrels in the lower mold; additional NCF layer forms part of the lower chord
1
2
3
4
4. NCF draping for upper chord 5. Infiltration and curing (not shown) 6. Profile separation 7. Final machining
6 7
5.40 Production of H-profiles with ‘stacked curing’.
‘stacked preforming’ or ‘stacked curing’ and offers a highly cost-effective production method for beam profiles with curvatures and variable crosssection geometries. The process starts with the overbraiding of rectangular contoured mandrels. These mandrels are then draped side by side on a mold which is already covered with a flat textile (weave or NCF). A second textile on top of this ‘stack’ of mandrels completes the operation. The whole stack is packed in a vacuum bag either for preforming or directly for infiltration and curing. As the final step, the H-profiles are formed by cutting the stack along the middle of each braiding mandrel on both sides. The mandrels became free and can be reused. Figure 5.40 shows the production process.
5.8
Typical applications
5.8.1
Profiles
Whereas pultruded profiles are already in use for stringers and floor beams, braided CFRP parts have not become established in the aerospace industry up to now. Thanks to advances in material properties, braided fuselage
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frames and floor beams are being considered as an alternative to conventional prepreg-based parts. There are three reasons for this development: ∑ ∑
The high degree of automation which offers significant cost reduction A nearly scrap-free process from the roving spool directly into the product ∑ High flexibility with respect to geometrical changes, e.g. in web height and curvature radius.
5.8.2
Contoured hollow bodies
As long as closed end caps need not be manufactured in one shot, overbraiding is a good production method for hollow parts. Complex air ducts, engine nozzles (see Fig. 5.15), multi-spar aircraft flaps (Fig. 5.41) and crash structures (see below) are just a small selection of a huge number of possible applications. Aircraft propeller blades are another example of contoured parts built by overbraiding. Here the foam mandrel stays in the part, acting as a shear core. Larger propeller blades can have a multi-spar design to increase stiffness.
5.8.3
Crash structures
Unlike structural parts, where straight fibers are important, crash structures (Fig. 5.42) are made of a coarse standard braid. In this case, fiber undulation and areas of neat resin act as a trigger mechanism to cause as many broken fibers as possible, because crushing the fibers is the most effective method of energy absorption. Such crash structures, manufactured by overbraiding, have been developed for Formula 1 racing cars and are also used for the Mercedes SLR front crash absorbers.
5.8.4
3D braids
Within COBRAID, a research project funded by the European Commission, preforms for blade axles of a variable turbine stator have been developed. The axles consist of a 3D braid to improve impact resistance. It changes
5.41 Multi-spar flap.
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5.42 Crash structures.
5.43 3D braided stator blade axle before cutting of the unused fibers.
its cross-sectional shape from rectangular (with rounded edges) to flat and circular and then back to rectangular. The rectangular part incorporates an angled lever and the circular a bearing pin. The flat region, later covered with a second preform for the fan blade, needs to be very thin. For that reason, some of the bobbins were placed on the side corners of the machine. These rovings passed the flat preform area unused and were cut manually before the preform was completed (see Fig. 5.43). Triangular 3D braided filler noodles (Fig. 5.44) are meanwhile a standard product. They offer unique properties for profiles with high bending loads because they fit perfectly in the gusset, they can be adjusted in size and they
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5.44 3D braided filler noodles.
do not attract loads to this critical area because they are completely without straight fibers.
5.9
Limitations and drawbacks
There are three main limitations for braiding: ∑
The number of yarns cannot be changed automatically during braiding of a preform. In the case of extreme changes in the mandrel diameter, the number of fibers has to be adjusted to the smallest extent to ensure that the braid closes around the mandrel. Hence the small number of fibers must also cover the larger mandrel diameters. This can only be achieved by accepting either an incidental fiber angle or a coverage rate below 100%. ∑ Whereas it is possible to achieve changes in wall thickness of two braiding layers by clamping the braid and changing the mandrel moving direction, it is still a problem if only one layer is intended. Devices to deal with this problem have only been demonstrated on a laboratory scale so far. ∑ There is still a lack of acceptance of the technology because the fiber architecture of a braided preform depends on a huge number of parameters. The numerical simulation of the process is still relatively new. Simple design rules are not yet available. Setup time is another problem. If the yarns on the bobbins of the braider
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run out, the production process stops and the braider must be manually equipped with a new set of full bobbins. The duration of this enforced stop depends on the machine size and the skill of the workers. Ideas to solve the problem with an external magazine of fresh bobbins have failed so far due to the high costs of such a device. The price is driven by the fact that a complete second set of both bobbin carriers and bobbins is needed. The set of carriers with the empty bobbins is guided to the magazine by a switch and then replaced by the full set.
5.10
Future trends
Braiding has made significant progress with the development of radial braiders and UD-braiding. The next steps will most likely concentrate on process automation, in situ preforming, mandrel handling, and the development of innovative mandrel materials. The second, more important field of work will be process simulation. Today, each part qualification is specific to that part and almost any design change leads to expensive work to perform the tests for a new qualification. Process simulation could significantly improve this situation.
5.11
Sources of further information and advice
Balzás Z (2005), Modeling of Braided Fiber Reinforced Compisites Crosslinked by Electron Beam, PhD Thesis, Budapest Eisenhauer E (2006), Characterization of UD-Braids, Master Thesis: European Masters in Advanced Textile Engineering Engels H (1992), Handbuch der Schmaltextilien Die Flechttechnologie, Teil 1, Fachhochschule Niederrhein, FB Textil- und Bekleidungstechnik Niu M (2005), Composite Airframe Structures, Hong Kong, Conmilit Press Roye A, Stüve J, Gries T (2005), Definition for the differentiation of 2D- and 3D-textiles, Technical Textiles 48 Useful Internet links A&P Technology: preform manufacturing; basic information about braiding http://www.braider.com/?a=24&b=braid-basics August Herzog Maschinenfabrik GmbH und Co. KG: manufacturer of braiding machines; basic information about braiding http://www.herzogonline.com/ RWTH Aachen University, Institut für Textiltechnik (ITA): basic research on preforming, including braiding http://www.ita.rwth-aachen.de/ita/ index.htm Stuttgart University, Institut für Flugzeugbau (IFB): basic research on preforming, including braiding http://www.ifb.uni-stuttgart.de/
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Dresden University, Institut für Leichtbau und Kunststofftechnik (ILK): basic research on preforming http://tu-dresden.de/die_tu_dresden/fakultaeten/ fakultaet_maschinenwesen/ilk/index_html SGL Kümpers: preform manufacturing http://www.sgl-kuempers.com TUM Technische Universität München, Institute for Carbon Composites (LCC): basic research on preforming, including braiding http://www.lcc. mw.tum.de/en/department/science/textile-preformtechnik/ Eurocarbon: manufacturer of braided preforms and semi-products; basic information about braiding http://www.eurocarbon.com
5.12
Reference
1. Rosenbaum J U (1991), Flechten: Rationelle Fertigung faserverstärkter Kunststoffbauteile, Köln, Verlag TÜV Rheinland, ISBN 3-88585-979-3
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6
Three-dimensional (3D) fibre reinforcements for composites
A. P. M o u r i t z, RMIT University, Australia
Abstract: An overview is presented into polymer matrix composites with three-dimensional fibre reinforcement. The applications, manufacturing processes, microstructure, mechanical and other properties of 3D fibre composites are reviewed. The materials examined are composites with threedimensional woven, stitched, braided, z-pinned, tufted or z-anchored fibre reinforcements. Progress in the modelling and experimental characterisation of the material properties of 3D fibre composites is described, including delamination toughness, impact damage tolerance, in-plane mechanical properties, and joint properties. Progress in the characterisation of other properties, including thermal conductivity, electrical resistivity and environmental durability, is also examined. Key words: three-dimensional fibre reinforcements, weaving, stitching, z-pinning, tufting, z-anchoring, manufacture, microstructure, mechanical properties, modelling.
6.1
Introduction
Three-dimensional fibre–polymer composites are an important class of advanced engineering materials containing a 3D network structure of planar and through-thickness reinforcing fibres. A variety of 3D fibre composites have been developed to overcome some of the problems that are inherent with conventional polymer laminates, such as their low delamination resistance and poor impact damage tolerance. Many types of 3D fibre composite materials have been created, and they are classified according to the method by which the three-dimensional fibre reinforcement is produced. The most common types are 3D woven composites (Bogdanovich and Mohamed, 2009; Dickinson et al., 1999a; Dickinson and Mohamed, 2000; Mohamed et al., 2001, 2005, Tong et al., 2002), stitched composites (Dransfield et al., 1994; Mouritz and Cox, 2000; Tong et al., 2002) and z-pinned composites (Mouritz, 2007; Partridge et al., 2003; Tong et al., 2002). Other important types are multi-layer non-crimp knitted composites (Lomov, 2010) and noncrimp orthogonal woven composites (Bogdanovich and Mohamed, 2009; Mohamed et al., 2001; Tong et al., 2002; Lomov, 2010). Other, less common types are 3D braided composites (Mungalov et al., 2007; Tong et al., 2002), 157 © Woodhead Publishing Limited, 2011
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tufted composites (Cartié et al., 2006; Dell’Anno et al., 2007) and z-anchor composites (Abe et al., 2003). The fibre architecture for the various types of 3D composites is basically the same, and is shown in Fig. 6.1. The fibre architecture is designed with a large percentage of in-plane continuous fibres (like a conventional laminate) to provide high in-plane mechanical properties. The novel architectural feature of 3D composites is the inclusion of high-stiffness, high-strength fibres in the through-thickness (or z-) direction. The through-thickness fibres, which are given the general term of z-binders, are used to provide 3D fibre composites with higher delamination fracture toughness and impact damage tolerance than conventional laminates. The volume fraction of z-binder reinforcement is usually low (typically under 5–10%), although this is sufficient to greatly increase the delamination resistance and damage tolerance. Z-binders are also used to improve the structural properties of composite joints, including higher ultimate strength, bearing strength, fatigue life and damage resistance compared to conventional bonded or bolted connections. In addition, 3D fibre reinforcements offer the possibilities of improving the manufacturing quality by minimising ply slippage and lowering the manufacturing cost by shortening the preform lay-up time. By appropriate selection of the z-binder material, 3D fibre composites can also have higher through-thickness thermal or electrical conductivity than conventional laminates. Many types of 3D fibre composite materials have been created over the past 30 years, although their applications are relatively few (Mouritz et al., 1999; Tong et al., 2002). A summary of current and potential applications for 3D woven, stitched, z-pinned and braided composites is presented in Table
z y
x
6.1 Basic fibre architecture of 3D composite with in-plane continuous fibres aligned in the x- and y-directions and a small volume fraction of through-thickness fibres in the z-direction. Adapted from image by 3Tex, Inc., Cary, North Carolina.
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6.1. Many of the current and most of the potential applications for 3D fibre composites are in aerospace structures to provide higher damage tolerance than conventional aircraft-grade laminates. The long-term prospects for the applications of 3D fibre composites in aircraft are uncertain. Some types of 3D fibre composites are being used for non-aerospace applications, such as 3D woven composites in civil structures and the use of z-pinned composites in Formula 1 racing cars. As in the aerospace field, the uptake of 3D fibre composites in automotive, civil, maritime and other non-aerospace fields has been slow despite the benefits they provide over conventional laminates. This chapter presents an overview of the science and technology of polymer composite materials with three-dimensional fibre reinforcement. The manufacturing processes, microstructure, through-thickness and in-plane properties of 3D fibre composites are described.
Table 6.1 Current and potential (demonstrator) applications for 3D woven, stitched, z-pinned and 3D braided composite materials 3D ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑
woven composites Missile nose cones and engine nozzles Stiffened panels for aircraft skins Leading edges and connections for aircraft wings Inlet blades, turbine rotors and other low-temperature components for gas turbine engines Engine mounts H-shaped connections on the Beech starship I-beams for civil infrastructure Ballistic and armour panels Manhole covers Prostheses
3D stitched composites ∑ Joint connections (lap, T-, J-, etc.) ∑ Stiffened panels for aircraft skins Z-pinned composites ∑ Air duct panel sections on the F-18 Superhornet ∑ Bay doors on C-17 Globemaster III ∑ Roll-over bars on Formula 1 racing cars ∑ Stiffened panels for aircraft ∑ Joints connections (lap, T-, J-, etc.) 3D ∑ ∑ ∑ ∑ ∑ ∑ ∑
braided composites Missile nose cones and engine nozzles Beams and trusses Shafts and connecting rods Ship propeller blades Bridge structures Automotive body panels, chassis and drive shafts Prostheses
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6.2
Manufacture of three-dimensional (3D) fibre composites
6.2.1
3D Weaving
Weaving is essentially the process of producing fabric by interlacing two sets of fibrous yarns (warp and weft) which are aligned at 90° to each other. Single-layer woven fabrics produced using conventional weaving looms have been used as the reinforcement in structural composite laminates for many years. 3D weaving is an adaptation of the single-layer weaving process to produce an integrated multi-layer fabric consisting of warp and weft yarns in the in-plane directions and z-binder yarns woven in the through-thickness direction. The z-binder yarns are interlaced with multiple layers of warp and weft yarns to create the 3D fibre reinforcement using hand-operated or (more often) computer-controlled looms. There are two main types of 3D woven reinforcement: 3D interlock woven and 3D orthogonal non-crimp woven fabrics (Bogdanovich and Mohamed, 2009; Dickinson et al., 1999a; Dickinson and Mohamed, 2000; Mohamed et al., 2001, 2005; Tong et al., 2002). 3D interlock fabrics are woven by interlacing the warp, weft and z-binder yarns to create a fully interlocked fibre structure. In these fabrics, each ply layer consists of interlaced warp and weft yarns like a conventional single-layer fabric. Multiple layers of fabric are interlocked with through-thickness z-binder yarns. 3D orthogonal non-crimp woven fabrics consist of warp and weft yarns that are stacked as separate ply layers (without interlacing) and held in place with z-binder yarns passing through the thickness in an orthogonal pattern. The non-woven structure avoids crimp of the warp and weft yarns, and thereby results in higher in-plane stiffness and strength than 3D interlock woven reinforcement. Any type of fibrous yarn can be used in 3D woven fabrics, with the most common being carbon and glass for structural composites and aramid for composites offering ballistic impact protection. The z-binder yarns are thin bundles of continuous filaments (typically 50–500 tex) which are woven into the fabric to a volume fraction in the range of 0.5–10%. The architecture of the z-binder reinforcement can be adjusted by the weaving process, with orthogonal and layer interlock patterns being the most widely used (Fig. 6.2). After the fabric has been woven, it is then infused with liquid resin using conventional composite processing methods such as resin transfer moulding or resin film infusion.
6.2.2
Stitching
Stitched fibre reinforcement for composites is produced by sewing highstrength thread through the thickness of a stack of dry fabric layers using a stitching machine (Fig. 6.3) (Dransfield et al., 1994; Mouritz and Cox, 2000; © Woodhead Publishing Limited, 2011
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z-binder
(a)
z-binder
(b)
6.2 Schematic representations of the (a) orthogonal and (b) layer interlock patterns for the z-binder reinforcement in 3D woven fabrics.
Tong et al., 2002). The machine contains a single or multiple needles which carry the stitch thread through the fabric layers to form a 3D fibrous structure. Machines range in capability from simple (household) sewing machines with a single needle to computer-controlled robotic machines with many stitching heads. Advanced machines have the capability to insert stitches at orthogonal and inclined angles in various densities and patterns. The three patterns most often used in 3D fibre reinforcements are the modified lock stitch, lock stitch and chain stitch (Fig. 6.3b), with the modified lock stitch being the most popular because it causes the least crimp to the in-plane fibres. Stitches are usually inserted in a grid pattern to an areal density of 1 to 25 stitches per cm2, which is equivalent to a volume content of z-binders of about 0.1 to 10%. The thread material is almost always high-stiffness, high-strength yarn (100–2000 tex) made of carbon, glass or aramid. Other high-strength thread materials, such as UHMW polyethylene, are sometimes used. Technical embroidery is a variant of stitching used to provide localised
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(a)
Modified lock stitch
Lock stitch
Chain stitch
(b)
6.3 (a) Schematic of the stitching process; (b) common stitch patterns.
through-thickness reinforcement together with in-plane fibre placement. In this process, fibrous yarn is fed into the path of a stitching needle which presses it through the fabric preform and thereby creates the z-binders. With computer-controlled embroidery machines it is possible to accurately place the yarns along complex paths, which allows high-stress regions of a composite component to be reinforced by fibres laid in the maximum stress direction. Technical embroidery is used sparingly as a through-thickness reinforcement technique for composites. As with 3D woven fabrics, the fabrics reinforced in the through-thickness direction by stitching or embroidery are infused with liquid resin to create the 3D fibre composite.
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6.2.3
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Z-Pinning
Z-pinned reinforced composites can be made using various methods, with the most common being the so-called UAZ® process that involves inserting thin rods using an ultrasonic device (Mouritz, 2007; Partridge et al., 2003; Tong et al., 2002). Z-pins are made from extruded metal wire or pultruded fibrous composite with a diameter in the range of 0.1 to 1.0 mm. The main steps in the UAZ® process are shown schematically in Fig. 6.4. Z-pins are held within a foam carrier which is placed over a stack of ply layers, which is usually uncured prepreg. The foam is used to ensure an even spacing between the z-pins and to provide them with lateral support during insertion. Z-pins are driven from the foam carrier into the preform by high-frequency compressive stress waves generated by an ultrasonic device. The stress waves also cause moderate heating of the prepreg that softens the resin matrix, which eases insertion of the z-pins. Z-pins are inserted progressively by moving the ultrasonic device over the foam carrier several times until it has completely collapsed and all the pins are embedded into the preform. The compressed foam carrier and any excess length of z-pin protruding the preform is shaved off and discarded. After z-pinning, the prepreg composite is cured using conventional processes such as vacuum bagging and autoclave. Z-pinning can be used for the through-thickness reinforcement of preforms consisting of multiple layers of dry fabric or (more often) uncured prepreg. An important aspect of z-pinning is that it is the only through-thickness reinforcement technique for prepreg composites; the other methods (e.g. 3D weaving, stitching, braiding) can only be applied to dry fabric which is later infused with liquid resin.
6.2.4
Structural non-crimp knitting
Structural non-crimp knitting is a manufacturing technique that combines aspects of weaving and knitting known as multi-axial warp knitting or stitch bonding to produce non-crimp fabric (Lomov, 2010). Conventional noncrimp fabric consists of multiple layers of straight (uncrimped) yarns that are linked together by z-binders inserted using a warp knitting machine. Figure 6.5 illustrates the insertion of z-binders by the knitting of thread through the thickness of dry fabric consisting of straight in-plane yarns oriented at different angles. Each yarn layer is separate from the other layers, and they are not interlaced, in order to avoid crimp. Mats of chopped fibres can be incorporated into non-crimp fabrics. The z-binders are inserted through the thickness of the fabric preform with sharp-head needles using a specialist knitting machine. During the knitting process the needles do not penetrate any yarns, but instead they pass to the sides to form knit loops between the yarns to avoid crimp and damage. Conventional non-crimp fabrics are knitted
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Preform on laminate ready for ultrasonically assisted z-pin insertion
Low density foam (upper half) of preform collapses allowing z-pins to penetrate the laminate
Reinforcement of laminate complete
Excess z-pin remaining in preform is sheared away
Z-pinned laminate
6.4 Main steps in the z-pinning of prepreg composite using the UAZ® process.
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0° 90° +45° 90° –45°
6.5 Non-crimp fabric.
using organic-based filaments with high flexibility to form a knit loop without breaking, such as polyester yarn. However, such filaments have low modulus and strength and therefore do not promote significant through-thickness reinforcement. Structural non-crimp fabrics are a variant of conventional non-crimp fabrics with the knitted z-binders being made of fibrous carbon, glass or aramid thread to provide high interlaminar properties.
6.2.5
Braiding
Braiding is a textile process that involves the counter-rotation of multiple fibrous yarns around a circular frame to intertwine the yarns into a braided fabric. Several variants of the braiding process are used to create fabrics with a 3D fibre structure, including the two-step 3D braiding (Mungalov et al., 2007). The architecture of a 3D braided fabric is illustrated in Fig. 6.6 and shows that the yarns are intertwined, causing a high degree of fibre crimp. Braided fabric composites have relatively low in-plane mechanical properties due to the crimp, but high crush resistance and delamination fracture toughness. Braiding is most suited to the manufacture of narrow-width flat fabric or tubular fabric. By control of the braiding process and the types of yarns used, it is possible to form intricate shapes including circular and square-shaped tubes, I-beams, C-channel sections, and solid rods.
6.2.6
Tufting
Tufting is a one-sided stitching process that involves the use of hollow needles to insert thread through a multi-layered fabric preform, as shown schematically in Fig. 6.7 (Cartié et al., 2006; Dell’Anno et al., 2007). The needle carries high-strength thread through the fabric, and then the needle is withdrawn along the path it entered, leaving behind tufted z-binder yarn. The z-binder is held in place as the needle is withdrawn by friction resistance
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6.6 3D braided fibre structure. Thread
Needle
Loops
6.7 Through-thickness reinforcement of a T-shaped fabric preform by tufting. Reproduced from Cartié et al. (2006).
imposed by the fabric. The end of the tufted z-binder forms a loop which can be cut or retained after the stitching process is complete. Threads with high flexibility such as aramid or other organic-based filaments are better suited for tufting than brittle yarns such as carbon or glass, which are more likely to break. The tufting process is simpler than conventional stitching because it does not require interlocking with a second (bobbin) thread. Despite the
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simplicity of the process, it is not commonly used to reinforce composite materials in the through-thickness direction.
6.2.7
Z-anchoring
Z-anchoring is a process which involves punching thin rods through a fabric preform to create z-binders (Abe et al., 2003). Pressing thin forming rods through multiple layers of fabric forces the in-plane fibres to bend towards the through-thickness direction, as illustrated in Fig. 6.8. Unlike the other through-thickness reinforcement techniques that use separate yarns or pins to create the z-binders, z-anchoring uses the in-plane yarns to the fabric preform to create the z-binders. The volume percentage of in-plane fibres that are deformed into z-anchors is controlled by the diameter and spatial density of the forming rods. After the rods have crimped the in-plane fibres they are removed, leaving behind through-thickness z-binders called z-anchors. When the z-anchoring process is complete, the fabric is then infused with liquid resin using conventional liquid moulding processes to produce a 3D fibre composite material.
Punch
Ply layers
(a)
(b)
6.8 Schematic of the (a) local ply deformation forming z-anchor and (b) composite with series of z-anchored preforms.
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6.3
Microstructure of three-dimensional (3D) fibre composites
6.3.1
Background
Understanding the microstructure of 3D fibre composites is the key to understanding their mechanical and other properties in the in-plane and through-thickness directions. As mentioned, 3D fibre composites are reinforced with a relatively high volume fraction of in-plane fibres (typically 50–60%) and a much lower fraction (under 5–10%) of z-binders. When z-binders are created, the in-plane fibre architecture can be disturbed, which introduces microstructural defects that affect the mechanical properties. The microstructural changes, which mostly occur at the sub-millimetre length scale (100–1000 mm) and fibre scale (5–15 mm), can be beneficial or detrimental to the mechanical properties of the 3D fibre composite material. This section describes the main microstructural features of 3D fibre composites based on experimental observation, with a focus on damage and defects created by the presence of the z-binders. Progress in the modelling of the microstructure of 3D fibre composites is also briefly outlined.
6.3.2
Waviness, crimp and damage to in-plane fibres
Despite the variety of manufacturing processes used to produce the different types of 3D fibre reinforcements, their microstructural features are remarkably similar. The insertion of z-binders through the thickness of dry fabric (in the cases of 3D weaving, stitching, tufting, braiding and z-anchoring) or uncured prepreg (in the case of z-pinning) can disturb the arrangement of the in-plane fibres as well as damage and break them. The z-binder and its carrier (e.g. needle, punch) push aside the in-plane fibres during the insertion process, which creates a defective region at each z-binder location as shown in Fig. 6.9. The fibres are forced to bend around the z-binders which create local regions of high fibre waviness. A resin-rich region next to the z-binders is often created where the fibres have been displaced. Fibre waviness can reduce those in-plane mechanical properties of the 3D composite that are sensitive to fibre misalignment, such as compressive strength. The severity of fibre waviness is dependent on the diameter of the z-binder and its carrier as well as the in-plane fibre architecture (e.g. fibre arrangement, fibre packing density). Fibre waviness angles can be as low as 2–3° and as high as 15–20°, with the angle increasing with the z-binder diameter. Fibre waviness can be avoided or minimised in 3D reinforcements by controlled insertion of the z-binders between the in-plane fibre yarns, as occurs in the manufacture of non-crimp orthogonal woven and non-crimp knitted fabrics.
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Through-thickness segment of stitch
Resin pocket
(a)
z-binder
Resin zone Wavy fibres
(b)
6.9 (a) Schematic and (b) photograph of fibre waviness around a z-binder.
An important characteristic of many of the defects in 3D fibre composites is their variability. The same type of defect can vary significantly in size and shape depending on its location within the 3D fibre reinforcement. For example, Fig. 6.10 shows examples of the scatter in fibre waviness, fibre crimp and z-binder strength within 3D fibre reinforcements. The variability is caused by a number of factors, including the fibre packing density of the in-plane yarns, the diameter of the z-binders, and whether the z-binders are located through the centre or to the sides of the in-plane yarns. The stochastic nature of many of the microstructural defects has significant impact on the variability of the mechanical properties of 3D fibre composites. In-plane fibres are crimped in the through-thickness direction under the force used to insert z-binders (Fig. 6.11). Crimp is particularly severe at the surface of 3D woven, stitched and tufted reinforced fabrics, where the surface segment of the z-binder presses into the in-plane fibres. Like fibre waviness, fibre crimp is localised to the region immediately surrounding the z-binder and is most likely to reduce mechanical properties sensitive to the misalignment of load-bearing fibres. Fibre crimp cannot be avoided using
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the 3D composite processing methods except those used to make non-crimp orthogonal woven and non-crimp knitted fabrics. It is also worth noting that while considered a defect, fibre crimp is used to create the z-binders in z-anchor composites. The in-plane fibres are deliberately crimped throughthe-thickness using the forming rods to create the z-anchors (Fig. 6.8). 60
Frequency distribution
50 40 30 20 10 0
0–5
5–10 10–15 Waviness angle (°) (a)
15–20
60
Frequency distribution
50
40
30
20
10
0
0–5
5–10
10–15 15–20 Crimp angle (°) (b)
20–25
25–30
6.10 Examples of the stochastic behaviour of defects and damage in 3D fibre reinforcements: (a) variation in fibre waviness angle in a stitched woven fabric (Mouritz and Cox, 2010); (b) variation in fibre crimp angle in a 3D woven composite (Mahadik et al., 2010); (c) variation in z-binder strength in a 3D woven fabric.
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Cumulative probability (%)
100
80
60
40
20
0 600
700
800 900 1000 Fibre strength (MPa) (c)
1100
1200
6.10 Continued.
In-plane fibres are also damaged and broken during production of 3D reinforcements (Fig. 6.12). Fibres are damaged and broken by the force used to drive the z-binders through the preform. Fibres are also broken by friction stresses generated as the z-binder is drawn through the preform. Friction between the in-plane fibres and the z-binder and its carrier creates sub-microscopic flaws on the fibre surface, which reduces the tensile strength. The damaged and broken fibres are located in the vicinity of the z-binders, and may weaken the composite when present in sufficient numbers. Again, damage and breakage of the fibres is avoided with non-crimp orthogonal woven and non-crimp knitted fabrics.
6.3.3
Resin-rich regions, voids and microcracks
Resin-rich regions and voids may form next to the z-binders due to changes in the fibre arrangement. In-plane fibres are displaced sideways during insertion of the z-binders, which creates a cavity that fills with polymer during the resin infusion of dry 3D fabric or the curing of z-pinned prepreg. The resin-rich region has an eyelet shape extending in the fibre direction for a distance of 0.5 mm or more (Fig. 6.9). When the z-binders are placed close together, continuous resin-rich channels are formed as shown in Fig. 6.13. Voids can also form in these regions when the resin gels before it has completely filled the cavity created by the z-binders. Microcracks can form at the interface between the z-binders and surrounding composite material (Fig. 6.14). These cracks are created due to the mismatch in the coefficients of thermal expansion of the z-binder and
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In-plane yarns
z-binder (a)
z-binder
(b)
6.11 (a) Schematic of crimping of surface fibres. (b) Photograph of fibre crimp (indicated by the arrows) through the thickness of a z-pinned composite.
surrounding material. Thermal stresses are generated during high-temperature curing of the composite, which induces tensile and shear strains at the z-binder interface that cause interfacial cracks to develop during cooling. This damage is impossible to avoid unless the z-binder and composite have similar coefficients of thermal expansion or high interfacial adhesion.
6.3.4
Damage to z-binders
The architecture of z-binders in the 3D fibre reinforcement is sometimes different from what was intended, and this can affect both the in-plane and
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500 µm
6.12 Broken fibres caused by stitching.
z-binder
Resin channel
z-binder
6.13 Resin-rich regions and voids within a z-pinned composite.
through-thickness properties. Imperfect z-binder architecture is created due to limitations with many of the 3D manufacturing processes. Even with the use of numerical control systems, most of the manufacturing processes cannot exactly insert the z-binder reinforcement and there is some departure (usually minor) from the pristine architecture. The z-binders can be crimped or displaced under the high compaction pressure used to consolidate the 3D reinforcement during processing. In addition, the z-binders can be damaged and weakened during their insertion
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z-binder
Interfacial cracking 20 µm
6.14 Microcracking around a z-binder.
into the preform. For example, reductions in the tensile strength of z-binders in 3D woven fabric of up to 50% and in the strength of stitches of 20–50% have been reported. It is believed that the high bending and friction stresses experienced by the z-binders during insertion causes damage to some of their fibres, which reduces the tensile strength. This damage has the potential to reduce the effectiveness of z-binders at increasing the through-thickness strength, delamination toughness and impact resistance of 3D fibre composites, which is a reason why flexible z-binders made of aramid and other organicbased fibres are often preferred over brittle z-binders made of carbon or glass fibres.
6.3.5
Swelling and compaction
Volumetric swelling or compaction caused by the z-binders is another feature of 3D fibre composites which can affect their mechanical properties. Swelling can occur with z-pinned prepreg composites, but not for other types of 3D fibre composites made using dry fabric preforms. The thickness of z-pinned composites increases with the volume fraction of z-binders with a corresponding reduction to the in-plane fibre volume fraction. It is believed that the high stiffness of the z-pins resists compaction of the prepreg preform during consolidation and curing, resulting in a slightly thicker material with a reduced fibre volume content. Swelling may also be caused by displacement of in-plane fibres to accommodate the z-binders within the 3D reinforcement. 3D fibre composites made using fabric preforms can display the opposite behaviour to z-pinned prepreg materials, with an increase to their fibre volume
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content due to compaction. The tension force applied to the z-binder yarns during 3D weaving, stitching, tufting and z-anchoring can draw the ply layers together into a compacted fabric preform, which reduces the thickness and increases the fibre volume content of the 3D composite. Increases in the fibre volume content of up to 10% due to compaction by the z-binders have been reported.
6.3.6
Microstructural modelling of 3D fibre composites
Advances in the design of 3D fibre composite structures for aerospace and other applications are dependent on geometric models which can accurately predict the microstructure and fibre architecture of the 3D reinforcement. Modelling is a challenging task because of the many microstructural features that must be considered (e.g. fibre crimp, fibre waviness, fibre breakage and so on). The task is further complicated by most of the microstructural defects being dependent on a multitude of factors, including the manufacturing process, z-binder properties, and the fibre architecture and fibre properties of the host composite. Progress has been made by Lomov and colleagues (2000, 2007) and others (Chen and Potiyaraj, 1999; Mahadik and Hallett, 2010) in the development of models to predict fibre waviness, fibre compaction, and resin-rich zones in 3D woven and other fabric preforms. Models for predicting interfacial cracking between the z-binders and host composite have also been developed. However, models for predicting other microstructural features such as fibre breakage, z-binder misalignment, and swelling are lacking. The development of a modelling capacity that can analyse all the microstructural features, including the stochastic nature of the defect geometry, is a key issue in the advancement of 3D fibre composite technology.
6.4
Delamination fracture of three-dimensional (3D) fibre composites
One of the main advantages of 3D fibre composites over traditional laminates is higher delamination toughness. Z-binder reinforcement increases the interlaminar fracture toughness (GIc, GIIc) under modes I and II loading, thereby resisting the propagation of delamination cracks. The delamination toughness properties of 3D woven, stitched, structural non-crimp, z-pinned, 3D braided composites and z-anchor composites have been determined theoretically and experimentally. There is no published information on the delamination toughness of 3D fibre composites made by tufting, although their interlaminar fracture properties are expected to be similar to those of stitched composites. Regardless of the type of 3D fibre composite, the z-binder reinforcement
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provides a large improvement to the interlaminar fracture toughness. For example, Fig. 6.15 shows the percentage improvements to the mode I delamination toughness of 3D woven and z-pinned composites with increasing volume content of z-binders. No single type of 3D fibre composite appears to show vastly higher delamination resistance than the other types; all materials exhibit similar interlaminar fracture toughness properties for the same volume fraction of z-binder reinforcement. The amount of delamination toughening achieved with 3D reinforcement is generally superior to that gained by other toughening techniques, such as using rubber toughened resin systems, nanoparticle toughened polymers, thermoplastic interleaving or fibre treatments to increase adhesion with the polymer matrix. The delamination resistance of 3D fibre composites is dependent on several parameters, which are listed in Table 6.2 for modes I and II conditions. An improvement to the delamination toughness is achieved by increasing the volume content or reducing the diameter of z-binders, which are the most common approaches to toughening. The delamination toughness also increases with the stiffness and strength of the z-binder, the embedded length of the z-binder (which increases with the material thickness), and the interfacial shear strength between the z-binder and the composite. The toughness can be affected by the end restraint of the z-binder, with continuous z-binders, such as those in 3D woven and stitched composites, providing better toughening than discrete z-binders, such as z-pins. Continuous z-binders are more resistant 600 3D woven composite z-pinned composite Percentage increase to mode I delamination toughness
500
400
300
200
100
0 0.0
0.5
1.0 1.5 Z-binder volume content (%)
2.0
6.15 Effect of z-binder content on the percentage increase to the mode I strain energy release rate (GIc) of 3D woven and z-pinned composites.
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Table 6.2 Summary of key factors controlling the mode I and II delamination toughness of 3D composites Parameter
Effect on delamination toughness
Increasing volume content of z-binders Increasing diameter of z-binders Increasing Young’s modulus of z-binders Increasing tensile strength of z-binders Increasing shear flow strength of z-binders Increasing interfacial shear strength of z-binders Increasing thickness of composite Increasing crush strength of composite Increasing crack length
Increases GI and GII Decreases GI and GII Increases GI and GII Increases GI Increases GII Increases GI and GII up to a limit Increases GI and GII up to a limit Increases GII Increases GI and GII up to a limit
to failure by pull-out due to the high end restraint imposed by the surface segment, thereby resulting in higher toughness. The amount of toughening is also dependent on the geometry of the z-binders: orthogonal z-binders aligned in the though-thickness direction provide high delamination toughness under mode I loading, while z-binders inclined at an angle of 45° against the direction of crack growth provide the highest mode II delamination toughness. The interlaminar fracture toughening processes are basically the same for the different types of 3D fibre composites, albeit with some minor differences between z-binder types. Z-binders increase the interlaminar toughness by creating a large-scale bridging traction zone along the delamination crack. The z-binders bridge the delamination crack over a large distance (in some cases many tens of millimetres), which creates traction forces that resist crack opening (under mode I) and crack sliding (under mode II). The bridging zone formed within a stitched composite under mode I loading is shown schematically in Fig. 6.16, and the bridging zone formed in the other types of 3D fibre composites is similar. The bridging processes responsible for improvements to the mode I and II delamination toughness of 3D fibre composites are similar, but not identical. The main toughening processes for mode I delamination fracture are elastic stretching at small crack opening and frictional pull-out or tensile rupture of the z-binders at large crack opening (Fig. 6.17). Friction generated during pullout of the z-binders contributes to the high delamination fracture toughness. Z-binders may rupture at the delamination crack without significant pull-out, and in this case the toughening effect is provided mostly by elastic deformation of the z-binders. Failure by tensile rupture is common for continuous z-binders with end restraint (e.g. 3D woven and stitched materials) or z-binders with a long embedded length where their tensile strength is less than the friction pull-out stress.
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Needle thread
Bobbin thread
6.16 Schematic of the crack bridging toughening process for a stitched composite under mode I loading. Reproduced from He and Cox (1998).
Similar toughening processes are operative under mode II loading with elastic shear deformation at small crack sliding displacements and pull-out or shear rupture of the z-binders at large crack displacements (Fig. 6.18). In addition to these toughening processes, the mode II delamination toughness is increased by snubbing of the z-binders at large crack sliding displacements. Snubbing is a toughening process in which the z-binder is pressed laterally into the surrounding composite material under high shear strain deformation. The composite material reacts against the snubbing force and thereby resists deformation of the z-binder, and consequently the mode II delamination toughness is increased. Under high snubbing stresses, the composite material next to the z-binder can be crushed. The z-binders under mode II loading eventually fail at high crack sliding displacements by pull-out or transverse shear rupture. An important feature of the delamination toughness properties of 3D fibre composites is their R-curve (crack growth resistance) behaviour. Examples of modes I and II R-curves are presented in Fig. 6.19, and they are characterised by a rapid rise in the interlaminar fracture toughness with increasing delamination crack length until a quasi-steady state toughness condition is reached at a long crack length (usually above 10–20 mm). This contrasts to the R-curve for conventional laminates which are flat or show a slight increase in toughness due to fibre bridging effects. The delamination toughness at crack initiation (i.e. Da = 0) for 3D fibre composites is not improved significantly by z-binder reinforcement. Z-binders are ineffective at resisting delamination crack growth over short lengths and have no influence on the
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Delamination
Stitch
200 µm (a)
200 µm (b)
1 mm (c)
6.17 Photographs of delamination crack bridging and failure of z-binders under mode I load: (a) elastic deformation of a stitch; (b) tensile fracture of a stitch; (c) complete pull-out of a z-pin.
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400 µm (a)
200 µm (b)
6.18 Photographs of delamination crack bridging and failure of z-binders under mode II load: (a) bridging of a z-pin; (b) complete fracture of a stitch.
critical strain energy release rate at crack initiation (Gi). As the delamination begins to grow the bridging zone is created by the z-binders, which raises the interlaminar fracture toughness. The toughness rises with crack length as more z-binders contribute to the bridging process. Maximum toughness is reached when the crack bridging zone is fully developed, which for 3D fibre composites with the z-binder content in the range of 1–5% typically occurs when the delamination is at least 10–20 mm long. Steady-state toughness occurs when the number of new z-binders joining the bridging zone at the
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7000
181
Mode I toughness Mode II toughness
6000
3D fibre composite
5000 4000 3000 2000 Conventional laminate 1000 0
0
5
10 15 20 25 Crack length, Da (mm)
30
35
6.19 Typical mode I and II R-curves for a 3D fibre composite material. The R-curve for a conventional laminate (without fibre bridging effects) is presented for comparison.
crack tip is equal to the number of z-binders at the end of the bridging zone that fail due to the large crack opening displacement (for mode I) or large crack sliding displacement (for mode II). Z-binders not only increase the delamination fracture toughness under static mode I and II loads, but they also increase the resistance against delamination crack growth under cyclic (fatigue) stress loading. Only a few studies into delamination fatigue properties of 3D fibre composites have been reported, although the work that has been published shows large reductions to the crack growth rate due to the formation of bridging tractions by the z-binders. Micromechanical models have been developed to calculate the mode I and II interlaminar fracture toughness properties of 3D woven, stitched, non-crimp knitted, and z-pinned composites. The models, which are solved using finite element analysis or finite difference methods, use the bridging traction law for the z-binder reinforcement together with the delamination fracture properties of the baseline composite material (without z-binders) to calculate the interlaminar fracture toughness of the 3D fibre composite. The models can be used to calculate the effect of various parameters of the z-binders, such as their volume content, diameter and material properties, on the delamination fracture toughness. The accuracy of the models is dependent on the theoretical bridging traction law giving an accurate description of the toughening processes and failure of the z-binders at different levels of crack opening (for mode I) and crack sliding (mode II) displacements.
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Figure 6.20 shows the accuracy of the delamination models developed by Jain and Mai (1994, 1995) for stitched composites, and the agreement between the calculated and measured mode I and II interlaminar fracture values is very good. Good agreement between calculated and measured delamination toughness properties has also been reported for 3D woven and z-pinned composites. Models have not been developed specifically for Calculated mode I toughness, Glc (kJ/m2)
6
5
4
3
2
1
0
0
1 2 3 4 5 Measured mode I toughness, Glc (kJ/m2) (a)
6
0
2 4 6 Measured mode II toughness, GIlc (kJ/m2) (b)
8
Calculated mode II toughness, GIIc (kJ/m2)
8
6
4
2
0
6.20 Comparison of the calculated and measured delamination fracture toughness (Gc) values for stitched composites for (a) mode I and (b) mode II crack growth. The theoretical toughness values were calculated using the models developed by Jain and Mai (1994, 1995).
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other types of 3D fibre composites (i.e. braided, tufted), although existing delamination fracture models can be modified to analyse the delamination toughness properties of these materials.
6.5
Impact damage resistance and tolerance of three-dimensional (3D) fibre composites
Poor impact damage resistance is a long-standing problem with conventional laminated composites because of their low interlaminar fracture toughness. One of the main drivers behind the development of 3D fibre composites has been their superior impact damage resistance and higher post-impact mechanical properties compared to conventional laminates. Numerous studies have proven that 3D woven, stitched and z-pinned composites have higher resistance than conventional laminates against barely visible damage created by low-energy impact loading. 3D tufted composites are also expected to have superior low-energy impact resistance, though published research demonstrating their impact performance is lacking. Figure 6.21 shows the typical reduction in delamination damage to a 3D fibre composite material due to the z-binder reinforcement. The impact damage resistance of 3D fibre composites increases with the volume content of z-binders, and this is due to the improvement in their delamination toughness (GIc, GIIc). In addition to z-binder content, other parameters which increase the impact damage resistance of 3D fibre composites are the use of finer 1200 No z-binders Impact damage area (mm2)
1000 0.5% z-binders 800 2% z-binders
600
400
200 0 0
5
10 15 20 Incident impact energy (J)
25
30
6.21 Improvement to the impact damage resistance of a 3D fibre composite with increasing z-binder content. The composite is a carbon/epoxy material reinforced with z-pins.
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diameter z-binders, high stiffness and strength z-binders, high interfacial shear strength between the z-binders and composite material, and long embedded length of the z-binder. These parameters promote greater delamination toughening which translates into higher impact damage resistance. It is important to note that 3D fibre composites do not always display superior low-energy impact damage resistance. Z-binders do not improve the impact damage resistance at very low energy levels (as shown, for example, in Fig. 6.21 at below 15 J), and improvements in damage resistance only occur when the impact energy exceeds a threshold level. At very low energies the amount of damage created by an impact event is small, and the impact-induced delamination cracks are too short for the z-binders to create a bridging traction zone. Improved damage resistance only occurs when the impact zone extends over a distance longer than 10–20 mm, which is sufficient for a bridging zone to develop and thereby promote high interlaminar toughening. In addition to low energy impact loading, it has been demonstrated that 3D fibre composites have superior damage resistance against ballistic impact, shock loading, explosive blast loading and high-energy collisions compared to traditional laminates. Again, the formation of a bridging traction zone across the delamination cracks is responsible for the improved damage resistance under these dynamic loading events. The superior impact damage resistance of 3D fibre composites can result in post-impact (damage tolerant) properties which are higher than those of conventional laminates. As examples, Fig. 6.22 shows the mechanical properties of z-pinned and stitched composites following low-energy impact and explosive blast loading, respectively. 3D fibre composites have higher post-impact damage tolerance than conventional laminates because they suffer less delamination damage from a dynamic loading event. Under certain loading conditions, the z-binders also resist unstable propagation of the impact-induced delamination cracks when the 3D fibre composite is under load, which results in higher post-impact properties. The post-impact mechanical properties which benefit the most from z-binder reinforcement are those sensitive to the presence of delamination cracks, such as compressive strength and interlaminar shear strength. It is the improvement to the post-impact mechanical properties gained by z-binder reinforcement which makes 3D fibre composites attractive materials to use in aircraft structures requiring high damage tolerance.
6.6
Through-thickness stiffness and strength of three dimensional (3D) fibre composites
Large improvements to the through-thickness elastic properties of composite materials can be achieved with z-binder reinforcement. Studies using finite
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Post-impact compression strength (MPa)
280
260
240
220
200
0
1
2 3 Volume content of z-pins (a)
4
Post-blast flexural strength (MPa)
350 300 250 200 150 100 50 0
0
3 Stitch density (stitches/cm2) (b)
6
6.22 Examples of damage tolerance of 3D fibre composites: (a) effect of volume content of z-pins on the compressive strength of carbon/ epoxy following low energy impact loading; (b) effect of areal density of stitches on the flexural strength of glass/vinyl ester following explosive blast loading.
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element analysis report that the through-thickness tensile modulus increases with the volume fraction and stiffness of the z-binders. For example, Dickinson et al. (1999b) predict that the through-thickness stiffness of a carbon/epoxy laminate is increased by 60% with a z-binder content of 4.9%. However, calculated improvements to elastic modulus have not been verified experimentally with through-thickness tensile tests performed on the different types of 3D fibre composite materials. Z-binder reinforcement has no influence on the through-thickness tensile strength, which is determined by the adhesion strength between the in-plane fibres and polymer matrix and by the presence of microstructural defects (e.g. voids), which are not affected by the z-binders. However, z-binders can stabilise the through-thickness tensile fracture process by resisting catastrophic failure. The z-binders create a bridging traction zone across the tensile crack and thereby provide residual strength after the composite has failed in through-thickness tension.
6.7
Through-thickness thermal properties of three-dimensional (3D) fibre composites
3D fibre reinforcements offer the capability of increasing the thermal and electrical conductivities of composite materials in the through-thickness direction. The through-thickness thermal conductivity of conventional composite materials is low because of poor heat conduction through the polymer matrix and resistance to heat flow at the fibre–matrix interfaces. As a result, composites lack the capacity to rapidly dissipate heat when used in elevated temperature applications. Similarly, composites have low throughthickness electricity conductivity due to the matrix, fibre–matrix interfaces and fibres (in the case of fibreglass). Sharp et al. (2008) demonstrated that the z-binders in 3D fibre composite materials can increase greatly the through-thickness thermal conductivity. It was found that inserting z-binders with high thermal conductivity (such as pitch-based carbon filaments or copper wire) into a 3D orthogonal woven carbon fibre composite increased the through-thickness heat conduction properties. The improvement to the through-thickness conductivity is controlled by the volume fraction and thermal conductivity of the z-binder reinforcement, as shown in Fig. 6.23. Other parameters also affect the through-thickness conductivity, including the thickness of the composite and the thermal isotropy of the z-binders. This work by Sharp et al. (2008) is the only reported demonstration of the use of z-binders to improve the throughthickness heat conduction of materials. The use of stitches, pins and other types of z-binders to increase the through-thickness thermal conductivity has been not evaluated.
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Through-thickness thermal conductivity (W/mK)
Three-dimensional (3D) fibre reinforcements for composites 10
187
Carbon pitch z-binder Annealed copper z-binder
8
6
4
2
0
1.8%
3.7% 5.5% Volume content of z-binder (%)
6.23 Effect of type and volume fraction of z-binder on the throughthickness thermal conductivity of 3D woven carbon/epoxy. Data from Sharp et al. (2008).
6.8
In-plane mechanical properties of threedimensional (3D) fibre composites
6.8.1
Elastic modulus
The in-plane elastic modulus of composite materials reinforced in the throughthickness direction by 3D weaving, stitching, z-pinning, tufting and z-anchoring have been characterised by numerical analysis and experimentation. Based on the large amount of published data on the elastic modulus of 3D woven and stitched composites, it can be inferred that their stiffness properties can be improved, degraded or unchanged by the z-binders, whereas the modulus of z-pinned composites is always reduced. It is not possible to infer the effect of braiding, tufting and z-anchoring on the elastic modulus of 3D fibre composites because of the small amount of published modulus data, and further evaluation of the stiffness properties of these materials is necessary. Mouritz and Cox (2000, 2010) performed a comprehensive assessment of a large body of published mechanical property data for 3D woven and stitched materials, and found that the z-binder reinforcement can improve or degrade the elastic modulus. For example, Fig. 6.24 shows the normalised tensile modulus of 3D woven and stitched composites plotted against their z-binder content. The normalised modulus is the tensile modulus of the 3D composite material (E3D) divided by the modulus of an equivalent 2D laminate (E2D), which has the same type, volume fraction and arrangement
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Normalised tension modulus (E3D/E2D)
1.50
1.25
1.00
0.75 3D 3D 3D 3D 3D 3D 3D
0.50
0.25
0.00
0
2
4
6
interlock carbon/epoxy orthogonal glass/epoxy interlock glass/epoxy orthogonal glass/vinyl ester interlock Kevlar/epoxy orthogonal glass/vinly ester orthogonal glass/poxy
8 10 12 14 Z-binder content (vol%) (a)
16
18
20
Normalised tension modulus (E3D/E2D)
1.50
1.25
1.00
0.75
0.50
0.25
0.00 0.00
CFRP/720 denier Kevlar CFRP/195 denier Kevlar CFRP/180 denier Kevlar CFRP/270 denier Kevlar CFRP/195 denier Kevlar CFRP/1350 denier cotton CFRP/900 denier carbon 0.02
CFRP/1500 denier Kevlar CFRP/100 denier Kevlar CFRP/3000 denier Kevlar CFRP/2840 denier Kevlar CFRP/2840 denier Kevlar CFRP/1200 denier Spectra
0.04 0.06 0.08 Z-binder content (stitches/mm2) (b)
0.10
0.12
6.24 Effect of z-binder content on the normalised tensile modulus of (a) 3D woven composites and (b) stitched composites. CFRP = carbon fibre reinforced polymer; GFRP = glass fibre reinforced polymer; KFRP = Kevlar fibre reinforced polymer; SFRP = Spectra fibre reinforced polymer. The denier and type of stitch material is given. Adapted from Mouritz and Cox (2010).
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of in-plane load-bearing fibres. With the exception of a few materials, the tensile modulus of 3D woven and stitched composites is always within 20% of that of the equivalent 2D laminate. Furthermore, the elastic modulus of the 3D woven and stitched composites shows no or a weak dependence on the z-binder content. Similar trends are observed for the elastic modulus of 3D woven and stitched composites measured under in-plane compression and bending (Mouritz and Cox, 2000, 2010). Mouritz and Cox (2000, 2010) attribute improvements to the elastic modulus of 3D woven and stitched composites to an increase in the fibre volume content caused by compaction of their fabric preforms by the z-binders. A tensile force is applied to z-binders during 3D weaving or stitching, and this causes compaction of the in-plane ply layers, resulting in higher fibre content and consequently higher stiffness. In cases when the elastic modulus of the 3D fibre composites is less than that of the equivalent 2D laminate, this is attributed to microstructural defects and, in particular, fibre waviness and fibre crimp caused by the z-binders. The distortion of the load-bearing fibres causes local softening of the material close to the z-binders, which results in a reduction to the bulk stiffness of the 3D fibre composite. This is supported by stiffness analysis of these materials using finite element analysis. The loss in stiffness caused by fibre waviness is minimal in non-crimp orthogonal woven composites and non-crimp knitted composites. Research has revealed, therefore, that with 3D woven and stitched composites a competition exists between an improvement to the elastic modulus (caused by preform compaction) and a deterioration to the modulus (caused by fibre waviness and crimp). Due to the different methods used to fabricate 3D fabric preforms, the amount of compaction varies between different types of materials. Also, due to the stochastic nature of microstructural defects in fabric preforms, and in particular the variable sizes and angles of the wavy fibre zones, the defects may have a small or large influence on the elastic properties. For these reasons, the elastic properties of 3D woven and stitched composites can be higher than, lower than or the same as the equivalent 2D laminate, and display little or no dependence on the volume content of z-binders. Mouritz and Cox (2000, 2010) found that z-pinning always reduces the elastic modulus of composite materials, and never improves their stiffness. This is because the z-pinning of prepreg does not cause compaction of the preform, unlike 3D weaving, stitching and other reinforcement techniques for fabrics. Figure 6.25 shows the effect of increasing z-pin content on the tensile modulus of carbon/epoxy composites with different fibre patterns. It is usually found that the elastic modulus decreases at a quasi-linear rate with increasing volume content of z-pins. The percentage reduction to the elastic modulus also increases with the percentage of load-bearing fibres in the composite; for example, a unidirectional laminate suffers a more rapid loss
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Normalised tension modulus (E3D/E2D)
1.50
1.25 Quasi-isotropic [0/+45/–45/90] 1.00
Cross-ply [0/90]
0.75
Unidirectional [0] 0.50
Unidirectional CFRP Unidirectional CFRP Unidirectional CFRP Cross-ply CFRP Quasi-isotropic CFRP Bias CFRP
0.25
0.00
0
1
2 3 Z-binder content (vol%)
4
5
6.25 Effect of z-binder content on the normalised tensile modulus of z-pinned composites. Adapted from Mouritz and Cox (2010).
in stiffness than a quasi-isotropic material with increasing z-pin content. The reduction to the elastic properties is attributed to waviness and crimping of the load-bearing fibres by the z-pins, which is the same softening mechanism responsible for the loss in stiffness of the other types of 3D fibre composite materials. The observation made by Mouritz and Cox (2010) that the elastic properties of 3D woven and stitched composites has little or no dependence on the volume content of z-binders has practical significance. The impact damage resistance and post-impact mechanical properties of 3D fibre composites increase with their z-binder content. Therefore, reinforcing 3D woven and stitched materials with a large amount of z-binders to provide high damage tolerance will not necessarily result in a large reduction to the elastic properties. However, increasing the z-binder content in z-pinned composites will always cause more elastic softening, and therefore a trade-off must be made between increased impact damage resistance against reduced stiffness.
6.8.2
Failure strength
The strength properties of 3D woven and stitched composites show similar behaviour to their elastic properties, in that they can be higher than, the same as or lower than those of the equivalent 2D laminate. This behaviour has been reported for the strength properties determined under different
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load conditions, including the tensile, compressive, flexural and interlaminar shear strengths. In contrast, the strength properties of 3D fibre composites are always reduced by z-pinning, and are never improved. The amount of published data for 3D braided, tufted and z-anchored composites is limited and insufficient to determine whether their strength properties are different from those of the equivalent 2D laminate. Figure 6.26 shows the effect of increasing z-binder content on the normalised tensile strength of stitched and z-pinned composites. The strength of the stitched composites shows little or no dependence on the areal density of z-binders, and is almost always within 20% of the equivalent material without through-thickness reinforcement. Mouritz and Cox (2010) report a similar effect for 3D woven composites. The tensile strength of z-pinned composites always decreases at a linear rate with increasing z-binder content, with the rate of strength loss increasing with the volume fraction of load-bearing fibres. The compressive and flexural strengths of z-pinned composites show a similar dependence on the z-binder content as the tensile strength. The improvement to the failure strength of 3D fibre composites is almost certainly caused by their higher fibre volume content. As with the improvement in elastic modulus, an increase in the strength properties is attributed to compaction of the fabric preform by the z-binders during manufacture, which raises the fibre content. Most of the manufacturing processes used to produce 3D fibre reinforcements, including weaving, stitching and z-anchoring, cause the fabric to compact due to the tensile stress applied to the z-binders. The only process not to cause preform compaction is z-pinning, which is the reason why z-pinned composites never have higher strength properties than their equivalent 2D laminate. In some 3D fibre composites, the increase in compressive strength may also be due to the suppression of delamination cracking by the z-binder reinforcement. Conventional laminates are susceptible to delamination damage under compression loading, which triggers failure. The resistance against large-scale delamination cracking by the z-binders may increase the compressive strength of 3D fibre composites, although this effect has not been extensively studied. The mechanisms responsible for the deterioration in the strength properties of 3D fibre composites are complex. The reduction to the tensile strength is attributed to damage and waviness of the load-bearing fibres. As the z-binders are inserted into 3D fibre reinforcements (with the exception of non-crimp orthogonal woven and non-crimp knitted fabrics) the in-plane fibres are damaged and broken, as shown, for example, in Fig. 6.12. The damaged fibres are clustered at each z-binder site, and when present in sufficient numbers they act as the site for tensile failure. The tensile strength of composite materials can be determined by a single defect, such as a region at a single z-binder weakened by fibre damage. Therefore, increasing the z-binder content and
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Normalised tension strength (s3D/s2D)
1.50
1.25
1.00
0.75
0.50
0.25 0.00 0.00
0.02
0.04 0.06 0.08 0.10 Z-binder content (stitches/mm2) (a)
0.12
CFRP/720 denier CFRP/195 denier CFRP/ unspec CFRP/180 denier CFRP/270 denier CFRP/1350 denier CFRP/900 denier CFRP/180 denier CFRP/1500 denier CFRP/2840 denier CFRP/2840 denier CFRP/1200 denier CFRP/180 denier CFRP/unspec CFRP/80 denier CFRP/40 denier CFRP/80 denier CFRP/40 denier
Normalised tension strength (s3D/s2D)
1.50
1.25
[0] carbon/epoxy
1.00
[0] carbon/epoxy [0] carbon/epoxy
[0] carbon/epoxy
Quasi-isotropic [0/+45/–45/90]
0.75 Unidirectional [0]
0.50
[0/90] carbon/epoxy [0/+45/–45/90] carbon/epoxy
Cross-ply [0/90]
[0/+45/–45/90] carbon/epoxy [0/+45/–45/90] carbon/epoxy [+45/–45] carbon/epoxy
0.25 0.00
[0/90] carbon/epoxy [0/90] carbon/epoxy
0
2
4 6 8 Z-binder content (vol%) (b)
10
6.26 Normalised tensile strength, defined as the tensile strength of the 3D composite, s3D, divided by that of the equivalent 2D laminate, s2D, plotted against z-binder content for (a) stitched composites and (b) z-pinned composites. CFRP = carbon fibre reinforced polymer; GFRP = glass fibre reinforced polymer; KFRP = Kevlar fibre reinforced polymer; SFRP = Spectra fibre reinforced polymer. The denier and type of stitch material is given. From Mouritz and Cox (2010).
consequently the density of defects does not necessarily lower the tensile strength below that of a material containing just one defect. The stochastic nature of defects in 3D fibre composites suggests that the one defect with
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the highest concentration of broken and damaged fibres will initiate tensile failure, and the other defects will not affect the failure stress. Fibre waviness and crimp may also contribute to the reduced tensile strength, although their influence is probably secondary to fibre damage and breakage. The loss to the compressive strength of 3D fibre composites is attributed to fibre waviness and crimp that causes microbuckling and kinking of the load-bearing plies. The compressive stress required to induce kinking decreases with increasing misalignment angle of the load-bearing fibres, and therefore kink initiation is expected to occur first at the most heavily distorted region around a z-binder. The kink bands that form in composite materials are typically only 100–200 mm wide, which is much less than the spacing between the z-binders. Thus the first kink band to form at the most severely misaligned region will develop independently of any kink bands that may form at neighbouring z-binders because the spacing is much larger than the kink band width. This explains the weak dependence of compressive strength on the volume content of z-binders in 3D fibre composites. The avoidance of fibre waviness and crimp in non-crimp orthogonal woven and non-crimp knitted composites results in these materials not suffering a significant loss in compressive strength due to the z-binder reinforcement.
6.8.3
Fatigue properties
The fatigue properties of 3D woven, stitched and z-pinned composites have been determined, whereas the fatigue behaviour of other types of 3D fibre composites, such as tufted and z-anchor materials, have not been evaluated. Like the static properties, the fatigue properties of 3D fibre composites can be inferior or superior to those of their equivalent 2D laminate. Numerous studies have shown that the fatigue life of 3D fibre composites is shorter than that of the equivalent 2D laminate, whereas other studies have reported better fatigue performance with 3D composites. Figure 6.27 shows two examples of 3D fibre composites with inferior fatigue performance compared to their equivalent 2D laminate. In both cases the fatigue life and fatigue strength of the 3D fibre composite is much lower than for the 2D laminate, and the loss in fatigue performance is attributed to microstructural defects caused by the z-binders. The reduction in tensile fatigue performance is attributed to fibre breakage and fibre waviness caused by the z-binders during manufacture of the 3D fibre reinforcement. Fibre waviness is also responsible for the reduction in the fatigue life of 3D fibre composites under cyclic compression. Kink bands develop in the wavy fibres under cyclic compression loading, which accelerates failure of the load-bearing yarns by microbuckling. Carvelli et al. (2010) and Aymerich et al. (2003) report respectively that 3D non-crimp orthogonal woven composites and stitched composites can
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Tensile fatigue stress (MPa)
650 Unstitched composite 600 550 500 Stitched composite 450 400 350 10–1
100
101
102 103 104 Load cycles to failure (a)
105
106
107
700
Bending fatigue stress (MPa)
650 No z-pins
600 550
2% z-pins 500
4% z-pins
450 400 350 10–1
100
101
102 103 104 Load cycles to failure (b)
105
106
107
6.27 (a) Tensile fatigue life curves for stitched and unstitched carbon/ epoxy composite (Aymerich et al., 2003). (b) Bending fatigue life curves for unpinned and z-pinned carbon/epoxy composite (Chang et al., 2007).
have superior fatigue properties than their equivalent 2D laminate. In cases when the 3D fibre composite has better fatigue performance, it is attributed to the z-binder reinforcement suppressing the growth of fatigue-induced delamination cracks. Conventional laminates are susceptible to delamination
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damage under cyclic compression, cyclic interlaminar shear or other matrixdominated cyclic loads. The through-thickness reinforcement of 3D fibre composites suppresses the development of delamination cracks, and thereby extends the fatigue life. 3D fibre composites also display superior fatigue performance when pre-existing delamination cracks are present, such as impact damage. Isa et al. (2011) recently showed that the compressive fatigue life of carbon/ epoxy containing barely visible impact damage is improved by z-pinning. The z-pins reduce the amount of delamination damage caused by impact loading, which results in higher post-impact fatigue performance. It may be expected that the post-impact fatigue performance of other types of 3D fibre composite materials is also improved by their higher damage resistance, although published research on the topic is lacking.
6.9
Joint properties of three-dimensional (3D) fibre composites
3D composite materials show enormous potential for the strengthening of structural joints. Conventional bonded composite joints are susceptible to delamination damage due to the weak interfacial bond strength between the adherends. The aerospace industry has sought solutions to the low strength of bonded joints, including improved designs and the use of toughened adhesives, but these often provide only incremental improvements. 3D fibre composites offer a potential solution by using the z-binder reinforcement to carry bond-line stresses which cause cracking and failure in conventional bonded joints. Large improvements to the structural properties of composite joints when reinforced by stitching, z-pinning and tufting have been reported. The joints made using 3D fibre composites include T-, C- and J-shaped designs as well as the single-lap configuration. The mechanical properties of bonded joints which are improved by z-binder reinforcement include the ultimate strength, failure strain limit, total work-of-fracture, and fatigue life. As examples, Fig. 6.28 shows the percentage improvement to the ultimate strength of a carbon/ epoxy T-joint with increasing volume content of z-pin reinforcement, and Fig. 6.29 shows the increase in fatigue life of a carbon/epoxy lap joint due to stitching. The joint properties are improved by the z-binders bridging between the adherends after the bond-line has fractured. The bridging z-binders are able to carry the applied load after the bonded region has failed, resulting in higher mechanical properties and damage tolerance than for conventional bonded joints. Several studies report that the strengthening provided by the z-binder can be high enough to cause failure of the composite adherends rather than the bond-line, indicating that maximum strengthening is achieved by the through-thickness reinforcement.
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100
80
60
40
20
0
0
1 2 3 Volume content of z-binders (%)
4
6.28 Effect of volume content of z-pins on the percentage improvement to the ultimate strength of T-joints. Data from Koh et al. (2010).
Applied force per unit width (N/mm)
500
400
300
Stitched lap joint
200 Unstitched lap joint 100
0 102
103
104 105 Load cycles to failure
106
107
6.29 Improvement to the fatigue life of a single lap joint due to stitching. Data from Tong et al. (1998).
6.10
Conclusions
Major advances in the science and technology of composite materials with 3D fibre reinforcement have been achieved since their original development
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in the 1970s. A variety of 3D fibre composites have been created, with the most important types being 3D woven, stitched, z-pinned and non-crimp knitted materials. The incentive to use 3D fibre composites is their superior delamination resistance, impact resistance and damage tolerance compared to traditional laminates. 3D composites also display other improved properties, such as high through-thickness thermal conductivity and joint strength. However, the through-thickness reinforcement may adversely affect in-plane properties such as stiffness, strength and fatigue life due to microstructural defects. Despite the impressive developments and demonstrated advantages of 3D fibre composites, their use in aerospace and other structural applications has been limited due to numerous economic, technical and certification (in the case of aircraft) issues. With the exception of a few applications, the use of 3D fibre composites has been modest compared to traditional laminates and their future usage is uncertain. Regardless of this uncertainty, it is important that research and development of 3D fibre composites continues in order to drive down cost, improve further the mechanical and other properties, and raise awareness of the potential benefits of these materials with the designers, manufacturers and users of composites. There are many important topics requiring further investigation, including better control and minimisation of microstructural defects, better understanding of the durability properties and long-term environmental ageing behaviour, closer control of changes to the in-plane mechanical properties, and the development of mechanistically based models for the analysis of microstructural and mechanical properties. The opportunity exists for the creation of the next generation of 3D fibre composites that use unique types of ultra-high-strength z-binder reinforcements such as carbon nanotubes and other advanced materials.
6.11
References
Abe T, Hayashi K, Sato T, Yamane S and Hirokawa T, 2003, ‘A-VARTM process and z-anchor technology for primary aircraft structures’, Proc. 24th SAMPE Europe Conf., 1–3 April, Paris. Aymerich F, Priolo P and Sun C T, 2003, ‘Static and fatigue behaviour of stitched graphite/epoxy composite laminates’, Comp. Sci. & Tech., 63, 907–917. Bogdanovich A E and Mohamed M H, 2009, ‘Three-dimensional reinforcements for composites’, SAMPE J., 45, 8–28. Cartié D D R, Dell’Anno G, Poulin E and Partridge I K, 2006, ‘3D reinforcement of stiffener to skin T-joints by z-pinning and tufting’, Eng. Frac. Mech., 73, 2532–2540. Carvelli V, Gramellini G, Lomov S V, Bogdanovich A E, Mungalov D D and Verpoest I, 2010, ‘Fatigue behavior of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass reinforced composites’, Comp. Sci. & Tech., 70, 2068–2076. Chang P, Mouritz A P and Cox B N, 2007, ‘Flexural properties of z-pinned laminates’, Comp., 38A, 224–251.
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Chen X and Potiyaraj P, 1999, ‘CAD/CAM of orthogonal and angle-interlock woven structures for industrial applications’, Text. Res. J., 69, 648–655. Dell’Anno G, Cartié D D, Partridge I K and Rezai A, 2007, ‘Exploring mechanical property balance in tufted carbon fabric/epoxy composites’, Comp., 38A, 2366–2373. Dickinson L C and Mohamed M H, 2000, ‘Recent advances in 3D weaving for textile preforming’, Proc. ASME Aerospace Division, pp. 3–8. Dickinson L C, Mohamed M H and Bogdanovich A, 1999a, ‘3D weaving: what, how, and where’, Proc. SAMPE Conf., 23–27 May, Long Beach, CA. Dickinson L C, Farley G L and Hinders M K, 1999b, ‘Prediction of effective threedimensional elastic constants of translaminar reinforced composites’, J. Comp. Mater. 33, 1002–1029. Dransfield K, Baillie C and Mai Y-W, 1994, ‘Improving the delamination resistance of CFRP by stitching – a review’, Comp. Sci. & Tech., 50, 305–317. He M and Cox B N, 1998, ‘Crack bridging by through-thickness reinforcement in delaminating curved structures’, Comp., 29A, 377–393. Isa M D, Feih S and Mouritz A P, 2011, ‘Fatigue properties of impact damaged composites reinforced with z-pins’, Proc. 14th Australian Int. Aerospace Congr., 28 February–3 March, Melbourne. Jain L K and Mai Y-W, 1994, ‘On the effect of stitching on mode I delamination toughness of laminated composites’, Comp. Sci. & Tech., 51, 331–345. Jain L K and Mai Y-W, 1995, ‘Determination of mode II delamination toughness of stitched laminated composites’, Comp. Sci. & Tech., 55, 241–253. Koh T M, Feih S and Mouritz A P, 2010, ‘Structural properties of composite T-joints reinforced with z–pins’, Proc. 21st Australasian Conf. on Mechanics and Materials, 7–10 December, Melbourne. Lomov S V, 2010, Non-crimp Fabric Composites: Manufacturing, Properties and Applications, Cambridge, Woodhead Publishing. Lomov S V, Gusakov A V, Huysmans G, Prodromou A and Verpoest I, 2000, ‘Textile geometry preprocessor for meso-mechanical models of woven composites’, Comp. Sci. & Tech., 60, 2083–2095. Lomov S V, Ivanov D S, Verpoest I, Zako M, Kurashiki T, Nakai H and Hirosawa S, 2007, ‘Meso-FE modelling of textile composites: Road map, data flow and algorithms’, Comp. Sci. & Tech., 67, 1870–1891. Mahadik Y and Hallett S R, 2010, ‘Finite element modelling of tow geometry in 3D woven fabrics’, Comp., 41A, 1192–1200. Mahadik Y, Robson Brown K A and Hallett S R, 2010, ‘Characterisation of 3D woven composite internal architecture and effect of compaction’, Comp., 41A, 872–880. Mohamed M H, Bogdanovich A E, Dickinson L C, Singletary J N and Lienhart R B, 2001, ‘A new generation of 3D woven fabric preforms and composites’, SAMPE J., 37, 8–17. Mohamed M H, Bogdanovich A E, Coffelt R, Schartow R W and Stobbe D, 2005, ‘Manufacturing, performance and applications of 3-D orthogonal woven fabrics’, Proc. 84th Annual Conf. of the Textile Institute, Raleigh, NC. Mouritz A P, 2007, ‘Review of z-pinned composite laminates’, Comp., 38A, 2383– 2397. Mouritz A P and Cox B N, 2010, ‘A mechanistic interpretation of the comparative inplane mechanical properties of 3D woven, stitched and pinned composites’, Comp., 41A, 709–728. Mouritz A P, Bannister M K, Falzon P J and Leong K H, 1999, ‘Review of applications for advanced three-dimensional fibre textile composites’, Comp., 30A, 1445–1461. © Woodhead Publishing Limited, 2011
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Mungalov D, Duke P and Bogdanovich A, 2007, ‘High performance 3-D braided fiber preforms: design and manufacturing advancements for complex composite structures’, SAMPE J., 43, 53–60. Partridge I K, Cartié D D R and Bonnington T, 2003, ‘Manufacture and performance of z-pinned composites’, in Advanced Polymeric Composites, eds Shonaike G and Advani S, CRC Press, Boca Raton, FL. Sharp K, Bogdanovich A, Tang W, Heider D, Advani S and Glowiana M, 2008, ‘High through-thickness thermal conductivity composites based on three-dimensional woven fiber architectures’, AIAA J., 46, 2944–2954. Tong L, Jain L K, Leong K H, Kelly D and Hertzberg I, 1998, ‘Failure of transversely stitched RTM lap joints’, Comp. Sci. & Tech., 58, 221–227. Tong L, Mouritz A P and Bannister M K, 2002, 3D Fibre Reinforced Polymer Composites, London, Elsevier.
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7
Modelling the geometry of textile reinforcements for composites: WiseTex S. V. L o m o v, Katholieke Universiteit Leuven, Belgium
Abstract: The WiseTex family of models and software provides a generic description of the internal geometry of textile reinforcements, covering a wide range of textile structures – woven (2D and 3D), braided, non-crimp fabric, knitted and stitched. The components of the WiseTex suite simulate properties of textile fabrics and composites on a unit cell level and are used for multilevel micro–meso–macro analysis. The chapter describes models of geometry and deformability of woven (2D and 3D), braided (bi- and triaxial) and non-crimp fabric (NCF) reinforcements, implemented in the WiseTex software package. The model description for each class of textile reinforcements includes coding of the corresponding textile architecture and principles of building a geometrical model of the textile, which provides a full description of the yarn placement. The WiseTex geometrical models serve as input data for models of mechanical properties of textile composites, deformability of the textile reinforcements in tension, shear and compression, and permeability of textiles. Key words: textile composites, geometrical models, deformability, mechanical properties, permeability.
7.1
Introduction
The internal geometry of a textile reinforcement is an important factor in the reinforcement performance during composite manufacturing and inservice life of the composite material. For the former, impregnation of the reinforcement is governed by its porosity (size, distribution and connectivity of pores). For the latter, load transfer from the matrix to the reinforcement is governed by the fibre orientation, which plays a paramount role in the composite stiffness; stress–strain concentration loci, determining the composite strength, are correlated with the resin-rich zones and fibre/matrix interfaces distributed in the composite volume in accordance with the reinforcement geometry. In a 3D shaped composite part the reinforcement is locally deformed (compressed, stretched and sheared), and the geometrical model should account for this deformation. A generalised description of the internal structure of a textile reinforcement has been developed in K. U. Leuven in the Composite Materials Group, led by Professor Ignaas Verpoest and Professor Stepan Lomov. It is the 200 © Woodhead Publishing Limited, 2011
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final result of a development that started in the mid-1990s, with the work of Vandeurzen et al. (Vandeurzen 1998; Vandeurzen et al. 1998) on woven fabric composites, and of Gommers et al. (1998) on knitted and woven fabric composites. Both authors based the description of the internal geometry of the textile on empirical data on the yarn orientation, obtained by optical or electron-microscopical observations on dry textiles and on cross-sections of textile composites. These geometrical descriptions were then used to calculate the homogenised mechanical properties of the textile-based composites. Vandeurzen (Vandeurzen 1998; Vandeurzen et al. 1998) used a multilevel homogenisation method, also called the cell method, which was later generalised by Prodromou (Prodromou 2004; Prodromou et al. 2011). Gommers et al. (1998) used a multi-inclusion Eshelby-type approach, in which each yarn segment is considered to be an inclusion embedded in a medium that has either the matrix properties (Mori–Tanaka method) or the already homogenised composite properties (self-consisting method). Predictions of homogenised stiffness properties were quite accurate for the knits and yarns considered. Some attempts were made to predict damage development, because both methods allow one to calculate, albeit in an approximate way, the stresses or strains at the lowest level (impregnated yarns and matrix), and hence to predict at which applied stress or strain level the local stresses or strains reach a critical value. The major drawback of these early models was their dependency on empirically obtained descriptions of the internal geometry of the textile. This means that the textile first has to be produced, and then measured, and that no predictions of the behaviour of yet non-existing textile composites can be made. In this way, the value of these ‘predictive’ models is rather limited. A step forward was achieved when models for the internal geometry of 2D and 3D weaves, initially developed for technical textiles (the CETKA model (Lomov and Truevtzev 1995; Lomov et al. 1995, 1997, 1998a, 1998b; Shtut et al. 1995; Gusakov and Lomov 1998)), were applied to composite reinforcements. These models are based on a minimum amount of topological data (weave style, inter-yarn distance) and yarn mechanical properties; they are mechanical models, as they apply a yarn deformation energy minimisation algorithm to predict the internal geometry of any 2D and 3D weave. Connecting this approach to the cell or inclusion models of Gommers and Vandeurzen resulted in a more versatile way to calculate the homogenised properties of textile-based composites (Huysmans et al. 1998, 2001; Lomov et al. 2000, 2001a). Since 1999, this approach has been systematically followed, extending the types of textiles to 2D and 3D woven (Lomov et al. 2000, 2001a, 2001b, 2001c, 2003, 2011b), two- and three-axial braided (Lomov et al. 2002b; Ivanov et al. 2009), weft-knitted (Moesen et al. 2003) and non-crimp warpknit stitched (Lomov et al. 2002a) fabrics and to laminates (Lomov et al.
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2002c). The mechanical models generate the internal geometry not only of relaxed textiles, but also of textiles deformed in tension, compression and shear. The models are implemented in the software package WiseTex. A further step was the integration of the geometrical models with other predictive models relevant for composites processing and performance prediction. Permeability tensors can be predicted by modelling the resin flow through the reinforcement (Belov et al. 2004; Laine et al. 2005; Verleye et al. 2008, 2010); the homogenised mechanical properties and the local stresses and strains can be generated by micro-mechanical calculations of properties of the composite (Lomov et al. 2000, 2001c; Bogdanovich et al. 2009; Perie et al. 2009), and can then be further linked to a micro–macro analysis of the composite parts (Lomov et al. 2004, 2007a; Van den Broucke et al. 2004; Verpoest and Lomov 2005), finite element models (Lomov et al. 2001c, 2005a, 2007b; Carvelli et al. 2004; Kurashiki et al. 2004; Ivanov et al. 2009) and virtual reality software (Lomov et al. 2007c). All these models use a unified description of the geometry of the reinforcement unit cell (Verpoest and Lomov 2005).
7.2
Generic data structure for description of internal geometry of textile reinforcement
The geometrical and mechanical model of textiles, implemented in the software package WiseTex, provides a full description of the internal geometry of a fabric: 2D and 3D woven, two- and three-axial braided, knitted, multi-axial multi-ply stitched (non-crimp fabric). Input data include (1) yarn properties: geometry of the cross-section (elliptical, lenticular or rectangular cross-section), compression, bending, frictional and tensile behaviour, and fibrous content; (2) yarn interlacing pattern; and (3) yarn spacing within the fabric repeat. Consider a fabric consisting of yarns only (fibrous plies in non-crimp fabrics will be discussed on pages 223–225). Figure 7.1 illustrates the description of the configuration of the yarns. The midline of a yarn is given by the spatial positions of the centres of the yarn cross-sections O: r(s), where s is the coordinate along the midline, and r is the radius-vector of the point O. Let t(s) be the tangent to the midline at point O. The crosssection of the yarn, normal to t, is defined by its dimensions d1(s) and d2(s) along axes a1(s) and a2(s). These axes are ‘glued’ with the cross-section and rotate around t(s) if the yarn is twisted along its path (such a twist can be the result of non-orthogonal intersection, as in braids or sheared woven fabrics: see page 222). Because of this rotation the system [a1a2t] may differ from the natural coordinate system along the spatial path [nbt] (t = dr/ds, n = dt/ds, b = t ¥ n). The shape of the cross-section can be assumed to be elliptical, lenticular, etc. The shape type does not change along the yarn, but dimensions d1 and d2 can change because of different compression of the
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a O
Z
r(s)
Y X
(a)
Vf
1
a2
O d2
a1
P f
d1
(b)
(c)
7.1 (a) Set of cross-sections defining a yarn shape in a unit cell; (b) parameters of a cross-section; and (c) properties of fibres in the vicinity of point P.
yarn in the contact zones and between them. The definition of a yarn with a given cross-sectional shape consists therefore of the five periodic functions: r (s) (then [nbt] vectors can be calculated), a1(s), a2(s), d1(s) and d2(s). When used in numerical calculations, the yarn description is given as arrays of values for a set of cross-sections Si along the yarn midline: ∑ Coordinates of the cross-section centre point O = (x, y, z) ∑ Tangent, normal and bi-normal to the yarn heart-line t, n, b ∑ Vectors of (orthogonal) axis of the contour a1, a2 ∑ Dimensions in the directions of the axis d1, d2 This description fully defines volumes of the yarns in a unit cell. The format is the same for orthogonal and non-orthogonal (angle a, Fig. 7.1) unit cells. The in-plane dimensions of the unit cell X, Y are given by the repeat size of the textile structure, the thickness Z, as the difference between the maximum and minimum z-coordinates of the cross-sections of all the yarns in the unit cell. The data presented above describe the volumes of the yarns. The fibrous
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structure of the yarns or, more generally, the fibrous structure of the unit cell is described as follows. Consider a point P and the fibrous assembly in the vicinity of this point (Fig. 7.1c). The fibrous assembly can be characterised by physical and mechanical parameters of the fibres near the point (which are not necessarily the same in all points of the fabric), the fibre volume fraction Vf and their direction f. If the point does not lie inside a yarn, then Vf = 0 and f is not defined. For a point inside a yarn, fibrous properties are easily calculated, providing that the fibrous structure of the yarn in the virgin state is known and its dependency on local compression of the yarn, bending and twisting of the yarn are given. Consider a point P. Searching the cross-sections of the yarns, crosssections Si = S(si) and Si+1 = S(si+1) (s is a coordinate along the yarn midline), containing between their planes the point P, are found (a binary search in the unit cell volume is employed to speed up the calculations). Then, using interpolation by s, the cross-section S(s), whose plane contains the point P, is built. Using the dimensions of the cross-section S, for a given shape of the cross-section, point P is identified as lying inside or outside the yarn. In the former case, with the position of the point P inside the yarn known, using the model of the yarn microstructure, the parameters of the fibrous assembly in the vicinity of the point P are calculated. The described data structure is generic and is used for any type of fabric. The simple organisation of the cross-sectional data allows easy transition of the description in the virtual reality representation (Lomov et al. 2007c), as illustrated in Fig. 7.2.
7.3
Geometrical description of specific types of reinforcements
7.3.1
Woven fabrics (2D and 3D)
Coding of the structure of a 2D/3D weave Consider a warp-interlaced 3D weave (or rather, a ‘multi-layered’ weave) as in Fig. 7.3. The topological coding of a multi-layered weave is based on the warp yarn paths. The ith warp path is coded by a sequence of intersection levels wij denoting either the index number of the weft layer situated above the warp yarn in its intersection with the jkth weft row, or 0, if the warp yarn lies on the face of the fabric. Let a fabric have L weft layers. The warp in intersections with the weft can occupy L + 1 levels, level 0 corresponding to the face of the fabric, level L to the back of the fabric. Each warp can now be coded as a sequence of level codes, and the entire weave as a matrix, as shown in Fig. 7.3. The matrix coding of a one-layer weave also represents a chequerboard pattern, if level 0 were identified with a black square and level 1 with a white square. In the pattern shown in Fig. 7.3a all the warps
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(b)
(c)
(d)
(e)
(f)
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7.2 Textile micro-VR worlds: (a) laminate of carbon/aramid woven composite reinforcement; (b) multi-layered fabric; (c) three-axial braided fabric; (d) weft-knitted fabric; (e) unidirectional laminate; (f) 3D woven fabric.
are situated side by side. It is very often in composite reinforcements that warps also are layered, as shown in Fig. 7.3b. Paths of the warps in this case also can be coded as a sequence of level codes. To represent their layered positioning, a notion of warp zones is introduced. A warp zone is a set of warp yarns layered one over another. The yarns going through the thickness of a fabric are called Z-yarns. Multilayered weaves for composite reinforcements are classified as orthogonal (Z-yarns going along columns of weft), through-the-thickness angle interlock (Z-yarns going across columns of weft, connecting the face and back of the fabric) and angle interlock (Z-yarns connecting separate layers of the fabric). These types of weaves are illustrated in Fig. 7.3b and 7.3c. The matrix coding is further developed (Lomov et al. 2011b) to represent weaves, which do not have necessarily the same number/placement of the weft yarns in the weft rows/layers (Fig. 7.3d). The simple solution to represent such weaves is to skip weft yarns in certain positions. This is done
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Warp 1 Warp 2 Warp 3 Warp 4
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0 1 2 3 4 1 2 3
4 1 2 3 0 1 2 3
1 2 3 4 Warp zones
(b)
(c)
Matrix code for 1st yarn: [0; –1; –2; –3; 3; –2; –1]
(d)
7.3 Matrix coding of a multilayered weave: (a) building the matrix; (b) warp in zones; (c) angle interlock; (d) complex placement of weft yarns in the layers: dashed circles show non-present weft yarns.
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by introducing Boolean values WElj, l = 1, …, L, j = NWe, where NWe is the number of weft rows, which are true if the weft yarn is present in the position and false if not. When processing the weave topology, the weft yarns with WElj = false are considered as not present. The coding based on the intersection levels can be further developed for more complex yarn paths. First, a modification of the matrix coding is needed to take into account the eventual ‘missing’ wefts. To describe cases where a warp yarn goes through a space where a weft yarn has been removed, negative values for the matrix coding have been introduced. Consider a missing weft on the first weft layer number L: if a warp goes though this empty space, the corresponding value of the matrix coding will be equal to –L. In this case the matrix coding value does not correspond directly to the supporting weft layer but indicates the position of the warp yarn in the weft network. Then a modified algorithm of the definition of supporting wefts is used to decide which of the above or below weft yarns is considered as supportive. The matrix weave coding for 3D fabrics was proposed in Lomov and Gusakov (1993, 1995) and is implemented in the software WiseTex with GUI for definition and editing the weave. Note that this approach differs from the approach used in Chen and Potiyaraj (1998, 1999a, 1999b), which is aimed at technological issues such as, for example, the shedding lifting plan for a loom. In Ping and Lixin (1999) the specific matrix coding is applied to produce 3D images of a fabric, and their approach is closer to the one described here. Building a geometric model of a woven fabric The topology of interlacing of yarns, set by the weave pattern, defines the waviness (or crimp) of the yarns inside the fabric. The approach described below was first formulated in Lomov and Truevtzev (1995) and Lomov et al. (1995), and further developed in Lomov et al. (2000, 2001a, 2001b, 2005b, 2011b). A waved shape of a yarn inside a woven fabric can be subdivided in intervals of crimp (between two intersections A and B, Fig. 7.4). Assume that we are considering a warp yarn (the case of weft is treated in the same way). Let p and h be distances between the points A and B in the direction of the yarn x and in the vertical direction z. The distance p is defined by the yarn spacing; the distance h is called a crimp height. The shape of the yarn on an elementary interval is described using a parameterised function z(x; h/p) (z and x are coordinates of the yarn middle line), which is computed using the principle of minimum bending energy of the yarn on the interval and has a form
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(a) 6 5 4
A
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7.4 Elementary crimp interval: (a) scheme; (b) characteristic functions; dotted line: F ª 6 (h/p)2, corresponding to the linearised spline solution.
( )
z = 1 (4xx 3 – 6x 2 +1) – AÊ hˆ x 2 (x – 1)2 x – 1 , x = x ÁË p˜¯ h 2 2 p
7.1
where the function A(h/p) is shown in Fig. 7.4. The value A = 3.5 provides a good approximation in the range 0 < h/p < 1. The first term of this formula, underlined in Equation 7.1, is a solution for a linearised minimum energy problem – a cubic spline. With this function known, the characteristic function F of the crimp interval is computed, representing the bending energy of the yarn (Fig. 7.4b):
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Modelling the geometry of textile reinforcements for composites p (z ¢¢ )2 B(k ) Ê hˆ W = 1 B (k ) Ú dx= dx x FÁ ˜ 2 5 /2 0 (1 + (z ¢ ) ) 2 p Ë p¯
209
7.2
where B(k) is the bending rigidity of the yarn (measured experimentally), which depends (non-linearly) on the local curvature k (x) or, after the integration, on an average curvature over the interval. Function F(h/p) is tabulated. With the function F known, the transverse forces acting on the interval ends can be estimated as Q = 2W = h
2B(k ) Ê hˆ FÁ ˜ ph Ë p¯
7.3
Having the coding of the structure, the next step will be to create a full description of the internal geometry of a multi-layered woven fabric. Throughout the section, subscript i, i = 1, …, NWa designates a warp yarn, subscripts j = 1, …, NWe and l = 1, …, L designate a weft yarn, NWa is the number of warps in the fabric, NWe is the number of wefts in each layer of the fabric and L is the number of weft layers. The following input data are given: 1. Fabric weave, given by a matrix of warp levels 2. Compression and bending behaviour of warp and weft yarns (there can be any number of different types of yarns in both warp and weft) 3. Spacing of warp and weft yarns (which can be non-uniform) 4. Shift between the weft layers in the warp direction. This is defined by the weft insertion and battening process. Two typical cases are zero shift (weft yarns one above another) and shift of 50% of the weft spacing (weft yarns of the upper layer in between yarns of the lower layer). Analysis of the weave matrix allows one to determine sets of crimp intervals on warp and weft yarns. Subscript k designates the interval number: for warp yarns k = 1, …, NWe, for weft yarns k = 1, …, Kjl, and the number of crimp intervals on different weft yarns Kjl may be different. For each interval the support yarns at the interval ends and signs of the yarn position relative to them (designated as P) are known from the analysis of the weave. If we consider a crimp interval k on a warp yarn i, then the indices of the weft yarn (and its crimp intervals) supporting the warp at the ends of the interval are designated as j¢, l¢, k¢ and j≤, l≤, k≤. If we consider a crimp interval k on a weft yarn (j, l), then the indices of the supporting warp yarns and intervals on them are designated by i¢, k¢ and i≤, k≤. Figure 7.5a explains the indexing. We assume that all the crimp intervals of a weft yarn (j, l) have the same crimp height hjlWe. The weft yarns deviate at the ends of the crimp intervals by hjlWe/2 in the z direction from the average planes of the weft layers. These average planes have z coordinates Zl, Z0 = 0. Dimensions of cross-sections © Woodhead Publishing Limited, 2011
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(a)
Weft crimp interval k”
Warp i
Warp crimp interval k
7.5 Crimp intervals for calculation of internal geometry.
Weft j ”, l ”
Weft j ’, l ’
Weft crimp interval k’
z
Weft j, l + 1; interval k2 (b)
Weft j, l; interval k1
Warp i
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can differ along a yarn. These dimensions at the ends of the crimp intervals Wa We We of warp and weft yarns are designated as d1Wa ,ik , d2,ik , d1,,jlk jlk and d2 , jlk . The parameters listed above allow building a full description of the internal geometry. Vertical positions of the centres of weft yarns at the ends of crimp intervals are given by We We zWe jlk = Z l + h jl ◊ Pjlk
7.4a
Vertical positions of the warp yarns at the ends of crimp intervals are given then as We ˆ Ê d1Wa d1j 1j ¢l ¢k ¢ ik Wa zikWa = z(We – + Á 2 ˜ Pik il ) ¢ k ¢ 2 Ë ¯
7.4.b
Positions in the x and y directions are determined by the yarn spacing. With dimensions of the yarns and positions of the yarns at the ends of crimp intervals and support yarns defined, Equation 7.1 is employed to generate fill paths of the yarns in the repeat. Wa Wa We We Now we calculate the unknowns hWe jl , Z l , d1,ik , d2 ,ik , d1,jlk and d2,jl jlk . Yarn jlk dimensions are defined by the laws of yarn compression: Wa Wa Ê Qij ¢l ¢ ˆ Wa Wa Wa Ê Qij ¢l ¢ ˆ d1Wa ik = d10i h 1i Á Wa ˜ , d2ik = d2 0 ih2i Á Wa ˜ Ë d2ik ¯ Ë d2ik ¯
7.5
Ê ˆ We Ê ˆ We We Qi ¢jjl We We Qi ¢jjl d1We jlk = d10jlh jl1 Á We ˜ , d2 jlk jlk = d2 0 jlkh2 2jl jl Á We ˜ Ë d2jlk Ë d2jlk jlk ¯ jlk ¯
7.6
Here and below subscripts with prime designate corresponding indexes of crimp intervals and yarns, which are the support yarns for the crimp interval under consideration. Qijl in Equations 7.5 and 7.6 are transverse forces of interaction of the warp yarn I and the weft yarn (j, l): Wa Ê BWa Ê hWa ˆ Ê hWa ˆ ˆ BWa Qijl = 1 Á Wai Wa F Á ikWa¢ ˜ + Wa i Wa FÁ ikWa¢ +1 ˜ ˜ 2 Ë pik ¢ hik ¢ Ë pik ¢ ¯ pik ¢ +1hik ¢ +1 Ë pik ¢ +1¯ ¯
Ê BWe Ê hWe ˆ Ê hWe ˆˆ BWe jl jl jl jl + 1 Á We We F Á We ˜ + We F Á ˜˜ We We 2 Ë p jlk ¢¢ h jl Ë p jlk ¢¢ ¯ p jlk ¢¢+1 Ë p jlk ¢¢ +1¯ ¯ ¢¢ +1h jl
7.7
The relation between crimp heights of the warp crimp intervals and the weft crimp is given by a constraint which describes the contact of the yarns in the adjacent layers: We We hikWa + 1 (hWe + d We ) = Zl ¢¢ – Zl ¢ + 1 (d1Wa + d1Wa ik +1 + d1j ¢l ¢k ¢ + d1 1jj ¢¢¢¢l ¢¢¢¢k ¢¢¢¢ ) 2 j ¢l ¢ 11jj ¢¢l ¢¢ 2 ik
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The coordinates of the weft layer planes Zl are calculated from the condition of tight packing of the woven structure in the z direction. Consider two weft yarns (j, l) and (j, l + 1) in two consecutive weft layers (Fig. 7.5b). k1 and k2 are indices of crimp intervals on these yarns which either have warp i as a support or are supports for the warp i at a certain crimp interval k¢. The z-coordinates of the centres of the cross-sections of the weft yarns are given by We z jlk1 = Zl + hWe jl Pjlk2 Wee We z j ,l +1,k2 = Zl +1 + hW j ,l +1 Pj ,l +1,k2
7.9
where P are the position codes of the weft yarns derived from the weave analysis. The distance D x between the centres of the weft yarns in the warp direction is defined by the spacing of the weft and possible shift between the weft layers. Consider a distance Dz between these centres. As the yarns are packed closely, then this is the distance defined by the condition: the distance between contours is equal to dikWa¢ . The solution of such a problem, designated as Dztight, depends on the shapes of the yarns, their dimensions and Dx. The value of Dztight defines by Equation 7.9 the distance between the layers. As we have assumed the existence of a common middle plane of a weft layer, then Zl +1 = Zl + maxx (Dzztight j ,k
We We We We We Wa ¥ (shapeWe jl , shape jl +1 , d1 1jlk jlk1 , d2 2jlk jlk1 , d1j ,l +1,,k21 , d2 2jj ,l + +1, 1,k21 , d1jk ¢ ) We We We – hWe j ,l + +1 1 Pj ,l + +1, 1,k2 + h jl Pjlk2 )
7.10 With the condition Z 1 = 0, Equation 7.10 defines all the weft layer positions. We Finally, weft crimp heights h jl are independent variables in the minimum problem, expressing the minimum of the bending energy: WS = S i ,k
We We BWe BikWa (k ikWa ) Ê hikWa ˆ jlk (k jlk ) Ê h jl ˆ F + S F Æ min Á We ˜ ÁË pWa ˜¯ j ,l , k pikWa pWe Ë p jlk ¯ ik jlk
7.11
where hWa are related to hWe by Equation 7.8. The minimum is reached when ∂WS =0 ∂hWe jl
7.12
The system of equations, which defines the parameters of the fabric internal geometry, is now complete (Table 7.1). It is solved by an iteration
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Table 7.1 Summary of the mathematical description of internal structure of a 3D woven fabric Unknown variables
Number
Dimensions of warp and weft yarns Wa Wa We We d1, ik , d 2, ik , d1, jlk , d 2, jlk
L NWe Ê ˆ 2 ÁNWa · NWe + S S K jl ˜ ÁË ˜¯ l =1 j =1
Equations 7.5–7.7
Vertical positions of mid-planes of weft layers Zl
L
7.10
Weft crimp heights hWe jl
L*NWe
7.12
procedure, doing calculations in the order given by Table 7.1, and checking the convergence by the convergence of weft crimp height values. Examples Figure 7.6 shows results of WiseTex geometrical modelling of three Twintex® glass/polypropylene fabrics. Fabric 1 is twill 2/2 fabric with areal density 1900 g/m2, yarn linear density 2 ¥ 2520 tex (glass and polypropylene together) and ends/picks count 2.6/0.76 yarns/cm; Fabric 2 is twill 2/2 fabric with areal density 1550 g/m2, yarn linear density 2050/2 ¥ 2520 tex and ends/picks count 4.1/1.9 yarns/cm; Fabric 3 is plain weave fabric with areal density 815 g/m2, yarn linear density 2110 tex and ends/picks count 1.9 yarns/cm. Figure 7.7 shows results of modelling the internal geometry of ply-to-ply interlock carbon fabric. Compression is applied on the WiseTex model to reach the fibre volume fraction of 58% measured on the composite samples (see Section 7.4). Comparison of the computed and measured parameters of the fabric shows good representation of the yarn crimp: calculated/measured crimp of the warp yarns is 1.50/1.44%, and of the weft yarns 1.50/1.49%, and the average interlock angle is 8.0°/9.2°.
7.3.2
Braided fabrics
Coding of a braided structure Interlacing of a two-axial braid is coded in the same way as for a 2D weave, using a chequerboard pattern which represents the matrix coding, described on page 204, with ‘black’ cells corresponding to intersection level 1 and ‘white’ cells to intersection level 0. If the braid is triaxial containing axial yarns (0° yarns, or inlays), then the position of the inlays can be arbitrarily chosen, as illustrated in Fig. 7.8.
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(d)
(c)
(e)
(f)
(h)
(g)
7.6 Examples of geometrical models of 2D woven fabrics: micrographs and WiseTex models of Twintex glass/ polypropylene fabrics: fabric 1, (a) warp and (b) weft cross-sections; fabric 2, (c) warp and (d) weft cross-sections; fabric 3, (e) warp cross-section; surface scans of fabrics 1 (f), 2 (g) and 3 (h).
(b)
(a)
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Warp direction
215
Weft direction
7.7 Cross-sections of ply-to-ply interlock reinforcement: calculated (above) and observed (below).
7.8 Examples of coding of triaxial braids.
Geometry of a biaxial braid The geometrical parameters of a two-axial braid are shown in Fig. 7.9a. The balance of crimp (crimp heights h1 and h2, Fig. 7.9a) in a flat braid is computed using the formulae for woven fabrics with the length of elementary crimp intervals computed along (non-orthogonal) yarns, described in the previous section. This assumption follows a generally accepted description
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p2
p1
2a h2 h1 (a) Middle line plane H
y n
a1 z
y na
t x
P
t
Middle line plane
P a2 x
a
Tangent plane
Middle line
z
q n
x
Support cylinder (b)
7.9 Geometry of a two-axial braid: (a) spacing of the yarns, braiding angle and crimp; (b) a scheme of non-orthogonal intersection of yarns.
of braided structures as analogous to woven structures. This model includes the crimp in the z-direction as well as side crimp s (in the fabric plane) for twill-like interlacing. The path of the yarn middle lines between intersections is described with the model for woven fabrics: a minimum energy solution, taking into account contact regions of yarns. The difference is in the treatment of positioning of the yarn cross-sections along the path. Consider a crosssection of a yarn in a braid with braiding angle a ≠ 45°. Let a1 and a2 be unit vectors of its shorter and longer axes, t be a tangent vector of the yarn middle line, and d1 and d2 be the dimensions of the shorter and longer axes (see Fig. 7.1b). Assume d1 and d2 to be constant along the yarn contact region. © Woodhead Publishing Limited, 2011
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If, as in a woven fabric (a = 45°), a1 lies in the plane formed by t and the z-axis, then the assumption of constant cross-section leads to penetration of the intersecting yarn volumes. To provide a realistic description one must introduce some deformation of the cross-section. In the present model this is done by imposing rotation on the yarns (which can be described by simple geometrical considerations) rather than by distorting the shape of the cross-section. The latter indeed implies a throughout description of the behaviour of the yarn cross-section in a complex deformed state (simultaneous compression and torque), something which is not readily available from the standard textile experimental equipment. Figure 7.9b illustrates the algorithm for computation of a cross-section position. Consider the intersection of a yarn with the middle line lying in the plane H with the second yarn at the angle 2a. Let P be a point on the middle line. Because of the intersection, it also lies on a support cylinder, which is the surface at a distance of d1/2 from the second yarn surface. Let n be a normal to the support cylinder at the point P, and na be a normal to the plane H. In the Cartesian coordinate system with the y-axis along the second yarn, n = (sin q, 0, cos q); na = (– cos a, sin a, 0) where q is the angle between n and the z-axis. Now the characteristic vectors of the cross-section of the first yarn are computed as t = na ¥ n; a2 = t ¥ n; a1 = a2 ¥ t Because of the change of n along the yarn axis, the vectors a1 and a2 change directions, introducing rotation of the triad (a1, a2, t) as in Fig. 7.9b. As a result of the twist, the penetration of the yarns in the model decreases considerably (the volume of the intermingled region for the flat braid model with rotations is about 6% of the intermingled volume of the model without the rotations). It is not completely eliminated, as the cross-sections are assumed to keep their shape along the yarn. The remaining penetration represents an error of the current model, which is caused by neglecting a change of the cross-sectional shape. For circular yarns there is no penetration at all, as the cross-sections, rotating along the yarn axis, stay inside the constant circular shape. To evaluate the maximum twist introduced in this manner, consider a somewhat extreme condition of the maximum angle of a contact region, qmax = 30°, and compute the full rotation t of the cross-section over two contact regions:
t =2
Ê
Úcontact t · ÁË a1 ¥
da 2 ˆ ds ds ˜¯
For three different braiding angles, given in parentheses, the rotations are t (45°) = 0, t (22.5°) = 35° and t (a Æ 0) = 40°.
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Geometry of a triaxial braid Inlays (axial, 0° yarns) can be placed with a certain offset (shift) in relation to the intersecting points of the braiding yarns (Fig. 7.10a) and may exhibit certain crimp. If the offset is zero and the braid is balanced, then the crimp of inlays is negligible. The crimp of the inlays in triaxial braid is largely affected by tensions during braiding and interaction of the yarns with the mandrel, hence the crimp balance no longer follows the equilibrium, defined by the minimum energy calculations, as for woven fabrics. Because of this the crimp of inlays in the present version of the WiseTex model must be defined by the user. The yarn paths are defined by the crimp intervals, which depend on the offsets of the inlays. If the offset is zero, then the inlay serves as support for the braiding yarns. If the offset is not zero, then the crimp intervals may go from the inlay to the neighbouring braiding yarn, as shown in Fig. 7.10b. The crimp height of the intervals is defined as the crimp height in biaxial braid without inlays plus the vertical displacement of the contact points defined by the thickness of the inlay and its prescribed crimp. The actual yarn paths are finally computed using this definition of the crimp intervals and the formulae on page 207 (woven fabrics) with the rotations introduced above for non-orthogonal interlacing. Examples A flat braided fabric with 2/2-intersection repeat was fabricated by using a flat braiding machine with 25 spindles. Two types of glass yarns were braided for this study, with each type of yarn composed of different numbers of fibres: ER575 (575 tex, 1000 fibres) and ER1150 (1150 tex, 2000 fibres). The fabric was impregnated with epoxy resin and subsequently a flat bar
Offset (a)
(b)
7.10 Geometry of a triaxial braid: (a) offset of the inlays; (b) support of crimp intervals (shown by arrows for one of the yarns) by inlays and braiding yarns.
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was manufactured by hand lay-up. The braiding yarn spacing was (ER575/ ER1150) 1.58/2.21 mm, and the braiding angle was 24°/35°. The reader is referred to Lomov et al. (2002b) for details of the experimental procedure. Figure 7.11 demonstrates good agreement between the measured and the simulated internal geometry of the braids. Figure 7.12 compares the surface image of a triaxial braid with braiding angle 60° and areal density 917 g/m2, made of carbon tows of 1600 tex in axial and 800 tex in the braiding yarns. The axial yarns in the photo are hidden behind the braiding yarns.
7.3.3
Non-crimp fabrics
For an in-depth discussion of non-crimp fabrics (NCF), also called multiaxial multi-ply warp knitted fabrics, the reader is referred to Lomov (2011). Here only the geometrical model of NCF, implemented in WiseTex (Lomov et al. 2002a 2007c; Loendersloot et al. 2006), is described. Coding of the knitting pattern The reader is referred to Spencer (1997) for description of the principle of warp-knitting. The knitting action of the needles creates connection of the loops in the machine (‘course’) direction (Fig. 7.13a). Movement of the guides across the needle bed creates the connection in the cross (‘wale’) direction, forming a warp-knit pattern. The positions of the gaps between the needles, where the guides pass in the subsequent knitting cycles, can be used to code the pattern (so-called Leicester notation). Consider a diagram of the guides’ movement, a so-called lapping diagram (Fig. 7.13b). Rows of dots represent needles in plane view. The numbering of needles assumes the pattern mechanism to be on the right side. As the guides position themselves in the spaces between needles, the positions between the vertical columns of dots represent shift of the guides. The pattern is coded by the sequence of these numbers:
s1 – S1/s2 – S2/…/sN – SN
where si and Si are the positions of the guides forming the ith loop and N is the number of knitting cycles in the pattern. The positions of the guides si and Si refer to gaps between needles, where the elementary movement of the guide starts. A pair si–Si represents an overlap movement of the guides (the guides move behind the needles), while Si–si+1 represents an underlap movement (the guides move in front of the needles). We consider only the patterns with one-needle overlap, which means that |si – Si| = 1. Figure 7.13c presents examples of warp-knitting patterns used for NCF reinforcements.
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A
(a)
B
(c)
d = 0.34 mm
d = 0.17 mm
d=0 d
Cross-section from the photo (b)
7.11 Comparison of the measured and simulated geometry of biaxial braid: cross-sections (a) along the production direction, ER575, and (b) along the braiding yarn axis, ER1150; (c) micro-CT image of the braid ER575.
A
Cross-section from the photo
B
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7.12 Comparison of the surface image of a triaxial braid with the WiseTex model.
Knitting cycles
Pattern on the face of the fabric
3
2
1 2
Loops on the back of he fabric (a)
1 needle # (b)
Tricot
Chain 1–0
1–0/1–2 (c)
Tricot + chain 1–0/1–2/2–1
7.13 Knitting pattern of NCF: (a) loops formed by the knitting process; (b) the diagram of the pattern; (c) typical patterns.
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Model of the stitching yarn The stitching yarn pierces the fibrous plies at the positions defined by the needle spacing in the needle bed (spacing in the cross direction, A) and by the speed of the material feeding in the knitting device (spacing in the machine direction, B). Normally B < A. The value of A is expressed also by the machine gauge, which is the number of needles per inch. The stitching yarn (polyester, polyamide, aramid, etc.) has normally low linear density (ca. 10 tex, number of filaments ca. 15) and low twist (<100 m–1). Hence it is easily compressible, which explains large variations of its dimensions. Three characteristic states of the yarn in the loop are considered (Fig. 7.14a):
0.15 d1min
d2max
d, mm
dinit 0.1 d0
ave, µm std. dev.
dinit
118
31
d 0
88
n/a
d1 min
30
7
d2 max
600
100
0.05 d0 d2max
w
0 0
d1min
10 Pressure, cN/m (b)
(a) z
J
I
y y
I
A B
H
20
y
M
C D F
E
d1min
d0
H
L
G
J
d0
K
A I
L
K
E
M
x
F
G
K
A x
h
d2max
L C
B
D J G
(c)
w B
E d0 F
7.14 Geometry of the stitching yarn in NCF: (a) dimensions of the different parts of the stitching yarn; (b) compression diagram of the stitching yarns and typical values of the dimensions; (c) anchor points defining the middle line of the yarn.
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1. Free: diameter of the circular cross-section dinit. The value of dinit is measured on the relaxed stitching yarn (off the bobbin) and represents the packing of its fibres before any interaction with the knitting device. 2. Compacted: diameter of the circular cross-section d0 =
4T prK
where T is the linear density of the yarn, r is the density of the fibres, and K is the packing coefficient. Experimentally observed values of K lie in the range 0.8–0.9. The value of d0 represents the ultimate compacted state of the yarn and does not depend on dinit. 3. Extreme flattened: dimensions of the cross-section d1min and d2max. These values are measured in compression test of the stitching yarn. Figure 7.14b provides typical values of these dimensions. The values dinit, d0, d1min and d2max characterise the relaxed state and the compression behaviour of the stitching yarn (measured on the standard textile equipment). They are not specific to the fabric architecture. The dimensions of the yarn cross sections in the straight parts of the yarn are kept constant; circular arcs connecting the straight parts introduce a linear transition from one dimension of the cross-sections to another. The geometry of the centre line of the yarn is defined by identifying a set of ‘anchor points’ A–M (Fig. 7.14c) along the loop, and approximating the loop centre line between the anchor points by straight lines or circular arcs. Positions of the anchor points on the centre line of the yarn are shown in Fig. 7.14c, where h is the total thickness of the plies (accounting for possible sinking of the stitching, see below), and all the arcs of the centre line (AB, CD, FE, GH, IJ and KL) are circular arcs with diameter d0. The width of the loop w depends on the knitting tension. Considerable tension produces longish loops with w = 3d0; lesser tension increases w to 4d0 to 5d0. Fibrous plies and fibre distortions by the stitching The stitching causes distortions of the fibre orientations in the fibrous mat. These distortions can be localised near the stitching sites, or collated into a linear channel (Fig. 7.15a, b). Different authors use different terminology to designate these distortions: ‘cracks’ and ‘channels’ (Lomov et al. 2002a), ‘SYD, yarn induced fibre distortion’ (Loendersloot et al. 2006), ‘fish eyes’ (Lekakou et al. 2004; Schneider et al. 2004) and ‘openings’ (Truong Chi et al. 2008). In WiseTex the term ‘fibre distortion’ is used, with the term ‘openings’ applied to the localised distortion and ‘channels’ to long separations of the fibrous plies. A localised distortion has a rhomboidal shape, with width b and length
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b l (c) DG E H
C B A (a)
(b)
F
(d)
7.15 Geometry of fibrous plies in NCF: (a) fibre distortions – ‘channels’; (b) fibre distortions – ‘openings’; (c) dimensions of the opening; (d) definition of the geometry in the model.
l (Fig. 7.15c). A channel is formed if localised distortions touch or overlap each other, and is more common for 0°/90° fabrics. Because the placement of the stitching sites has different spacing in the length and the width direction (B < A), the distortions in +45°/–45° fabrics do not overlap even if they are long enough to reach the vicinity of the neighbouring stitching site. Formation of channels in 0°/90° NCF is not a general rule. It can be expected that the width of fibre distortions is proportional to the thickness of the stitching yarn. Choosing the compacted diameter d0 as a representative parameter, one can introduce empirical coefficients k = b/ d0 to calculate width b = kd0. The length of the openings is related to the width by a coefficient l = l/b. The variation of the experimentally observed values of k is quite large (Lomov et al. 2002a; Loendersloot et al. 2006, Lomov 2011). In non-powdered fabrics channels tend to be wider then cracks, with averages for k of about 4 (openings) and about 6 (channels), and l for openings of about 21. It seems that the fibres in a powdered fabric are more difficult to push away by the stitching, which explains lower values of k (2–3) and l (6–8) for these fabrics. The dimensions of the openings/ channels are also influenced by knitting tension. The complex geometry of a fibrous ply with ‘cracks’ and ‘channels’ is represented in the model as a set of ‘slabs’. A slab is a volume formed by two parallel polygons. Vertices of all the slabs are stored in the counter-clockwise order. Figure 7.15d shows two examples of the slabs: ABCDEF and FEGH. The former (ABCDEF) represents the part of the unit cell, bounded by fibre directions BC and EF, lines between the centres of the stitching sites AF and CD and boundaries of openings in the fibrous plies AB and DE. The latter (FEGH) is bound by the fibre direction EF, part of the line between
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the centres of the stitching sites FH, the edge of the unit cell GH and the boundary of an opening EG. Note that line EF belongs to both plies. The vertices of the slab are therefore (1) corners of the unit cell (H); (2) lines of intersection of boundaries of ‘openings’ and ‘channels’ with edges of the unit cell (A, G); (3) intersections of the fibre directions, starting from the corners of the openings, with lines connecting centres of the stitching sites (C, F); and (4) corners of the openings (B, E). A set of ‘slabs’ constitutes a set of volumes, forming the fibrous ply. Example Figure 7.16 shows the WiseTex model of a quadriaxial carbon NCF with orientation of the plies 0°/–45°/90°/45°, areal density of the fabric 629 g/ m2, and tricot-chain polyester stitching of 6 tex, gauge 5.
7.4
Geometrical model as a pre-processor for prediction of mechanical properties of the reinforcement
7.4.1
Deformability of the reinforcement
The mechanical nature of the model of internal geometry of woven and braided fabrics allows upgrading them into analytical models for calculation
(a)
(b)
(c)
7.16 WiseTex model of quadriaxial NCF: (a) full model; (b) stitching yarn; (c) fibrous plies.
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of resistance of these fabrics to deformation during forming (compression, biaxial tension and shear) and prediction of geometry of a deformed unit cell of the fabric. Due to the more descriptive nature of the model of NCF, only a description of the geometry of the deformed NCF unit cell is available in WiseTex, based on the empirical data. Here, in accordance with the chapter topic, only a short outline of the models of internal geometry of deformed fabrics is given. The reader is referred to (Lomov et al. 2000, 2002c, 2003); Lomov and Verpoest 2006; Lomov 2007; 2011) for the detailed descriptions of the models. Note that the deformed configuration of the unit cell is described in the geometrical model using the same data structure as for undeformed fabric (Section 7.2). Compression A model of compression of woven and braided fabrics implemented in WiseTex (Lomov and Verpoest 2000) accounts for two physical phenomena associated with the fabric compression: change of the yarn crimp and compression of the individual yarns. The internal structure of laminated preforms after compression is affected also by relative shift and nesting of the layers in lay-up (Lomov et al. 2002c). The model of NCF after compression is built based on the empirical compression laws for the fibrous plies. Biaxial tension Biaxial tension of a woven fabric of a braid (Lomov and Verpoest 2006) is characterised by change of the fabric dimensions in the warp (x-axis) and weft (y-axis) directions X = X0 (1 + ex), Y = Y0 (1 + ey), where X and Y are sizes of the fabric repeat, subscript ‘0’ designates the undeformed state, and ex and ey are technical deformations of the fabric. As discussed in Section 7.2, the internal structure of the fabric is described based on weft crimp heights and weft and warp cross-section dimensions at the intersections, and these values change after the deformation. Tension of the yarns induces transverse forces which compress the yarns, changing their dimensions. The same transverse forces change the equilibrium conditions between warp and weft, which leads to a redistribution of crimp and a change of crimp heights. When the crimp and the yarn dimensions values in the deformed configuration are computed, the internal geometry of the deformed fabric is built as explained in Sections 7.2 and 7.3. Shear A model of the shear of woven fabrics and braids (Lomov and Verpoest 2006) accounts for the following mechanisms of yarn deformation, determining
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the shear resistance: friction, (un)bending, lateral compression, torsion and vertical displacement of the yarns. The geometrical model of the sheared fabric is similar to the model of a biaxial braid (non-orthogonal unit cell), with the additional complication of the change of yarn cross-section dimensions induced by the lateral compression of the yarns during the fabric shear. When a fabric is sheared, the deformation is resisted by friction between yarns, and by bending and compression of the yarns. Frictional forces are estimated in the model using normal forces of the yarn interaction, tension being a pretension normally employed in the shear test. The transverse forces are increased by the internal pressure developed inside yarns due to their lateral compression in the sheared structure. This is taken into account using the experimental compression diagrams of the yarns. Resistance due to bending is estimated using the difference in bending energies in deformed and undeformed configurations, the latter computed with algorithms for non-orthogonal structures. When NCF is sheared, the orientation of the fibres in the plies is changed. Also the density of the fibres in the plies increases, and the width of the ‘openings’ and ‘channels’ decreases. After a shear angle of 30°, the width reaches its minimum. This change of the width in shear is described by empirical formulae (Loendersloot et al. 2006, Lomov 2011). After recalculation of the fibre directions, the fibre volume fraction and the dimensions of the distortions, the geometrical model of the deformed fabric is built as explained in Section 7.3.3.
7.4.2
Permeability: FlowTex software
Calculation of permeability is based on a voxel representation of the unit cell volume (Fig. 7.17a). A voxel is either empty (pore) or filled with fibres. The flow of the fluid in the pores is governed by Navier–Stokes equations (NSvoxels), and inside the permeable tows by Brinkmann equation (B-voxels). In the latter case local permeability (micro-level) is calculated with the formulae of Gebart and Berdichevsky for a unidirectional array of fibres. These equations are solved by numerical schemes based on lattice Boltzmann (Belov et al. 2004) or finite difference (Verleye et al. 2008, 2010) solvers of the Navier–Stokes or Stokes equation. The homogenised permeability of a unit cell is then determined using an average flux of the fluid through the unit cell under periodic boundary conditions for the given pressure difference on the unit cell facets (Fig. 7.17b). These algorithms are implemented in FlowTex software. Figure 7.17c shows the comparison of the experimental and predicted permeability of reinforcements of different structure: woven, NCF, and random fibres. The unit cell scale permeability of textile reinforcements is an important input parameter for the simulation of the impregnation of the preform. The
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speed of the Stokes solver in FlowTex is fast enough to allow efficient creation of look-up tables (for example, depending on shear angle and compaction of the reinforcement), providing input for macro-scale Darcy solvers (Verleye et al. 2010).
7.4.3
Analytical models of mechanical properties of textile composites and integration with structural analysis of composite parts
The analytical models for the calculation of textile composite stiffness based on the WiseTex geometrical model use an approach based on the theory of inclusions. To apply the method of inclusions, implemented in the TexComp software (Huysmans et al. 1998, 2001; Lomov et al. 2000, 2001a; Verpoest and Lomov 2005), the yarns in the unit cell are subdivided into a number of smaller segments, where each yarn segment is geometrically characterised by its total volume fraction, spatial orientation, cross-sectional aspect ratio and local curvature (all these parameters are readily provided by the geometrical model). Next, Eshelby’s equivalent inclusion principle is adopted to transform each heterogeneous yarn segment into homogeneity with a fictitious transformation strain distribution. The solution makes use of a short fibre equivalent, which physically reflects the drop in the axial loadcarrying capability of a curved yarn with respect to an initially straight yarn. Every yarn segment is hence linked to an equivalent short fibre, possessing an identical cross-sectional shape, volume fraction and orientation as the original segment it is derived from. The length of the equivalent fibre on the other hand is related to the curvature of the original yarn. For textiles with smoothly varying curvature radii, a proportional relationship between the short fibre length and the local yarn curvature radius is the most straightforward choice and sufficiently accurate for the present purpose. The interaction problem between the different reinforcing yarns is solved in the traditional way, by averaging out the image stress sampling over the different phases. If a Mori–Tanaka scheme is used, the stiffness tensor CC of the composite is hence obtained as:
CC = [cm Cm + ·cs Cs AsÒ] [cmI + ·cs AsÒ]–1
where the subscripts and superscripts m and s denote the matrix and a yarn segment respectively, ci is the volume fraction of phase i (i = m, s), and the angle brackets denote a configurational average. As follows from this brief description, the homogenisation procedure does not depend on the configuration of the unit cell, and can be applied to both non-deformed and deformed reinforcement configurations and tackle geometries which are difficult to transfer to finite element mesh. As an example of application of the inclusion model consider the angle
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interlock reinforcement shown in Fig. 7.7 (Lomov et al. 2011b). Based on the good comparison between the geometrical model and the experimental observations, a parametric study has been performed. WiseTex models of this interlock fabric have been created with evolution of the pick spacing. In order to compare mechanical properties of these different configurations, the fibre volume fraction has to remain constant. Compression was then applied to maintain the fibre volume fraction at the level of 58%, which is the one measured on the samples used for tensile tests. The thickness of WiseTex models and samples decreases with pick spacing. The average interlock angle decreases, and the warp/weft ratio in the unit cell increases in the warp direction and decreases in the weft direction. The undulations in the warp path imposed by the tightened weft network for low pick spacing are attenuated when the fabric becomes looser. The effect of these changes in the geometry on the elastic mechanical properties is depicted in Fig. 7.18, which demonstrates good correlation between the theory and the experiment. Being integrated with a geometrical processor WiseTex, the model allows fast calculation of local homogenised properties of deformed reinforcement and integration into multi-level micro–meso–macro calculations (Lomov et al. 2007a).
7.4.4
Transformation of geometrical model into meso-FE
The reader is referred to a paper (Lomov et al. 2007b) which treats in detail meso-FE modelling of textile composites. Here we will focus on one aspect of the problem which applies to models of dry fabrics as well as to models of composite unit cells: interpenetration of the yarn volumes. If the geometry for the meso-FE model is acquired by direct measurement of the yarn shapes (for example, using micro-CT) then there are no defects in mutual placement of the yarn volumes. However, such an approach has limited predictive capabilities. General-purpose geometrical models, like the models discussed here, use several simplifying assumptions. One of these assumptions in WiseTex is a fixed shape (but maybe changing dimensions) of the yarn cross-sections. The shape of the yarn middle line prescribes the positions of the centres of the cross-sections. The model calculates dimensions of the cross-sections, ensuring that the distance between the contacting yarn centre lines is equal to the sum of their dimensions. For quite a wide class of 2D woven fabrics, such a treatment is sufficient to create a geometrical model that can be meshed in an FE package. However, the condition of point or line contact does not guarantee that the surfaces of the contacting yarn never penetrate one another, and interpenetration may occur. Interpenetrations in modelling of 3D fabrics can be treated by solving an intermediate FE problem: artificial separation of the contacting parts of the yarn and then compressing them together, using a compressible medium as
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a filler of the space around the yarns. This approach was proposed by M. Zako (Zako et al. 2003) and is explained in detail in Lomov et al. (2007b); it was used successfully for 2D and 3D woven and braided composites (Ivanov et al. 2009; Lomov et al. 2009; Xu et al. 2009). It is implemented in MeshTex software, integrated with WiseTex geometrical simulations of woven (2D and 3D) fabrics: see Lomov et al. (2007b). Other approaches to solve the interpenetration problem starting from WiseTex models include the use of contact algorithms, used in Lomov et al. (2011a) for knitted fabric, or superimposed mesh techniques, for example for NCF and structural stitching (Koissin et al. 2009; Kurashiki et al. 2009). Recently D. Durville has applied his Multifil method (Durville 2005) of simulating multiple contacts inside a fibrous assembly, with individual fibres represented by beam elements, to build up consistent (not interpenetrating) models of woven fabrics based on WiseTex description of the geometry (Durville 2007). The WiseTex model of NCF was used successfully to create FE models which neglect non-structural stitching but account for fibre distortions in the plies created by the stitching (Mikhaluk et al. 2008; Truong Chi et al. 2008). Figure 7.19 represents a gallery of FE models based on WiseTex geometry of reinforcements of different classes (Lomov et al. 2008).
7.5
Conclusion
The WiseTex family of models and software (Fig. 7.20) provides a generic description of the internal geometry of textile reinforcements, covering a wide range of textile structures – woven (2D and 3D), braided, NCF, knitted and stitched. The components of the WiseTex suite simulate properties of textile fabrics and composites on a unit cell level and are used for multilevel micro–meso–macro analysis. WiseTex constitutes an integrated approach to the modelling of textile composites, which allows: ∑
Creation and easy variation of weave architectures, (almost) without restriction of the number of the yarns, layers, interlacing pattern or other complexity factors of the fabric weave ∑ Creation of geometrical models of the internal structure of reinforcements, adequately representing yarn paths (hence crimp factors, hence overall parameters of the fabric, such as areal density, tightness, porosity, etc.) ∑ Calculation (with certain reservations vis-à-vis precision) of the mechanical response of the fabric to compression, tension and shear ∑ Modelling of the geometry of deformed fabric ∑ Translation of the fabric geometry model into a finite element model for simulation of local stresses and strains during deformation ∑ Calculation of effective (homogenised) properties of textile composites
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7.19 A gallery of meso-FE models based on WiseTex-created geometry: (a) plain weave laminate; (b) 3D woven composite; (c) triaxial braided composite; (d) NCF; (e) structural stitching. Left: WiseTex geometry; right: FE mesh.
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with precision conforming to requirements of the macro-structural analysis of the composite part Building meso-level FE models of the unit cell of 3D woven composites and approaching the problem of damage prediction.
7.6
References
Belov, E. B., S. V. Lomov, I. Verpoest, T. Peeters, D. Roose, R. S. Parnas, K. Hoes and V. Sol (2004). Modelling of permeability of textile reinforcements: Lattice Boltzmann method. Composites Science and Technology 64: 1069–1080. Bogdanovich, A. E., D. Mungalov, S. V. Lomov, D. S. Ivanov and I. Verpoest (2009). A combined theoretical and experimental study of progressive failure of non-crimp 3D orthogonal weave composite. 17th International Conference on Composite Materials (ICCM-17), Edinburgh, IOM Communications Ltd. Carvelli, V., T. Truong Chi, M. Larosa, S. V. Lomov, C. Poggi, D. Ranz Angulo and I. Verpoest (2004). Experimental and numerical determination of the mechanical properties of multi-axial multi-ply composites. Proceedings ECCM-11, Rodos, CD edition. Chen, X. and P. Potiyaraj (1998). CAD/CAM for complex woven fabrics. Part I. Backed cloths. Journal of the Textile Institute 89, part I(3): 532–545. Chen, X. and P. Potiyaraj (1999a). CAD/CAM of orthogonal and angle-interlock woven structures for industrial applications. Textile Research Journal 69(9): 648–655. Chen, X. and P. Potiyaraj (1999b). CAD/CAM for complex woven fabrics. Part II. Multilayer fabrics. Journal of the Textile Institute 90, part I(1): 73–90. Durville, D. (2005). Numerical simulation of entangled materials mechanical properties. Journal of Materials Science 40: 5941–5948. Durville, D. (2007). Finite element simulation of textile materials at mesoscopic scale. Proceedings of Symposium Finite element modelling of textiles and textile composites, St Petersburg. Gommers, B., I. Verpoest and P. Van Houtte (1998). The Mori–Tanaka method applied to textile composite materials. Acta Materialia 46(6): 2223–2235. Gusakov, A. V. and S. V. Lomov (1998). Parametric studies of the internal structure of 3D woven fabrics. Fibres & Textiles in Eastern Europe 6(2): 60–63. Huysmans, G., I. Verpoest and P. Van Houtte (1998). A poly-inclusion approach for the elastic modelling of knitted fabric composites. Acta Materialia 46(9): 3003–3013. Huysmans, G., I. Verpoest and P. Van Houtte (2001). A damage model for knitted fabric composites. Composites, part A 32(10): 1465–1475. Ivanov, D. S., S. V. Lomov, F. Baudry, H. Xie, B. Van den Broucke and I. Verpoest (2009). Failure analysis of triaxial braided composite. Composites Science and Technology 69: 1372–1380. Koissin, V., J. Kustermans, S. V. Lomov, I. Verpoest, H. Nakai, T. Kurashiki, K. Hamada, Y. Momoji and M. Zako (2009). Structurally stitched woven preforms: Experimental characterisation, geometrical modelling and FE analysis. Plastics, Rubber and Composites: Macromolecular Engineering 38(2): 98–105. Kurashiki, T., M. Zako, S. Hirosawa, S. V. Lomov and I. Verpoest (2004). Estimation of a mechanical characterization for woven fabric composites by FEM based on damage mechanics. Proceedings ECCM-11, Rodos, CD edition. Kurashiki, T., K. Hamada, S. Honda, M. Zako, S. V. Lomov and I. Verpoest (2009). Mechanical behaviors of non-crimp fabric composites based on multi-scale analysis.
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17th International Conference on Composite Materials (ICCM-17), Edinburgh, IOM Communications Ltd. Laine, B., G. Hivet, P. Boisse, F. Boust and S. V. Lomov (2005). Permeability of the woven fabrics: a parametric study. Proceedings of the 8th ESAFORM Conference on Material Forming, Cluj-Napoca, Romania: 995–998. Lekakou, C., S. Edwards, G. Bell and S. Amico (2004). Computer modeling for the prediction of the in-plane permeability of non-crimp stitch bonded fabrics. Flow Modelling in Composite Materials (FPCM-4), Newark, Delaware. Loendersloot, R., S. V. Lomov, R. Akkerman and I. Verpoest (2006). Carbon composites based on multiaxial multiply stitched preforms. Part 5: Geometry of sheared biaxial fabrics. Composites, part A 37: 103–113. Lomov, S. V. (2007). Virtual testing for material formability. In Composite Forming Technologies, ed. A. Long. Cambridge, UK, Woodhead: 80–116. Lomov, S. V. ed. (2011). Non-crimp Fabric Composites: Manufacturing, Properties and Applications. Cambridge, UK, Woodhead Publishing. Lomov, S. V. and A. V. Gusakov (1993). Coding of carcasse-layered weaves. Technologia Tekstilnoy Promyshlennosty (4): 40–45. Lomov, S. V. and A. V. Gusakov (1995). Modellirung von drei-dimensionalen gewebe Strukturen. Technische Textilen 38: 20–21. Lomov, S. V. and A. V. Gusakov (1998a). Computation of the porosity of one and multilayered woven synthetic fabrics. Chimicheskie Volokna (5): 52–55. Lomov, S. V. and A. V. Gusakov (1998b). Mathematical modelling of 3D and conventional woven fabrics. International Journal of Clothing Science & Technology 10(6): 90–91. Lomov, S. V. and N. N. Truevtzev (1995). A software package for the prediction of woven fabrics geometrical and mechanical properties. Fibres & Textiles in Eastern Europe 3(2): 49–52. Lomov, S. V. and I. Verpoest (2000). Compression of woven reinforcements: a mathematical model. Journal of Reinforced Plastics and Composites 19(16): 1329–1350. Lomov, S. V. and I. Verpoest (2006). Model of shear of woven fabric and parametric description of shear resistance of glass woven reinforcements. Composites Science and Technology 66: 919–933. Lomov, S. V., A. V. Gusakov and C. Cassidy (1995). 3D fabrics: Technology, structure properties and mathematical simulation. EUROMECH-334, Textile Composites and Textile Structures, Lyon: 187–200. Lomov, S. V., B. M. Primachenko and N. N. Truevtzev (1997). Two-component multilayered woven fabrics: weaves, properties and computer simulation. International Journal of Clothing Science & Technology 9: 98–112. Lomov, S. V., A. V. Gusakov, G. Huysmans, A. Prodromou and I. Verpoest (2000). Textile geometry preprocessor for meso-mechanical models of woven composites. Composites Science and Technology 60: 2083–2095. Lomov, S. V., G. Huysmans, Y. Luo, R. Parnas, A. Prodromou, I. Verpoest and F. R. Phelan (2001a). Textile composites: Modelling strategies. Composites, part A 32(10): 1379–1394. Lomov, S. V., G. Huysmans and I. Verpoest (2001b). Hierarchy of textile structures and architecture of fabric geometric models. Textile Research Journal 71(6): 534–543. Lomov, S. V., I. Verpoest, S. V. Kondratiev and A. I. Borovkov (2001c). An integrated model strategy for processing and properties of textile composites: New results. Proceedings of the 22nd International SAMPE Europe Conference, ed. G. R. Griffith and R. F. J. McCarthy, Paris, SAMPE: 379–389. © Woodhead Publishing Limited, 2011
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Lomov, S. V., E. B. Belov, T. Bischoff, S. B. Ghosh, T. Truong Chi and I. Verpoest (2002a). Carbon composites based on multiaxial multiply stitched preforms. Part 1: Geometry of the preform. Composites, part A 33(9): 1171–1183. Lomov, S. V., A. Nakai, R. S. Parnas, S. B. Ghosh and I. Verpoest (2002b). Experimental and theoretical characterisation of the geometry of flat two- and three-axial braids. Textile Research Journal 72(8): 706–712. Lomov, S. V., I. Verpoest, T. Peeters, D. Roose and M. Zako (2002c). Nesting in textile laminates: Geometrical modelling of the laminate. Composites Science and Technology 63(7): 993–1007. Lomov, S. V., T. Truong Chi, I. Verpoest, T. Peeters, V. Roose, P. Boisse and A. Gasser (2003). Mathematical modelling of internal geometry and deformability of woven preforms. International Journal of Forming Processes 6(3–4): 413–442. Lomov, S. V., B. Van den Broucke, F. Tumer, I. Verpoest, P. De Luka and L. Dufort (2004). Micro–macro structural analysis of textile composite parts. Proceedings ECCM-11, Rodos, CD edition. Lomov, S. V., E. Bernal, D. S. Ivanov, S. V. Kondratiev and I. Verpoest (2005a). Homogenisation of a sheared unit cell of textile composites: FEA and approximate inclusion model. Revue Européenne de Mécanique Numérique (formerly Revue Européenne des Éléments Finis) 14(6–7): 709–728. Lomov, S. V., I. Verpoest and F. Robitaille (2005b). Manufacturing and internal geometry of textiles. In Design and Manufacture of Textile Composites, ed. A. C. Long. Cambridge, UK, Woodhead: 1–60. Lomov, S. V., L. Dufort, P. De Luca and I. Verpoest (2007a). Meso–macro integration of modelling of stiffness of textile composites. Proceedings of the 28th International Conference of SAMPE Europe, Paris: 403–408. Lomov, S. V., D. S. Ivanov, I. Verpoest, M. Zako, T. Kurashiki, H. Nakai and S. Hirosawa (2007b). Meso-FE modelling of textile composites: Road map, data flow and algorithms. Composites Science and Technology 67: 1870–1891. Lomov, S. V., T. Mikolanda, M. Kosek and I. Verpoest (2007c). Model of internal geometry of textile composite reinforcements: Data structure and virtual reality implementation. Journal of the Textile Institute 98(1): 1–13. Lomov, S. V., D. S. Ivanov, V. Koissin, I. Verpoest, M. Zako, T. Kurashiki and H. Nakai (2008). FEA of textiles and textile composites: a gallery. 9th International Conference on Textile Composites (TexComp-9), Newark, DE, DEStech Publications. Lomov, S. V., A. E. Bogdanovich, D. S. Ivanov, K. Hamada, T. Kurashiki, M. Zako, M. Karahan and I. Verpoest (2009). Finite element modelling of progressive damage in non-crimp 3D orthogonal weave and plain weave E-glass composites. 2nd World Conference on 3D Fabrics, Greenville, SC. Lomov, S. V., M. Moesen, R. Stalmans, G. Trzcinski, J. Van Humbeeck and I. Verpoest (2011a). Finite element modelling of SMA textiles: superelastic behaviour. Journal of the Textile Institute 102(3): 232–247. Lomov, S. V., G. Perie, D. S. Ivanov, I. Verpoest and D. Marsal (2011b). Modelling three-dimensional fabrics and three-dimensional reinforced composites: Challenges and solutions. Textile Research Journal 81(1): 28–41. Mikhaluk, D. S., T. C. Truong, A. I. Borovkov, S. V. Lomov and I. Verpoest (2008). Experimental observations and finite element modelling of damage and fracture in carbon/epoxy non-crimp fabric composites. Engineering Fracture Mechanics 75(9): 2751–2766. Moesen, M., S. V. Lomov and I. Verpoest (2003). Modelling of the geometry of weftknit fabrics. TechTextil Symposium, Frankfurt, CD edition. © Woodhead Publishing Limited, 2011
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Perie, G., S. V. Lomov, I. Verpoest and D. Marsal (2009). Meso-scale modelling and homogenisation of interlock reinforced composite. 17th International Conference on Composite Materials (ICCM-17), Edinburgh. IOM Communications Ltd, CD edition. Ping, G. and D. Lixin (1999). Algorithms for computer-aided construction of double weaves: Application of the Kronecker product. Journal of the Textile Institute 90, part I(2): 158–176. Prodromou, A. (2004). Mechanical modelling of textile composites utilising a cell method. Department MTM, K. U. Leuven, 149. Prodromou, A., S. V. Lomov and I. Verpoest (2011). The method of cells and the mechanical properties of textile composites. Composite Structures 93(4): 1290–1299. Schneider, M., K. Edelmann and U. Tiltmann (2004). Quality analysis of reinforcement structure for composites by digital image processing. Proceedings of the 25th International SAMPE-Europe Conference, Paris, 267–272. Shtut, I. I., P. B. Bezin and S. V. Lomov (1995). Optimizing the fibrous content and properties of blended SVM-cotton yarns. Chimicheskie Volokna (4): 31–33. Spencer, D. J. ed. (1997). Knitting Technology. Cambridge, UK, Woodhead Publishing. Truong Chi, T., D. S. Ivanov, D. V. Klimshin, S. V. Lomov and I. Verpoest (2008). Carbon composites based on multiaxial multiply stitched preforms. Part 7: Mechanical properties and damage observations in composite with sheared reinforcement. Composites, part A 39: 1380–1393. Van den Broucke, B., F. Tumer, S. V. Lomov, I. Verpoest, P. De Luka and L. Dufort (2004). Micro–macro structural analysis of textile composite parts: Case study. Proceedings of the 25th International SAMPE Europe Conference, 30 March – 1 April, Paris: 194–199. Vandeurzen, P. (1998). Structure–performance modelling of two-dimensional woven fabric composites. Department MTM, K. U. Leuven. Vandeurzen, P., J. Ivens and I. Verpoest (1998). Micro-stress analysis of woven fabric composites by multilevel decomposition. Journal of Composite Materials 32(7): 623–651. Verleye, B., R. Croce, M. Griebel, M. Klitz, S. V. Lomov, G. Morren, H. Sol, I. Verpoest and D. Roose (2008). Permeability of textile reinforcements: simulation; influence of shear, nesting and boundary conditions; validation. Composites Science and Technology 68(13): 2804–2810. Verleye, B., S. V. Lomov, A. C. Long, I. Verpoest and D. Roose (2010). Permeability prediction for the meso–macro coupling in the simulation of the impregnation stage of Resin Transfer Moulding. Composites, part A 41: 29–35. Verpoest, I. and S. V. Lomov (2005). Virtual textile composites software Wisetex: Integration with micro-mechanical, permeability and structural analysis. Composites Science and Technology 65(15–16): 2563–2574. Xu, J., S. V. Lomov, I. Verpoest, S. Daggumati, W. Van Paepegem and J. Degriek (2009). Meso-scale modeling of static and fatigue damage in woven composite materials with finite element method. 17th International Conference on Composite Materials (ICCM-17), Edinburgh. IOM Communications Ltd. Zako, M., Y. Uetsuji and T. Kurashiki (2003). Finite element analysis of damaged woven fabric composite materials. Composites Science and Technology 63: 507–516.
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8
Modelling the geometry of textile reinforcements for composites: TexGen
A. C. L o n g and L. P. B r o w n, University of Nottingham, UK
Abstract: This chapter provides an overview of TexGen, the open source software package for 3D modelling of textiles and their composites developed at the University of Nottingham. The modular design of the crossplatform software and its modules (Core, Renderer, Export, Python Interface and Graphical User Interface) are described. The underlying modelling theory is then discussed, followed by descriptions of applications utilising TexGen in the fields of textile mechanics, textile composite mechanics and permeability. Finally, future developments, including further automation of the modelling process, improvements to methods of dealing with yarn interpenetration and issues of variability, are considered. Key words: TexGen software, 3D geometric textile modelling.
8.1
Introduction: rationale and background to TexGen
Realistic geometric representation of fabrics is essential for modelling of mechanical and physical properties of textiles and textile composites. If this can be achieved, a reduction in expensive physical iterations required during a textile design is possible. TexGen (2010) is software developed by the team at the University of Nottingham as a modelling pre-processor for textiles simulation for a variety of applications, including mechanics of dry textiles (to simulate forming and compaction), permeability (to simulate resin infusion) and mechanics of composite materials (to predict subsequent performance). TexGen is used to generate predictive 3D geometric models of textiles and their composites which are then implemented within macroscopic models used for analysis. The initial development of the software took place between 1998 and 2002 (Robitaille et al., 1999, 2000), resulting in a textile schema with a user interface to generate a variety of textiles relevant to composites processing. Sherburn (2007) developed Version 3 during the course of his PhD study, completely redesigning the software to make it platform-independent, modular and flexible. With the release of Version 3 in 2006 the decision was taken to release TexGen as an open-source project under General Public Licence. 239 © Woodhead Publishing Limited, 2011
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The decision to develop TexGen as an open-source project was taken for several reasons: making TexGen freely downloadable encourages thirdparty use and even casual downloads may lead to collaboration. Having the code open to scrutiny gives a better level of knowledge transfer and verification and also enables users to extend the code for their own uses, aided by the better understanding gained by being able to access the code. The executable and source code can be downloaded from the Sourceforge website (TexGen, 2010). By October 2010 over 5000 downloads had been recorded, resulting also in a number of new collaborative projects for the group at Nottingham. Taking a modular approach, particularly with the use of third-party libraries for functions such as rendering and file export, has enabled development and research effort to be concentrated on the core 3D modelling function. In particular the use of the Simplified Wrapper and Interface Generator (SWIG) library to wrap the C++ functions in the Core and Renderer modules gives great flexibility to the system by enabling the TexGen functions to be called from Python scripts. The Python interface allows integration with other packages which have a Python capability, e.g. the Abaqus Finite Element Analysis package, and also allows specific functionality to be developed separately from the main development tree (e.g. where commercial confidentiality is required). Documentation for the project is available via a wiki which gives access to a user guide, compilation instructions for different operating systems, a guide to writing Python scripts and information on recent research utilising TexGen. Either the software can be downloaded as a bundle ready for installation, or the source code can be downloaded for subsequent building in either a Windows or a Linux environment. A user forum also enables TexGen users to discuss issues relating to use and applications of the software and to exchange ideas. The work has been funded primarily by the UK Engineering and Physical Research Council (EPSRC), in collaboration with various industrial partners. The awarding of a ‘Platform Grant’ by EPSRC in 2005, which has been renewed to 2013, has ensured the continuation of the development of TexGen to this date.
8.2
Implementation
At the time of writing the most recent version of TexGen (version 3.3.2) has been implemented using a cross-platform design, tested on Windows and Linux operating systems. It consists of several modules: Core, Renderer, Export, Python Interface and Graphical User Interface (GUI). The relationship between these modules is shown in Fig. 8.1 and a brief description of each module follows.
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TexGen GUI
wxWidgets
Python interface Core
Export
Core
Triangle
TinyXML
HXA7241 Octree
Export
Renderer
Python
Renderer
OpenCASCADE
VTK
8.1 TexGen module diagram.
8.2.1
Core
The bulk of the software is contained in the core which contains the modelling described in Section 8.3. The module depends on three third-party libraries – Triangle (Shewchuk, 2002, 2005), used for the triangulation of sections used to create tetrahedral volume meshes; TinyXML (Thomason, 2007), used for reading and writing XML files; and HXA7241 Octree Component C++ (Ainsworth, 2007), used for optimisation within the program. Figure 8.2 shows the structure of the core module in more detail in the form of a Unified Modelling Language (UML) class diagram. The CTexGen class contains a database of the textiles and domains created. Any textile model created should contain both a geometric description of the textile and a domain to specify the area being examined. It also contains an instance of the CLogger class which handles output messages. The CTextile class contains the yarns, specified by the CYarn class, which make up that textile and which in turn contain the classes to create the detailed models described in Section 8.3. The CTextileWeave class contains a specialisation of the CTextile class which enables weave patterns to be described in a more automatic way. It is further specialised into CTextileWeave2D and CTextileWeave3D which specify the number of threads in the warp and weft directions and also a matrix which specifies how these cross over each other. The actual yarns are then generated automatically using this information. The ‘refine’ algorithm described in Section 8.3.5 can be applied to textiles generated using the CTextileWeave2D class using the information available from this class about the relative up/down positions of the yarns. These classes are also implemented in the user interface as described in Section 8.2.3.
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CSectionellipse
*
CSectionhybrid
CYarnsectioninterpnode
CLogger
CDomainplanes
CLoggerGUI
CSectionpowerellipse
* CSlavenode
* Plane
Slavenodes
CInterpolationcubic
* CNode
Masternodes
CLoggerscreen
CSectionpolygon
CInterpolationbezier
CSectionlenticular
CSection
CDomain
*
CInterpolation
CYarnsection
* CYarn
CYarnsectionconstant
CTextileweave2D
8.2 UML class diagram for the Core module.
CSectionbezier
CYarnsectioninterpposition
CYarnsectioninterp
CTextileweave3D
CTextileweave
CTextile
CTexgen
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Renderer
The renderer module allows for 3D visualisation of the model. This is achieved using Visualization Toolkit (VTK) (Schroeder et al., 2007), an open-source package for 3D computer graphics, image processing and visualisation. The TexGen core module can be used with or without the renderer module.
8.2.3
Graphical User Interface (GUI)
The GUI module contains the code for the user interface and depends on wxWidgets (Smart, 2007), an open-source, cross-platform widget toolkit which provides a library of elements for building a graphical user interface. It provides the facility to produce user interfaces for different operating systems from the same initial code. TexGen has currently been implemented in Windows and Linux operating systems. An example screenshot is shown in Fig. 8.3. The GUI provides a means of constructing textiles without the need to be familiar with the underlying modelling theory. A model can be constructed either by creating an empty textile and then using the Modeller menu to create individual yarns, or by using the ‘Weave Wizard’, which provides a more automated method of generating textiles.
8.3 Screenshot of TexGen Windows user interface.
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The initial weave wizard dialog allows input of parameters. The number of warp and weft yarns and their spacing are input, and additional options may be selected, including the ‘refine’ utility which provides intersection correction for 2D weave models (see Section 8.3.5) and also the option to create a 3D weave. The following weave wizard dialog allows for the specification of weave patterns. In the case of 2D weaves, crossovers can be specified as ‘up’ or ‘down’ simply by clicking on them. Widths, heights and spacing of individual yarns may also be specified at this stage. Specifying the weave pattern for 3D textiles is more complex, as is predicting the actual behaviour of yarns within these textiles, particularly through thickness binder (or warp weaver) yarns. The implication of this is that, even if the up/down pattern of yarns is known, the final positions of those yarns within the final fabric may be different from those suggested simply by the weave pattern. Currently a cross-section is selected and then the yarns can be dragged vertically to define a weave pattern, as shown in Fig. 8.4. Extra layers of yarns may also be added at this point. Further work is underway on the 3D wizard to improve its user-friendliness and to further implement geometric ‘rules’ for the behaviour of yarns in 3D textiles.
8.4 Weave wizard weave pattern dialog.
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Export
The export module allows export to file formats widely used by CAD systems. These include Initial Graphics Exchange Format (IGES) and Standard for the Exchange of Product model data (STEP). The module uses another third-party module, OpenCascade (OPEN CASCADE S A S, 2010), to achieve this.
8.2.5
Python interface
Three wrapper modules provide Python interfaces to the core, renderer and export modules described above. These are generated automatically by SWIG (Beazley and Matus, 2007), another open-source library, which wraps the original C++ functions and classes, making them available to be called from Python scripts. These functions can be utilised in stand-alone scripts run independently of the GUI or in scripts run from the GUI, or they can be called from the Python console within the GUI. There is also a facility within the GUI to record the executed operations as a script which can then be rerun as required.
8.2.6
TexGen usage
TexGen can be downloaded from the Sourceforge host website either as a bundled version with Python interpreter included or as a standard installation package which assumes that Python is already installed on the destination computer. If a user wishes to build the package from the source code then it can be downloaded from the SVN repository. These options give users the maximum flexibility in how TexGen can be used: either the package can be used in its released form or it can be built and adapted to suit a user’s own requirements. The TexGenCore.dll can also be included in C++ programs, enabling the TexGen functions to be called directly from a user’s own program. A simple Python script to create a plain weave textile is shown below using the specialised CTextileWeave2D class. The resulting weave is shown in Fig. 8.5. # Create a 4x4 2d woven textile with yarn spacing of 5 and thickness 2 Textile = CTextileWeave2D(4, 4, 5, 2); # Set the weave pattern Textile.SwapPosition(3, 0); Textile.SwapPosition(2, 1); Textile.SwapPosition(1, 2); Textile.SwapPosition(0, 3); # Adjust the yarn width and height Textile.SetYarnWidths(4); Textile.SetYarnHeights(0.8); # Add the textile AddTextile(Textile) © Woodhead Publishing Limited, 2011
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8.5 Twill weave created from Python script.
8.3
Modelling theory
The core of the TexGen program has been written to give maximum flexibility to the textile model that can be produced, thus allowing accurate modelling of as wide a range of textiles as possible. The geometry of any textile fabric is generated in a generic way by independent specification of yarn path and yarn cross-sections. This approach allows easy modelling of any textile fabric structure, e.g. woven, knitted, braided and non-crimp (Fig. 8.6). TexGen was written primarily to enable textiles to be modelled at the level of the unit cell. Due to the flexibility of the model, however, both larger and smaller sections of the fabric can be modelled as required. The fabric is built up from a number of yarns brought together to form a self-supporting structure. The yarns are modelled as solid volumes which bound the fibre bundle and, for each yarn, the smallest repeating section is specified. A domain can then be specified for the fabric to give the area to be investigated. In many cases this will correspond to the textile unit cell, but this is not a constraint as more or less than one repeating unit can also be specified by the domain. A brief outline of the theory used to create the model is given in the rest of Section 8.3. For a comprehensive description the reader is referred to Sherburn (2007).
8.3.1
Yarn path
The yarn path is modelled by specifying its centre line, defined by a position in three-dimensional space as a function of the distance along the yarn. The smallest repeatable section of the yarn is specified by means of a series of points along its length, known as master nodes, and an interpolation function to give the exact path between those nodes. The interpolation function must give at least C1 continuity. That is, there
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8.6 Variety of fabric geometries generated by TexGen.
should be no gaps in the yarn path and the tangent should vary smoothly. Splines are used to address this criterion and in TexGen three spline interpolation functions are available: cubic Bezier, natural cubic and periodic cubic. A simple linear interpolation function gives a fourth option. In its most general form a polynomial spline, S, is represented by the set of equations
S(t) = S0(t), if t0 ≤ t < t1
S(t) = S1(t), if t1 ≤ t < t2
.....
S(t) = Sk–2(t), if tk–2 ≤ t ≤ tk–1
8.1
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The k given points ti are called knots. The splines used within Texgen are all polynomials of degree 3. The natural and periodic cubic splines both have continuity C2 while the Bezier generally does not. A detailed analysis of the spline functions as used in Texgen is given by Sherburn (2007).
8.3.2
Cross-sections
The yarn cross-section is defined as the 2D shape of the yarn when cut by a plane perpendicular to the yarn path tangent. As previously stated, yarns are treated as solid volumes and the cross-section is approximated to be the smallest region that encompasses all of the fibres within the yarn (generally this will be convex). The cross-section may be constant along the entire length of the yarn, specified at selected points along the yarn or specified at the master nodes. An interpolation function, either smooth or polar, is selected to specify how the shape changes from one section to another. This will be described in more detail in the next section. There are five options available to describe cross-sections: ellipse, power ellipse, lenticular, hybrid and polygon. The last two of these give the most flexibility to the shape, which is particularly useful where yarns are in contact at cross-sections, causing the yarn shape to be deformed and necessitating modifications to the cross-section in order avoid intersections between yarns in the model. Elliptical, lenticular, power ellipse and hybrid cross-sections can be selected from both the gUI and scripts. The polygon cross-section can be accessed via the API, allowing individual points to be specified around the circumference of the cross-section. The ellipse is the simplest cross-section available, simply specified by width, w, and height, h. In the power ellipse the elliptical cross-section is slightly modified, assigning a power n to the y-coordinate such that the section resembles a rectangle with rounded edges when n < 1 or a shape similar to a lenticular cross-section when n > 1. The x-and y-coordinates are defined by the following parametric equations: C (t )x = w cos (22 p t ) 0≤t ≤1 2 ÏÔ h2 (sin (2 p t ))n if 0 ≤ t ≤ 0.5 C (t )y = Ì h n ÔÓ– 2 (– sin (2 p t )) if 0.5 ≤ t ≤ 1
8.2
Two examples with different values of n are shown in Fig 8.7. The lenticular cross-section is described by the intersection of two circles of radii r1 and r2 each offset vertically by distances o1 and o2 respectively.
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n =1 2
h 2 y
0
n=2
–h 2w – 2
0 x
w 2
8.7 Power elliptical cross-sections. h 2 y 0 –h 2 h 2 y 0 –h 2w – 2
d=0
d
d =h 4 0 x
w 2
8.8 Lenticular cross-sections.
A detailed description of these is given by Sherburn (2007). Two typical sections are shown in Fig. 8.8. The hybrid section utilises any combination of the ellipse, power ellipse and lenticular sections. As shown in Fig. 8.9, different cross-sections can be assigned to a selected sector of the cross-section. The user interface gives interactive display of the currently selected section, allowing the boundary between sections to be adjusted to give smooth transition between sections. The polygon cross-section gives a high degree of flexibility, allowing individual points to be specified around the outline of the section. This section cannot be accessed via the user interface but is called using functions in the API (from either a C++ program or a Python script). It is used in the ‘refine section’ option where the points around the sections are moved in order to reduce the yarn interference, as shown in Fig. 8.10. Further flexibility can be achieved by utilising functions which allow cross-sections to be scaled and rotated. Rotation is particularly useful to create realistic models at yarn crossovers where a yarn may rotate to accommodate crossing yarns, as can be seen in Fig. 8.10.
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8.9 Selection of sections for hybrid cross-sections.
8.10 Example hybrid cross-sections.
8.3.3
Interpolation between yarn sections
As described in Section 8.3.2, the interpolation function describes the way in which the cross-section varies along the length of the yarn. Yarns are easily deformed even under low loads and, typically, woven yarns will be compacted at crossover points. The ability to create smooth transitions between the cross-sections at different points on the yarn is therefore essential. The simplest approach is of course to model yarns with an idealised, constant cross-section. This allows models to be created quickly for simple textiles with little yarn deformation. Where cross-sections have been specified at defined positions along the yarn they are then interpolated between these points. For two cross-sections A(t) and B(t) the intermediate cross-section C(t) is defined as:
C(t, m) = A(t) + (B(t) – A(t)) m 0 ≤ t ≤ 1, 0 ≤ m ≤ 1
8.3
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where m varies linearly with distance between cross-sections A(t) and B(t). This method gives a smooth transition with C0 continuity between crosssectional shapes. It requires that A(t) and B(t) describe similar positions on the cross-section for all values of t. The linear interpolation can also be modified with a cubic function to give C1 continuity. Fig 8.11 shows an example of a yarn with varying cross-section.
8.3.4
Yarn repeats and domain
For each yarn the smallest repeating section of yarn is given and then repeats are specified, giving a potentially infinite textile. Typically a textile will have two repeat vectors (x, 0, 0) and (0, y, 0) as shown in Fig. 8.12. Having specified repeat vectors, thus giving a potentially infinite textile, the area to be modelled is given by selection of a domain. This is described by a set of planes forming a convex polygon. Typically this will be a rectilinear box with six planes but it may be any shape, e.g. trapezoidal for a sheared fabric. A typical domain may specify a unit cell, but the method used allows for specification of any area, both smaller and larger than the unit cell, as required.
8.11 Yarn formed with varying cross-section.
C0 = –1 C1 = 1
C0 = 0 C1 = 1
Æ
C0 = –1 C1 = 0
R1
P C0 = 0 C1 = 0
C0 = 1 C1 = 1
C0 = 1 C1 = 0
Æ
R0
8.12 Yarn repeated with two repeat vectors.
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8.3.5
Intersections
The method of modelling textiles used in TexGen can lead to yarn intersections (i.e. part of one yarn volume intersecting with or penetrating either a crossing or a parallel yarn volume). It is easy to visualise where the yarn centre lines will intersect but not so straightforward to predict where intersections will occur once the cross-sections have been applied to those yarn paths, particularly where the cross-section changes along the length of the yarn. The problem can be addressed in two stages: firstly by identifying where the intersections occur, and secondly by modifying the textile in order to eliminate or reduce the intersections to an acceptable level. The TexGen GUI allows visual identification of areas within a textile where intersections occur, indicating both their location and severity (depth of the intersections). The maximum intersection depth can also be determined. To achieve this, points on the surface meshes are interrogated using a PointInsideYarn function which determines whether those points are inside any of the yarn volumes. When the function is called from the GetPointInformation function it also returns information about which yarn the point is inside, the yarn tangent at that point and the yarn volume fraction at that point. Several strategies have been implemented to address the issue of yarn intersections. The strategy adopted may depend on the size of the intersections and also how far the textile is along the modelling and output process. Using the TexGen GUI a woven fabric can be automatically created as described in Section 8.2.3. In the first instance this is created with elliptical cross-sections. The problem in this case is that intersections often occur in all but the very simplest weaves and the elliptical sections do not accurately reflect the true geometry of the textile. Sherburn (2007) describes a method for refining a yarn mesh by first adjusting the yarn width and rotation at the crossover points in the weave. Subsequent adjustment of the yarn crosssection by taking advantage of the flexibility of the polygon cross-section gives further refinements which minimise the intersections. This model was validated for Chomarat 800S4-F1 satin weave fabric. Figure 8.13 shows a comparison of the final refined model with the actual textile. There are some cases where small intersections in the model may be acceptable. If the model is to be discretised directly within TexGen (see
8.13 Comparison of actual textile and model corrected using the ‘refine model’ option.
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Section 8.4), it may be exported as either a tetrahedral or a voxel mesh. In both cases the mesh is adapted to eliminate intersections as it is generated. This does lead to some loss in yarn volume, which may affect the accuracy of (for example) Computational Fluid Dynamics (CFD) simulations to predict fabric permeability. To address this, a method has been developed which is suitable for making fine adjustments to the dry fibre volume mesh. The method is based on interference depth and an iterative method is used to adjust the node position in the volume mesh until the intersection is reduced to a given tolerance. The local volume fraction of the yarn is recalculated at the adjusted points to take into account the local change in cross-section. Figures 8.14 and 8.15 show solid and mesh views of a small section of intersecting yarns with Fig. 8.15 showing the rendered intersection points. The fine adjustments made to the final volume mesh are shown in Fig. 8.16.
8.3.6
Yarn properties
Yarn properties can be assigned to each yarn as follows: yarn linear density, fibre density, fibre diameter, fibres per yarn, Young’s modulus and Poisson’s ratio. These are currently used in the calculation of the fibre volume fraction and within the export functions to calculate yarn or composite properties as required for the relevant model.
Z Y
X
8.14 Section of textile with intersections.
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Y
Z
X
8.15 Surface mesh showing intersection depths.
8.16 Volume mesh adjusted for small intersections.
8.4
Rendering and export of model
Once a model has been specified as described in Section 8.3, it can be meshed in order both to visualise the textile and to enable it to be output in various formats as required for further processing. This may also lead to the requirement for adjusting the model or the mesh in order to avoid intersections between the yarns. The magnitude of the intersection will dictate the stage at which the adjustment is made.
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Surface mesh
The easiest way to form a surface mesh which can then be rendered is to represent it by a series of polygon elements. In order to generate this surface, mesh slave nodes are generated between the master nodes along the length of the yarn. By forming a cross-section at the slave node, perpendicular to the yarn tangent at that point and interpolating as described in Section 8.3.3, a set of node points are formed which can then be assembled into quadrilateral elements generated by joining section points from the crosssections at adjacent slave nodes. To render the surface mesh the elements are passed to the renderer functions to be displayed as either a solid or a mesh. The surface generated can also be used for checking whether a given point is inside the yarn as described in Section 8.3.5. Figure 8.17 shows a yarn rendered in ‘X-ray mode’, showing the mesh elements.
8.4.2
Dry fibre volume mesh
Volume meshes are created in two stages. Firstly a two dimensional mesh is created at each slave node using a rectangular/triangular mesh generation technique as illustrated in Fig. 8.18. The mesh points between subsequent sections are then joined to form hexahedral and wedge elements as illustrated in Fig. 8.18. This mesh may be exported to external packages for further
8.17 Surface mesh for single yarn. 1
2
3
4
5
6
7
8
0 1 2 3 4
8.18 Cross-section meshed with rectangular/triangular mesh generating technique.
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8.19 Volume mesh for single yarn.
processing and is used when exporting the dry fibre volume mesh for mechanical analysis. When the volume mesh is exported as an Abaqus input file, the textile is interrogated to find the yarn tangents and volume fractions at the centre points of the elements. Textiles defined using the CTextileWeave class use the information available to define contact pairs at the yarn crossover regions. Material properties will also be set, taken either from the overall yarn properties or from those set for individual yarns within the textile.
8.4.3
Volume mesh of yarns and matrix
Where both the yarns and resin matrix are required to be meshed, a tetrahedral mesh is created. First the separate yarn areas are projected onto a 2D surface (Fig. 8.20). Each area is triangulated using open-source software Triangle (Shewchuk, 2005). These triangles are then projected in the z-direction through the thickness to create elements with boundaries being created at the yarn/matrix transitions. The resulting wedge shapes are then further subdivided to give tetrahedral mesh elements. Views of the resulting yarn and combined yarn/matrix meshes are shown in Figs 8.21 and 8.22. This method is successful for two-dimensional weaves but less so for weaves containing vertical or near-vertical yarns, as the method of projection in the z-direction gives misshaped elements in these cases. This is an area for further development, with external meshing techniques based on an exported geometric model (IGES, STEP) preferred at present.
8.4.4
Voxel mesh
Generation of conformal meshes as described above can be challenging for textile geometries. In particular, close to yarn crossovers it is often difficult or impossible to generate elements of acceptable quality that conform to the local yarn surfaces. Here a voxel mesh may be preferred, which is formed by dividing the domain area into cuboids or rectangular bricks. By interrogating the model at the centre point of each cuboid, the volume fraction and yarn
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8.20 Triangulated yarn areas for plain weave.
8.21 Tetrahedral yarn mesh for plain weave.
direction can be obtained for each element and output with the position information to an input file for subsequent analysis, selecting yarn, matrix or both to be output. Figure 8.23 shows an example of a voxel mesh.
8.5
Applications
TexGen has been used in a variety of different applications, some of which are outlined in the following sections.
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8.22 Tetrahedral yarn/matrix mesh for plain weave.
8.23 Example of voxel mesh for plain weave.
8.5.1
Textile mechanics
Analysis of textile mechanics can provide materials data to predict forming and compaction behaviour for dry textiles and pre-impregnated fabrics. Accurate models can be created when detailed measurements taken from actual fabrics are used as input data. An example of this is described by Lin et al. (2009a) where models of commercial fabrics are created using accurate input information and results of subsequent validation studies are shown. The application of the TexGen model to mechanical modelling of a commercial polyester/cotton plain weave fabric is described, with the yarn being meshed and exported to the Abaqus finite element analysis (FEA) package as described in Section 8.4.2. Deformations were predicted for the plain weave unit cell in tension, compression, shear and bending, utilising their measured equivalents for individual yarns as input data. The predictions were
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then validated against experimental data for fabric samples, which showed that the model was able to represent the fabric behaviour very accurately, particularly during tensile and shear loading. The results from this study provide a fundamental understanding of textile deformation and show that predictions based on TexGen models can be used as inputs to simulate complex deformation of fabrics in actual applications. A further study by Lin et al. (2009b) describes the development of a TexGen model for fabric shear behaviour. The fabric modelled was a Chomarat 150TBN plain woven E-glass. The fabric unit cell and corresponding TexGen model are shown in Fig. 8.24. A Python script was developed which read the TexGen unit cell geometry, retrieved data defining the outer surfaces of the yarns, trimmed the yarns to a single unit cell, output material boundary conditions and contacts between yarns, and also generated a mesh. The study concentrated on the accurate specification of boundary conditions for the accurate prediction of deformation mechanisms and shear force. Two boundary conditions were used: pure rotation and unified displacementdifference. When compared with experimental data (Fig. 8.24) it could be seen that the simple rotation boundary condition was adequate for predicting shear force at large deformations where most of the energy is dissipated at higher shear angles due to yarn compaction. The second boundary condition improved the shear force prediction at low shear angles where the energy dissipated by both shearing of the unit cell and relative inter-yarn motion at crossovers and intra-yarn shearing were modelled. The study concluded that for small deformations the fabric should be modelled at meso-scale for yarn rotation and micro-scale for fibre slippage within the yarn. Accurate prediction at this level is of significant importance for clothing in order to predict comfort, wear and aesthetics. 0.25 EXP FE result
Shear force (N/mm)
0.2
0.04
0.15
0.03 0.02
0.1
0.01 0
0.05 0
0
0
10
20
20
40
60
30 40 Shear angle (°)
50
60
70
8.24 Chomarat150TB (top left), modelled unit cell (bottom left) and validation of finite element prediction of shear force versus shear angle (rotation boundary condition, right).
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8.5.2
Textile composite mechanics
Extensive use has been made of TexGen to predict mechanical properties of textile composites, including elastic and failure behaviour. Models generated are used to permit FEA of the repeating unit cell so that effective macroscopic properties of the composite can be determined. Voxel meshes generated from TexGen models have been used as input for FE analysis using Abaqus. The technique was modified to incorporate adaptive mesh refinement (AMR) such that localised mesh refinement occurs around material boundaries and other stress concentrations while leaving coarse meshes in non-critical regions (Crookston et al., 2006). This method was further developed by Ruijter (2008) to automatically link directly into TexGen at each step to interrogate for orientation and local volume fraction data for use in the next iteration during mesh refinement. The method was demonstrated to give satisfactory agreement between predictions and experimental data for stiffness and strength for selected woven composites loaded in both the fibre and bias directions. An example is shown in Fig. 8.25 for the case of bias direction loading. Here a simple continuum damage model is used to degrade elastic properties for failed elements. The example also shows the effect of modelling nesting, which results in a more accurate representation of stress–strain behaviour after initial failure. 0.12
Stress (GPa), volume averaged
0.10 Experimental data 0.08
0.06
0.04 Single-layer model
0.02
0.00 0.00
0.02
0.04
Nested twolayer model
0.06 0.08 0.10 Strain (–), volume averaged
55539 dof 7803 dof 19008 dof 0.12
0.14
8.25 Finite element analysis with continuum damage model for ±45° plain weave glass/polyester composite.
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Smitheman et al. (2009) describe a voxel-based homogenization technique for predicting warpage of a textile-reinforced composite during manufacture. TexGen is used to provide the geometric model of the representative volume element (RVE) which is then divided into voxels. The composite is modelled as being homogeneous in each voxel. Subdivision into subcuboids and interrogation by TexGen give the local fibre volume fraction, from which effective thermomechanical properties are determined using micromechanics. The thermomechanical properties of the bulk composite are then obtained using FEA based on the RVE. On application of appropriate periodic boundary conditions, the method was shown to give good agreement with experimental data for a carbon/epoxy composite. This work has been modified and extended to sheared textile composites (Smitheman et al., 2010). A rigorous framework for applying periodic boundary conditions to the unit cells is described, with predictions in very good agreement with experimental measurement achieved.
8.5.3
Textile permeability
TexGen has been used to generate models to predict reinforcement permeability using a variety of techniques. Broadly these aim to predict in-plane and/or through-thickness permeability as input data for macro-scale models of resin infusion during composites manufacture. Early work using TexGen utilised commercial CFD software to relate fluid velocity to pressure drop for steady-state flow through a textile model (Robitaille et al., 2003). Several problems were encountered with this method, the main one being that it was not always possible to generate a conformal CFD mesh using commercial software. In light of this a number of approaches have been developed to simplify the flow domain (Wong, 2006). In the 2D grid average method the flow domain is collapsed onto a 2D grid, with each node assigned a thickness-weighted average permeability based on the values for the individual yarns and ‘gaps’ that occur through the thickness. This method results in very fast computation times (consistently less than 30 s) compared to 3D CFD approaches. The 2D grid average approach was compared to commercial CFD for a fairly open textile structure, providing predicted permeabilities typically within 30% of each other (Wong et al., 2006). The approach was then used to assess effects of fabric architecture, particularly intra-ply shear (from draping) and variability in tow path on permeability (Wong and Long, 2006). The latter allowed predicted permeability distributions to be produced using a stochastic (Monte Carlo) approach, thus allowing variability in flow to be simulated. More recently a voxel mesh approach has been taken for flow simulation of 3D textile fabrics facilitated by the software to export voxel meshes to Ansys CFX CFD software (Zeng et al., 2010). Figure 8.26 shows the textile model
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analysed, and also compares predicted permeabilities along warp and weft to experimental data. Using the average values for width and height of warp, weft and binder yarns gives reasonably close agreement with experimental data. Predictions based on the lower bounds from measured dimensions of all yarns (corresponding to a reduction in around 10% in yarn width and height) lead to a doubling in the predicted permeabilities. This illustrates the sensitivity of permeability predictions to small changes in the geometric model, indicating the importance of accuracy in geometric parameters for flow simulation.
8.6
Future trends
Predicted permeability (10–9m2)
Clearly the accuracy in predicting the properties of textiles and their composites depends on the accuracy of the fabric geometric model. This appears to be particularly true for fluid flow problems as illustrated by the last example on CFD modelling. Hence one of the key areas for current and
2.5E–09
Weft
Warp
2.0E–09 1.5E–09 1.0E–09 5.0E–10 0.0E+00
Experiment
CFD average CFD lower CFD lower CFD lower yarn cross- bound binder bound bin- bound all yarn section yarn cross- der and warp cross-section section wadding yarn cross-section
8.26 Computational fluid dynamics predictions (right) for in-plane permeabilities of angle-interlock 3D woven carbon fabric model (left).
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future development of TexGen is further improvement of model accuracy. Broadly speaking, models generated automatically by TexGen are very accurate for 2D weaves, whereas accurate modelling for 3D weaves requires fairly extensive experimental characterisation of the textile geometry. The ultimate goal is to be able to predict the textile geometry with sufficient accuracy based on simple geometric parameters easily determined by the fabric manufacturer. The issue of yarn intersection is one which is often present when modelling densely packed fabrics which are typical in composites applications. In some cases these intersections can be ignored or removed artificially during discretisation as described in Section 8.3.5. Better understanding of the way in which yarns interact, when they move, when they deform and in what way, will allow for improvement of the textile model. Such understanding is now available routinely for 2D textiles via the ‘refine model’ option. Current work is focused on developing a similar methodology for 3D textiles. Another issue which reflects the realism of the textile model is that of variability. A unit cell may be accurately modelled for one repeat of the textile but in reality there will be small variations in this due to yarn waviness as it is repeated to form a textile. Variation may also occur due to layer placement in laminates (nesting), which can significantly affect the mechanical properties of the material. This is an area for further research and possible developments within TexGen. Whilst this has already been addressed for the example of flow (Wong and Long, 2006), the models analysed had a relatively large fibre spacing to avoid problems of yarn intersections. Hence development of improved intersection correction/avoidance algorithms should allow a more thorough analysis of variability. Developments in this area should allow modellers to develop a fundamental understanding of the sources of variability and of their relative influence on the distribution of textile composite properties.
8.7
References
Ainsworth H (2007), Octree C++ General Component. Available from http://www.hxa. name/articles/content/octree-general-cpp_hxa7241_2005.html (accessed 23 October 2010). Beazley D and Matus M (2007), SWIG. Available from http://www.swig.org (accessed 23 October 2010). Crookston J J, Ruijter W, Long A C and Jones I A (2006), Modelling mechanical performance including damage development for textile composites using a gridbased finite element method with adaptive mesh refinement, Proceedings of the 8th International Conference on Textile Composites (TexComp 8), Nottingham, October 2006, Paper T09. Lin H, Sherburn M, Long A C, Clifford M J, Jones I A and Rudd C (2009a), Modelling and simulations of fabrics using TexGen, Proceedings of the Fibre Society’s Spring 2009 Conference, Shanghai.
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Lin H, Clifford M J, Long A C and Sherburn M (2009b), Finite element modelling of fabric shear, Modelling and Simulation in Materials Science and Engineering, 17(1). OPEN CASCADE S A S (2010), OpenCASCADE. Available from http://www.opencascade. org (accessed 23 October 2010). Robitaille F, Clayton B R, Long A C, Souter B J and Rudd C D (1999), Geometric modelling of industrial preforms: Woven and braided textiles, Journal of Materials: Design and Applications, Proc. Inst. Mech. Eng. (Part L), 213, 69–84. Robitaille F, Clayton B R, Long A C, Souter B J and Rudd C D (2000), Geometric modelling of industrial preforms: Warp-knitted and multiple layer textiles, Journal of Materials: Design and Applications, Proc. Inst. Mech. Eng. (Part L), 214, 71–90. Robitaille F, Wong C C, Long A C and Rudd C (2003), Systematic predictive permeability modeling using commercial CFD and dedicated calculation method, Proceedings of the 14th International Conference on Composite Materials (ICCM-14), San Diego, July 2003. Ruijter W (2008), Analysis of mechanical properties of woven textile composites as a function of textile geometry, PhD thesis, University of Nottingham. Schroeder W, Martin K and Lorensen B (2007), Visualization Toolkit. Available from http://www.vtk.org (Accessed 23 October 2010). Sherburn M (2007), Geometric and mechanical modelling of textiles, PhD thesis, University of Nottingham. Shewchuk J R (2002), Delaunay refinement algorithms for triangular mesh generation, Computational Geometry: Theory and Applications, 22(1–3). Shewchuk J R (2005), Triangle: A two-dimensional quality mesh generator and Delaunay triangulator. Available from http://www.cs.cmu.edu/~quake/triangle.html (Accessed 23 October 2010). Smart J (2007), wxWidgets. Available from http://www.wxwidgets.org (accessed 23 October 2010). Smitheman S A, Jones I A, Long A C and Ruijter W (2009), A voxel-based homogenization technique for the unit cell elastic and thermoelastic analysis of woven composites, Proceedings of the 17th International Conference on Composite Materials (ICCM17), Edinburgh, July 2009. Smitheman S A, Fontana Q, Davies M G, Li S, Jones I A, Long A C and Ruijter W (2010), Unit cell modelling and experimental measurement of the elastic and thermoelastic properties of sheared textile composites, Proceedings of the 10th International Conference on Textile Composites (TexComp 10), Lille, France, October 2010. TexGen (2010), TexGen. Available from http://www.texgen.sourceforge.net (accessed 23 October 2010). Thomason L (2007), TinyXml. Available from http://www.grinninglizard.com/tinyxml (accessed 23 October 2010). Wong C C (2006), Permeability prediction for multi-layer textile preforms, PhD thesis, University of Nottingham. Wong C C and Long A C (2006), Modelling variation of textile fabric permeability at mesoscopic scale, Plastics Rubber and Composites, 35(3), 101–111. Wong C C, Long A C, Sherburn M, Robitaille F, Harrison P and Rudd C D (2006), Comparisons of novel and efficient approaches for permeability prediction based on the fabric architecture, Composites Part A, 37(6), 847–857. Zeng X S, Endruweit A, Long A C and Clifford M J (2010), CFD flow simulation for impregnation of 3D woven reinforcements, Proceedings of the 10th International Conference on Textile Composites (TexComp 10), Lille, France, October 2010.
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9
In-plane shear properties of woven fabric reinforced composites
J. C a o, Northwestern University, USA, J. C h e n, University of Massachusetts at Lowell, USA and X. Q. P e n g, Shanghai Jiao Tong University, P. R. China
Abstract: This chapter discusses the in-plane shear properties of textile composites. Shear deformation is the dominate deformation mode for woven fabrics in forming, therefore trellis-frame (picture-frame) and bias extension tests for both balanced and unbalanced fabrics have been conducted and compared. Tests were conducted by seven international research institutions on three identical woven fabrics. Both the variations in the setup of each research laboratory and the normalization methods used to compare the test results are presented and discussed. With an understanding of the effects of testing variations on the results and the normalization methods, numerical modeling efforts can commence and new testing methods can be developed to advance the field. Key words: woven fabric, mechanical properties, shear, testing.
9.1
Introduction
Woven-fabric reinforced composites (hereafter referred to as textile composites) have attracted a significant amount of attention from both industry and academia due to their high specific strength and stiffness, as well as their supreme formability characteristics. Woven fabrics are created by weaving yarn into a repeating pattern. Yarn is made of continuous or stretchable fibers with diameters typically on the order of micrometers or microns (mm). The manufacture of components from woven fabrics involves a forming stage in which the fabric is deformed into a desired shape either (a) by a punch with the fabric being subjected to a binder holding force; or (b) by machine or manual laying up where the fabric can be subjected to either complex edge stretching or no stretching. This step can be performed at room temperature for dry fabric or at elevated temperatures (i.e., thermoforming) for fabrics made of glass/carbon fibers commingled with thermoplastic fibers. The formed fabric can further be injected with resin and consolidated in a resin transfer molding (RTM) process or a liquid composite molding (LCM) process.[1–3] Commercial applications for textile composites include products for energy 267 © Woodhead Publishing Limited, 2011
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absorption (e.g., helmets),[4] aerospace and defense applications (e.g., engine inlet cowlings, fuselage sections, rotor blade spars and fuel pods),[5] and automotive and structural applications (e.g., battery trays, seat structures, front end modules and load floors).[6] However, robust process simulation methods and adaptive design tools are needed to shorten the design cycle and reduce manufacturing cost, creating reliable final products and expanding the applications of textile composites. Some examples of the fundamental questions from practitioners are: (1) Is it possible to form to a specific threedimensional geometry without wrinkles or fiber breakage? (2) What are the final fiber orientations? (3) What is the final fiber distribution? and (4) What process parameters should be used to form such a part? Recognizing these requirements, a group of international researchers gathered at the University of Massachusetts Lowell for the Workshop on Composite Sheet Forming sponsored by the US National Science Foundation in September 2001. The main objectives of the workshop were to better understand the state-of-the-art and identify existing challenges in both materials characterization and numerical methods required for the robust simulations of forming processes. One direct outcome of the workshop, and the effort to move towards standardization of material characterization methods, was a web-based forum exclusively for research on the forming of textile composites, which was established in September 2003.[7] Other outcomes of the workshop are in the form of publications,[8–57] such as this one, highlighting recommended practices for experimental techniques and modeling methods. Standard material testing methods are necessary for researchers to understand the formability of the material and the effects of process variables on formability, and to provide input data and validation data for numerical simulations. Thus, international researchers embarked on a benchmarking project to comprehend and report the results of material testing efforts currently in use around the world for textile composites to make recommendations for best practices. Three different commingled fiberglass–polypropylene woven-composite materials were used in this collaborative effort. The materials were donated by Vetrotex Saint-Gobain and were distributed to the following research groups: Hong Kong University of Science and Technology (HKUST) in Hong Kong, Katholieke Universiteit Leuven (KUL) in Belgium, Laboratoire de Mécanique des Systèmes et des Procédés (LMSP), INSA-Lyon (INSA) in France, Northwestern University (NU) in the USA, University of Massachusetts Lowell (UML) in the USA, University of Twente (UT) in the Netherlands, and University of Nottingham (UN) in the UK. Intra-ply shear is the most dominant deformation mode in woven composite forming, therefore the trellis‑frame (picture-frame) test (Fig. 9.1) and the bias extension test (Fig. 9.2) were identified for material shear-property characterization. A summary and comparison of the test methods and the findings from all participating research groups will be presented. The rest
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Crossheand mount
Sliding link
Toggle clamps
B
269
Lf
Fabric
C
A F Amplifier
La (a)
(b)
9.1 Trellising-shear test apparatus: (a) starting position; (b) deformed position. (Note that the top hinge has traveled from the bottom of the slot in the undeformed position (a) to the top of the slot in the deformed position (b).)
Hydraulic grips Fabric sample
Attachment points for a suitable loading frame
9.2 Bias extension test apparatus.
of the chapter is organized as follows: a summary of the properties of the materials used in this study is presented in Section 9.2, trellis-frame setups at each participating institution and data processing procedures are discussed in Section 9.3, and trellis-frame test results and the normalization analysis
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are presented in Section 9.4. Similarly, the experimental setup procedure and analytical calculation of bias extension are presented in Section 9.5, while Section 9.6 has the results from the bias extension tests. Finally, the comparison of normalized shear force vs. shear angle obtained from trellis-frame and bias extension tests are compared in Section 9.7, followed by a discussion on how the data from these tests can be used to advance the benchmarking effort related to numerical modeling of thermostamping simulations.
9.2
Fabric properties
The three types of woven fabrics were donated by Vetrotex Saint‑Gobain (Fig. 9.3). The fabric properties are listed in Table 9.1. Each fabric comprises yarns with continuous commingled glass and polypropylene (PP) fibers. These fabrics were chosen because of their ability to be formed using the thermostamping method.
9.3
Experimental setups of the trellis-frame test
A trellis frame, or picture frame, shown in Fig. 9.1, is a fixture used to perform a shear test for woven fabrics.[58–60] In Fig. 9.1, a fabric sample was loaded in the picture frame and was shown both in the starting position and also in the deformed position. Using this test method, uniform shearing of the majority of the fabric specimen is obtained. Displacement and load data are recorded to aid in the characterization of pure shear behavior. In the picture frame tests, the fabric sample is initially square and the tows are oriented in the 0/90 position to start the test. By using varying procedures, researchers could study and recommend methods for data comparison. Table 9.2 shows the fabrics tested by each participant. All researchers reported load histories and global shear-angle data for picture-frame tests conducted at room temperature.
Plain weave
Balanced twill weave
Unbalanced twill weave
9.3 Woven fabrics used in this study.
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Table 9.1 Fabric parameters (as reported by the material supplier unless specified otherwise) Manufacturer’s stylea
TPEET22XXX
Plain Weave type Glass/PP Yarns Plain Weave 743 Area density, g/m2 Yarn linear density, tex 1870 Thickness, mmb 1.2 (NU)
TPEET44XXX
TPECU53XXX
Balanced twill Glass/PP Twill 2/2 1485 1870 2.0 (NU)
Unbalanced twill Glass/PP Twill 2/2 1816 2400 3.3 (NU)
Yarn count, picks/cm or ends/cm:
Warp
1.91 (KUL) 1.93 (HKUST) 1.95 (NU)
5.56 (KUL)
3.39 (KUL)
Weft
1.90 (KUL) 1.93 (HKUST) 1.95 (NU)
3.75 (KUL)
1.52 (KUL)
Yarn width in the fabric, mm:
Warp
4.18 ± 0.140c (KUL) 1.62 ± 0.107c (KUL) 2.72 ± 0.38c (KUL) 4.20 (HKUST) 4.27 (NU)
Weft
4.22 ± 0.150c (KUL) 2.32 ± 0.401c (KUL) 3.58 ± 0.21c (KUL) 4.20 (HKUST) 4.27 (NU)
a b c
Designated by Twintex. ASTM Standard D1777 (applied pressure = 4.14 kPa). Standard deviation. Table 9.2 Tested fabrics in picture-frame tests used by participating researchers Group
Plain weave
Balanced twill weave
Unbalanced twill weave
HKUST KUL LMSP UML UT UN
Y Y Y Y Y Y
N Y N Y Y N
N Y N Y N Y
Y = data reported; N = data not reported.
9.3.1
Frame design and clamping mechanism
Figure 9.4 illustrates the different frame designs used in five research groups. Although the frames used are not identical, all of them have common features. For example, the corners of each frame are pinned. When the fabric is loaded
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Hkust
kul
uml
ut
Shear frame
Fabric
Camera
lmsp
un
9.4 Picture frames designed, fabricated and used by the research groups.
into the frame, it is clamped on all edges to prevent slippage. The corners of a sample are cut out to allow the tows to rotate without wrinkling the fabric. Thus, it appears that each sample has four flanges (Fig. 9.4). It was assumed that all clamping mechanisms held the fabric rigidly in the frame and there was no slippage.1 Thus, differences in clamping mechanisms are not taken 1
This assumption was later eliminated in the work of Boisse’s group,[28] in which they found clamping force plays an important role in the measurement of shear behavior using picture frame tests.
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into account in the analysis of the results and the friction effect between the fabric and grips can be ignored. With the fabric properly aligned and tightly clamped in the frame, the distance between two opposing corners is increased with the aid of the tensile testing machine, and therefore, the tows begin to reorient themselves as they shear (Fig. 9.1(b)). Note the inclusion of a sliding slot in the KUL and UML frames. While the shear frames of HKUST, UT and LMSP were displaced at the opposing two joints of their frames, the mechanism by which the fabric deforms is aided by linkages in the frames used by KUL and UML. UML’s linkage was added to allow the frame to displace at a greater speed than that which could be achieved by the tensile test machine alone. It was found that the frame could travel at a rate 4.25 times faster than what was possible through the specified crosshead displacement rate. In addition to amplifying the distance traveled, these linkages amplify the measured force, and this amplification factor must be accounted for when the results from all the groups are analyzed and compared. A detailed discussion of the amplification factor associated with the inclusion of the linkages and the various normalization techniques is included in the discussion of results (Section 9.4). This section focuses on the similarities and differences of the test methods used by each group.
9.3.2
Sample preparation
The sample size is noted as the area of the fabric without the flanges, because this area represents the amount of fabric that is deformed during the test. It is the area that encompasses the tows which must rotate at the crossover points during the test. Table 9.3 lists the frame size and the maximum fabric size in one direction and the testing speeds used by different research groups. In addition to the difference in the sample size, additional differences among the groups were related to sample preparation. For example, to eliminate the potential force contribution from shearing of the yarns in the edge (arm) parts of the sample, HKUST removed all of the unclamped fringe yarns (Fig. 9.5(a)). UT reported that they removed some of the yarns adjacent to the center area of the sample to prevent the material from wrinkling during Table 9.3 Frame size and test parameters Group
Frame (mm)
Fabric (mm)
Speed (mm/min)
Specimen temperature
HKUST KUL LMSP UML UT UN
180 250 245 216 250 145
140 180 240 140 180 120
10 20 75–450 120 1000 100
Room temperature
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(b)
(a)
9.5 Specimens with yarns removed from arm regions: (a) HKUST; (b) UT.
testing (Fig. 9.5(b)). In the previous research by Lussier[43], it was reported that care must be taken not to alter the tightness of the weave or the local orientation of the remaining yarns when removing some yarns prior to testing the fabric. This statement was further supported by HKUST who noted that theoretically in an obliquely oriented or misaligned specimen in the frame, one group of yarns would be under tension while the other would be under compression. Because a yarn cannot be compressed in the longitudinal direction, a misalignment would indicate that the yarn buckles out of the original plane and the onset of wrinkling in the fabric occurs at lower shear angles than when the specimen is properly aligned in the frame. UT terminated their tests at the onset of wrinkling, as the shear deformation is no longer uniform once wrinkling occurs. UML noted that by ‘mechanically conditioning’ the specimen, i.e., by shearing the fabric in the frame several times before starting the test, the variability in tension due to local deviations in orientation could be eliminated. This occurrence indicates the importance of the precise handling of both the sample and test fixture.
9.3.3
Determination of shear angle
Fabric conforms to its final geometry mostly by yarn rotation, i.e., shearing between weft and warp yarns, denoted as the shear angle g. The shear angle g is commonly assigned as zero at the initial stage when weft and warp yarns are perpendicular to each other. We will use this shear angle as a common parameter for comparison in the analysis of tests from different research groups. Figure 9.4 showed photos of the different frames used in various groups. In the labs of HKUST, UT and LMSP, a displacement transducer in the
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tensile machine measures the vertical displacement, d, of point a. Through trigonometric relations, the angle of the frame, q, is calculated: coss q =
2 Lfra frame me + d 2Lframe
9.1
where Lframe is the frame length indicated in Fig. 9.4. The shear angle, g, is calculated from the geometry of the picture frame: g = 90° – 2 q
9.2
This value, g, is also called the global shear angle. Note that this value is taken to be an average shear value over the entire specimen. The actual shear angle at any point on the fabric may vary. For the picture frames used at UML and KUL, respectively, where both had a linkage in their frames, the displacements at point a in the corresponding figures were reported. Therefore, instead of Lframe in eq. 9.1, the length of amplifier link, La, as indicated in Fig. 9.1(a), was used in eq. 9.1 to calculate the angle of the frame. Equation 9.2 applied to all five cases. optical methods which can aid in the determination of the shear angle at any particular point on the fabric specimen also exist. hKUST used a camera to capture arrays of images during the loading process. They then processed these images with autoCaD, as shown in Fig. 9.6. They found that the maximum deviation between the measured shear angle and the calculated shear angle (Eq. 9.2) is about 9.3% and that the maximum deviation typically occurs at larger shear angles. KUL incorporated an image mapping system (aramis) into their experiment. After photos were taken by a CCD camera, displacement and strain fields were identified by the Aramis software by analyzing the difference between two subsequent photos. Figure 9.7 shows a distribution of the total equivalent strain over an image of a fabric sample during testing. By averaging the local shear angles produced by aramis and comparing them with the global shear angles calculated from the crosshead displacement using eqs 9.1 and 9.2, KUL generated the graph in Fig. 9.8, which shows that the shear angle
9.6 Shear angle measurement on a photo (HKUST).
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9.7 Image of the fabric and the central region with the von Mises strain field (KUL). 50
= min max ave
45
Gamma, fabric (°)
40 35 30 25 20 15 10 5 0
0
10
20 30 Gamma, frame (°)
40
50
9.8 Typical relationship between the optically measured shear angle and shear angle of the frame, unbalanced twill weave (KUL).
values obtained using the two methods are comparable. However, for the unbalanced twill weave, the difference becomes larger after a shear angle of 33°. LMSP also incorporated an image mapping system into their experiments. Optical measurements were made with a zoom lens that covered the entire specimen area. A typical measured displacement field is shown in Fig. 9.9 and the equivalent strain field distribution can then be obtained.[22] Note that there exists a variation of logarithmic shear strain, exy, in the picture frame specimen. Figure 9.10 compared this logarithmic shear strain obtained by
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0
50
Y (mm)
100
150
200
250
0
50
100
150
X (mm)
200
250
300
9.9 Typical experimental displacement field of a picture frame measured at LMSP.
optical measurement to those given by the frame kinematics. Similar to the findings reported by KUL, the difference between the calculated shear angle and the measured shear angle is negligible before 33° but becomes larger as the frame displacement advances. as the difference between the global shear angle and the local shear angle is small, especially at the initial deformation zone as confirmed by HKUST, KUL and LMSP, the shear angle reported in the rest of this chapter is the global one calculated from the crosshead displacement.
9.3.4
Determination of shear force
The force needed to deform the fixture must be measured accurately to determine the actual force required in shearing the fabric. The shear force, Fs, can be calculated from the measured pulling force and the current frame configuration as Fs =
F = F ¢¢ – F ¢ 2 cos q 2 cos cos q
9.3
where F is the net load and q is calculated using eq. 9.1. To eliminate the error caused by the weight and inertia of the fixture, the net load F should
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0.4 Optical measurement Reference
Deformation exy
0.3
0.2
0.1
0 0
10
20
30 40 Shear angle (°)
50
60
70
9.10 Comparison between global strain component exy measured with an optical device and the theoretical strain corresponding to frame kinematics (LMSP).
be obtained by subtracting an offset value F¢ from the machine-recorded value F≤ when the fabric is being deformed in the picture frame. The offset value F¢ can be determined by two means. HKUST and UML conducted several tests on their frames without including a fabric sample to record the force required to deform the frame, F¢. KUL used a different method to measure the force required to deform the fixture, F¢. Their method required a hinge to balance the initial weight and calibrate the results under various loading speeds. Recall that UML and KUL have a linkage in their frame design (Fig. 9.4), which introduces an amplification factor in the force calculation. To calculate the shear load, the kinematics of the picture frame must be studied. Let’s first examine UML’s picture frame. The free body diagrams of the side frame BC and BAF are shown in Fig. 9.11. From Fig. 9.11, note that joint C is free for motion. Using symmetry, it can be determined that the force applied on joint C from link CD and BC is zero. Thus, performing a static analysis using the free body diagram of link BC (Fig. 9.12(a)),
FB – Fs = 0 or FB = Fs
9.4
where FB is the force on joint B between link BC and link BAF, and Fs is the shear force the fabric sample applied to link BC. Then, from the free body diagram of link BAF in Fig. 9.12(b),
MA = 0 or FB Lframe sin (2q) – Fsa La sin (2q) = 0
9.5
where MA is the moment at point A, Lframe is the length of link BC, Fsa is the
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Lframe
Lf
279
q
D
B P, da A E
O
F
La
9.11 Schematic diagram of the picture frame at UML. C B Fs FB
B
q
FB
Lframe AY
Lf RAX
A
La F
Fsa (a)
(b)
9.12 Free body diagrams of (a) link BC, and (b) link BAF.
shear force applied on the amplifier frame from the tensile machine, and q is the angle between link BC and the vertical direction as seen in Fig. 9.12(b). Solving eq. 9.5 for FB, FB =
Fsa La Lframe
9.6
Defining an amplification factor as
a=
Lframe La
9.7
from the geometry of the amplifier frame, the shear force of the amplifier, Fsa, can be calculated: Fsa =
F 2 cos q
9.8
where F is the force measured on the load cell in the crosshead or F≤ – F¢ in eq. 9.3.
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Substituting eqs 9.4, 9.7 and 9.8 into eq. 9.6, we can obtain Fs =
F 2a ccos os q
9.9
Thus, in processing the picture frame test data at UML, eq. 9.9 is used to calculate the shear load. after comparing eq. 9.9 with eq. 9.3, it should be noted that the shear force equation is only altered through the inclusion of the amplification factor in the denominator on the right-hand side of the equation. Equation 9.9 will reduce to Eq. 9.3 if the amplification factor, a, approaches 1. a similar analysis can be performed on the frame used by KUL. however, some differences exist because the amplification linkage in the KUL frame is inverted when compared to the amplification linkage in the UML frame (Fig. 9.4). The geometry of the linkage in KUL’s frame is shown in detail in Fig. 9.13. Note that none of the angles of the KUL amplification linkage are equal to the shear angle for the fabric as the crosshead moves in the vertical direction. Thus, the amplification factor for the KUL frame is not a constant value as it was for the UML frame. however, upon performing a kinematic analysis, an equation (as opposed to a constant value) can be determined for the amplification factor, a, and substituted into eq. 9.9.
9.4
Experimental results of the trellis-frame test
In the previous section, frame design, specimen preparation and the calculation of shear angle and shear force were discussed. here, in this section, results from these six research groups (i.e., hKUST, KUL, LMSP, UML, UT and UN) will be compared. Note that the capacities of the load cells were varied (1–50 kN) from each group. Since the tensile machine with the high load
Fr
am
e
le
ng
th
3
5
4
2q
1 a 2q 180 250 2
A
9.13 Geometry of picture frame (KUL).
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capacity induced noisy load data, the force data were smoothed in advance of the results comparison.
9.4.1
Behavior of the plain-weave fabric
Figure 9.14 shows the comparison of shear-force data from Eq. 9.9 as a function of the calculated shear angle for plain-weave fabric at room temperature. Here, the amplification factor introduced by the linkages in the frames used at UML and KUL was considered, i.e., La was used in the shear angle calculation in Eq. 9.1 and a was used in the shear force calculation in Eq. 9.9 for data from UML and KUL. In the other cases, Lframe was used in Eq. 9.1 to calculate the shear angles and Eq. 9.3 was used in obtaining the shear forces. As can be seen from Fig. 9.14, the results showed similar behaviors within a certain small angle (35°) except for the LMSP result. To demonstrate the importance of considering the amplification factor in the calculations, Fig. 9.15 shows the comparison when the amplification factor was not included, which resulted in a quite different figure. Also, in order to examine how results varied for each set of data, error bars were plotted in Fig. 9.15 for the curves of several groups and the variations in the shear force behavior were below 5 N within a 35° shear angle range. HKUST and UML mechanically conditioned the samples prior to testing (see discussion in Section 9.3.2). KUL reported data for each of three repetitions of the test on a single sample. Examining the data from the third repetition of the test on a single sample can be equated to mechanical conditioning, 120 hkust lmsp uml ut kul 1st. KUL 3rd. UN
Shear force (N)
100 80 60 40 20 0 0
10
20
30 40 Shear angle (°)
50
60
70
9.14 Shear force vs shear angle with linkage amplification removed from UML and KUL results.
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hkust lmsp uml ut kul 1st. KUL 3rd. UN
Shear force (N)
30
20
10
0 0
5
10
15
20
25 30 35 Shear angle (°)
40
45
50
55
9.15 Shear force vs shear angle where the amplification factor was not considered.
as the sample has deformed twice. KUL noted that the data from the second and third repetitions on a single sample were comparable with each other, but both were below the data from the first time the sample was deformed in the shear frame (see Fig. 9.15). UT performed their tests using ‘as is’ samples without prior mechanical conditioning, which also explained why their data was slightly higher than the other results. The discussion in the remainder of this subsection will focus on the region of the plot before the shear angle reached 60°. It is at approximately 45° where locking began to occur for this fabric. Locking refers to the point at which the tows are no longer able to freely rotate and they begin to exert a compressive force on each other as the fabric is further deformed. The force required to deform the fabric begins to increase significantly as the locking angle is reached and surpassed. When the compression of the tows reaches a maximum, wrinkling begins to occur and the fabric begins to buckle out of plane. Wrinkling in a formed part is considered a defect and thus is undesirable. Figure 9.16 shows the shear force vs. shear angle obtained from different groups up to 60° of shearing angle where the amplification factor was considered. Careful readers must have noted one deficiency in the data format presented in Fig. 9.16, i.e., the total shear force was presented. However, each group has a different frame size as reported in Table 9.3 One proposed method for normalization was to use the frame length as presented in Harrison et al. based on an energy method.[25] Their assumption was that the frame length was equal to the fabric length. Figure 9.17 presents the shear force results
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40 hkust lmsp uml
Shear force (N)
30
ut kul 1st. KUL 3rd.
20
UN
10
0 0
5
10
15
20 25 30 35 Shear angle (°)
40
45
50
55
9.16 Shear force vs shear angle comparison for the plain-weave fabric where the amplification factors in picture frames were considered.
Normalized shear force (N/mm)
0.30 0.25
hkust lmsp uml
0.20
ut kul 1st. KUL 3rd.
0.15
UN
0.10 0.05 0.00 0
5
10
15
20 25 30 35 Shear angle (°)
40
45
50
55
9.17 Shear force normalized by the frame length vs shear angle.
normalized by the length of the frame used by each group. As seen, the normalization brought curves closer, but noticeable deviations still exist. Frame length could be indicative of sample size, i.e. a larger frame may indicate a larger sample size which in turn would indicate the deformation of a greater number of crossovers. However, as there is no standard ratio for the length of a test sample to the length of the frame, this method is not the best method for normalization.
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The investigation continued by comparing the data when normalized by the fabric area. Here, the fabric area was defined as the inner square area of the sample, i.e., the arm areas were neglected. The fabric area is directly related to the number of crossovers in the material. a larger sample would have more yarns, resulting in more crossovers between the yarns. With an increased number of yarns and crossovers, a larger force is required to shear the sample. Figure 9.18 shows the results when the data were normalized by the inner fabric area. again, this normalization technique brought the curves closer together. The normalization by the inner fabric area is quite straightforward and reasonable if part of the yarns in the arm area was pulled out so that no shear occurred in the arm area. however, when yarns in those arm areas were not pulled out as shown in Fig. 9.1, additional contributions of shear force from the arm area must be considered. Peng et al. proposed a normalization method based on an energy method.[44] They studied the case where the length of the fabric sample was not necessarily equal to the length of the frame. The shear force data can be normalized using the following equation: Fnormalized normalized = Fs ·
Lframe L2fabr fabric bric
9.10
where Fnormalized is the shear force normalized according to the energy method, Fs is the shear force obtained from eq. 9.9, Lframe is the side length of the frame and Lfabric is the side length of the fabric. The above equation reduces to the method proposed by harrison et al.[25]
Normalized shear force (N/mm2)
0.0020 HKUST LMSP UML
0.0015
UT KUL 1st. KUL 3rd.
0.0010
UN
0.0005
0.0000 0
5
10
15
20 25 30 35 Shear angle (°)
40
45
50
55
9.18 Shear force normalized by the inner fabric area vs shear angle.
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for the case when the fabric length is equal to the frame length, in which case Lframe = Lfabric and Fnor normalized malized = Fs ·
1 =F · 1 s Lframe Lfabr fabric bric
9.11
as proposed in harrison et al.[25] Figure 9.19 presents the normalized data using Eq. 9.10. In summary, for the plain-weave fabric tested here, we demonstrated the importance of recognizing different frame designs used in the benchmark, and therefore obtaining the correct shear force is the very first step in material characterization. Figure 9.14 presented the calculated total shear force versus the calculated global shear angle in each test, while Fig. 9.16 showed the same data but up to a shear angle of 60° for better illustration. Furthermore, the total shear force was first normalized by only the frame size in Fig. 9.17, by only the inner fabric area in Fig. 9.18 and finally by the combination of frame size and fabric size in Fig. 9.19. From Fig. 9.19, it can be concluded that the testing results from different groups using different shear frames can be compared for plain-weave fabrics.
9.4.2
Behavior of the balanced twill-weave fabric
Similarly, picture frame tests for the balanced 2 ¥ 2 twill-weave fabric were conducted in KUL, UML and UT as indicated in Table 9.2. The properties of the fabric were listed in Table 9.1 Following the same procedure outlined
Normalized shear force (N/mm)
0.30 HKUST LMSP
0.25
UML UT KUL 1st.
0.20
KUL 3rd. 0.15
UN
0.10 0.05 0.00
0
5
10
15
20
25 30 35 Shear angle (°)
40
45
50
55
9.19 Shear force normalized using the energy method vs shear angle.
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in Section 9.4.1, here only the final results of the normalized shear forces using Eq. 9.10 are plotted against the global shear angle (Eq. 9.2) as shown in Fig. 9.20 considering the amplification factor resulting from the specific shear frame design. Note that the scattering for this fabric is much greater than that of the plain-weave one.
9.4.3
Behavior of the unbalanced twill-weave fabric
Finally, picture frame tests for the unbalanced 2 ¥ 2 twill-weave fabric were conducted in KUL, UML and UN as indicated in Table 9.2. Due to the relatively large yarn size of this fabric compared to the picture frame and the difficulties of handling this fabric, only two groups submitted their testing results. The properties of the fabric were listed in Table 9.1. Following the same procedure outlined in Section 9.4.1, here only the final results of the normalized shear forces using Eq. 9.10 are plotted against the global shear angle (Eq. 9.2) as shown in Fig. 9.21 considering the amplification factor resulting from the specific shear frame design. Between these three groups, the comparison is acceptable.
9.5
Experimental setups of the bias extension test
The bias extension test involves clamping a rectangular piece of woven material such that the warp and weft directions of the tows are orientated initially at 45° to the direction of the applied tensile direction.[25] Figure 9.22 shows a sample set where the specimen was placed in an oven so that high-
Normalized shear force (N/mm)
1.2 uml ut kul 1st. KUL 3rd.
1.0 0.8 0.6 0.4 0.2 0.0
0
5
10
15
20 25 30 35 Shear angle (°)
40
45
50
55
9.20 Normalized shear force vs global shear angle of the balanced twill weave fabric.
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Normalized shear force (N/mm)
0.30 uml kul 1st. KUL 3rd. UN
0.25 0.20 0.15 0.10 0.05 0.00
0
5
10
15
20 25 30 35 Shear angle (°)
40
45
50
55
9.21 Normalized shear force vs shear angle of the unbalanced twill weave fabric.
9.22 A bias extension sample in an oven.
temperature tests could be conducted. In the bias extension test, when the initial length of the sample (L0) is more than twice the width of the sample (w0), there exists a perfect pure shear zone in the center of a sample (zone C
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in Fig. 9.23). It has been shown that the shear angle in region C is assumed to be twice that in region B, while region A remains undeformed, assuming yarns are inextensible and no slip occurs in the sample.[61] Therefore, the bias extension test is considered to be an alternative to the picture frame test to study the material behavior of fabrics. Several research groups conducted the bias extension tests and reported load histories and global shear-angle data conducted at room temperature as listed in Table 9.4. In the remaining paragraphs of this section, sample preparation, shear angle and shear force calculation will be presented.
9.5.1
Sample preparation
Unlike the picture frame tests, the clamping device and sample preparation in bias extension tests is much simpler. Taking NU’s tests, for example, a W
W A W
x A
B
A
W cos q0
y
q
q0 C
H
Leff
C B W/2
y
B
B
B C
x A
Xbe
C
B A
A
B
(H – W) cos q0
B
(H + W) cos q0
A
A
W(sin q/sin q0)
9.23 Illustration of a fabric specimen under a bias extension test. Table 9.4 Tested fabrics in bias extension tests used by participating researchers Group
Plain weave
Balanced twill weave
Unbalanced twill weave
HKUST INSA-Lyon/NU NU UN
Y
N
N
Y
Y
Y
Y
Y
N
Y
N
Y
Y = data reported; N = data not reported.
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pair of grippers can be easily fabricated as shown in Fig. 9.24. One critical aspect in the bias extension test is to ensure fiber yarns are oriented at ±45° to the edges of the grippers before testing. As shown in Fig. 9.25, for an unbalanced weave, extra care must be taken. The black solid lines indicated the fiber yarn directions while an initial visual inspection might wrongly take the dashed lines as initial fiber yarn directions. A twist of fabric might be noticed in the bias extension test if the initial fiber yarn directions are not orientated exactly. An aspect ratio of 2 was used to prepare samples at HKUST, NU, UN and INSA-Lyon/NU. Lines were drawn from the central point of the sample’s edge at A to point B along the warp yarn and through other points as shown in Fig. 9.26, which shows a marked plain woven fabric with a yarn width of 4 mm. These lines would be used to help measure the shear angle variation during the test. The areas of CIGJ and KHLF are clamped by the grippers.
Half of the grip
Assembled
9.24 The grips used at NU’s bias extension tests.
9.25 The unbalanced twill weave fabric in a bias extension test. Note that the thin solid lines, not the thick solid lines, followed the yarn directions (INSA-Lyon/NU).
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H
D
E
L
C
I
G
J
B
F
A
9.26 A 16-yarn sample. Table 9.5 Sample size and process condition used in the bias extension tests Group
Material
Length (mm) Width (mm) Speed (mm/min)
Temp. (∞C)
HKUST
Plain weave
230
115
10
20
INSA-Lyon/NU Plain weave
230 300 450 300 300 450 400
115 100 150 150 100 150 200
10 10 10 10 10 10 10 10 10
20 20 20 20 20 20 20 20 20
Balanced twill weave Unbalanced twill weave NU-New
Plain weave
230
115
10
20
NU-Old
Balanced twill weave
240 300
120 150
10 10
20 20
UN
Plain weave
200 250 300 200 250 300
100 100 100 100 100 100
50 50 50 50 50 50
20 20 20 20 20 20
Unbalanced twill weave
Setting the initial length precisely helps to improve the repeatability of tests. When the sample is under loading, the extension, the tensile force, the width of the sample in the middle, the angle between DE and EA, and the angle between HE and EF can be recorded. For samples using other aspect ratios, one can prepare a mold to copy the pattern onto the fabric. Table 9.5 lists the sample sizes and process conditions used in the various groups.
9.5.2
Determination of shear angle
For a general bias extension test, Lebrun et al. developed formulae to calculate the global shear angle based on the assumption that there exist three © Woodhead Publishing Limited, 2011
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distinct areas in terms of shear deformation and each area has a uniform shear deformation.[62] Figure 9.23 illustrated the notations used in Eq. 9.12 to calculate the global shear angle: (H + d ) – W 2(H – W ) cos q 0 d = coss q 0 + 2(H – W ) cos q 0 2(H
coss q =
9.12
where d is a displacement during the test. The shear angle in region C (Fig. 9.23) can also be measured using image processing software based on pictures taken during a test. a joint effort between researchers at INSa-Lyon and NU was made to investigate bias extension tests. The true shear angle in the fabric was measured using two methods. The solid line with diamond-shaped symbols in Fig. 9.27 resulted from the manual measurements of angles based on images taken. The line of circles resulted from the image correlation software, IcaSoft developed at INSa-Lyon.[62, 63] as can be seen, the shear angle from the software showed good agreement with the actual (measured) shear angle. Furthermore, it was concluded that the theoretical shear angle calculated by Eq. 9.12, represented by the solid black line, can accurately reflect the true shear angle in the fabric until the shear angle reaches a value of 30°. In Fig. 9.27, the shear angle curves from the image analysis and by manual measurement were compared with the theoretical one for the plain weave 60 Theoretical Image analysis Measured
Shear angle (°)
50 40 30 20 10 0 0
10
20
30
40 50 60 70 Displacement (mm)
80
90
100 110
9.27 Plot of shear angle in Zone C vs displacement of the plain weave fabric with a sample size of 150 mm ¥ 450 mm in a bias extension test (INSA-Lyon/NU).
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with dimensions of 150 mm ¥ 450 mm. Note that the difference between the theoretical shear angle and the true shear angle was below 5° until 30° shear angle was reached and became large after this angle. Similar behavior was observed for other fabrics. The optical measurement was also used to examine another assumption used in the calculation of bias extension test, i.e., the shear angle in zone B (Fig. 9.23) is half of that in zone C. Figure 9.28 shows a contour of shear angle in a plain weave fabric with a sample size used in a bias extension test. The contour was obtained from the optical measurement software Icasoft at INSA-Lyon. It can be seen that the assumption held well for the majority of areas.
9.5.3
Determination of shear force
As discussed in Section 9.5.2 and illustrated in Figs 9.23 and 9.28, there exist three distinct deformation zones in a bias extension test. In this subsection, we will illustrate how to obtain the normalized shear force vs. shear angle from a bias extension test following the four basic assumptions, i.e., (a) shear angles in each zone are considered uniform; (b) the shear angle in zone C is twice that in zone B; (c) there is no shear deformation in zone A; and (d) the initial fabric has a perfect orthogonal configuration, i.e., qo = 45°. A simple kinematic analysis of a bias extension sample in Fig. 9.23
9.28 Contour of shear angle in a plain weave fabric with a sample size in a bias extension test obtained from the optical measurement software Icasoft (INSA-Lyon/NU).
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gives the shear angle in zone C, g, as a function of fabric size and the end displacement, d, as Ê Lef ˆ Ê L + dˆ g = 90° – 2q = 90° – 2 cos –1 Á eff = 90° – 2 cos –1 Á 0 9.13 ˜ Ë 2 L0 ¯ Ë 2L0 ˜¯ where L0 = H – W and H and W are the original height and width of the specimen, respectively. The power made through the clamping force, F, is dissipated in two zones, zone B and zone C: Ê Êg ˆ gˆ F · d = CS (g ) · Ag · g + Á CS Á ˜ · Ag /2 · ˜ 2¯ Ë Ë 2¯
(
)
9.14
where Ag is the original area of zone C, which is subjected to a shear angle of g, Ag/2 is the original area of zone B, and CS (g) is the torque per original unit area that is needed to deform the fabric in shear.[64] From eq. 9.13, we can obtain
g = – 2q =
2 d L0 sin q
9.15
2 Substituting eq. 9.15 and geometrical parameters of Ag = 2HW – 3W and 2 Ag/2 = W2 into eq. 9.14, we can obtain
CS (g ) =
=
Ê 2 Êg ˆˆ 1 (H – W ) F sinq – W · CS Á ˜ ˜ (H 22H H – 3W ÁË W Ë 2¯ ¯
( )
Ê H g gˆ Ê Êg ˆˆ 1 – 1 F Á cos – sin sin ˜ – W · CS Á ˜ ˜ 2H – 3W ÁË W 2 2¯ Ë Ë 2¯ ¯
9.16
The unit torque CS (g) can be related to shear force Fsh as Êg ˆ Êg ˆ Êg ˆ CS (g ) = Fsh (g ) · cos (g ) and CS Á ˜ = Fsh Á ˜ · cos Á ˜ Ë 2¯ Ë 2¯ Ë 2¯
9.17
hence, Fsh (g ) =
1 (2H – 3W ) cos g ÊÊ H ˆ ¥ Á Á – 1˜ · F ¯ ËËW
g gˆ gˆ Êg ˆ Ê · Á cos – sin ˜ – W · Fsh Á ˜ cos ˜ 9.18 2 2¯ 2¯ Ë Ë 2¯
The above derivation can also be found in.[28] Note that here Fsh is the
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normalized shear force per unit length, the same quantity as Fnormalized in Eq. 9.10 from the picture frame test. Therefore, the experimental clamping force vs. displacement curve can be first converted to the curve of clamping force vs. shear angle using Eq. 9.12, then using an iterative process the normalized shear force per unit length Fsh can be represented against the shear angle.
9.6
Experimental results of the bias extension test
In the previous section, specimen preparation and the calculation of shear angle and shear force were discussed. Here, in this section, results from four research groups (i.e., HKUST, INSA-Lyon, NU, and UN) will be compared for tests conducted at room temperature. Data labeled as HKUST, NU-old and UN were obtained in 2004, while NU-new were tested in 2007. INSALyon and NU collaborated in the summer of 2007 in performing those tests marked ‘INSA-NU’. UT also submitted their testing data, but it was for consolidated fabric[12] and therefore will not be included in this chapter. As in the picture-frame test, the force data were smoothed for the highly noised load data.
9.6.1
Behavior of the plain-weave fabric
Figure 9.29 illustrates the raw experimental data with error bars reported from each group. Note that the specimen sizes and ratios are different among those tests. Using Eqs 9.12 and 9.18, normalized shear forces per unit length 70 60 50 Force (N)
¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥
230 300 300 450 230 230 200 250 300
20
30
115 150 100 150 115 115 100 100 100
40 30
INSA-NU INSA-NU INSA–NU INSA-NU NU-New HKUST UN UN UN
20 10 0
0
10
40 50 60 70 80 Displacement (mm)
90 100 110 120
9.29 Tensile force vs cross-head displacement of plain weave for bias extension tests.
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are plotted in Fig. 9.30. As shown in Fig. 9.30 all data showed consistent results within a certain shear angle in the 35° range except for 100 ¥ 200 UN and 100 ¥ 300 INSA-NU data. The difference might be due to the handling of the specimens. Note that the normalized shear force curves were almost similar regardless of the different ratio of width to length and showed good agreement with the results from the picture frame tests as shown in Fig. 9.36, which will be discussed in the next section. Figure 9.31 compares the high-temperature and room-temperature tests as well as the two different loading speeds. One can clearly see that the tensile load is much lower when the temperature increases. Experiments at high temperature are conducted by heating the sample in an oven to the expected processing temperature before the test (Fig. 9.22). During the test, the oven is maintained at a specified temperature. As for the loading speed, it can be seen that the deformation rate does not affect the tensile force.
9.6.2
Behavior of the balanced twill-weave fabric
Similarly, the balanced twill-weave fabric was tested in different laboratories. Figure 9.32 illustrates the raw experimental data reported from each group. Note that the specimen sizes and ratios are different. Using Eqs 9.12 and 9.18, normalized shear forces per unit length are plotted in Fig. 9.33. Although several curves showed similar behavior for the normalized shear force, they still showed large deviations between each other. However, when the data are compared with the results from the picture-frame tests, the bias extension results are within the range of the picture-frame test results as shown in Fig. 9.37 below.
Normalized shear force (N/mm)
0.30 115 150 100 150 115 115 100 100 100
0.25 0.20 0.15 0.10
¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥
230 300 300 450 230 230 200 250 300
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9.32 Tensile force vs cross-head displacement of balanced twill weave for bias extension tests.
9.6.3
Behavior of the unbalanced twill-weave fabric
For the unbalanced twill-weave fabric, only the INSA-NU and UN groups reported the test result for the bias extension test. Both groups performed four tests for each dimension and all results showed almost similar behavior. In Fig. 9.34, raw experimental data were illustrated. Using Eqs 9.12 and 9.18,
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the normalized shear forces per unit length are plotted in Fig. 9.35. Even the dimensions of the samples are different from each other, three UN data showing similar behavior while INSA-NU data showed large deviations from UN data.
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9.7
Conclusions
The properties of woven fabrics are very different from those of conventional materials, such as bulk metals and polymers. This phenomenon leads to the interest among the materials community of woven-fabric composites to conduct benchmark tests. It has been shown that picture-frame tests are able to produce valuable experimental data for characterizing shear behavior of textile composites. Mechanically conditioning the sample can also improve repeatability as demonstrated in the laboratories. This was shown through results from UML and KUL. UML’s samples were all mechanically conditioned and appeared very repeatable. While KUL did not mechanically condition their samples, they conducted the shear test three times on each fabric blank and noted a large difference between the first run and the second and third runs. However, there was not a large difference in the results when only comparing the second and third runs. Mechanical conditioning may equalize tow tensions left in the fabric from the weaving process and therefore reduces the variability and increases the repeatability. However, mechanical conditioning might not be feasible in industry unless the material handling system is modified. The optical methods showed that determining the shear angle mathematically from the crosshead displacement was a reasonable method before the plainweave fabric reaches the 35° shear angle in picture-frame tests and 30° in the bias extension tests. Therefore, data beyond these angles are recommended to be interrupted using optical measurements instead of theoretical calculation of the shear angle.
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Normalization methods were presented for comparing test data from various groups. It is interesting to compare the normalized shear force vs. shear angle from picture-frame tests and bias extension tests. Figures 9.36–9.38 compare the results from bias extension tests (solid curves) with those from picture-frame tests (dash-dotted curves). As shown in Fig. 9.36, test data showed similar behaviors, even though the testing method and aspect ratios were different. For the balanced twill weave, both test results were located within the same range, as shown in Fig. 9.37, even though the deviation was not small. Therefore, it can be concluded that the suggested normalization methods by Eqs 9.10, 9.12 and 9.18 for picture-frame and bias extension tests give consistently the shear force behavior for isotropic and homogeneous fabrics. As for the unbalanced twill weave in which the anisotropy and directionality are quite large, the two normalization methods did not give consistent results, as shown in Fig. 9.37. Test results provided by different groups show consistency but still have some deviations. Further studies are underway to help develop a standard test setup and procedure for obtaining accurate and appropriate material properties. For example, the effect of clamping force on shear behavior in the picture-frame tests needs to be taken into account in material characterization. Material responses under different speeds and temperatures will be further investigated. Calibration, sample preparation, and other important techniques 0.30
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to increase the accuracy of the tests will be collected and shared among the community. High-temperature tests present challenges to researchers as they limit the use of optical devices and require higher sensitivity of the testing equipment.
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Since precise descriptions for mechanical material properties in the numerical simulations are required to predict the accurate responses for the composite sheet forming, well-defined normalization methods are necessary. Even though there are still deviations, the suggested normalization methods can give consistent shear behaviors, especially for homogeneous fabrics. Therefore, further numerical simulations are needed to use the suggested normalized shear property in order to obtain consistently formed results and predictions. For non-homogeneous fabrics, such as unbalanced-twill weave, careful material description is required and further studies of standardization methods need to be investigated. The next phase of this benchmark activity is to focus on the predictability of various material models and simulation methods in modeling the thermostamping of woven composites in terms of shear angle prediction, force and deformed shape predictions. Interested readers can refer to the benchmark website[7] for the latest updates.
9.8
Acknowledgments
The authors would like to thank the National Science Foundation, Saint-Gobain, Inc., Hong Kong RGC (under grant HKUST6012/02E), the Netherlands Agency for Aerospace Programmes, a Marie Curie Fellowship of the EC (HPMT-CT-2000-00030) and the Engineering and Physical Sciences Research Council (under grant GR/R32291/01) for their support of this work.
9.9
References
1. Advani S G, Flow and Rheology in Polymeric Composites Manufacturing, Amsterdam, Elsevier, 1994. 2. Parnas R S, Liquid Composite Molding, Munich, Hanser Publishers, 2000. 3. Hivet G, Modelisation mésoscopique pour le comportement et la mise en forme des renforts de composites tissés, Ph.D. thesis, University of Orléans, 2002 (in French). 4. Yu T X, Tao X M, Xue P, ‘The energy-absorbing capacity of grid-domed textile composites’, Composites Science and Technology, 2000, 60, 785–800. 5. Rudd C D, Turner M R, Long A C, Middleton V, ‘Tow placement studies for liquid composite moulding’, Composites Part A: Applied Science and Manufacturing, 1999, 30, 1105–1121. 6. Long A C, Wilks C E, Rudd C D, ‘Experimental characterisation of the consolidation of a commingled glass/polypropylene composite’, Composites Science and Technology, 2001, 61, 1591–1603. 7. http://nwbenchmark.gtwebsolutions.com/ 8. ESAFORM 2004, Proceedings of the 7th ESAFORM Conference on Material Forming, 27–30 April 2004, Trondheim, Norway, ed. Sigurd Stören, ISBN: 8292499-02-04. 9. ESAFORM 2005, Proceedings of the 8th ESAFORM Conference on Material Forming, 27–29 April 2005, Cluj-Napoca, Romania, ed. D. Banabic, ISBN: 973-27-1174-4.
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10. ESAFORM 2006, Proceedings of the 9th ESAFORM Conference on Material Forming, 26–28 April 2006, Glasgow, UK, ed. N. Juster and A. Rosochowski, ISBN: 83-89541-66-1. 11. ESAFORM 2007, Proceedings of the 10th ESAFORM Conference on Material Forming, 18–20 April 2007, Zaragoza, Spain, ed. E. Cueto and F. Chinesta, ISBN: 978-0-7354-0414-4. 12. Cao J, Cheng H S, Yu T X, Zhu B, Tao X M, Lomov S V, Stoilova Tz, Verpoest I, Boisse P, Launay J, Hivet G, Liu L, Chen J, de Graaf E F, Akkerman R, ‘A cooperative benchmark effort on testing of textile composites’, ESAFORM 2004, 305–308. 13. Akkerman R, Lamers E A D, Wijskamp S, ‘An integral model for high precision composite forming’, European Journal of Computational Mechanics, 2006, 15, 359–377. 14. Akkerman R, Ubbink M P, De Rooij M B, ten Thije R H W, ‘Tool-ply friction in composite forming’, Proceedings of the 10th International ESAFORM Conference on Material Forming, Zaragoza Spain, 2007, 1080–1085. 15. Badel P, Vidal-Sallé E, Boisse P, ‘Computational determination of in plane shear mechanical behaviour of textile composite reinforcements’, Computational Material Science, 2007, 40, 439–448. 16. Boisse P, Zouari B, Daniel J L, ‘Importance of in-plane shear rigidity in finite element analyses of woven fabric composite preforming’, Composites Part A – Applied Science and Manufacturing, 2006, 37, 2201–2212. 17. Boisse P, ‘Meso–macro approach for composites forming simulation’, International Journal Material Science, 2006, 41, 6591–6598. 18. Boisse P, Gasser A, Hagege B, Billoet J L, ‘Analysis of the mechanical behaviour of woven fibrous material using virtual tests at the unit cell level’, International Journal of Material Science, 2005, 40, 5955–5962. 19. Boisse P, Zouari B, Gasser A, ‘A mesoscopic approach for the simulation of woven fibre composite forming’, Composites Science and Technology, 2005, 65, 429–436. 20. Cao J, Xue P, Peng X Q, Krishnan N, ‘An approach in modeling the temperature effect in thermo-forming of woven composites’, Composite Structures, 2003, 61, 413–420. 21. Cheng H S, Cao J, Mahayotsanun N, ‘Experimental study on behavior of woven composites in thermo-stamping under nonlinear temperature trajectories’, International Journal of Forming Processes, 2006, 8, 1–12. 22. Dumon F, Hivet G, Rotinat R, Launay J, Boisse P, Vacher P, ‘Field measurements for shear tests on woven reinforcements’, Mécanique & Industries, 2003, 4, 627–635. 23. Gorczyca J, Sherwood J, Liu L, Chen J, ‘Modeling of friction and shear in thermo-stamping process – Part I’, Journal of Composite Materials, 2004, 38, 1911–1929. 24. Harrison P, Hua L, Ubbink M, Akkerman R, van de Haar K, Long A C, ‘Characterising and modelling tool-ply friction of viscous textile composites’, Proceedings of the 16th International Conference on Composite Materials, ICCM-16, 2007, Kyoto, Japan. 25. Harrison P, Clifford M J, Long A C, ‘Shear characterization of viscous woven textile composites: a comparison between picture frame and bias extension experiments’, Composites Science and Technology, 2004, 4, 1453–1465. 26. Hivet G, Boisse P, ‘Consistent 3D geometrical model of fabric elementary cell.
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43. Lussier D, ‘Shear characterization of textile composite formability’, Master’s Thesis, 2000, Department of Mechanical Engineering at the University of Massachusetts Lowell. 44. Peng X Q, Cao J, Chen J, Xue P, Lussier D S, Liu L, ‘Experimental and numerical analysis on normalization of picture frame tests for composite materials’, Composites Science and Technology, 2004, 64, 11–21. 45. Peng X Q, Cao J, ‘A continuum mechanics based non-orthogonal constitutive model for woven composites’, Composites Part A, 2005, 36, 859–874. 46. Peng X Q, Cao J, ‘A dual homogenization and Finite Element approach for material characterization of textile composites’, Composites Part B, 2002, 33, 45–56. 47. Potter K, ‘Bias extension measurements on cross-plied unidirectional prepreg’, Composites Part A, 2002, 33, 63–73. 48. Wijskamp S, Lamers E A D, Akkerman R, ‘Effects out-of-plane properties on distortions of composite panels’, in Proceedings of Fibre Reinforced Composites 2000, Newcastle, UK, pp. 361–368. 49. Xue P, Cao J, Chen J, ‘Integrated micro/macro mechanical model of woven fabric composites under large deformation’, Composite Structures, 2005, 70, 69–80. 50. Xue P, Peng X Q, Cao J, ‘A non-orthogonal constitutive model for characterizing woven composite’, Composites Part A, 2003, 3, 83–193. 51. Yu W R, Chung K, Kang T J, Zampaloni M A, Pourboghrat F, Liu L, Chen J, ‘Sheet forming analysis of woven frt composites using the picture-frame shear test and the nonorthogonal constitutive equation’, International Journal of Materials and Product Technology, 2004, 21, 71–88. 52. Zhu B, Teng J, Yu T X and Tao X M, ‘Theoretical modelling of large shear deformation and wrinkling of plain woven composite’, Journal of Composite Materials, 2009, 43, 125–138. 53. Zhu B, Yu T X, Tao X M, ‘Research on the constitutive relation and formability of woven textile composites’, Advances in Mechanics, 2004, 34, 327–340. 54. Zhu B, Yu T X, Tao X M, ‘An experimental study of in-plane large shear deformation of woven fabric composite’, Composites Science and Technology, 2007, 67, 252– 261. 55. Zhu B, Yu T X, Tao X M, ‘Large deformation and failure mechanism of plain woven composite in bias extension test’, Key Engineering Materials, 2007, 334–335, 253–256. 56. Zhu B, Yu T X, Tao X M, ‘Large shear deformation of E-glass/polypropylene woven fabric composite at elevated temperatures’, Journal of Reinforced Plastics and Composites, 2009, 28, 2615–2630. 57. Zhu B, Yu T X, Tao X M, ‘Large deformation and slippage mechanism of plain woven composite in bias extension’, Composites Part A, 2007, 38, 1821–1828. 58. Nestor T A, Obradaigh C M, ‘Experimental investigation of the intraply shear mechanism in thermoplastic composites sheet forming’, Advances in Engineering Materials, 1995, 99, 19–35. 59. Mohammed U, Lekakou C, Dong L, Bader M G, ‘Shear deformation and micromechanics of woven fabrics’, Composites Part A, 2000, 31, 299–308. 60. Nguyen M, Herszberg I, Paton R, ‘The shear properties of woven carbon fabric’, Composite Structures, 1999, 47, 767–779. 61. Lebrun G, Bureau M N, Denault J, ‘Evaluation of bias-extension and picture-frame test methods for the measurement of intraply shear properties of PP/glass commingled fabrics’, Composite Structures, 2003, 61, 341–352.
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62. http://www.techlab.fr/strain/htm#icasoft 63. Toucha S, Morestin F, Brunet M, ‘Various experimental applications of digital image correlation method’, in Proceedings of CMEM 97 (Computational Methods and Experimental Measurements VIII), 1997, Rhodes, pp. 45–58. 64. De Luycker E, Boisse P, Morestin F, in: SNECMA Maia meeting, Villaroche, 2006.
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10
Biaxial tensile properties of reinforcements in composites
V. C a r v e l l i, Politecnico di Milano, Italy
Abstract: The deformability of a composite reinforcement depends on its internal structure and has a relevant influence on the quality of the composite part. Among the mechanical features of a composite reinforcement, the biaxial behaviour plays an important role in its capability to drape a complex shape. In this chapter the methodologies employed to measure and to predict the biaxial behaviour of composite reinforcements are discussed. The features of some experimental devices adopted to investigate the response to biaxial loading of composite reinforcements are detailed. A prediction tool based on an analytical model of a plain weave reinforcement is briefly described and compared to experimental data. Finally, the characteristics of reliable numerical models, based on the finite element method, are highlighted in terms of information obtained at the macro- and meso-scale. Key words: composite reinforcements, mechanical properties, biaxial tensile behaviour.
10.1
Introduction
The mechanical behaviour of a composite reinforcement has an important role during the forming and infusion processes. This chapter focuses on one of the deformation mechanisms involved in the shaping of a composite reinforcement, i.e. biaxial tensile behaviour. In this context, investigations available in the literature are mainly dedicated to textile reinforcements, these being particularly suitable at producing complex three-dimensional shapes. In this chapter, therefore, the biaxial tensile behaviour of textile reinforcements is mainly discussed and some data for non-crimp fabrics are also mentioned. Textile reinforcements are produced by interlacing two or more sets of tows. Depending on the degree of interlacing, a variety of weaves can be produced. The interlacing provides a stable structure that is easy to handle and has the flexibility to be draped around complex double-curvature surfaces (Potluri et al.1), i.e. a textile preform has excellent formability characteristics. The woven fabric drapability allows the manufacture of a component involving a forming stage in which the dry fabric is deformed into a desired shape. The formed fabric is then injected with resin and consolidated. The 306 © Woodhead Publishing Limited, 2011
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deformability of the reinforcement depends on the fabric structure and plays a key role in the quality of a composite part. During the draping and moulding processes of a dry reinforcement, different deformation mechanisms influence the internal geometry (Potluri et al.1): ∑
Uniaxial tensile loading causes crimp interchange, leading to reduced tow undulation in the loading direction and an increase in tow undulation in the transverse direction. ∑ Biaxial tensile loading provides a reduction of the tow undulation in both directions. ∑ In-plane shear or bias extension changes the in-plane tow orientation. Once the lock angle is reached, transverse compaction leads to a slight reduction in the tow width and an increase in the tow thickness (see Chapter 9). ∑ Transverse compression results in tow flattening, a decrease in tow undulation and an increase in the fibre volume fraction (see Chapter 11). ∑ Out-of-plane bending gives a slight reduction in tow undulation, evident mainly for 3D fabrics (see Chapter 12).
The combination of the above-mentioned deformation mechanisms during the forming stage of a preform allows variation of the internal geometry of the reinforcements, i.e. variation of the parameters which describe the spatial orientation of the fibres in the fabric and/or the stacking sequence of fabrics in a multilayer. The first consequence is the variation of the preform permeability to the resin flow, and the second is the variation of mechanical features and thermal conductivity of the composite component. In recent decades many studies have been performed to generate reliable models to predict the final shape of the preform during a forming stage (see, e.g., References 2 and 3). These models generally require accurate knowledge of the mechanical behaviour of the reinforcement. Therefore, the characterization of the deformability of reinforcements is extremely important with both experimental tests and meso-scale models. The information obtained has to be reliable as input for macro-scale models of drapability used to optimize the preform and the process conditions (Lomov et al.4). As one of the main factors responsible for reinforcement deformability, biaxial tensile mechanical behaviour is discussed in this chapter by means of different approaches. Experimental devices and procedures employed in several research activities are presented in Section 10.2. The theoretical background of an analytical model is detailed and some predictions are compared in Section 10.3. The main characteristics of numerical models based on the finite element method are highlighted in Section 10.4.
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10.2
Experimental analysis
This section presents and discusses some experimental devices employed for biaxial tensile tests of composite reinforcements, some techniques for measurement of deformations during loading, and some experimental experiences, taken from the literature, for two-dimensional textiles and non-crimp fabrics.
10.2.1 Biaxial tensile devices Different experimental devices have been used to produce biaxial tensile loading of composite reinforcements. The experimental techniques may be classified into two categories: (1) single loading system; and (2) two or more independent loading systems. In the following, experimental devices belonging to these categories will be briefly described to highlight some peculiarities. Their description is not exhaustive and does not cover several systems available worldwide for biaxial testing of reinforcements and/or composite materials.5–9 To the first category belongs the loading device employed in several research works of Boisse and co-workers10–15 (see Fig. 10.1). A similar device was used for biaxial testing of metal sheets.16 The device in Fig. 10.1 is based on two hinged lozenges that impose a biaxial strain state to a cross-shaped specimen (Boisse et al.15). The device does not need servo
10.1 Biaxial tensile device in Buet-Gautier and Boisse11 and Boisse et al.12
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control between the two tensile axes and is set directly on a classical traction/ compression uniaxial testing machine. When the machine crosshead is raised, it compresses the whole system. A displacement is then generated in each direction of the cross-shaped specimen set in the middle of the device. One of the parallelograms has different lengths, which allows one to impose various strain ratios between the two perpendicular directions. Load sensors placed just behind the specimen measure the total load in each direction. A regulation system included in the biaxial tensile device allows one to fix an angle between the yarn directions different from 90° (Buet-Gautier and Boisse11). To the second category belongs the loading device employed in Luo and Verpoest,17 Lomov et al.18 and Willems et al.19,20 This biaxial tensile machine (Fig. 10.2) is equipped with four independently controlled axes, which can all be driven separately. The machine has two force transducers, one for each direction (Lomov et al.18). A different concept to apply biaxial loads was developed for the device employed in Carvelli et al.21 and Quaglini et al.22 (see Fig. 10.3). This is a home-made device equipped with two independent orthogonal axes. Along each direction the load is applied by a pair of cross-bars that can translate parallel to the edges of the specimen placed in the centre of the device. Each bar can slide on two rails by means of two worm-screws (Fig. 10.3). Two independent servo-motors are linked to the worm-screws to apply
10.2 Biaxial tensile device in Luo and Verpoest17 and Lomov et al.18
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10.3 Biaxial tensile device in Carvelli et al.21 and Quaglini et al.22 F2
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10.4 Biaxial tensile test of cruciform specimen with (a) free ends; (b) two fixed ends.
displacements with a controlled speed. During the movement, the two crossbars of each loading direction are always equidistant from the centre of the device, i.e. from the orthogonal loading axis. All the above-mentioned biaxial loading devices aim to produce the ideal situation in which the centre point of the specimen has no displacement (Fig. 10.4a for a cruciform specimen), i.e. with equal and collinear forces in each load direction, and in addition the forces F1 have to be perpendicular to the forces F2. It should be underlined that an arrangement where two ends of the specimen are fixed while the two opposite ends are loaded is completely unsatisfactory (Fig. 10.4b). The centre of the specimen will move during a test, introducing undesired shear strain components. The measurement of the shear strain field assesses the quality of the biaxial strain state of the
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specimen. The shear strain component should be ideally zero in a biaxial tension test. Typical distributions of the strain components during a biaxial tensile test of a plain weave technical textile are reproduced in Fig. 10.5.21 The pictures show the intensity of the strain components in the warp and weft directions (Figs 10.5a, b) and the shear component in the textile plane (Fig. 10.5c) in the central part of the specimen. The negligible shear strain 15
Weft direction (mm)
10 5 0 –5 –10 –15 –15
–10
–5 0 5 Warp direction (mm) (a)
10
15
–10
–5 0 5 Warp direction (mm) (b)
10
15
15
Weft direction (mm)
10 5 0 –5 –10 –15 –15
10.5 Maps of the strain components during a biaxial tensile test of a plain weave technical textile: (a) strain in warp direction; (b) strain in weft direction; (c) shear strain (%).
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Weft direction (mm)
10 5 0 –5 –10 –15 –15
–10
–5 0 5 Warp direction (mm) (c)
10
15
10.5 Continued.
demonstrates the correct application of the biaxial loading in the principal direction of the textile. However, the expected biaxial stress state could not be produced even when using a loading device with four actuators. A small displacement of the specimen centre might occur in a real situation and an imbalance arises in the forces. As four load cells are used, it is possible to measure this small load difference and to use it as a control signal. In the literature, the shape generally used for a biaxial tensile specimen of composite reinforcements is cruciform4,10–15 or rectangular according to the available grip length.17 Cruciform specimens have, as a standard,4 the unloaded yarns in the arm parts removed (see Fig. 10.6). The loaded yarns in these parts allow the tensile loads to be transmitted while permitting the transverse deformation of the specimen (Boisse et al.15).
10.2.2 Clamping systems Different clamping systems have been designed for the available biaxial tensile devices. They may be collected in two main categories. In the first category the clamps produce a uniform pressure in the grip zones of the specimen by rigid mechanical4,10 or pneumatic systems. These clamps prevent the transverse contraction of the specimen in the grips. In consequence, a stress concentration appears in the specimen zones close to the grips that could lead to premature failure. The clamping systems in the second category consist of freely moveable rigs (mechanical22 or pneumatic18) mounted on
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Arm part
Woven general part
10.6 Composite reinforcement specimen for biaxial tensile tests.4
each side to have a discontinuous grab (see Fig. 10.7). Each clamp is free to slide orthogonally to the load direction and to rotate about a pivot. 21,22 This system allows free lateral contraction and, moreover, the free rotation reduces the misalignment of the loads transmitted in each clamp.
10.2.3 Strain field measurement The shape of the specimens for biaxial tensile testing aims to have a biaxial strain field in the centre part not affected by the distribution close to the grip zones. In order to assess the biaxial strain state and to relate it to the applied forces, suitable measurement techniques have to be used. The discontinuous structure and the negligible value of some stiffnesses of the composite reinforcements do not allow an accurate measurement of strains by contact instruments or extensometers. In the last decade, digital image correlation (DIC) methods have been commonly used for optical measurement of full-field strain in reinforcement deformability investigations.4,23–26 DIC methods refer to the class of non-contacting methods that acquire images of
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(a)
(b)
10.7 Clamping systems in (a) Carvelli et al.21 and Quaglini et al.,22 and (b) Lomov et al.18
an object, store images in digital form, and perform image analysis to extract full-field shape and deformation measurements. Digital image correlation (i.e. matching) has been performed with many types of object-based patterns, including lines, grids, dots, and random arrays. One of the most commonly
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used approach employs random grey speckle patterns (natural or artificial) and compares sub-regions throughout the image to obtain a full field of measurements.27 For composite reinforcements the pattern can be natural texture or additionally applied paint. Two-dimensional digital image correlation (2-D DIC) uses one camera positioned perpendicular to a flat surface. The displacement field obtained is only reliable under the assumption that out-of-plane deformation can be neglected. Relatively small out-of-plane motions can change the magnification and introduce errors in the measured in-plane displacement field when using a single camera in 2-D DIC, a limitation that is eliminated by 3-D DIC.27 Three-dimensional digital image correlation (3-D DIC) requires a two-camera stereo vision system to accurately measure the full three-dimensional shape and deformation of a curved or planar object. The measurements of the in-plane deformation of a composite reinforcement do not improve by adopting 3D-DIC. In-plane deformation of a reinforcement is characterized by the strains of the fabric middle surface.4 3D-DIC measurement gives strains in the tangential plane to a local point on the surface which deviates largely from the middle surface. Hence, strains measured by 3DDIC cannot be used to characterize in-plane deformations of a composite reinforcement.4 Image analysis software can be used repeatedly to obtain surface deformations and strain field mapping. Different home made and commercial software is used for reinforcement deformability investigations.20,22,23,25,26 Some commercial software packages are employed and compared in Lomov et al.4 for full-field strain measurements of textile deformability during shear and tensile testing, observation of textile deformation on the scale of the textile unit cell and of the individual yarns, measurement of the 3D-deformed shape, and local deformations of a textile reinforcement after draping. The authors’ main conclusion is that ‘optical full-field strain techniques are the preferable (sometimes the only) way of assuring correct deformation measurements’ during mechanical testing of composite reinforcements.4
10.2.4 Some experimental results The experimental results of biaxial tensile tests of composite reinforcements available in the literature are mainly devoted to woven textile reinforcements; an exception (to the author’s knowledge) is in Reference 18. To highlight the peculiarities of the response during biaxial loading of composite reinforcements, some experimental results available in the literature are hereafter discussed. A twill 2 ¥ 2 carbon balanced reinforcement (0.35 untwisted yarns/mm, 0.35% fabric crimp) was studied in Refs 11 and 23. The tensile versus strain curves in the warp direction are reproduced in Fig. 10.8 for different ratios
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k=1 k = 0.5
Tension (N/yarn)
200 Yarn
150 k=2
Weft free
100 50 0 0
0.2
0.4 Strain (%)
0.6
0.8
10.8 Biaxial tensile tests of a twill 2 ¥ 2 carbon textile reinforcement: tension vs strain in the warp direction (Buet-Gautier and Boisse,11 Launay et al.23).
k (k = warp strain/weft strain). The curves are very nonlinear at low loads and then linear at higher loads. This nonlinear behaviour is a consequence of nonlinear phenomena occurring at lower scales (undulation variations and yarn flattening). The nonlinear zone depends on the imposed strain ratio k, which points out the biaxial aspect of fabric behaviour, that behaviour in each direction influences behaviour in the other. The nonlinear zone has the maximum extension for tests in which the other direction is free of displacement. In the load direction, yarns tend to reach a totally straight state under very low loads. When the yarn is straight, the fabric behaviour is similar to that of the yarn alone. The strain corresponding to this transition is representative of the fabric crimp in this direction and of the textile architecture (see, e.g., the response of different textile reinforcements in Ref. 11). In Fig. 10.8, the nonlinear zone for k = 1 reaches approximately the strain of 0.3%. These biaxial tensile test results demonstrate that during loading, e.g. in the forming process, a reinforcement has highly nonlinear behaviour. The biaxial tensile response of non-crimp reinforcements is reported in Lomov et al.18 The authors detailed experimental mechanical tests on multiaxial multiply carbon stitched preforms. The reinforcements loaded in two orthogonal directions (see Ref. 18 for a complete list of the preform parameters) are a bi-diagonal carbon fabric with two layers oriented at +45° and –45° (named B1) and a biaxial carbon fabric with two layers oriented at 0° and 90° (named B2). Orientations are relative to the production direction. Preforms B1 and B2 have a tricot and a tricot-chain stitching pattern, respectively, five needles
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per inch and PES stitching yarns. In these reinforcements, the fibres in plies are almost straight and are not interlaced. They are, however, linked by the stitching. The influence of the link is different for different fabrics and different directions of the test. The curves reproduced in Fig. 10.9 show the response for biaxial loading in the bias direction (0°/90° for fabric B1 and +45°/–45° for fabric B2) for different strain ratios k (k = strain X/strain Y, X and Y being the loading directions), setting different speeds of the axes. Further loading conditions are extensively detailed in Ref. 18. Loading in 25 B1
Tension (N/mm)
20
15
k = 0.2
k = 0.5
k=2
k=1
k=5
10
5
0
0
0.5
1
1.5 Strain X (%) (a)
2
2.5
3
25 B2 20 Tension (N/mm)
k=5 15
k = 0.2
k=1
k = 0.5
k=2
10
5
0
0
0.5
1
1.5 Strain X (%) (b)
2
2.5
3
10.9 Multiaxial multiply carbon stitched preforms (Lomov et al.18). Biaxial loading in bias directions: (a) 0°/90° for fabric B1 and (b) +45°/–45° for fabric B2.
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the bias direction produces deformation in the direction between the fibres, and stitching ensures a strong link between the two tension directions.18 The test reflects the combination of shear and tension, which may happen in the actual forming. It is interesting to observe that the nonlinear ranges are slightly different for the two multiaxial multiply carbon stitched preforms for the same loading ratio k. Willems et al.19,20 investigated experimentally a twill 2 ¥ 2 glass–PP reinforcement under different load conditions. This is an unbalanced textile with 9.7% crimp in the warp and 0.1% in the weft direction and a yarn linear density of 1870 and 3740 tex.20 The mechanical response of the reinforcement during biaxial tension of a square sample is depicted in Fig. 10.10 for both warp and weft loading directions,19,20 assuming the strain ratio k = 1. The curves in Fig. 10.10 are based on the measurement of the strain components by DIC on the central part of the sample. When applying tension to yarns that are crimped the yarns need to be straightened before being stretched. The large initial warp crimp (9.7%) explains the larger low-stiffness part of the warp tensile curve in comparison to the weft curve. The crimp exchange in the weave basically leads to a displacement shift of the linear high-stiffness part of the curve (Fig. 10.10).
10.3
Analytical model
The difficulties and the costs of experimental tests suggest the use of alternative approaches to predict the mechanical behaviour of composite reinforcements starting from the knowledge of the nominal internal architecture and the characteristics of the fibres and yarns. 100 k=1
Tension (N/mm)
80 Weft 60 40 Warp 20 0
0
0.5
1
1.5 Strain (%)
2
2.5
3
10.10 Biaxial tensile tests of a twill 2 ¥ 2 glass–PP textile reinforcement: tension vs strain in the warp and weft directions (Willems et al.19,20).
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Analytical models are fast and economic tools to predict the mechanical behaviour of some fabrics and textile reinforcements and to evaluate the influence of some geometrical and material parameters. The main purpose of these models is to determine the behaviour of a homogeneous material equivalent, from the mechanical point of view at the macro-scale, to the real reinforcement. The analytical models provide ‘closed-form’ solutions of the loading problems under several approximation hypotheses. Restrictions are imposed on the geometry of the reinforcements, on the mechanical behaviour of the fibres and yarns, etc. The accuracy of the predicted mechanical behaviour depends strongly on the initial hypotheses. The attention of researchers available in the literature is mainly centred on analytical models dedicated to woven textile reinforcements and in particular to the plane weave interlacement.5,28–34 The model presented in Ref. 33 is briefly described here to highlight the theoretical background, the initial hypotheses, and the accuracy in prediction the biaxial properties of plane weave composite reinforcements.
10.3.1 Theoretical background The analytical model described in Ref. 33 is based on a three-dimensional theory of curved beams. The main assumptions of the model are as follows (see Fig. 10.11): 1. The regular distribution of the yarns in the plain weave textile. This allows one to analyse the periodic ‘unit cell’ (meso-scale), i.e. the smallest element that reproduces the real textile by periodic repetition. 2. The representation of the yarn paths by parabolic arcs.
Hy
K K Py
Hz
K K
x
y
z Pz
10.11 Geometry of the unit cell for the analytical model in Ref.33.
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3. The yarn cross-sections are circular with different diameters for unbalanced fabric (this hypothesis is easy to remove). 4. The materials of the yarns are isotropic. 5. The yarn interaction in the direction orthogonal to the textile plane is simulated by elastic springs. The longitudinal shape of yarns is modelled by a set of parabolic beams (four per filament in the unit cell of Fig. 10.11). The solution of the curved beam problem in three-dimensional space is based on the theory of the transmission matrix.35 in the following, bold symbols represent vectors or matrices. Setting a global (x, y, z) and a local (x¢, y¢, z¢) reference system (x¢ being tangential to the beam longitudinal axis and y¢ the z¢ principal axes of the cross-section), the matrix linking the global and local reference systems is (a being the angle between z and x¢): È sinn a 0 cos cos a Í T0 (s ) = Í 0 1 0 Í coss a 0 –s –sin in a Î
˘ ˙ ˙ ˙ ˚
10.1
The vectors S and S¢ collect the mechanical features of the sections in the global and local reference systems, respectively. For a general cross-section n (see Fig. 10.12), located in the position of abscissa s, they are: È n Í Í fn Sn = Í M Í n ÍÎ Fn
˘ ˙ ˙ ˙, ˙ ˙˚
È ¢n Í Í fn¢ Sn¢ = Í M¢ Í n Fn ÍÎ F¢
˘ ˙ ˙ ˙ ˙ ˙˚
10.2
The vectors in eqns [10.2] contain sub-vectors in which are listed the three displacement components , the three rotation components f, the three internal X
Z¢ X¢ a
n S
y¢
Z
Xn
Zn
y
k
10.12 Global (x, y, z) and local (x¢, y¢, z¢) reference systems.
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moments M and the three internal forces F. The link between the global and local quantities is n = T0(s) ¢n and analogously for f, M and F. The vectors in eqns [10.2] for a general cross-section n, as a function of the cross-section k in the origin s = 0 (see Fig. 10.12), are: Sn(s) = H(s)Sk
10.3
The matrix H, called the transmission function, is defined as: È I A12 –A13 –A14 Í 0 I –A 23 –A 24 H(s ) = Í Í0 0 I A 34 Í 0 0 0 I ÍÎ
˘ ˙ ˙ ˙ ˙ ˙˚
10.4
Sub-matrices Aij depend on the mechanical features and geometry of the current cross-section as obtained below. The elastic relations, moments–rotations and forces–displacements for the current cross-section n are: dfn¢ = – V1¢n M¢n ds, d ¢n = – V2¢n Fn¢ ds
10.5
where ds = dx 2 + ddyy 2 + dz 2 , and È 1/GJ n 0 0 Í 0 V1¢n = Í 0 1//EI y ¢n Í 0 0 1/ 1/EI EI z ¢n ÎÍ
˘ È 1/EA 1/EAn 0 0 ˙ Í 0 1//GAy ¢n 0 ˙, V2n 2¢ = Í ˙ Í 0 0 1/GAz ¢n ˚˙ ÎÍ
˘ ˙ ˙ ˙ ˚˙ 10.6
In eqns [10.6], E is the Young’s modulus and G is the shear modulus of the material; I y¢ and I z¢ are the second-order inertia moments with respect to the local axes; Jn is the torque inertia moment; Ay¢ and Az¢ are the shear areas; and An is the cross-section area. The transformation of eqns [10.5] in the global reference system allows one to write eqn [10.3] as (refer to Ref. 33 for details): Ï s Ô n = k + U n f k – Ú0 (U n – U i )V1i Mi ds – Ô s Ô fn = fk – Ú0 V1i Mi ds Ô Ì Ô M n = M k + U n Fk Ô Ô Ô Fn = Fk Ó
s
Ú0
V2i Fi ds 10.7
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where index i represents a cross-section between k and n, and È 0 zn – yn Í U n = Í – zn 0 x n Í y –x 0 n Î n
˘ ˙ T T ˙, V1n = T0 V1¢n T0 , V2n = T0 V2¢n T0 ˙ ˚
10.8
The definition of the components of matrix [10.4] is obtained from [10.7]. These equations give the mechanical response of a textile reinforcement applying suitable kinematic boundary conditions to the unit cell to reproduce the periodicity. The boundary conditions are similar to those employed in Carvelli et al.21 The analytical model requires as input the geometric features of the unit cell (Pz, Py, Hz, Hy: see Fig. 10.11); the diameters of the yarns (Dz, Dy); and the tensile behaviour of each yarn (load vs. strain). Moreover, to describe the interaction of yarns in the direction orthogonal to the textile plane during loading, elastic springs of stiffness K are supposed in the yarn contact points (see Fig. 10.11). The springs stiffness K is obtained assuming the exact solution of the two spherical isotropic bodies in contact (see Ref. 33). This assumption of interaction does not consider the anisotropic nature of yarns in reinforcements for composite materials and, therefore, the real compaction of the yarns in the crossover zones. as experimentally observed in ref. 5, the bending stiffness varies with increasing the axial loading in the yarns. in the presented analytical model the variation of the bending stiffness Bi of the yarns is described with the empirical relation: Bi = Bei [bi exp (Fi/Fui)], i = y, z
10.9
In eqn [10.9] Bei is the stiffness of the isotropic yarn; Fi are the axial forces in the yarn directions; Fui are the ultimate tensile forces of the yarns; and bi are constant parameters identified from biaxial experimental tests. a step-by-step procedure was adopted to predict the behaviour of textile reinforcements during biaxial loading as described in ref. 33. in each step, a small increment of displacements on the boundary of the unit cell is imposed. The solution [10.7] of the current step is used to update the unit cell configuration and to prepare the next step.
10.3.2 Predictions and comparisons The predictions of the analytical model are compared to experimental results for two textile reinforcements: a balanced and an unbalanced plain weave glass fabric. The experimental data for biaxial tests are detailed in ref. 11. The balanced plain weave glass fabric has a density of 0.22 yarns/mm and a crimp of 0.4%. The experimental biaxial responses of the fabric for © Woodhead Publishing Limited, 2011
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two values of the ratio k (k = warp strain/weft strain) are reproduced in Fig. 10.13 (continuous lines). Assuming the yarns’ axial mechanical behaviour in Fig. 10.13, the analytical model, based on curved beams, predicts the biaxial tension vs. strain curves (dashed lines) depicted in Fig. 10.13. The unbalanced plain weave glass fabric has warp and weft density of 0.22 and 0.16 yarns/mm, respectively, and warp and weft crimp of 0.4 and 0.8%, respectively. The good agreement between experimental11 data and predictions of the analytical model is detailed in terms of tension–strain curves for different values of k in Fig. 10.14. For the two reinforcements considered, the analytical model shows accuracy and efficiency. The analytical model provides the mechanical response of 150 k = 0.5
Tension (N/yarn)
Yarn k=1 100
50 Experimental (BuetGautier and boisse11) Analytical 0 0
0.2
0.4 Strain (%)
0.6
0.8
10.13 Biaxial behaviour of a balanced plain weave glass fabric: experimental11 vs analytical. 40 Weft
Tension (N/yarn)
30 k=1
20
Experimental (buetgautier and boisse11)
10 k=2
Analytical
0 0
0.2
0.4 Strain (%)
0.6
0.8
10.14 Biaxial behaviour of an unbalanced plain weave glass fabric: weft tension-strain, experimental11 vs analytical.
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the reinforcement to biaxial loadings in a few seconds. Therefore, this and other analytical models, e.g. from Ref. 30–32, can be considered appropriate tools to investigate, in a preliminary design analysis, the influence of yarn material and geometrical parameters (yarn density, crimp, etc.) on the biaxial tensile properties of plain weave textile reinforcements.
10.4
Numerical modelling
Numerical models, mainly based on the finite element method (FEM), are widely used to predict the mechanical properties of composites. That applications of the FEM in the simulation of the mechanical properties of dry reinforcements under biaxial loading have been less investigated in the literature (see, e.g., Refs 10, 12, 13, 15, 36 and 37) despite this knowledge is an important aspect in the manufacturing of composite components. The three-dimensional FE analysis allows one to consider some peculiar aspects of textile reinforcements that are not included in the above-mentioned analytical models. The purpose of FE models is to determine the behaviour of an equivalent homogeneous material at the macro-scale, as for the analytical models, and to provide information on local phenomena (meso-scale) and their influence on the global mechanical behaviour of the reinforcement. One of the advantages of the FEM consists in the possibility of modelling complex 3D interlacing geometries, whereas only simple textile geometries (i.e. plain weave) have been considered in the available analytical models. The periodic arrangement of the yarns in woven reinforcements allows one to define a model domain as the representative ‘unit cell’ (meso-scale model). An accurate geometrical description of the yarn paths in the unit cell is the first step in modelling. Several mathematical descriptions have been proposed for the internal geometry of some woven reinforcements. In Ref. 38, the authors proposed analytical expressions to completely define the internal geometry of 2D fabrics (including plain weave, twill and satin) whose yarns have constant cross-section along their length. Kuhn and Charalambides39 developed a model for plain weave fabric in which the yarn section varies along the trajectory, taking into account the reorganization of fibres near the contact zones. Hivet and Boisse40 defined a 3D geometrical model of different types of 2D woven fabrics which describes the contact zone between yarns accurately to ensure a consistent contact surface. The mesh of the obtained 3D geometry is produced by commercial FE software.40 Integrated modelling and design tools for textile reinforcements and textile composites were implemented in WiseTex41–44 (see Chapter 7) and TexGen45 software (see Chapter 8) in which the translation to FE models is done automatically. In the first software, the prediction of the yarn geometry is based on the principle of minimum energy of a fibrous material; in the
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second, it is obtained as an assembly of topologically simple volumes that encompass either part of a tow or part of an empty volume. The unit cell represents the textile reinforcement through adequate boundary conditions. These have to reproduce the periodicity of the interlacing and eventually the symmetries, as in plain weave. The periodicity is enforced prescribing kinematic relations between the displacement components on the boundary of the unit cell. Kinematic conditions are detailed in Ref. 46 and applied to in-plane shear of plain weave reinforcements. In Refs 21 and 47 the periodic boundary conditions are obtained in the context of the homogenization theory for periodic textiles and textile composites. Tang and Whitcomb48 derived general formulas for boundary conditions enforcing periodicity and symmetries based on the concept of the ‘equivalent subcell’. To simulate biaxial loading, or other loading conditions, suitable displacement fields have to be enforced on the unit cell. Kinematic boundary conditions that allow the simulations of macroscopic strain components are detailed in Ref. 47 and used for biaxial loading of technical textiles in Ref. 21. The relative displacement of the yarns during loading simulation have to respect the no-penetration condition and the friction contact. In the numerical simulations of reinforcements these are commonly introduced with the master–slave approach.12 The main task in the numerical simulation of the mechanical behaviour of composite reinforcements is the description of the constitutive behaviour of the yarn material. The yarn is made of thousands of very flexible fibres that can slide on each other. In numerical FE modelling the yarns are considered as a continuum whose constitutive behaviour has to take into account the fibrous nature of the reinforcement. The equivalent continuum material must reproduce: ∑ ∑
the very low stiffness in any direction except for that of the fibres; the dependency of transverse stiffness on the yarn compaction.
The first feature is achieved using an elastic orthotropic constitutive model with negligible shear modulus (G12, G13, G23) and Poisson’s ratios (n12, n13, n23), being the reference frame with axis 1 in the longitudinal direction of the fibres. Moreover, the transverse Young’s moduli (E2, E3) are very small in comparison to the longitudinal one (E1). A material with very small stiffnesses, like the equivalent used for yarn simulation, could lead to numerical problems, i.e. spurious zero energy modes. The use of finite elements with reduced integration, based on the hourglass control method,49 avoids this problem. The nonlinear behaviour of a textile reinforcement is mainly geometrical. This important aspect is considered in a FE simulation using the finite strain
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theory. in this context, it is important to assign the mechanical properties in the correct directions during the analysis. in refs 36 and 46 the authors suggest prescribing the stiffnesses in the material directions, i.e. linked to the finite elements in the current position.12 The second feature of the yarn equivalent material is to reproduce the effect of yarn compaction on the deformation of the fabric. The yarns in a textile increase the transverse rigidity, increasing the axial tension and the compression force acting on the crossover zones, because the voids between fibres are reduced. The transverse stiffness increases up to a value much larger than the one at low tension. This is an important aspect in the biaxial tension of the fabric. Kawabata5 performed experimental measurements to have the variation of the yarn diameter as a function of the contact force. The effect of the compaction is represented in the FE model as an evolution of the transverse modulus E3. Boisse and coworkers10,12,36 assume an increase of E3 increasing the transverse and longitudinal strains, as: n m E3 = E0 + E|e 33 |e11
10.10
In eqn [10.10], E0 is the transverse Young’s modulus in the unloaded state, almost zero; and E, m and n are material parameters. In Refs 10 and 12 the identification of the three parameters was obtained with an inverse method using biaxial tests with strain ratio k = 1. The above-mentioned aspects allow the adoption of refined FE models for textile composite reinforcements. FEM has been used in the literature to investigate several aspect of composite reinforcements at the meso-scale: compaction (see, e.g., Refs 50 and 51), mechanical behaviour for different load conditions (see, e.g., Refs 32 and 46), resin flow (see Chapter 19), etc. To highlight some potentialities of the FEM in the simulation of different reinforcements under biaxial loading, some numerical analyses, available in the literature, are hereafter discussed. a glass yarn reinforcement having twill architecture 3 ¥ 1 was numerically investigated by Gasser et al.10 The FE mesh of the unit cell has 24,500 degrees of freedom (Fig. 10.15). The simulation took into consideration the yarn compaction through eqn [10.10]. Figure 10.16 shows the tension vs strain curves of the yarn and of the textile for two values of the strain ratio k. The macroscopic response obtained with numerical simulations is in agreement with the experimental tests for both the applied load conditions. an advantage of a FE model is the possibility to investigate local phenomena like stress and strain concentrations. Figure 10.17 reproduces the results of the FE analysis in terms of transverse strain e33 (logarithmic) distribution and highlights the elevated level of transverse compaction at yarn crossovers.10 The influence of some parameters on the biaxial behaviour of textile reinforcements was investigated by numerical models in the paper of Boisse
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Warp
327
Weft
10.15 Glass twill 3 ¥ 1 reinforcement: FE mesh of the unit cell (Gasser et al.10). 300 Experimental FEM prediction
Tension (N/yarn)
250 200
Yarn
k=1
150 k=2 100 50 0 0
0.2
0.4 Strain (%)
0.6
0.8
10.16 Glass twill 3 ¥ 1 reinforcement: biaxial tension vs strain curves (Gasser et al.10).
10%
22%
30%
10.17 Glass twill 3 ¥ 1 reinforcement: map of transverse strain for biaxial loading with k = 1 (Gasser et al.10).
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et al.12 The variation in the crimp of the yarn produces the biaxial (k = 1) behaviour in Fig. 10.18. The plain weave reinforcements have the same fibres and yarns but differents distance between yarns. The comparison shows an increase of the nonlinear behaviour when the crimp increases, i.e. the reinforcement with larger crimp has the lower stiffness at low strain. The biaxial behaviour of three textile reinforcements having the same glass yarns but different weave pattern (plain weave, twill 2 ¥ 2 and twill 3 ¥ 1) are compared in Fig. 10.19 (Boisse et al.12). The picture shows a similar behaviour of the twill reinforcements while the plain weave is more nonlinear. This is due to the higher crimp of the plain weave with respect to the two twills. Other numerical comparisons to assess the influence of the material properties on the biaxial behaviour of composite reinforcement are described in Ref. 10. These results show that a FE model provides several pieces of information (at both macro- and meso-scales) that are useful during the design of a composite reinforcement for a specific application. The choice of the material, the yarn distribution and the interlacing pattern can be decided before manufacture, reducing the number of experimental tests. This aspect is very important for complex 3D textile reinforcements for which analytical models are not available.
10.5
Conclusions
The biaxial behaviour of a composite reinforcement is an important aspect in understanding the deformation of its internal geometry during draping
125 (a)
(b)
(c)
Tension (N/yarn)
100 (c)
(b)
(a)
75
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0 0
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0.3 Strain (%)
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10.18 Biaxial tension vs strain curves (k = 1) of three plain weave reinforcements with different crimp (Boisse et al.12).
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125 (1)
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10.19 Biaxial tension vs strain curves (k = 1) of three different woven reinforcements (Boisse et al.12): (1) plain weave; (2) twill 2 ¥ 2; (3) twill 3 ¥ 1.
and moulding. In this chapter, some features of the commonly adopted approaches to investigate the biaxial behaviour of composite reinforcements have been discussed, i.e. experimental, analytical and numerical modelling. In the author’s opinion, to improve knowledge of the biaxial behaviour of composite reinforcements the above-mentioned approaches should be focused on the following: ∑ Experimental tests to deeply understand the mechanical behaviour of the yarn as a set of thousands of fibres, mainly in the transverse direction ∑ Experimental biaxial tests for complex interlacements, e.g. 3D weave, 3D braided, etc. ∑ Analytical models for weave patterns frequently employed (e.g. twill, 2D braided), including the orthotropic behaviour of the yarns and the frictional contact between yarns ∑ Numerical simulation of the biaxial behaviour of 3D textile and 3D non-crimp reinforcements, including refined geometric modelling and constitutive behaviour of the yarns to better catch the compaction and local phenomena under biaxial loading conditions.
10.6
References
1. Potluri P, Parlak I, Ramgulam R, Sagar T V, ‘Analysis of tow deformations in textile preforms subjected to forming forces’, Composites Science and Technology, 2006, 66, 297–305. 2. De Luycker E, Morestin F, Boisse P, Marsal D, ‘Simulation of 3D interlock composite preforming’, Composite Structures, 2009, 88, 615–623.
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3. Peng X Q, Cao J, ‘A continuum mechanics-based non-orthogonal constitutive model for woven composite fabrics’, Composites: Part A, 2005, 36, 859–874. 4. Lomov S V, Boisse P, Deluycker E, Morestin F, Vanclooster K, Vandepitte D, Verpoest I, Willems A, ‘Full-field strain measurements in textile deformability studies’, Composites: Part A, 2008, 39, 1232–1244. 5. Kawabata S, ‘Nonlinear mechanics of woven and knitted materials’, in Chou T W and Ko F K (eds), Textile Structural Composites, Amsterdam, Elsevier, 1989. 6. Smits A, Van Hemelrijck D, Philippidis T P, Cardon A, ‘Design of a cruciform specimen for biaxial testing of fibre reinforced composite laminates’, Composites Science and Technology, 2006, 66, 964–975. 7. Welsh J S, Mayes J S, Biskner A C, ‘2-D biaxial testing and failure predictions of IM7/977-2 carbon/epoxy quasi-isotropic laminates’, Composite Structures, 2006, 75, 60–66. 8. Lecieux Y, Bouzidi R, ‘Experimental analysis on membrane wrinkling under biaxial load – Comparison with bifurcation analysis’, International Journal of Solids and Structures, 2010, 47, 2459–2475. 9. Mailly L, Wang S S, ‘Recent development of planar cruciform experiment on biaxial tensile deformation and failure of unidirectional glass/epoxy composite’, Journal of Composite Materials, 2008, 42, 1359–1379. 10. Gasser A, Boisse P, Hanklar S, ‘Mechanical behaviour of dry fabric reinforcements. 3D simulations versus biaxial tests’, Composites Science and Technology, 2000, 17, 7–20. 11. Buet-Gautier K, Boisse P, ‘Experimental analysis and modeling of biaxial mechanical behavior of woven composite reinforcements’, Experimental Mechanics, 2001, 41, 260–269. 12. Boisse P, Gasser A, Hivet G, ‘Analyses of fabric tensile behavior: determination of the biaxial tension–strain surfaces and their uses in forming simulations’, Composites: Part A, 2001, 32, 1395–1414. 13. Boisse P, Buet K, Gasser A, Launay J, ‘Meso/macro-mechanical behaviour of textile reinforcements for thin composites’, Composites Science and Technology, 2001, 61, 395–401. 14. Boisse P, Zouari B, Gasser A, ‘A mesoscopic approach for the simulation of woven fibre composite forming’, Composites Science and Technology, 2005, 65, 429–436. 15. Boisse P, Gasser A, Hagege B, Billoet J L, ‘Analysis of the mechanical behavior of woven fibrous material using virtual tests at the unit cell level’, Journal of Materials Science, 2005, 40, 5955–5962. 16. Teaca M, Charpentier I, Martiny M, Ferron G, ‘Identification of sheet metal plastic anisotropy using heterogeneous biaxial tensile tests’, International Journal of Mechanical Sciences, 2010, 52, 572–580. 17. Luo Y, Verpoest I, ‘Biaxial tension and ultimate deformation of knitted fabric reinforcements’, Composites: Part A, 2002, 33, 197–203. 18. Lomov S V, Barburski M, Stoilova Tz, Verpoest I, Akkerman R, Loendersloot R, Thije R H V, ‘Carbon composites based on multiaxial multiply stitched preforms. Part 3: Biaxial tension, picture frame and compression tests of the preforms’, Composites: Part A, 2005, 36, 1188–1206. 19. Willems A, Lomov S V, Verpoest I, Vandepitte D, ‘Drape-ability characterization of textile composite reinforcements using digital image correlation’, Optics and Lasers in Engineering, 2009, 47, 343–351.
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20. Willems A, Lomov S V, Verpoest I, Vandepitte D, ‘Optical strain fields in shear and tensile testing of textile reinforcements’, Composites Science and Technology, 2008, 68, 807–819. 21. Carvelli V, Corazza C, Poggi C, ‘Mechanical modelling of monofilament technical textiles’, Computational Materials Science, 2008, 42, 679–691. 22. Quaglini V, Corazza C, Poggi C, ‘Experimental characterization of orthotropic technical textiles under uniaxial and biaxial loading’, Composites: Part A, 2008, 39, 1331–1342. 23. Launay J, Lahmar F, Boisse P, Vacher P, ‘Strain measurement in tests on fibre fabric by image correlation method’, Advanced Composites Letters, 2002, 11, 7–12. 24. Hivet G, Boisse P, ‘Consistent mesoscopic mechanical behaviour model for woven composite reinforcements in biaxial tension’, Composites: Part B, 2008, 39, 345–361. 25. Lomov S V, Willems A, Verpoest I, Zhu Y, Barburski M, Stoilova Tz, ‘Picture frame test of woven composite reinforcements with a full-field strain registration’, Textile Research Journal, 2006, 76, 243–252. 26. Zhu B, Yu T X, Tao X M, ‘Large deformation and slippage mechanism of plain weave composite in bias extension’, Composites: Part A, 2007, 38, 1821–1828. 27. Sutton M A, ‘Digital image correlation for shape and deformation measurements’, in Sharpe W N (ed.), Handbook of Experimental Solid Mechanics, New York, Springer, 2008. 28. Peirce F T, ‘The geometry of cloth structure’, Journal of the Textile Institute, 1937, 28(3), 45–96. 29. Hearle J W S, Shanahan W J, ‘An energy method for calculations in fabric mechanics, Part I: Principles of the method’, Journal of the Textile Institute, 1978, 69, 81–91. 30. Kawabata S, Niwa M, Kawai H, ‘The finite-deformation theory of plain-weave fabrics. Part I: The biaxial-deformation theory’, Journal of the Textile Institute, 1973, 64, 21–46. 31. Sagar T V, Potluri P, Hearle J W S, ‘Mesoscale modelling of interlaced fibre assemblies using energy method’, Computational Materials Science, 2003, 28, 49–62. 32. Long A C, Boisse P, Robitaille F, ‘Mechanical analysis of textiles’, in Long A C (ed.), Design and Manufacture of Textile Composites, Cambridge, UK, Woodhead Publishing, 2005. 33. Carvelli V, ‘Monofilament technical textiles: an analytical model for the prediction of the mechanical behaviour’, Mechanics Research Communications, 2009, 36, 573–580. 34. Parsons E M, Weerasooriya T, Sarva S, Socrate S, ‘Impact of woven fabric: Experiments and mesostructure-based continuum-level simulations’, Journal of the Mechanics and Physics of Solids, 2010, 58, 1995–2021. 35. Toniolo G, La Teoria della Matrice di Trasmissione, Milan, Masson, 1979, ISBN: 8821400522. 36. Boisse P, Zouari B, Daniel J L, ‘Importance of in-plane shear rigidity in finite element analyses of woven fabric composite preforming’, Composites: Part A, 2006, 37, 2201–2212. 37. Tarfaoui M, Drean J Y, Akesbi S, ‘Predicting the stress–strain behaviour of woven fabrics using the finite element method’, Textile Research Journal, 2001, 71, 790–795. 38. Adumitroaie A, Barbero E J, ‘Beyond plain weave fabrics – I. Geometrical model’, Composite Structures, 2011, 93, 1424–1432.
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39. Kuhn L, Charalambides P G, ‘Modeling of plain weave fabric composite geometry’, Journal of Composite Materials, 1999, 33, 188–220. 40. Hivet G, Boisse P, ‘Consistent 3D geometrical model of fabric elementary cell. Application to a meshing preprocessor for 3D finite element analysis’, Finite Elements in Analysis and Design, 2005, 42, 25–49. 41. Lomov S V, Gusakov A V, Huysmans G, Prodromou A, Verpoest I, ‘Textile geometry preprocessor for meso-mechanical models of woven composites’, Composites Science and Technology, 2000, 60, 2083–2095. 42. Lomov S V, Huysmans G, Luo Y, Parnas R S, Prodromou A, Verpoest I, Phelan F R, ‘Textile composites: modelling strategies’, Composites: Part A, 2001, 32, 1379–1394. 43. Lomov S V, Truong Chi T, Verpoest I, Peeters T, Roose D, Boisse P, Gasser A, ‘Mathematical modelling of internal geometry and deformability of woven preforms’, International Journal of Forming Processes, 2003, 6, 413–442. 44. Verpoest I, Lomov S V, ‘Virtual textile composites software Wisetex: integration with micro-mechanical, permeability and structural analysis’, Composites Science and Technology, 2005, 65, 2563–2574. 45. Robitaille F, Long A C, Jones I A, Rudd C D, ‘Automatically generated geometric descriptions of textile and composite unit cells’, Composites: Part A, 2003, 34, 303–312. 46. Badel P, Vidal-Sallé E, Boisse P, ‘Computational determination of in-plane shear mechanical behavior of textile composite reinforcements’, Computational Materials Science, 2007, 40, 439–448. 47. Carvelli V, Poggi C, ‘A homogenization procedure for the numerical analysis of woven fabric composites’, Composites: Part A, 2001, 32, 1425–1432. 48. Tang X, Whitcomb J D, ‘General techniques for exploiting periodicity and symmetries in micromechanics analysis of textile composites’, Journal of Composite Materials, 2003, 37, 1167–1189. 49. Flanagan D P, Belytschko T, ‘A uniform strain hexahedron and quadrilateral with orthogonal hourglass control’, International Journal for Numerical Methods in Engineering, 1981, 17, 679–706. 50. Potluri P, Sagar T V, ‘Compaction modelling of textile preforms for composite structures’, Composite Structures, 2008, 86, 177–185. 51. Mahadik Y, Hallett S R, ‘Finite element modelling of tow geometry in 3D woven fabrics’, Composites: Part A, 2010, 41, 1192–1200.
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11
Transverse compression properties of composite reinforcements P. A. K e l l y, The University of Auckland, New Zealand
Abstract: This chapter discusses transverse compression, one of the dominant deformation modes arising in fibrous reinforcement materials during composites forming and manufacture. After a brief review of the experimental procedure to determine fabric compressibility, the standard compaction curve is introduced, together with associated numerical models. The chapter then discusses the distinctive inelastic properties and response of fibrous materials, including their viscoelasticity and plasticity. The concept of locked energy is introduced, and modelling strategies for its incorporation within a general thermomechanical framework of reinforcement compression is discussed. The chapter ends with a short review of current challenges within transverse compression and possible future trends. Key words: fibrous materials, compression, mechanical properties, relaxation, viscoplasticity.
11.1
Introduction
The performance and integrity of any composites part depends intimately on the arrangement of its embedded reinforcement material. This arrangement of the reinforcement will have been brought about by, in general, a complex forming involving a range of deformation modes, such as in-plane tension and shear, and out-of-plane bending. Regardless of the fabric type or application, one of the most important deformation modes, in many cases the dominant mode, is transverse compression, where the fabric is compacted through its thickness. An understanding of fabric transverse compression is required in order to understand reinforcement formability in general. This understanding then assists with issues of structure design, the design of new reinforcement materials themselves, and with the optimisation of manufacturing processes. Transverse compression is particularly important in the liquid composites moulding (LCM) family of manufacturing processes. In these processes, a reinforcement is placed in a mould and the mould is closed, causing the transverse compression of the fabric. This compression is vital since, prior to manufacture, the fibrous material is not yet at the optimum fibre volume fraction; compression increases the volume fraction to that required of the part. A polymeric resin is then injected into the mould and forced under 333 © Woodhead Publishing Limited, 2011
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pressure to fully impregnate and wet-out the fabric, and curing takes place, during or after filling. Transverse compression of reinforcements also occurs during consolidation of prepregs, although the changes in reinforcement volume fraction are not as significant as in the LCM processes. Here, as well as increasing the initial volume fraction, the transverse compression helps to drive out voids from the prepreg. Similarly, transverse compression is a dominant deformation mode in resin film infusion, where resin sheets and reinforcement are compacted in order to bring about integration. Knowledge of transverse compression is essential to an understanding of all these processes. In this chapter, a review of the liquid composites moulding (LCM) processes is first given, since transverse compression is particularly relevant here. This discussion highlights a number of different facets to fabric compression and how, in many situations, knowledge of transverse compression is critical to an overall understanding. This is followed by a discussion of the typical experimental method to determine fabric compressibility, the standard compaction curve and numerical models and analysis of the compaction curve. The latter part of the chapter deals with properties of reinforcements which become important when they are subjected to anything but a simple compaction, and the distinctive inelastic response of composite reinforcements to compression. The important concept of locked energy is also considered, and modelling strategies for its incorporation within a general thermomechanical framework of reinforcement compression are discussed. It should be noted that fibrous materials are subjected to compression in a variety of industrial and technical applications (Jaganathan et al., 2009), and knowledge of their mechanical response is of interest to more than just the composites community (Kothari and Das, 1992). In particular, transverse compression of fabrics is studied extensively by the textile materials community. These investigations complement well the many investigations into fibrous materials carried out by the composites community; both sets of investigations are drawn upon in this chapter.
11.2
Transverse compression of composite reinforcements
As mentioned in the introduction, transverse compression plays a major role in the LCM processes. In the basic LCM process, resin transfer moulding (RTM), the fabric is first compressed in a two-sided relatively rigid mould to the volume fraction required of the composite part, and only then is resin injected, and so involves compression of a dry fabric. Analysis and prediction of resin injection and fabric wet-out can be carried out with little or no knowledge of fabric transverse compression properties; this knowledge, however, is necessary if one wants to know the requirements of the mould,
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that is, what forces are required to close and maintain closure of the mould during the complete process (Bickerton et al., 2003). In compression resin transfer moulding (CRTM), resin is injected before the fibrous material is fully compacted (Wirth and Gauvin, 1998). The fabric can be compacted partly before injection (Pham et al., 1998; Verleye et al., 2011) or a gap can be left between the upper mould and fabric into which the resin can be injected (Chang, 2006; Bhat et al., 2009). Once injection is complete, a ‘wet compression’ takes place, compacting the reinforcement to the final volume fraction and simultaneously driving the resin through any remaining dry fabric. Here, then, compression of both dry fabric and wet fabric is involved. As with the initial compaction, the secondary compression can be brought about by specifying either the applied force or the mould closing velocity. Since, following Terzaghi’s law (Terzaghi, 1943), the force applied to close the mould is taken up by both the fluid (fluid pressure) and the fabric (the compaction stress, the stress carried by the reinforcement as it is transversely compressed), knowledge of fabric compression is critical in predicting how the resin will flow (Gutowski et al., 1987). In another LCM variant, RTM-Light, the moulds are made of relatively compliant material, for example a fibre reinforced plastic, in order to reduce tooling costs. In this case, significant deformations occur in the mould walls themselves (MacLaren et al., 2009). Since the mould-wall deformation is coupled with the compliance properties of the fabric, again the complete mould closure and the manner in which the reinforcement wets out depend on the transverse compression properties of the fabric. As a final example, in the resin infusion (RI) (or vacuum assisted resin transfer moulding (VARTM)) process, one mould is near-rigid whereas the other is a flexible bag (Kessels et al., 2007). A vacuum is pulled within the mould cavity, so that the bag compacts the dry fabric under (usually near to) atmospheric pressure. Resin is then injected: the fluid now takes up some of the (constant) applied pressure, so that the compaction stress taken up by the fabric decreases, causing the volume fraction to decrease. On completion of resin injection, the part will in general contain large gradients in resin pressure and volume fraction. These are most often controlled in a post-filling stage, where excess resin is drawn out of the mould. During this latter phase, the resin pressure decreases and so the fabric compression stress increases (Govignon et al., 2010). In summary then, the RI process can involve transverse compression of dry fabric and then the unloading and a reloading of wet fabric. Other circumstances arise in LCM and other manufacturing processes: for example, a fabric is often loaded and unloaded before resin injection, such as during preform manufacture (Long and Clifford, 2007), or to ‘de-bulk’ it, making a high final volume fraction more attainable. These examples from manufacturing illustrate the range of circumstances
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in which transverse compression arises: transverse compression of dry fabrics and of wet fabrics, single loading and cyclic loading/unloading of dry and wet fabrics, fabrics compressed and kept at a constant volume fraction for a length of time, and movement due to changes in applied force or applied mould closing velocity. The reinforcements themselves can have a wide variety of architectures, preforms can have different numbers of fabric layers, orientation and layup of layers can vary, and these all affect transverse compression response. In this chapter, the response of reinforcements to all these circumstances will be examined.
11.2.1 Experimental procedure for material characterisation The transverse compression response of reinforcements is most easily determined by placing a preform between two fairly thick flat metallic plates installed in a standard universal testing machine. The plates must be kept as parallel as possible, for example using a spherical alignment unit; it should be possible to keep plate misalignment to less than 0.01° (Walbran, 2011). Machine compliance needs to be carefully accounted for (Kruckenberg and Paton, 2004). Wet reinforcement samples can be studied using confined moulds. Test fluid can be injected from a pressure pot into the fabric-containing mould (Govignon et al., 2010). Newtonian test fluids, i.e. mineral oils, can be used instead of standard resins in order to save experimental time. The fluid pressure at the injection point can be monitored using a pressure transducer, if required. A displacement sensor is used to monitor actual cavity thickness during the experiments, taking into account compliance of the facility. From conservation of mass, the thickness h of a sample can be related to the volume fraction Vf through Vf =
Am N rs h
11.1
where Am is the areal mass of the fabric, N is the number of layers and rs is the density of the fibres. In the case of saturated samples, the total applied force taken up by the fluid can be calculated using the theory of Darcy flow and, following Terzaghi’s law, this can be subtracted from the force applied through the mould to obtain the reinforcement compaction force. For example, for an isotropic preform compacted with constant velocity h , with fluid occupying an annular region with inner radius ri (where fluid is injected) and an outer radius rf, the total fluid force is (Bickerton and Buntain, 2007) Ffl = p
˘ mh 1 È 4 Ê rf ˆ 1 2 2r ln Á ˜ + (rrf – ri2 ) (rf2 – 3ri2 )˙ Kh 4 ÍÎ i Ë ri ¯ 2 ˚
11.2
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where h is the current preform thickness, K is the (thickness-dependent) fabric permeability, and m is the fluid viscosity. As can be seen from Eqn 11.2, this force can be neglected provided the viscosity is very low or the sample is compressed very slowly. As an alternative to Eqn 11.2, if one measures the pressure pinj at the injection gate, the fluid force can be expressed in terms of pinj, ri and rf alone, obviating the necessity of measuring the permeability. A number of standard tests can be carried out. In the stress relaxation test, the material is compacted (usually at constant velocity) to a given volume fraction and then that volume fraction is held constant, allowing the stress to relax (reduction in stress at fixed volume fraction). In the creep test, the material is compacted using a (usually linearly ramped) force until a final force is achieved. This force is maintained, allowing the material to creep (continuing compaction at constant stress). The cyclic test involves repeated compaction and unloading to some fixed volume fraction or to some fixed level of applied force. Experiments should be repeated a number of times: even with a perfect experimental set-up and procedure, there will inevitably be some variability in results due to material variability; this variability should be determined and quantified (see further below). The response of a sample of 10 layers of E-glass plain weave fabric, areal mass 232.8 g/m2, to a cyclic test, repeatedly loaded to a fixed volume fraction of 60%, is illustrated in Fig. 11.1, plotted on a graph of compaction stress versus volume fraction. This response is typical of any fibrous material, whatever its architecture or constituents. A notable feature here is the cyclic softening, that is, the reduction in the stress required to reach the target volume fraction with each subsequent loading cycle. The unloading curve 0.4
Stress (MPa)
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Cycle 5 Cycle 6
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11.1 Cyclic compaction of a plain weave fabric to 60% volume fraction.
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falls below the compaction curve for any given cycle, forming a hysteresis loop. With each subsequent cycle, the hysteresis loops decrease in size and grow closer, until eventually a single asymptotic stabilised equilibrium hysteresis loop is reached and repeats indefinitely; this usually occurs after about 10 cycles, but for some materials this can take up to 20 cycles or more to achieve (of course, this depends on the tolerance and accuracy specified for two loops to be ‘close together’). The free volume fraction, that is, the volume fraction at zero stress, for this material was initially measured to be Vf0 = 0.251. This free volume fraction increases with cycling, as indicated in Fig. 11.1, until it reaches an equilibrium free volume fraction of Vf0 = 0.377 corresponding to the equilibrium hysteresis loop. Shown in Fig. 11.2 (after Fig. 3a of Kelly, 2011) are the results of stress relaxation experiments carried out on samples of 10 layers of continuous filament mat, areal mass 450 g/m2, compacted to a volume fraction of 35% at three different speeds. These tests illustrate the increase in compaction stress with compaction speed and the significant stress relaxation which can occur (close to 30% in less than one minute for the 5 mm/min test).
11.2.2 The compaction curve The compaction curve for a fibrous material is the compaction stress versus volume fraction curve for a single (usually the first) compaction. The compaction curves for samples of various materials are shown in Fig. 11.3. These are for (1) random mat (OCF8610) at 5 mm/min; (2) stitched unidirectional (Cofab-A0108) – data points are for stresses after one minute of relaxation; (3) plain weave fabric (Wovmat 1124) at 2 mm/min; (4) as (3) but saturated; and (5) non-crimp bidirectional (NCS 81053) at 2 mm/ v = 5 mm/min
600
v = 0.5 mm/min
Stress (kPa)
v = 2 mm/min 400
200
0
1000 Time (s)
2000
11.2 Stress relaxation experiments on a continuous filament mat; material was compacted at different speeds v and held at 35% volume fraction.
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1.2 Stress (MPa)
339
UD-stitched PWF-dry
0.8
PWF-wet NCS
0.4 0 0
0.2
Vf
0.4
0.6
11.3 Compaction stress vs volume fraction compaction curves for a random mat, a stitched unidirectional fabric, dry and wet plain woven fabric, and a non-crimp bidirectional (NCS).
min. (1), (3), (4) and (5) are experiments numbers 185, 308, 309 and 13 from Robitaille and Gauvin (1999) respectively; (2) is from Trevino et al. (1991). A large volume of similar data is available in the literature (see the references to many of these studies in what follows). The initial phase of the compaction curve is more or less linear. This is followed by a second phase where the gradient increases rapidly. Finally, there is a third phase where the stress again increases more or less linearly with steep gradient. During the first, low-stress, phase, fibres within yarns become closer together and fibres and yarns undergo a small amount of bending. Low-resistance nesting of layers also occurs (Saunders et al., 1998). During the second phase, the number of fibre–fibre contacts increases and significant resistance is provided by fibre/yarn bending. All the while, friction between fibres must be overcome as fibres slide over one another. With woven fabrics, yarns will flatten significantly. Finally, in the final phase, the fibres are in close contact and further bending is difficult, so that the resistance to load is dominated by individual fibre compression (Potluri and Sagar, 2008). Reinforcement architecture affects the compaction curve. In general, the more room for fibre movement, the more compressible the material (Kim et al., 1991). For example, unidirectional fabrics are usually more compressible than plain weave fabrics (Endruweit et al., 2002), because the yarn contacts in the latter provide resistance to fibre movement. Similarly, the stitching in stitched fabrics will inhibit movement and an increase in stitching density will reduce compressibility (Ogale and Mitschang, 2007). Random mats, with their initially low free–volume fraction, are highly compressible; however, when the volume fraction is increased significantly, they become less compressible due to the large number of fibre–fibre contact points and the resistance to fibre movement provided by a compacted entangled mass of fellow fibres. One should distinguish between the compressibility, that is, the change in
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volume fraction with change in stress, and the actual applied stress; even though a random mat is generally more compressible than a woven fabric (see Fig. 11.3), the compaction stress of the former will in general be much larger than that of the latter at any given volume fraction. The number of layers or plies of fabric in a lay-up affects the compaction curve. Generally, the more layers there are, the lower the compaction stress, with this increase in compressibility becoming less significant with each newly added layer (Saunders et al., 1999). Although the stress will be lower, the compressibility will in general be lower too. The explanation for this is that an increase in the number of layers increases the opportunity for nesting between layers (Mogavero and Advani, 1997), and fibre movement in general. This can happen with all reinforcement types, for example the loops of a layer of knitted fabric can intermingle with those of an adjacent layer (Luo and Verpoest, 1999). By the same token, lay-ups and stacking sequences which inhibit nesting between layers result in stiffer materials (Vidal-Sallé et al., 2010). (See Fig. 3 of Pearce and Summerscales, 1995, and Fig. 8 of Robitaille and Gauvin, 1998, for counterexamples of this behaviour.) The compressibility of stacks of layers consisting of different materials can be related to the compressibility of the individual material layers using a type of ‘rule of mixtures’ (see, for example, Trevino et al., 1991; Luo and Verpoest, 1999). As mentioned, there will be some variability in results for compaction stress. In general, variability will be more evident for samples which allow significant nesting between layers. An example of this variability is evident in Fig. 15(b) of Bickerton et al. (2003), which shows a scatter of results for compaction stress as a function of compaction speed for a sample of plain weave fabric, rather than the expected smooth increase in stress with speed (as seen in Fig. 11.2). Even when nesting is controlled for, variability can still be significant, due to the variability in structure between samples (ComasCardona et al., 2008). It should also be noted that, whereas the compaction stress is determined by dividing a total applied force by an area (of sample), there are wide variations in compaction stress over the surface area of any sample of reinforcement; Comas-Cardona et al. (2008) and Walbran et al. (2009) have used a Tekscan pressure measurement system to show that architecture variations lead to wide spatial variations in stress in any one sample. As with the compaction curves for the PWF material in Fig. 11.3, saturated fabric requires a lower stress to compact to a given volume fraction than the corresponding dry fabric; this has been observed many times, e.g. by Kim et al. (1991). This has been attributed to a lubricating effect of the fluid on the fibres. For materials with low fibre mobility such as non-crimp stitched fabrics, this wet/dry lubrication effect will be less significant than for highmobility fabrics (Endruweit et al., 2002). For similar reasons, Walbran (2011)
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reports a low lubrication effect for a chopped strand mat (CSM), believed to be due to the resistance to fibre movement brought about by the binding material present in the CSM. Micro-mechanics models Theoretical work on the compression response of fibrous materials began, essentially, with the celebrated paper by Van Wyk (1946). Considering a random assembly of fibres, and assuming that the fibres displace under force according to the elementary elastic beam theory, Van Wyk derived the power law relation sf = A(V f3 – V3f0)
11.3
where sf is the compaction stress, Vf is the fibre volume fraction, Vf0 is the free fibre volume fraction and A is a material parameter related to the stiffness of the fibres. Many other researchers have followed Wan Wyk’s methodology, which involves specifying a probability density function giving the probable orientation of fibres spatially, so estimating the number of fibre-to-fibre contact points in an assembly, determining how the number of contacts increases with compaction, and hence relating stress, via the beam theory, to fibre volume fraction. For example, the increasing fibre alignment with compaction, allowances for different material symmetries and other refinements have been made (Stearn, 1971; Komori and Makishima, 1977; Lee and Carnaby, 1992; Toll and Manson, 1995). Note that this formulation and analysis has many similarities to that in the extensive literature on network models, that is, on models of fibrous masses with bonded/linked contact points (see, for example, Åström et al., 1994; Narter et al., 1999; Head et al., 2003; Heussinger and Frey, 2006). The methodology cannot be easily used to analyse reinforcements with more organised architectures, for example woven and knitted fabrics, and has not been used widely in more recent times; however, it is still used in applications related to composites reinforcement deformation (Rawal, 2009). The approach just described is a micromechanical one, where the fibre diameter, fiber rigidity, etc., are specified and used in the model. Another micromechanical model, one of the first developed specifically for composite reinforcements, and applicable to fabrics consisting of bundles of aligned fibres, was developed by Gutowski (Gutowski et al., 1987; Gutowski, 1997). Using the elementary beam theory, and assuming the fibres undulate slightly, leads to the result
sf = k
(Vf /Vff0 – 1) (1/Vf – 1/Vf max )4
11.4
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volume fraction and Vf0 is the free volume fraction – since these are difficult to determine experimentally, they are often treated as adjustable parameters in the model. Although developed strictly for aligned fibres, this model is sometimes used now to capture the compaction curve for any reinforcement, regardless of architecture. Another micromechanical approach, particularly useful in the study of knitted and woven fabrics, is to analyse the response of individual yarns and tows, or ‘unit cells’ of tows. These models take the precise fabric structural geometric details, for example yarn cross-sectional shape, and predict the deformation under load. The yarns are treated as deforming continuum solids, so no allowance is made for the movement of individual fibres within yarns. In this respect, they are more precisely known as meso-scale models (neglecting as they do the fibres of the micro-scale). The more simple of these models assume yarns which bend according to the beam theory, and result in relations of the form sf = f (Vf), sometimes implicit relations, which need to be solved numerically (see, for example, Lekakou et al., 1996; Chen and Chou, 1999; Batch et al., 2002; Choi and Tandon, 2006). Lomov, in the 1990s, developed a more comprehensive computer model which can deal with any textile architecture and solves for the yarn deformations by applying energy conservation principles (Lomov and Verpoest, 2000; Lomov et al., 2001), out of which the WiseTex software, detailed in Chapter 7, arose. This model is now very detailed, for example involving variable volume fraction within yarns, and the unit-cell models can be incorporated into full-scale finite element (FE) analyses using homogenisation techniques (Lomov et al., 2007). The TexGen software, discussed in Chapter 8, can also be used to generate fabric unit-cell geometries; this geometry can then also be incorporated into full-scale FE software to model textile compression (Lin et al., 2008). Nesting, although extremely difficult to predict because of its random nature (Potluri and Sagar, 2008), can be accounted for by examining the extreme cases of zero nesting and maximum nesting, and using a parameter to then model intermediate degrees of nesting (Chen and Chou, 2000). Lomov et al. (2003) have investigated randomness of nesting using the WiseTex software; this information can then be used to predict scatter and variability of compaction-dependent properties such as fabric permeability and, ultimately, component strength. Empirical models The early probability-orientation elastic models, e.g. Eqn 11.3, are often not of sufficient accuracy for many fabrics, and the meso-scale models are rather complex for the modelling and prediction of the simple compaction curve. For these reasons, a number of compaction-curve relations have been established without recourse to physical considerations and as such
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are simply very useful and easy-to-implement empirical relations. The most commonly used relation is the general power law (Sebestyen and Hickie, 1971; Trevino et al., 1991; Robitaille and Gauvin, 1998; Kruckenberg and Paton, 2004):
s f = mVfn
11.5
where m and n are material parameters. This relation, a generalisation of Eqn 11.3, has been used to model a wide variety of materials, from random mats to textiles. The power index n typically takes a value in the range 3–19 (Robitaille and Gauvin, 1998), depending on the particular architecture of the material, though for most fabrics it lies in the range 5–10. In general, the larger is m, the smaller is n (Correia, 2004). The main advantages of Eqn 11.5 are that it fits data from a wide range of materials and that it can easily be inverted so as to express volume fraction in terms of compaction stress. One physically unappealing drawback is that the function does not pass through the data point (sf, Vf) = (0, Vf0), where Vf0 is the free volume fraction. Further, there is the possibility that Vf can become larger than 1 (Neckárˇ, 1997). A more problematic feature is that the relation is often not of sufficient accuracy when used over a wide range of volume fractions. In fact, the compressibility b is seen to be dVf b= 1 = 1 Vf ds ns
11.6
which is a rather constrained model for the observed changeable compressibility of fabrics during compaction over a wide range of volume fraction. One way to overcome this problem is to use a number of piecewise functions, but this can be cumbersome (Govignon, 2009). An attractive option is the simple polynomial (Kelly, 2011)
s f = S Ai (Vf – Vf dat )i , A0 = s fdat i =0
11.7
where Vf dat is a datum volume fraction at which the corresponding datum compaction stress is sf dat. The datum data pair (sf dat, Vf dat) can be taken to be (0, Vf0); however, whereas Vf0 is difficult to determine, Vf dat can be determined by applying a small nominal compaction stress, for example sf dat = 10 kPa. Polynomials of this type have been found to be extremely accurate for almost any material over any range of volume fraction, when the polynomial is of order 4 (Bickerton and Buntain, 2007; Kelly and Bickerton, 2009). Although Eqn 11.7 can have many material parameters, this is not a disadvantage, since the key to the effectiveness of a material model is the number of experiments and labour required to find these parameters; as with Eqn 11.5, but a single experiment is required, and the parameters can be found from a simple least squares fit. Problematic polynomial ‘wiggle’,
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which can cause numerical problems, can be avoided by minimising the square of the least squares residuals subject to constraints which maintain a continuously increasing gradient with increasing volume fraction. The main disadvantage of Eqn 11.7 is that it is not easily inverted. A number of other relations are sometimes used, for example involving exponential relations such as those used by Kothari and Das (1992) and Kang et al. (2001). Finally, Merotte et al. (2010) have proposed a useful four-parameter model involving the hyperbolic tangent function, which also features the advantage of being able to capture the compaction response over a wide range of volume fractions. Moreover, it also incorporates the fact that the volume fraction cannot exceed some maximum possible volume fraction Vf max, with the stress sf Æ • as Vf Æ Vf max (as does the model of Gutowski, Eqn 11.4; see also the model of Neckárˇ, 1997). The lubrication effect of saturated fabrics can be modelled simply by using two sets of parameters in the various relations (Eqns 11.3–11.5, 11.7), one set for a dry fabric and one set for the corresponding wet fabric. Alternatively, the relations can be adjusted by introducing a parameter which accounts for the drop in stress with lubrication (see, for example, Andersson et al., 2003).
11.3
Inelastic response of fibrous materials
The models associated with Eqns 11.3–11.7, and all the other models mentioned above, are (non-linear) elastic, in that the current stress depends on the current volume fraction and not on the loading history (and similarly for many other models of deforming fibrous materials, e.g. Badel et al., 2008). Physically, one can imagine the reinforcement fibres to be undergoing but one micro-level deformation mode, that of bending, as in the aforementioned micromechanical models of Van Wyk, Gutowski, and Chen and Chou. All energy input is stored as elastic strain energy within the bending fibres (see, for example, Potluri and Sagar, 2008). During transverse compression, however, there are undoubtedly a number of micro-level deformation modes occurring apart from fibre bending (or elastic torsion), leading to inelastic effects. One could argue that, even if these inelastic effects are large (and they are – see below), the compaction curve can be modelled using a purely elastic model, with suitably adjusted parameters. The reason for this is that the inelastic effects cannot be revealed from a single compaction curve, but only become evident upon halting or reversing the compaction. There are a number of observable effects of reinforcement inelasticity. One way to see these is to perform a recovery test. Here, the material is compacted at some compaction speed to some fixed volume fraction. After holding the material for a length of time, the load is released rapidly. One
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will observe, in the terminology of Susich and Backer (1951), an immediate elastic recovery (IER), delayed recovery (DR) and a permanent set (PS). The IER, or ‘spring-back’, is usually taken to be the recovery observed within a few seconds, whereas the DL is the recovery observed after a lengthy period of time; the PS is the deformation which still remains, so that
Total deformation = IER + DR + PS
Susich and Backer (1951) carried out recovery tests on 16 different types of fibre in tension, including glass (IER = 78%, DR = 19%, PS = 3%), polypropylene (IER = 35%, DR = 58%, PS = 7%) and acrylic (IER = 30%, DR = 47%, PS = 23%). The importance of these tests in the present context is that they illustrate the possible significance of inelastic effects due to the inherent material properties of the reinforcement fibers themselves and their usually polymeric bindings and sizings (as opposed to inelasticity due to the arrangement of the fibres within a fabric). Somashekar et al. (2006) carried out similar recovery tests on glass fibre reinforcements. These clearly show the large inelastic effects which arise when composite reinforcements are subjected to compression. For example, in one test on a plain woven fabric, IER = 39%, DR = 25% and PS = 36%. It is sometimes stated that ‘frictional’, i.e. inelastic, effects are not important in many analyses of fibrous materials. However, this must be argued alongside the fact that extremely large amounts of inelastic effects can occur; apart from the significant DR and PS just mentioned, further below in Section 11.3.2 it will also be demonstrated that up to 75% (as opposed to 0% for an elastic material) of the energy expended on deforming a fibrous material is not returned upon unloading. These inelastic phenomena of delayed recovery and permanent set are the result of the viscoelasticity, plasticity and, more generally, viscoplasticity of fibrous materials. These material characteristics are discussed next.
11.3.1 Viscoelasticity, plasticity and viscoplasticity An important inelastic property of reinforcements is their viscoelasticity (time-dependence). Fibrous materials have long been known to respond in a viscoelastic manner when loaded (Chapman, 1970; Phillips and Ghosh, 2005). Viscoelastic phenomena have been observed in fabrics of many different architectures and consisting of many different fibre types, including, to mention just some, acrylic (Korkmaz and Kocer, 2010), polypropylene (Dayiary et al., 2010), elastane (Šajn et al., 2006) and glass (Kim et al., 1991). The viscoelastic effects are believed to be due to time-dependent movement and rearrangement of, at the fibre level, fibre molecules and, at the yarn scale, individual fibres within yarns, or inherent viscoelastic properties of fibre sizings, although the precise natures of the dominant mechanisms
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are still unclear. Viscoelastic effects are often attributed to ‘fibres sliding over one another’ during a rearrangement – if this is true, this sliding must be time-dependent and so may imply time-dependent frictional properties. One manifestation of the viscoelasticity is the aforementioned timedependent delayed recovery upon load removal after compaction (see also the work of Korkmaz and Kocer, 2010). A second manifestation of viscoelasticity is stress relaxation, as seen in Fig. 11.2. Again, this has been observed many times, examples being the relaxation observed in E-glass stitched and random mats studied by Trevino et al. (1991) and the relaxation observed in knitted fabrics documented by Matsuo and Yamada (2009). In general, the larger the volume fraction at which the fabric is held (or, equivalently, the larger the stress reached at the onset of relaxation), the larger the reduction in stress. Further, the amount of relaxation is strongly influenced by the rate of compaction (Bickerton et al., 2003). Fabrics also creep, that is, when subjected to a constant compaction stress, they undergo a continuing increase in volume fraction over time (Kruckenberg and Paton, 2004). A final manifestation is that different results are obtained at different compaction speeds, for example the increase in stress required to compact a fabric to a given volume fraction with an increase in rate of compaction, as seen in Fig. 11.2 (Pearce and Summerscales, 1995). Some investigators have found this latter effect to be not significant; for example Saunders et al. (1998), observed that the maximum compressive stress required to compact a plain woven fabric to 50% volume fraction increased by less than 5% for a compaction speed increase from 0.05 mm/min to 1 mm/min. On the other hand, Bickerton et al. (2003) showed that an increase in compaction speed from 0.5 to 10 mm/min effects a 13% increase in compaction stress for a continuous filament mat compacted to 41.5%. Viscoelastic phenomena often become more significant for fabrics with less restrained fibres, fabrics with larger numbers of layers, and fabrics which are saturated (Kruckenberg and Paton, 2004). Following on from the discussion of fabric compressibility, this seems to give weight to the idea that viscoelastic effects are associated with fibre mobility and rearrangement. However, in the study by Somashekar et al. (2006), increasing the number layers of a continuous filament random mat appeared to reduce viscoelastic effects; it was hypothesised that this was due to the intertwining of fibres from adjoining layers, restricting movement and thus the possibility of viscoelastic rearrangement. Regardless, it is difficult to compare viscoelastic effects across different fabrics in a systematic way, since the phenomena vary so much for any given fabric. For example, depending on the compaction speed and the volume fraction at which the fabric is held, Bickerton et al. (2003) report stress relaxation reductions of 17–44% for a chopped strand mat, 21–31% for a continuous filament mat and 47–55% for a plain weave. Fibrous materials exhibit hysteresis when subjected to a cyclic load and
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unload, as seen in Fig. 11.1, indicating inelastic energy loss during the deformation (see Section 11.3.2 below) (Gao et al., 2010; Dayiary et al., 2010). This hysteresis is a result of two properties, and can result from either alone; the first is the aforementioned viscoelasticity, the second is the ability of fibrous materials to undergo rate-independent, i.e. non-viscous, permanent deformation (Olofsson, 1967; Dunlop et al., 1974; Dunlop, 1983; Zhang et al., 2000). This permanent deformation is the permanent set (PS) mentioned earlier in connection with the recovery test. This can be quantified by measuring the free volume fraction at the end of each cycle of loading. As discussed in connection with Fig. 11.1, this free volume fraction will increase with cycling and eventually converge on an equilibrium value (at which the permanent set is maximised). Some other notable features of this permanent deformation are that it is unaffected by a length of holding at constant volume fraction, most of it occurs during the early cycles of loading, and it generally increases with the number of layers (Somashekar et al., 2006). Note the widely held assumption here that any permanent deformation is the result of a rate-independent micromechanism, assumed to be time-independent frictional sliding between fibres. This discounts the possible occurrence of permanent deformations arising due to viscous effects alone, as in a Maxwell fluid, for example. The term plastic is used to denote a material which undergoes rateindependent permanent deformation. The term is strongly associated with engineering materials such as metals; however, fibrous materials respond quite differently to the standard engineering materials. For example, metals ‘flow’ with ever-increasing strain when stressed highly enough; fibrous materials, on the other hand, display the opposite effect of the strain reaching a limit with the associated stress increasing indefinitely. Fibrous materials are in this sense locking materials (see, for example, Prager, 1957). Plasticity theory also invariably involves a distinct yield point, at which elastic behaviour ends and permanent deformations begin (but see, for example, Lubliner, 1980); however, it is difficult to discern any distinct yield point in the compaction curve of a fibrous material. Another distinguishing property of fibrous materials is that they undergo hysteresis with ever-increasing permanent set with cycling, with no reversal of load; engineering materials require load-reversal (tension–compression) in order to develop a cyclic hysteresis pattern similar to that shown in Fig. 11.1. Further, rate-independent plasticity theory is almost universally taken to mean that the unloading is purely elastic. However, with fibrous materials, there is the possibility of significant plastic straining (energy loss) during unloading, due to the frictional sliding of fibres as they unbend. Also, the cyclic softening observable in Fig. 11.1 is dissimilar to the response of most metals (though not fully hardened metals), which display the opposite effect of (limited) cyclic hardening. In respect of these inelastic responses, fibrous materials behave similarly in
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some ways to soils and sands, which also exhibit a frictional-type response. Also, unsurprisingly, there are many similarities to the response of materials consisting of fibrous networks, for example biological soft tissue, fibrous connective tissue and bio-artificial tissues. For example, fibroblast populated collagen matrices (FPCMs) (Wagenseil et al., 2003) display large hysteresis accompanied with strain softening behaviour. See also the study by Liu et al. (2007) on leather (fibrous collagen material) and Lobosco and Kaul (2001) on paper. More generally, a fibrous material is viscoplastic, exhibiting as it does both viscous and plastic effects. To be more specific, consider the schematic of a typical force–displacement curve for a fibrous material under cyclic compaction shown in Fig. 11.4. The solid line(s) is the equilibrium curve: this is the theoretical response of the material were it to be compacted ‘infinitely slowly’ (no rate effects). The dotted line is the response at some non-zero compaction velocity – the greater the viscoelastic properties, the greater the deviation is this actual response curve from the equilibrium curve (the response curve may be above or below the corresponding equilibrium curve at any given instant, whether loading or unloading, depending on the ratedependence and loading history). If the compaction is velocity-controlled, and the compaction is paused so that the volume fraction is held fixed, the
1st cycle A
Stress
B
2nd cycle
C
D
Permanent deformation during 2nd cycle Volume fraction
11.4 Schematic of the stress–volume fraction response of a fibrous material to two load/unload cycles to a given maximum volume fraction. The solid line is the equilibrium curve (at ‘zero’ compaction velocity). The dotted line is the response at some non-zero compaction velocity. The movement from point A to point B shows stress relaxation and the movement from C to D shows creep.
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stress will relax towards the equilibrium curve, as from point A to point B in Fig. 11.4. If, on the other hand, the compaction is force-controlled, and the stress is held constant, the material will creep toward the equilibrium curve, as from point C to D in Fig. 11.4. A convenient way to classify material response is now as follows (see, for example, Haupt, 2000). An elastic material is one whose equilibrium curve has no hysteresis and there are no rate effects (a single solid line). A rate-independent plastic material is one whose equilibrium curve has hysteresis but there are no rate effects (solid line(s) only). A viscoelastic material is one whose equilibrium curve has no hysteresis but there are rate effects. Finally, the viscoplastic material has both the hysteresis equilibrium curve and rate effects. Accordingly, a fibrous material is significantly viscoplastic. This can be seen in Fig. 11.5, which shows results for cyclic tests carried out on a continuous filament mat at two different speeds, 1 mm/min and 25 mm/min, to volume fraction 40%.
11.3.2 Locked energy When fibrous materials are compacted and then unloaded, the fibres will not unbend to their original configurations due to the frictional constraints at the contacts. This implies that the elastic strain energy due to bending is not all released during unloading; some of this energy is locked into the sample. This concept of locked strain energy was noted and used by Grosberg (1963) in a calculation of the fibre withdrawal force from a sample of fibres.
25 mm/min (cycle 1)
800
25 mm/min (cycle 20) 1 mm/min (cycle 1) 1 mm/min (cycle 20)
Stress (kPa)
600
400
200
0 0.0
0.1
0.2 Volume fraction
0.3
0.4
11.5 Cyclic loading of a continuous filament random mat to 38% volume fraction; cycles 1 and 20 are shown for compaction speeds of 1 mm/min and 25 mm/min.
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Carnaby and Pan (1989) also note that, when the compaction stress is reduced to zero upon unloading, a fibrous sample will still contain strain energy locked into the assembly of bent fibres by virtue of the internal friction. (The presence of locked energy has been noted in other classes of material (e.g. Maugin, 1992); it has been used more recently in the analysis of soils (Collins, 2005) and asphalt (Drescher et al., 2010), wherein it is referred to as frozen energy.) With negligible change to preform cross-sectional area, and assuming a quasi-static process, the work done W1 (per unit uncompacted volume V0) in compacting a sample from an initial thickness h0 to a final thickness hmin can be calculated from W1 = – 1 V0
hmin
Úh
0
F dh = Vf0
Vf max
ÚV
f0
s dV f Vf2
11.8
Here, Vf0 is the initial uncompacted, free volume fraction, Vf max is the final volume fraction, F is the varying compaction force and s is the compaction stress (F, s > 0 in compression). A similar calculation gives the work W2 done during unloading (this work is negative but, for clarity, the W’s in what follows are taken to be positive). The percentage work differences (W1 – W2)/ W1 ¥ 100 for samples of 10 layers of chopped strand mat (CSM) compacted to 42.5% volume fraction at two different compaction speeds, and also for the plain weave featured in Fig. 11.1 (compacted to a volume fraction of 60%), are shown in Table 11.1. As can be seen, there is a reduction in the work difference as the number of cycles proceeds, but there is still a significant work difference even after 20 cycles for both materials. For example, with the CSM material compacted at 25 mm/min, there is a 78% work difference during cycle 1 and a 64% work difference in cycle 20. The work difference W1 – W2 is conventionally considered to be the energy loss (per unit uncompacted volume) during the cycle (see, for example, Kothari and Das, 1992; Luo and Verpoest, 1999; Robitaille and Gauvin, 1999; Das and Pourdeyhimi, 2010). However, this is not appropriate for fibrous materials, because some of this energy is not irreversibly lost. As mentioned above, some of this energy remains locked in the fibres, and cannot be accessed without applying tension to the material. Thus,
Table 11.1 Percentage work difference (W1 – W2)/W1 ¥ 100, where W1 is the work done CSM (0.05 mm/min) CSM (25 mm/min) Plain weave (5 mm/min) First cycle 61.1% Fifth cycle 44.5% Twentieth cycle n/a
77.7% 68.9% 64.0%
72.9% 59.1% 58.4%
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11.9
W1 – W2 = (W1s – W2s) + (W1d – W2d)
where the subscript s denotes ‘stored’ and the subscript d denotes ‘dissipated’. The term inside the first pair of parentheses on the right-hand side of Eqn 11.9 is the locked energy. The term inside the second pair of parentheses gives the energy loss. The energy loss during compaction is W1d and, as mentioned, there is also the possibility of energy loss during unloading W2d. Most analyses of fibrous materials have been carried out from a force perspective, rather than an energy perspective, including the two studies which mention locked energy cited at the start of this subsection. This has made it difficult to incorporate the locked energy concept, even if one were aware of it, into modelling methodologies in a systematic way. Towards the end of the next section is discussed a straightforward means by which this can be achieved.
11.4
Inelastic models of reinforcement compression
When the inelastic effects need to be accounted for, inelastic models of reinforcement compression are required. Some of these models are discussed briefly in this section.
11.4.1 Viscoelastic models There are not many micro- or meso-scale mechanical models which incorporate viscous effects, mainly because the precise micro-level viscous mechanisms are simply unknown. Even were the mechanisms to be postulated, this would most likely require one to consider a yarn to be an assembly of many thousands of contacting individual fibres moving relative to one another in some way, and would, at present, be very expensive computationally to simulate. For this reason, it is almost universal to treat fibrous materials as continua when analysing their viscoelasticity, that is, microstructural geometry is not accounted for in any detail, but only perhaps indirectly. The most oft-addressed viscoelastic effect is the stress relaxation, because it arises most often in applications and it is not too difficult to determine experimentally. Fabrics can be classified as to their tendency to relax through the use of simple relations of the general form sf = f (t), where t is the time since the onset of relaxation. As mentioned, it is important to record the conditions under which the relation is valid, e.g. the compaction speed, the volume fraction at which the sample is held, the number of layers and so on. One can use stress reductions according to power laws in time (Robitaille and Gauvin, 1998) or logarithms of time (Luo and Verpoest, 1999), but the most commonly used expressions involve exponentials (Pearce and Summerscales, 1995).
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Spring/dashpot models are useful for modelling viscoelasticity because, although relatively simple, they offer some intuitive feel for the physics involved. Examples of such models in studies of a range of fibrous materials under various test conditions can be found in Gotlih (1998), Manich et al. (1999) and Westenbroek et al. (1999). More specifically, as an example, Kim et al. (1991) use an assembly of five Maxwell units (spring and dashpot in series) in parallel to capture the relaxation response of a range of reinforcement fabrics. This results in a relaxation expression of the form 5
R
s f = A S e i e – t / ti i =1
11.10
where A is a constant related to the volume fraction at which relaxation takes place and the stress at the onset of relaxation; the Ei are the moduli of the springs, and the relaxation times are tiR ∫ hi /Ei , where hi are the viscosities of the dashpots. Whereas the models just cited are all linear, Ghosh et al. (2005) and Šajn et al. (2006) find the necessity of including non-linear springs into the spring/dashpot arrangements in order to capture viscoelastic effects accurately. When more than one mode of deformation, or more than one viscoelastic phase or phenomenon, such as relaxation, is required to be modelled, more involved models are necessary. One easily implemented approach to this problem is to use a limiting elastic model. Here, one uses two limiting compaction curves, one obtained at a relatively slow speed and one at a relatively fast speed. The complete response due to any compaction speed can then be taken to be some interpolation of these (Bickerton and Buntain, 2007). Relaxation is then modelled in a simple manner by allowing the stress to drop instantaneously to the stress recorded at that volume fraction for the ‘slow’ compaction curve. el Oudiani et al. (2009) use an arrangement of three springs and two dashpots to model both creep and relaxation of fibrous material, yet it is still difficult to capture the initial, rapid, relaxation phase. Cai (1995) developed a model involving a non-linear spring and also a non-linear dashpot, in the analysis of fibrous seals under compression. The model viscosity depends on the volume fraction, it being argued that the larger the volume fraction, the larger the effective friction between moving fibres and so the higher the effective viscosity. Bréard et al. (2003) use two linear viscoelastic models, one for the compaction phase and one for the relaxation phase. This model incorporates a ‘double dynamic’, with the rapid initial relaxation assumed to be due to a rearrangement of yarns, the subsequent slower relaxation due to intra-tow movement of individual fibres. In the model of Kelly et al. (2006), the compaction stress s is given by the sum of an equilibrium stress s• and a viscous stress q. The equilibrium stress is the compaction stress when the material is compressed very slowly
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(or when all rate effects have been given time to die away); one can use any of the expressions discussed earlier, e.g. Eqns 11.3–11.5, 11.7, for this purpose. It models the equilibrium curve of Fig. 11.4. The viscous stress is determined from the differential equation 1/n
q1–1/n ] q = – nE [(emax – ee))m q 2–1/ n – heeq h
11.11
where E, n, h, m and emax are material parameters. Here, e is the measure of strain, defined here to be e = ln(Vf/Vf0)
11.12
which is the standard logarithmic strain (see Kelly, 2011, for a discussion of appropriate strain and strain-rate measures to use in this context). The model can be interpreted as a non-linear Maxwell unit in parallel with a non-linear spring, with the viscous stress being the stress taken up the Maxwell unit. Figure 11.6 (after Fig. 4 of Kelly et al., 2006) displays results using this model for compaction to different volume fractions of dry and wet preforms of continuous filament mat at 2 mm/min. Although the results of this model are good, it is very difficult to satisfactorily predict the response of a reinforcement over a wide range of volume fractions and compaction speeds, during more than one phase of viscoelasticity. This is true for all the viscoelastic models mentioned above. However, when stress is plotted against time, the stress and relaxation 1500 Experiment Model
Stress (kPa)
1000 (dry) Vf = 0.45
(wet) Vf = 0.45 500 (dry) Vf = 0.35 (wet) Vf = 0.35 0
(dry) Vf = 0.25 (wet) Vf = 0.25 200
600 Time (s)
1000
11.6 Stress relaxation for dry and lubricated continuous filament mats, compacted at 2 mm/min to different volume fractions.
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curves for reinforcements often collapse, or nearly do so, onto a universal master curve, when normalised appropriately. For example, shown in Fig. 11.7 are the compaction curves of the continuous filament mat of Fig. 11.2, but with the stress normalised with respect to the maximum stress (or the stress at some reference volume fraction) and the time normalised with respect to the time taken to reach the maximum stress. Also shown is the curve corresponding to a compaction speed of 0.035 mm/min (not shown in Fig. 11.2 for clarity). It is seen that the curves collapse onto a single master curve. This collapsing of data simplifies the modelling task, in the same way as the concept of the master curve simplifies the modelling of temperaturedependent thermorheological viscoelastic materials (Findley et al., 1976). This behaviour implies that the compaction stress should be expressible in the form of a multiplicative decomposition (Kelly, 2011):
s (Vf, v) = fa (v) fb (Vf)
11.13
where v is the compaction speed. It should be noted that this formulation immediately excludes the standard viscoelastic models, e.g. Maxwell, Voigt, Burgers, and similar rheological models with linear and/or non-linear spring and dashpot elements. The derivation of the functions fa and fb in Eqn 11.13 is explained in Kelly (2011) and the characterisation of the model can be completed from five separate compaction tests at different compaction speeds (Walbran, 2011). Results are shown in Fig. 11.8 for compaction and relaxation of continuous filament mat over a range of speeds and volume fractions. The model results are an excellent fit to the data over both compaction and relaxation phases. 1.00
0.035 mm/min 0.5 mm/min
0.80
s smax
0.60
2 mm/min
0.40
5 mm/min ( )
0.20 0.00
0.0
0.5
1.0 t/T
1.5
2.0
11.7 Compaction stress for continuous filament mat (CFM) compacted to a volume fraction of 35% at four different compaction velocities; stress normalised with respect to the maximum stress, as a function of time normalised with respect to the time T taken to reach the maximum stress.
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Stress (¥ 105 Pa)
8
Model Experiment
Vf = 0.45 2 mm/min
12
Vf = 0.35 5 mm/min
4
355
Vf = 0.40 1 mm/min
Vf = 0.35 0.5 mm/min
Vf = 0.35 2 mm/min Vf = 0.25 2 mm/min
0 0
500
t
1000
1500
11.8 Stress during compaction and relaxation of continuous filament mat (CFM) for different maximum compaction volume fractions and compaction velocities.
11.4.2 Plasticity models One of the first analyses of permanent deformation in fibrous materials was carried out by Olofsson (1967), who developed elasto-plastic models based on various arrangements of springs and sliding friction blocks; cyclic hysteresis was obtained using load reversal. Dunlop (1983) showed that arrangements of (non-linear) springs and friction blocks of various attributes could account for some of the observed phenomena. One of the few probability orientation models to incorporate rate-independent inelastic effects was that of Carnaby and Pan (1989), who considered a micromechanical model of a random assembly involving fibre-to-fibre friction. They carried out unidirectional compression simulations and obtained the much-observed hysteresis curve. Šimá�ek and Karbhari (1996) developed an elasto-plastic model of fibre bundles using the methodology of Gutowski (1997). They incorporated frictional sliding between fibres and applied their model to a single compaction. They noted that this formulation resulted in a dependence of yield on the applied bulk (cylindrical) stress. Comas-Cardona et al. (2007) developed an elasto-plastic model based on experimentation. An interesting feature of this study is that no yield stress is assumed or necessary. By assuming a purely elastic unloading, they were able to determine elastic moduli experimentally. This data was then used in loading phases, from which plastic moduli could be extracted. The plastic strain was seen to increase linearly with the total strain during cyclic loading (with the permanent set reaching 70% of the total deformation). Lin et al. (2008) assumed frictional contact between yarns in a finite element model of deforming fabric, but did not account for fibre movement within yarns.
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They showed that this approach leads to a large underestimation of the energy loss (or magnitude of the hysteresis loop), giving support to the importance of intra-yarn fibre movement. Note that a reference state from which permanent deformation is reckoned must be defined. This is most usually simply the initial, unstressed, state. However, although this is an intuitive choice, it is not without problems. For example, any sample of fibrous material will have undergone handling and deformation before testing and use, that is, the reference state is not static but easily disturbed and, crucially, is dependent on the loading history. This problem can be overcome by choosing the reference state to be ‘in the future’, when the material is compressed to its theoretically maximum extent. This concept has been used, for example, in the analysis of cellular solids (Gibson and Ashby, 1988) and in the study of the compressibility of wet fibre networks (Vomhoff and Norman, 2001).
11.4.3 Viscoplasticity There are very few viscoplastic models of fibrous materials available in the literature. One of the earliest analyses to consider viscoelasticity of fibres with frictional aspects was that of Chapman and Hearle (1972), who used a five-element model of springs, dashpots and a sliding friction block to examine the deformation and recovery of a planar sheet of fibres under various modes of deformation. Zhang et al. (2000) considered a four-element model with small strains in the analysis of ‘bagging’ of woven fabrics. Lobosco and Kaul (2001) study the wet pressing of fibre networks using the conventional Perzyna viscoplastic model, although here the rate effects are due to flow of fluid. In the following subsection, a recently developed comprehensive thermomechanical approach to modelling fibrous materials is discussed.
11.4.4 Thermomechanics A convenient approach to take when modelling fibrous materials is to use a thermomechanical framework, i.e. to build a model up from thermodynamic principles (Kelly, 2008). This is because (1) the laws of thermodynamics will automatically be satisfied, and (2) the framework is simple, in the sense that only a number of principles are necessary, and it is not necessary to develop an ad-hoc theory involving separate yield criteria, flow rules, etc. On the contrary, one need only specify two energy potentials and apply a principle of maximum energy dissipation. Following the thermomechanical procedure outlined in, for example, Ziegler (1977) and Holzapfel (2000), the rate at which work is done by the external forces, in the case of isothermal deformations, can be expressed as
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+F :: d = Y
357
11.14
where is the Cauchy stress, d is the rate of deformation, Y is the Helmholtz free energy, a measure of the energy stored, and F is the dissipation, the rate at which energy is lost. One is required to specify the forms of these two energy functions. The free energy will be of the general form Y = Y (e, ), where e represents a strain measure, for example the Green–Lagrange strain, and represents a set of internal variables which describe the dissipative, inelastic, mechanisms in some way. The dissipation will be a function the rate of change of the internal variable(s). The dissipation may of , also depend on internal variables which do not appear in the free energy function. For rate-independent permanent deformations, the dissipation must be a homogeneous function of degree 1 in these rates. The free energy and dissipation then act as potentials from which the stresses and strains can be determined (Collins and Houlsby, 1997; Houlsby and Puzrin, 2000). Regarding the hypothesising of functional forms for Y and F, the following two key points are to be made concerning fibrous materials in general: 1. They have an ability to acquire locked energy, as discussed earlier; this energy cannot be accessed without a reversal of any permanent deformation. 2. They are frictional materials, i.e. energy is dissipated through a frictional mechanism. Point 1 implies that the free energy can be expressed in the general form Y (e, ) = Y1 (e, ) + Hˆ ( )
11.15
with the second term here being the locked energy, a function of permanent deformations, as described by , but not of the current strain. Point 2 implies that the dissipation is of the form F (, , )
11.16
that is, it depends explicitly on the stress. This stress dependence of the dissipation will lead in general to non-associated flow rules (Collins and Kelly, 2002); classical metal plasticity with its associated flow rules is thus not appropriate for fibrous materials. Viscoplasticity can be accommodated by specifying a dissipation function of the form F = F1 + F 2 =
∂F1 ∂F 2 a + 1 a ∂a 2 ∂a
11.17
where F1 represents the rate-independent effects and F2 is a homogeneous function of degree 2, representing the rate-dependent energy losses (Cheng et al., 2009). This approach can be used to build up models of increasing complexity.
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For example, Kelly (2008) describes a model with power-law free energy functions and with the dissipation of the form F = f sa , where f is constant, playing the role of a friction coefficient. This leads to plastic deformation hysteresis. Cheng et al. (2011) have developed a single-internal variable model for the equilibrium hysteresis loop. Here, dissipation is continuous during loading and unloading, with the amount of locked energy and permanent deformation returning to their original values after a complete loop. Model results are shown in Figs 11.9(a–c) for the equilibrium loops of a twill weave,
Model Experiment twill weave
Compaction stress (MPa)
0.08
0.06
0.04
0.02
0 0.5
0.55
Volume fraction Vf (a)
0.6
0.65
0.05 Model Experiment CSM
Compaction stress (MPa)
0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0.3
0.32
0.34
0.36 0.38 Volume fraction Vf (b)
0.4
0.42
11.9 Equilibrium hysteresis loops: experimental and model data for (a) twill weave, (b) chopped strand mat, and (c) plain weave fabric.
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0.2 0.18 Compaction stress (MPa)
0.16
Model Experiment PWF
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.35
0.4
0.45 0.5 Volume fraction Vf (c)
0.55
0.6
11.9 Continued.
a chopped strand mat and a plain weave, respectively; these compare well with cyclic experimental results for these materials subjected to at least 20 cycles of loading.
11.5
Future trends
One of the most beneficial developments in the field of fibrous materials in recent times has been the advance in imaging and measurement techniques (Lomov et al., 2008; Jaganathan et al., 2009; Vidal-Sallé et al., 2010). This will undoubtedly lead to a necessary better understanding of the micromechanical deformation modes taking place during deformation (when and where). In particular, it should become possible to identify with more precision the nature of viscoelastic and plastic micromechanisms occurring in fibrous materials. Another important topic which needs to be addressed in more depth is that of frictional properties of fibrous materials. Frictional behaviour has been analysed and quantified for fibres of many different materials, including deviations from Amonton’s Law and the distinction between kinetic and static friction (see, for example, Morton and Hearle, 1993; Valizadeh et al., 2008). However, more data needs to be acquired for different types of fibre used in typical applications (Lin et al., 2008). A prominent topic which needs to be addressed further is the issue of variability – variability in experiments, variability in the in-plane structure of fibre reinforcements, variability due to nesting and so on – so that results and predictions can be used with more certainty. Other issues which need
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to be examined include the effect of excessive draping on compressibility, and analyses involving compression of new types of reinforcement material as they increase in use (Lomov et al., 2011a). Regarding modelling, some of the challenges facing further development in meso-scale modelling are outlined by Lomov et al. (2011b), for example the numerical issues involved with loosely supported yarns. Regarding inelastic modelling, as computational power increases, comprehensive micromechanical models incorporating sliding friction and rate effects cannot be discounted. Until these arise, the thermomechanics approach outlined earlier seems to offer the most promise for accurate and realistic modelling of complex viscoplasticity, incorporating as it does much of the physics involved. The focus of this chapter has been on transverse compression of reinforcements. The use of transverse compression analysis, independent of the other deformation modes which arise in composites forming, is possible only because of the near-one-dimensional nature of fabric compression (Pham et al., 1998). When coupling between the different deformation modes is important, it is likely that models will deal with this in a fairly simple way: regarding the pursuit of a comprehensive three-dimensional rate-dependent model of fibrous material deformation, Tschoegl (1997) gives pause for thought when he states that, referring to large-strain viscoelastic models (not even with any plasticity): ‘it is questionable whether a practically useful equation for the description of time-dependent behaviour in large deformations will ever become available’. Fortunately, it is true that fabric compression often arises with little accompanying shear or in-plane deformation, but nonetheless it still offers plenty of challenges to the analyst.
11.6
References and further reading
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porous media. Part II: Deformation of a double-scale fibrous reinforcement’, Polym Comp, 24, 409–421. Buntain M J and Bickerton S (2007), ‘Modeling forces generated within rigid liquid composite molding tools. Part A: Experimental study’, Comp Part A, 38, 1729– 1741. Cai Z (1995), ‘A nonlinear viscoelastic model for describing the deformation behaviour of braided fiber seals’, Text Res J, 65, 461–470. Carnaby G A and Pan N (1989), ‘Theory of the compression hysteresis of fibrous assemblies’, Text Res J, 59, 275–284. Chang C-Y (2006), ‘Simulation of mold filling in simultaneous resin injection/compression molding’, J Reinf Plast Comp, 25, 1255–1268. Chapman B M (1970), ‘Observations on the viscoelastic behaviour of Lincoln-wool fibres in water’, J Text Inst, 61, 408–411. Chapman M and Hearle J W S (1972), ‘The bending and creasing of multicomponent viscoelastic fibre assemblies. Part II: The mechanics of a two-dimensional assembly of long straight fibres of different types’, J Text Inst, 63, 404–412. Chen B and Chou T-W (1999), ‘Compaction of woven-fabric preforms in liquid composite molding processes: single-layer deformation’, Comp Sci Tech, 59, 1519–1526. Chen B and Chou T-W (2000), ‘Compaction of woven-fabric preforms: nesting and multi-layer deformation’, Comp Sci Tech, 60, 2223–2231. Cheng J, Kelly P A and Bickerton S (2009), ‘A thermomechanical constitutive model for fibrous reinforcements’, ICCM17, Edinburgh, 27–31 July. Cheng J, Kelly P A and Bickerton S (2011), ‘A rate-independent thermomechanical constitutive model for fibrous reinforcements’, J Comp Mat, in press. Choi K F and Tandon S K (2006), ‘An energy model of yarn bending’, J Text Inst, 97, 49–56. Collins I F (2005), ‘Elastic/plastic models for sands and soils’, Int J Mech Sci, 47, 493–508. Collins I F and Houlsby G T (1997), ‘Application of thermomechanical principles to the modelling of geotechnical materials’, Proc R Soc Lond A, 453, 1975–2001. Collins I F and Kelly P A (2002), ‘A thermomechanical analysis of a family of soil models’, Geotechnique, 52, 7, 507–518. Comas-Cardona S, Le Grognec P, Binétruy C and Krawczak P (2007), ‘Unidirectional compression of fibre reinforcements. Part 1: A non-linear elastic–plastic behaviour’, Comp Sci Tech, 67, 507–514. Comas-Cardona S, Bickerton S, Deléglise M, Walbran W A, Binétruy C and Krawczak P (2008), ‘Influence of textile architectures on the compaction and saturated permeability spatial variations’, in Recent Advances in Textile Composites, Proceedings of the 9th International Conference on Textile Composites (TEXCOMP9), ed Advani G and Gillespie J W, DEStech Publications, Lancaster. Correia N (2004), Analysis of the vacuum infusion moulding process, PhD thesis, University of Nottingham. Das D and Pourdeyhimi B (2010), ‘Compressional and recovery behaviour of highloft nonwovens’, Ind J Fibre Text Res, 35, 303–309. Dayiary M, Shaikhzadeh Najar S and Shamsi M (2010), ‘An experimental verification of cut-pile carpet compression behaviour’, J Text Inst, 101, 488–494. Drescher A, Kringos N and Scarpas T (2010), ‘On the behaviour of a parallel elastovisco-plastic model for asphaltic materials’, Mech Mater, 42, 109–117. Dunlop J I (1983), ‘On the compression characteristics of fibre masses’, J Text Inst, 74, 92–97. © Woodhead Publishing Limited, 2011
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Lubliner J (1980), ‘An axiomatic model of rate-independent plasticity’, Int J Sol Struct, 16, 709–713. Luo Y and Verpoest I (1999), ‘Compressibility and relaxation of a new sandwich textile preform for liquid composite molding’, Polym Comp, 20, 179–191. MacLaren O, Gan J M, Hickey C M D, Bickerton S and Kelly P A (2009), ‘The RTMLight manufacturing process: experimentation and modelling’, ICCM17, Edinburgh, 27–31 July. Manich A M, Ussman M H and Barella A (1999), ‘Viscoelastic behavior of polypropylene fibers’, Text Res J, 69, 325–330. Maugin G A (1992), The Thermomechanics of Plasticity and Fracture, Cambridge, UK, Cambridge University Press. Matsuo M and Yamada T (2009), ‘Hysteresis of tensile load–strain route of knitted fabrics under extension and recovery processes estimated by strain history’, Text Res J, 79, 275–284. Meltem A N, Batra S K and Buchanan D R (1999), ‘Micromechanics of three-dimensional fibrewebs: constitutive equations’, Proc R Soc Lond A, 455, 3543–3563. Merotte J, Šimá�ek P and Advani S G (2010), ‘Resin flow analysis with fiber preform deformation in through-thickness direction during compression resin transfer molding’, Comp Part A, 41, 881–887. Mogavero J and Advani S G (1997), ‘Experimental investigation of flow through multilayered preforms’, Polym Comp, 18, 649–655. Morton W E and Hearle J W S (1993), ‘Fibre friction’, in Physical Properties of Textile Fibres, 3rd edn, Manchester, UK, The Textile Institute. Narter M A, Batra S K and Buchanan D R (1999) ‘Micromechanics of three-dimensional fibrewebs: constitutive equations’, Proc R Soc Lond A, 455, 3543–3563. Neckárˇ B (1997), ‘Compression and packing density of fibrous assemblies’, Text Res J, 67, 123–130. Ogale A and Mitschang P (2007), ‘Compaction behaviour of assembled fiber reinforced preforms’, J Ind Text, 37, 15–29. Olofsson B (1967), ‘A study of inelastic deformations of textile fabrics’, J Text Inst, 58, 221–241. Pearce N and Summerscales J (1995), ‘The compressibility of a reinforcement fabric’, Comp Manufact, 6, 15–21. Pham X-T, Trochu F and Gauvin R (1998), ‘Simulation of compression resin transfer molding with displacement control’, J Reinf Plast Comp, 17, 1525–1556. Phillips K J and Ghosh T K (2005), ‘Stress relaxation of tufted carpets and carpet components: analysis of the tufted carpet structure’, Text Res J, 75, 485–491. Potluri P and Sagar T V (2008), ‘Compaction modelling of textile preforms for composite structures’, Comp Struct, 86, 177–185. Prager W (1957), ‘On ideal locking materials’, Trans Soc Rheol, 1, 169–175. Rawal A (2009), ‘Application of theory of compression to thermal bonded non-woven structures’, J Text Inst, 100, 28–34. Robitaille F and Gauvin R (1998), ‘Compaction of textile reinforcements for composites manufacturing. 1: Review of experimental results’, Polym Comp, 19, 198–216. Robitaille F and Gauvin R (1999), ‘Compaction of textile reinforcements for composites manufacturing. III: Reorganisation of the fiber network’, Polym Comp, 20, 48–61. Šajn D, Geršak J and Flajs R (2006), ‘Prediction of stress relaxation of fabrics with increased elasticity’, Text Res J, 76, 742–750. Saunders R A, Lekakou C and Bader M G (1998), ‘Compression and microstructure of
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fibre plain woven cloths in the processing of polymer composites’, Comp Part A, 29, 443–454. Saunders R A, Lekakou C and Bader M G (1999), ‘Compression in the processing of polymer composites 1. A mechanical and microstructural study for different glass fabrics and resins’, Comp Sci Tech, 59, 983–993. Sebestyen E and Hickie T S (1971), ‘The effect of certain fibre parameters on the compressibility of wool’, J Text Inst, 62, 353–360. Šimá�ek P and Karbhari V M (1996), ‘Notes on the modelling of preform compaction: I – Micromechanics at the fiber bundle level’, J Reinf Plast Comp, 15, 86–122. Somashekar A A, Bickerton S and Bhattacharyya D (2006), ‘An experimental investigation of non-elastic deformation of fibrous reinforcements in composites manufacturing’, Comp Part A, 37, 858–867. Somashekar A A, Bickerton S and Bhattacharyya D (2007), ‘Exploring the non-elastic compression deformation of dry glass fibre reinforcements’, Comp Sci Tech, 67, 183–200. Stearn A E (1971), ‘The effect of anisotropy in randomness of fiber orientation on fiberto-fiber contacts’, J Text Inst, 62, 353–356. Susich G and Backer S (1951), ‘Tensile recovery behaviour of textile fibers’, Text Res J, 21, 482–509. Terzaghi K (1943), Theoretical Soil Mechanics, New York, John Wiley & Sons. Toll S and Manson J-A E (1995), ‘Elastic compression of a fiber network’, J Appl Mech, 62, 223–226. Trevino L, Rupel K, Young W B, Liou M J and Lee L J (1991), ‘Analysis of resin injection molding in molds with preplaced fiber mats. I: Permeability and compressibility measurements’, Polym Comp, 12, 20–29. Tschoegl N W (1997), ‘Time dependence in material properties: an overview’, Mech Time-Dependent Mater, 1, 3–31. Valizadeh M, Ravandi S A H, Salimi M and Sheikhzadeh M (2008), ‘Determination of internal mechanical characteristics of woven fabrics using the force–balance analysis of yarn pullout test’, J Text Inst, 99, 47–55. Van Wyk C M (1946), ‘Note on the compressibility of wool’, J Text Inst, 37, T285– T292. Verleye B, Walbran W A, Bickerton S and Kelly P A (2011), ‘Simulation and experimental validation of force controlled compression resin transfer moulding’, J Comp Mat, 45, 815–829. Vidal-Sallé E, Nguyen Q T, Charmetant A, Bréard J, Maire E and Boisse P (2010), ‘Use of numerical simulation of woven reinforcement forming at mesoscale: influence of transverse compression on the global response’, Int J Mater Form, 3, Suppl 1, 699–702. Vomhoff H and Norman B (2001), ‘Method for the investigation of the dynamic compressibility of wet fibre networks’, Nordic Pulp and Paper Res J, 16, 57–62. Wagenseil J E, Wakatsuki T, Okamoto R J, Zahalak G I and Elson E L (2003), ‘One dimensional viscoelastic behaviour of fibroblast populated collagen matrices’, J Biomech Engng, 125, 719–725. Walbran W A (2011), Modelling and development of rigid tool liquid composite moulding processes, PhD thesis, University of Auckland. Walbran W A, Bickerton S and Kelly P A (2009), ‘Measurements of normal stress distributions experienced by rigid liquid composite moulding tools’, Comp Part A, 40, 1119–1133.
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Walbran W A, Verleye B, Bickerton S and Kelly P A (2011), ‘Prediction and experimental verification of normal stress distributions on mould tools during liquid composite moulding’, Comp Part A, in press. Westenbroek A P H, van Roekel G J, de Jong E, Weickert G and Westerterp G W (1999), ‘Compressibility of hemp bast fibres’, Nordic Pulp and Paper Res J, 14, 336–344. Wirth S and Gauvin R (1998), ‘Experimental analysis of mold filling in compression resin transfer molding’, J Reinf Plast Comp, 17, 1414–1430. Zhang X, Li Y, Yeung K W and Yao M (2000), ‘Relative contributions of elasticity and viscoelasticity in bagging of woven wool fabrics’, J Text Inst, 91, 577–589. Ziegler H (1977), An Introduction to Thermomechanics, Amsterdam, North-Holland.
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12
Bending properties of reinforcements in composites E. d e B i l b a o, Institut Universitaire de Technologie d’Orléans, France
Abstract: This chapter discusses out-of-plane bending deformations in the shaping of composite reinforcements, because taking them into account would give more accurate simulations of forming, especially for stiff and thick textiles. Bending behaviour is specific because the reinforcements are structural parts and out-of-plane properties cannot be directly deduced from in-plane properties, as is the case for continuous material. On the other hand, the standard tests used for clothing textiles are not suitable for stiff reinforcements with non-linear behaviour. Therefore, a new flexometer using optical measurements has been developed for testing such reinforcements. Direct identification and inverse identification were applied to characterise carbon fabrics, and the results are discussed. Key words: composite reinforcements, bending behaviour, experimentation, modelling.
12.1
Context
12.1.1 Bending behaviour during shaping Shaping of the dry woven preform is the first step in the liquid composite moulding (LCM) process before resin injection (Rudd et al., 1997; Potter, 1999; Parnas, 2000). In prepreg draping (Rudd et al., 1997) or in continuous fibre reinforcements and thermoplastic resin (CFRTP) forming (Maison et al., 1998, Soulat et al., 2006), the matrix is present but is not hardened and the deformations of the structure are driven by those of the woven reinforcement. Textile preforms undergo biaxial tensile deformations, inplane shear deformations, transverse compaction and out-of-plane bending deformations during shaping. Due to the fibrous nature of the material, the tensile stiffness for each fibre direction is much larger than the bending rigidity, so that the latter is neglected, which is equivalent to considering the woven fabric as a membrane (Badel et al., 2009; De Luycker et al., 2009; Hamila and Boisse, 2008; Boisse et al., 2005, 2006). That is why the membrane components of the behaviour have been widely studied (Launay et al., 2008; Cao et al., 2008). However, several studies (Prodromou and Chen, 1997; Hamila, 2007; Wang et al., 1999) demonstrated the relative 367 © Woodhead Publishing Limited, 2011
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importance of bending behaviour during composites forming and especially for the simulation of wrinkle phenomena during the forming stage. Classically, the bending behaviour of a continuous shell is derived from the in-plane properties of the material. This is not the case for composite reinforcements because they are structurally heterogeneous materials. Yu et al. (2005) considered the bending behaviour of a woven preform through a cantilever experiment and simulation where the deflection was only due to gravity. The authors showed the discrepancy between the experimental and numerical results and concluded that bending rigidity derived from in-plane properties gives an unrealistically high value compared to the experimental bending rigidity of the woven preform. During deformations, a part of the yarn will have its curvature increasing while another part will have its curvature decreasing, involving interactions between filaments and between yarns with high sliding. Bending behaviour depends, among other things, on the geometrical configuration of the yarns, their mechanical properties, and the contact behaviour. This is then a structural multi-scale specific problem which cannot be directly deduced from the in-plane material properties. As a result, considering the bending behaviour has become necessary.
12.1.2 Bending behaviour of composite reinforcements Composite reinforcements studied in this chapter have small thicknesses when compared with other dimensions. As a result they are assumed to be shells on a macroscopic scale and bending of shells can be described using a structural mechanics approach. The problem can be formulated with bending moment and changes in curvature. The study of the bending behaviour for simulation on a macroscopic scale consists in defining the relationship between the components of the moment and the curvature tensors in the general case:
M = f (k – k0)
12.1
where M is the bending moment and k and k0 are the current and the initial curvature tensors respectively. Considering the problem in one direction, the bending behaviour can be restricted to a relationship between the moment component and the change in corresponding curvature. Hence, assuming the moment component depends only on the corresponding curvature, equation 12.1 is transformed into
M = f (k – k0)
12.2
where M is the bending moment and k and k0 are the current and the initial curvature respectively. Figure 12.1 presents the result of an assumed pure bending test carried out on a woven composite fabric in the weft direction. In the pure bending
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M (gf) ad
ing
g
Lo
Un
lo
ad
in
5
–2
–1
Un
Lo
a lo
ad
di
0
1
2 K (cm–1)
ng –5
ing –10
12.1 Fabric bending response recorded during a pure bending test.
test, the specimen undergoes a bending moment in only one direction obtained with a couple of forces. Bending moment and changes of curvature in the corresponding direction are recorded during the test. In this example, bending moment is plotted against curvature. Bending moment is defined per unit width in gf.cm/cm = gf (1 gf ≈ 1 cN) and curvature is in cm–1. Initial curvature was equal to 0. The curve describing the cycle of the test performed under curvature control can be broken down into four parts. In the first part, that is to say at the beginning, the curvature starts from zero to increase up to 2.5 cm–1 and the moment increases with the curvature. The second portion corresponds to the unloading stage. The bending moment decreases to zero and the curvature decreases to a non-zero value (around 1.0 cm–1) which would indicate permanent deformation if the test were stopped. Carrying on the test up to –2.5 cm–1 for the curvature, the moment becomes negative during the third loading stage. Finally, the curvature increases again back towards zero in the fourth part and the moment increases also. It can be observed that the curve follows distinct paths denoting non-elastic behaviour. Two approaches can be applied to model the bending behaviour. The first
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method is based on modelling of the structural configuration and mechanical behaviour of the yarns and on the contact behaviour between these. Macroscopic bending behaviour can be deduced from these considerations. Many studies have investigated the bending behaviour of woven fabrics in the field of clothing for a long time and Ghosh et al. (1990a) presented a critical review of them. Assuming a linear elastic behaviour, the constant bending rigidity was defined from the properties of the fibres and their arrangement (Livesey and Owen, 1964; Platt et al., 1959). Taking into account the friction between fibres, non-linear non-elastic models were proposed by Grosberg (1966), Abbott et al. (1971) and Huang (1979). Bending properties of the fabrics were defined next from those of yarns, trying to improve the modelling of the yarn section (Abbott et al., 1973; Kawabata et al., 1973; Kemp, 1958; Peirce, 1937; Shanahan and Hearle, 1978) and the contact conditions between the yarns (Ghosh et al., 1990b, 1990c). However, results obtained in this field cannot be directly applied to composite reinforcements because the dimensions, structural configurations and mechanical properties of the yarns are different. Nevertheless, such approaches have been developed for tensile and shear behaviour in the composite field using the finite element method (Gasser et al., 2000). Apart from by Lomov et al. (2000), Yu et al. (2005) and de Bilbao et al. (2010), bending behaviour of composite reinforcements has not yet been explored, and the relationships between yarn properties and arrangements and bending properties of reinforcements remain difficult to define. The second method consists in investigating directly the bending behaviour on a macroscopic scale. The bending model parameters are defined by means of bending tests. The next section is devoted to this macroscopic approach, including both of the usual macroscopic bending models and bending tests. This kind of work was carried out by Lomov et al. on multiaxial multiply stitched preforms (Lomov et al., 2003) using the bending test of the Kawabata Evaluation System (KES) (Kawabata, 1980).
12.1.3 Macroscopic bending models and bending tests Two standard tests are commonly used for evaluating fabric bending stiffness: the standard cantilever test (ASTM, 2002) and the Kawabata bending test (KES-FB2) (Kawabata, 1980). The first is based on elastic linear behaviour and enables the determination of only one parameter: the bending rigidity. The second test was designed by Kawabata and enables one to record the moment versus the curvature during a bending cycle. Peirce was the first to present both a macroscopic measurement of the bending behaviour (Peirce, 1930) and a mesoscopic approach (Peirce, 1937) to model the geometric configuration of the mesh. Assuming an elastic linear
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behaviour between the bending moment and the curvature of the strip, he proposed to assess the bending stiffness thanks to a cantilever test (Fig. 12.2) where the fabric is cantilevered under gravity. The bending moment M is hence assumed to be a linear function of the curvature k: M=G¥b¥k
12.3
where G (N.m) is the flexural rigidity per unit width and b (m) is the width of the strip. Peirce defined the ratio S of the flexural rigidity to the weight w per unit area (N.m–2): S = G/w
12.4
Assuming the fabric being ‘elastic’ and small strains but large deflections, he defined the relation between the ratio S, the angle q of the chord with the horizontal axis and the length l of the bent part of the sample (Fig. 12.2): 3 cos s q //2 S=l · 8 tan q
12.5
Taking the cube root of S makes it possible to compare the fabrics. it has the unit of a length and was called by Peirce the ‘bending length’. a device designed by Peirce enabled one to measure the angle for a chosen length of overhang. Another way is to fix the angle by means of a tilted plate and to push the specimen until it touches the plate. The length of overhang is then evaluated. The standard commercial devices in use today are based on this latter method with a specific angle equal to 41.5° (ASTM, l 3 coss q //2 2002; ISO, 1978). With this value, the equation S = · [12.5] can 8 tan q be written more simply:
Length of overhang: l Embedded section
Bent section
Horizontal board
q Chord
Strip
12.2 Cantilever test.
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12.6
Performing a bending test is thus easy with the standard devices but only the bending stiffness of a linear elastic model can be evaluated. Moreover, pushing the specimen makes it slide. The mesh structure is likely to be modified and thus its bending properties. Kawabata’s Evaluation System was originally designed to measure basic mechanical properties of fabrics (Kawabata, 1980). it has become a set of standard tests for tensile, shear, compression, surface roughness and bending behaviour. The pure bending tester KES-FB2 (Fig. 12.3) allows taking into account non-elastic behaviour. it enables one to record directly the evolution of the bending momentum per unit width versus the curvature during a loading– unloading cycle but also to assess bending properties. The specimen is 1 cm wide in the bending direction and can be up to 20 cm long for flexible fabrics. It is clamped between a fixed clamp (A) and a moving one (B). The fixture setting of the sample in the clamps ensures pure bending deformation. during the test, the moving clamp (B) rotates round the fixed one (A) ensuring a constant curvature through the sample length. The movement is made with a constant rate of curvature equal to 0.5 cm–1.s–1 from –2.5 to 2.5 cm–1. Figure 12.4 represents a typical result of the KES-FB2 test carried out on a 2.5d carbon fabric in the weft direction. The area weight of the fabric was 630 g.m–2 and it was 1 mm thick. The representation of Grosberg’s
A
20 cm
B
12.3 Kawabata Evaluation System – Fabric Bending test (KES-FB2).
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5
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M (gf)
B h1
s1 M0
K (cm–1) –2
–1
0
1
2
Grosberg’s model: B, M0
–5
h2 s2
–10
12.4 KES-FB2 test result carried out on 2.5D carbon fabric and Grosberg’s parameters.
bending model (Grosberg, 1966) is drawn over. This model takes into account frictional restraint due to the friction of the fibres and thus: k = 0 if M < M0
12.7
M = B ¥ k + sign (k)·M0 if M ≥ M0
12.8
as indicated in the manual for the device, the bending rigidity B and the bending hysteresis M0 of Grosberg’s model are computed as follows. The slopes are computed respectively between k = 0.5 cm–1 and k = 1.5 cm–1 for s1 and between k = –0.5 cm–1 and k = –1.5 cm–1 for s2 where: s1 =
DM M = M (k = 1.5) – M (k = 0.5) Dk = 1
s2 =
DM M = M (k = – 0.5) – M (k = –1.5) Dk = 1
12.9 12.10
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B = (s1 + s2)/2
12.11
The bending hysteresis, that is the frictional restraint force M0, is the halfaverage of the two hysteresis values h1 and h2 computed respectively at k = 1 cm–1 and k = –1 cm–1: h1 = Ml (k = 1) – Mul(k = 1)
12.12
h2 = Ml (k = –1) – Mul(k = –1)
12.13
2 · M0 = (h1 + h2)/2
12.14
where Ml and Mul are the moment for loading and unloading curves respectively. Grosberg’s parameters computed for this test are presented in Table 12.1. Using KES bending test results performed on cloth, Ngo Ngoc et al. (2002) fitted Dahl’s model adapted to bending behaviour. Dahl’s model is obtained by generalising Coulomb’s friction model. adapted to the fabric bending problem, it is defined as follows: dM (k ) M (k ) Ê ˆ = B Á1 – sign (k )˜ dk M0 Ë ¯
n
12.15
where B is the original bending module, n is a shape parameter and M0 is the limit of the moment. later, lahey et al. (lahey and Heppler, 2004; lahey, 2002) developed a more complicated model to reproduce the fabric bending measurements taken by the Kawabata Evaluation System (KES). Based upon the studies of Bliman and Sorine (1995), the model took into account non-linear elastic behaviour, friction and viscous behaviour. Results of the model and its components were in varying degrees of agreement with experimental results for fabrics composed of different weaves and yarn types. Finally, the KES bending test has been developed for flexible textiles, and the bending moment–curvature curve obtained depends strongly on the boundary conditions set by the clamp. Moreover, reducing the length of the specimen is required when the reinforcement is stiff or thick. In this case, required hypotheses of macroscopic characterisation are not satisfied as the specimen dimensions are close to the characteristic dimension of the mesh. Finally, the apparatus does not make it possible for stiffer, multiply reinforcements to be tested nor the behaviour in the thickness to be analysed. Table 12.1 Grosberg’s parameters computed from the KES-FB2 test performed on 2.5D carbon fabric s1 (gf.cm)
s2 (gf.cm)
B (gf.cm)
h1 (gf)
h2 (gf)
M0 (gf)
3.95
3.40
3.68
6.44
5.04
2.87
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12.2
375
Improved cantilever test
The cantilever principle has been retained to develop a new device because of its simplicity and flexibility in testing different reinforcements. However, it has been developed with the aim of characterising the non-elastic bending properties for stiff or multiply reinforcement. Improvements have already been made to Peirce’s bending test based on the cantilever test. Grosberg and Abbott (1966a, 1966b) proposed a cantilever test to evaluate the parameters of his model (k = 0 if M < M0 [12.7], M = B ¥ k + sign (k) · M0 if M ≥ M0 [12.8]. Considering two specific values of the angle (q = 40° and q = 20°) and assuming the load as a concentrated and a distributed load, the parameters were computed using functions derived from known solutions. Lastly, Clapp et al. (1990) developed an indirect method of experimental measurement of the moment–curvature relationship for fabrics based on recording the coordinates of the deformed sample. Applying least–squares polynomial regression and numerical differentiation techniques, the moment– curvature relationship is computed from coordinate data and weight per unit width. This method allows taking into account the non-linear behaviour but assumes elastic behaviour. Combining both approaches, a new cantilever test has been developed (de Bilbao et al., 2010). The next section is devoted to its description.
12.2.1 Device description This new cantilever test, the so-called flexometer, is made up of two modules: a mechanical module and an optical module. The mechanical module enables one to place the specimen in a cantilever configuration under its own weight. It is also possible to add a mass at the free edge of the sample to reach larger curvatures. The optical module takes pictures of the shape of the bent sample. The sample can be a yarn, a monoply or a multiply reinforcement. It is about 300 mm long and up to 150 mm wide. The thickness can reach several millimetres. At the beginning of the test (Fig. 12.5), the sample (S) is placed upon a fixed board (F) and a special plane made of laths (B) inserted in the frame (E). The length direction of the sample must be parallel to the bending direction and its free edge must be aligned with the lath (L1). A translucent plate (C) is fixed upon both the specimen and the board to ensure the embedding condition. Thus the sample (S) will not slide. During the test, thanks to the translation of the drawer (T), the laths are successively released, beginning with lath L1, and the length of overhang increases. The test is regularly broken and a picture of the bent sample is taken. At each stop, the length of overhang is also measured. The complete test is hence a series of quasi-static tests with different loading cases. While a single cantilever test provided only one © Woodhead Publishing Limited, 2011
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Composite reinforcements for optimum performance (C) plate (B) laths Dir ect
(E) frame
irec gd
ase rele ath of l tion
din
ion
Ben
(A) axis
(T) drawer
100 mm
(S) sample (F) board
12.5 New flexometer. Mechanical module with a sample of non-crimp fabric.
configuration, the new flexometer, with its set of loading cases associated with the different bent shapes, enables one to identify a non-elastic behaviour model. Full-field strain measurements are applied to measure the deformed shape of the bent sample. The digital camera takes a picture for each length and the images are processed to extract the shapes of the bent sample (Fig. 12.6). A previous step of pixel calibration (Bailey, 1995) is required so that pixel measurements can be translated into real dimensions by scaling. Then, the image of the bent sample profile is captured, filtered (Bailey, 1990), and converted to binary form. The last step in image processing consists of extracting the borders of the binary object and deducing the mean line (Fig. 12.7).
12.2.2 Post-processing of the mean lines Once extracted, each mean line, corresponding to the bent shape, is defined by the bending length L and by the data points defined in a coordinate system.
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150 mm
377
100 mm
200 mm
12.6 Bending test. Pictures taken by CCD camera during the test for four bending lengths.
The next stage of post-processing aims to compute the bending moment and the curvature along the line (de Bilbao et al., 2008b). The line is defined by a noisy set of XY data points and has to be smoothed by a series of exponential functions to make possible the calculus of the curvature. A first-order polynomial is added to ensure the boundary conditions: n
iKxx z (x ) = S pi eiK + k1 x + k0 i =1
12.16
where K is an amplification factor to fit accurately the points; the degree n of the exponential polynomial is defined to optimise the parameters with the least-squares criterion. For a current point P defined by its coordinates (XP, ZP) (Fig. 12.8) its curvilinear coordinate is given by
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0
0
50
100
X (mm)
150
200
250
Z (mm)
–50
–100
–150
L100 L150 L200 L240
–200
12.7 Mean lines extracted from bent shapes for four bending lengths (100, 150, 200 and 240 mm). z x
E n(u)
P(s)
Q(u)
j(u) t(u)
F
12.8 Bending moment computing along the mean line; E is the embedded point and F corresponds to the free edge.
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ÚX
s (P ) =
E
1 + z ¢ (x) x )2 ddxx
379
12.17
where XE is the abscissa of the embedded point. The length of the deformed line is equal to Lb =
XF
ÚX
1 + z ¢ ((xx )2 dx
E
12.18
where XF is the abscissa of the free point. The curvature is
k (P ) =
z ¢¢ (1 + z ¢ 2 )3/2
12.19
The bending moment can be computed as follows: M (P ) = W
Lb
Ús
(u – s ) cos(ϕ(u))du
12.20
where W is the weight per unit length (N.m–1), and u and ϕ are the Frenet’s coordinates. The total strain energy Ut is the summation of the bending energy Ub, the membrane strain energy Um, and the transverse shear energy Uts: Ut =
Lb
Ú0
M ( s ) · k (s ) ds +
Lb
Ú0
N ( s ) · e (s ) ds +
Lb
Ú0
T ( s) s · g (s ) ds 12.21
where the bending moment M(s), the axial stress N(s) and the transverse shear T(s) are the only non-zero stress components and are assumed to be a function only of the curvilinear abscissa s. Moreover, assuming that the membrane strain energy and transverse shear energy (Um + Uts) are insignificant compared with the bending energy, membrane strains are negligible and the length of the deformed line Lb is equal to the initial length L. For each length of bending test, the bending moment and the curvature have to be computed along the profile and the moment–curvature graph can be drawn.
12.2.3 Test interpretation as the bending test is a series of cantilever tests with increasing length of overhang L, bending moment and curvature can be computed at each point along the line and plotted against L (Fig. 12.9). Each point P of the line is defined by its own length Lp equal to the distance it has to the free point. P is within the embedded part of the line while the length of overhang L is lower than Lp. When L is equal to Lp the bending moment reaches spontaneously © Woodhead Publishing Limited, 2011
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z
P
x
ˆ M P
MP(L)
MQ(L)
d
LQ
Q
Lp
ˆ M Q
loa
z
P
x
Loa
Un
z
Lp
M = f(L) d
M (N.m)
x
P LP
LQ
L(m)
12.9 Interpretation of bending test carried out with flexometer on non-elastic fabric.
Lb
the maximum value, defined by M (P ) = W Ú (u – s ) cos(ϕ(u))du [12.20]. s Then, it is obvious that the bending moment decreases while L goes on increasing. The same analysis can be done for the curvature. Each point of the sample hence undergoes a loading only once, when L = Lp, followed by an unloading. in this way bending moments and curvatures computed at the embedded points allow the loading curve Ml (k) (Fig. 12.10) to be plotted. in the same way, following a point after it has become an embedded point allows plotting of an unloading curve Mul(k). it is then obvious that a point close to the free point F gives a lower bending moment and a lower curvature than those given by a point close to the permanent embedded point E. if the composite reinforcement is elastic, but not necessarily linear, both loading and unloading curves are superposed. This explanation points out that the identification of non-elastic bending behaviour requires performing bending tests with different lengths of overhang. It also points out the difference between the new flexometer with its complete test and the simple standard cantilever test. The standard test enables one to evaluate only bending rigidity for the linear elastic model if only one point of the shape is exploited. it also enables one to evaluate the parameters of a non-linear but elastic model if the complete shape is processed. But it does not enable one to provide parameters for a non-elastic model, contrary to the new flexometer, which takes into account the history of the deformations. The calculus of the curvature along the mean line depends strongly on the degree of the interpolation functions. Defining unloading behaviour is difficult because of the inaccuracy in computing the low curvatures when the curve is not smoothed enough. Moreover, choosing a suitable series of interpolation functions is similar to choosing implicitly the shape of the bent sample, although the bending behaviour is unknown. That is why the
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Loa
MQ
381
d
MQ(L)
Un loa
MP(L)
d
MP
M (N.m)
k (m )
MP
d
kQ
P
d oa
a Lo
U
kP
Un
kQ k (m–1)
loa
kP(L)
nl
kO(L)
kP
Q Non elastic
L (m)
LQ
–1
Lo
LP
ad
El
as
tic
MQ
d
LP
LQ
L (m)
12.10 Construction of the bending response from the curvature and moment computing.
present development of the previous method is limited and an alternative method has been developed. The next section is devoted to it.
12.2.4 Inverse identification The inverse method is built on experimental results and the results obtained by the numerical simulation of the bending test using a finite element method. The aim of the inverse method is indeed to match the experimental profiles to the finite element method model by optimising the parameters of the chosen model (de Bilbao et al., 2008a). The first step in the inverse identification consists of simulating the test by means of commercial FEA software. The sample was modelled as a shell. The shell section response is directly defined by generalised stresses (membrane forces per unit length, bending moments per unit length) and by the generalised section strains in the shell (reference surface strains and curvatures). The specific bending model is defined using a user subroutine. Assuming the complete test to be a series of successive quasi-static cantilever configurations and perfect contact between the sample and the mechanical
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module, a set of steps has been defined in which boundary conditions change to simulate the increasing bending length. There are as many different strategies to carry out inverse identification as different possibilities to process the experimental shapes. all are based on a non-linear least-squares method to fit simulated results with experimental data. The strategy presented here uses one shape for the identification while the other shapes are used to verify the behaviour model. in this aim, the experimental shape is meshed in a set of nodes corresponding to the model of simulation. The optimisation of the bending model involves adjusting the bending model parameters in the finite element model until the calculated profile Ns matches the experimental profile Nd in the least-squares sense. The residual vector between Ns and Nd is defined by (Fig. 12.11) 12.22 di = N Nddi N Nssi = (Xs Xsi – X Xddi , Zs Zsi – Z Zd di ) 12.23 ri = di · ni where i is the number of the point (i = {1, 2, . . ., n}; n points) and ni is the normal of the experimental line at the point. Using a modified Levenberg– Marquardt method (Levenberg, 1944; Marquardt, 1963; Schnur and Zabaras, 1992; Sun and Yuan, 2006), optimisation is accomplished by minimising the error function: n
e (m ) = 12 S [ri (m )]2 = 12 r Tr
12.24
i =1
ni Ê Xs ˆ Nsi Á i ˜ Ë Zs i ¯ ri di
Ê Xd i ˆ Nd i Á ˜ Ë Zd i ¯
12.11 Definition of the residual vector.
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Results and discussion
12.3.1 Bending test on an interlock carbon fabric The first results discussed in this section have been obtained with a test performed on an interlock carbon fabric (Fig. 12.12). It is a laminate of four layers. Its area weight is 600 g/m2, the thickness is 0.6 mm and it has 7.4 yarns/cm in the warp direction and 7.4 yarns/cm in the weft direction. A first series of five tests was performed under the own weight only in the weft direction. The samples were 100 mm wide and 350 mm long. The usable bending length varied from 100 mm to 260 mm. Figure 12.13 shows the bent lines extracted from the samples for three bending lengths of 150, 200 and 250 mm, and Fig. 12.14 represents the relative standard deviation of the maximal deflection versus the bending length. It appears that the higher the bending length the smaller the relative standard deviation. The latter varies between 1% for the largest lengths and 19% for the smallest lengths. Another test of intrinsic repeatability of the flexometer gave a relative standard deviation of the maximal deflection less than 0.2% for the largest lengths and 5% for the smallest lengths. As a result, the observed repeatability is essentially due to the material scattering. A first way to analyse the behaviour consists of computing the Peirce’s rigidity at each bending length. Results reveal that the behaviour is not linear-elastic (Fig. 12.15). The curvature (Fig. 12.16) increases with the
1 cm
12.12 Interlock carbon fabric.
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0
0
50
100
X (mm)
150
200
250
L = 150 mm
–50
L = 200 mm
Z (mm)
–100
L = 250 mm –150
–200
Sample Sample Sample Sample Sample
1 2 3 4 5
–250
12.13 Repeatability of profiles and deflections. Mean lines for bending lengths 150, 200 and 250 mm. 20 18
sR(d) = s (d )/ d
16 14 sR (d )%
12 10 8 6 4 2 0
80
100
120
140
160
180 L (mm)
200
220
240
260
280
12.14 Repeatability of profiles and deflections. Relative standard deviation of the maximal deflection versus the bending length (d = z-coordinate of free edge F).
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13.00 G (N.mm)
12.00
Gmin
11.00
Gmax
G (N.mm)
10.00 9.00 8.00 7.00 6.00 5.00
80
100
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180 L (mm)
200
220
240
260
280
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260
280
12.15 Peirce’s modulus G vs bending length. 0.040 0.035
k (mm–1)
0.030 0.025 0.020 0.015 0.010 0.005 0.000 80
100
120
140
160
180 L (mm)
200
220
12.16 Curvature vs bending length.
length, increasing with a more marked visible asymptotic behaviour from 210 mm. For the largest length it reaches around 0.036 mm–1. For the moment (Fig. 12.17), a change of slope can be observed around L = 190 mm. To complete the repeatability of the test, a study of the scatterings of curvature indicated that the relative standard deviation is length independent and comprised between 5 and 21%. These large variations could be explained by the numerical double derivative of the smoothing exponential functions series (with wiggles) to compute the curvature. On the contrary, the variation in the relative standard deviation of the moment, between 1 and 5%, is much less extensive because the moment is computed by integration. In a second series of tests, metal strips were stuck on the free edges of the samples to reach larger curvatures. This added mass increases the moment, especially at the beginning of the test for small bending lengths,
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0.12 0.10
M (N)
0.08 0.06 0.04 0.02 0
80
100
120
140
160
180 L (mm)
200
220
240
260
280
12.17 Moment vs bending length.
0.12 Elastic behaviour
0.10
0.08 M (N)
Non elastic behaviour
0.06 Area of change of behaviour 0.04
0.02
0.00 –0.02
0.00
0.02
0.04
0.06 0.08 k (mm–1)
0.10
0.12
0.14
0.16
12.18 Bending test on an interlock fabric with added mass (glued strip): moment vs curvature.
and the deformation of the samples was accelerated. Tests were performed with strips of weight equal to two-thirds of the sample weight. This time, the bending length varied from 50 mm to 240 mm. Figure 12.18 presents the moment versus the curvature computed along the profiles for all the lengths for one of the samples. Using the mass, the maximum curvature can reach
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the significant value of 0.15 mm–1. The curves are divided into two sets. The curves of the first set are superposed, suggesting an elastic behaviour. On the contrary, the curves split off in the second set. This could be explained by the non-elastic behaviour. The behaviour changes for k between 0.04 and 0.045 mm–1. It must be stressed that computation of curvature for low values gives unstable results with large variation due to the numerical computation of the second derivative. Finally, Fig. 12.19 shows the average loading curves for the two sets of tests performed on interlock carbon fabric. For the tests carried out under own weight only, the curvature reaches 0.036 mm–1 and the moment 0.11 N. The behaviour is only elastic all over the range of curvature. For the tests carried out with added mass, the curvature reaches 0.10 mm–1 and the moment 0.12 N. The marked change of slope confirms the material change from an elastic behaviour to an inelastic at curvature between 0.04 and 0.045 mm–1. It is worth noting the good continuity between both series.
0.12
0.10
Change of behaviour
M (N)
0.08
Gravity 0.06
Mass
0.04
Elastic behaviour
0.02
0.00 0.00
0.01
0.02
0.03
Non-elastic behaviour
0.04
0.05 0.06 k (mm–1)
0.07
0.08
0.09
0.10
12.19 Bending test on an interlock fabric. average bending response under gravity and with added mass.
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12.3.2 Bending test on non-crimp fabric This second investigation examines the bending behaviour of a carbon noncrimp fabric (NCF) (Lomov et al., 2002). It is a laminate of two unidirectional plies with yarn orientation at ±45° and plies tied by warp stitching (Fig. 12.20a
1 cm
(a)
(a)
12.20 Carbon non-crimp fabric (NCF): (a) top; (b) bottom.
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(top), 12.20b (bottom)). The area weight is 568 g/m2 and the thickness is 1.1 mm. Such reinforcement is generally characterised by the number and the orientation of plies, the dimensions of the yarns, the pattern of the stitch (chain or tricot) and its dimensions. The sample was 50 mm wide and 350 mm long. The test was performed with the bent strip under its own weight only. The results are presented for bending lengths varying from 80 mm to 200 mm by 10 mm. Bending moment and curvature were computed at each embedded point and results are shown in Fig. 12.21. The set of points can be broken into two parts. In the first part the moment increases sharply, and in the second part the moment remains stable. This plateau could be explained by the wide slipping of fibres and yarn once the friction restraint level is reached. With such apparent non-elastic behaviour, even from small curvatures, dahl’s model n dM (k ) M (k ) Ê ˆ = B Á1 – sign (k )˜ [12.15] could be suitable to fit with the Ë ¯ dk M0 moment–curvature curve using a least-squares method. The optimisation performed with this model gave B = 3.38 N.mm and M0 = 0.046 N. At a second time, inverse identification was performed. Optimisation was carried out on the profile with bending length equal to 200 mm, and it converged, giving dahl’s parameters B = 2.68 N.mm and M0 = 0.052 N. For the simulated shape, the maximal curvature computed at the embedded 0.05
0.04
M (N)
0.03
0.02
0.01 Exp. Dahl 0
0
0.01
0.02
0.03 k (mm–1)
0.04
0.05
0.06
12.21 Bending test on NCF. Loading curve – experimental result and Dahl’s model fitting.
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edge is equal to kL200 = 3.18 ¥ 10–2 mm–1. The average of the residues is r = 1.05 mm and the maximal value is 2.36 mm. Both shapes (Fig. 12.22a) are in fairly good agreement in the first part where the curvature is high. On the other hand, the shapes are not close in the second part, which indicates that the model is not really adapted to simulate accurately the bending behaviour at lower curvature. In this part of the shape, curvatures are low and correspond to small bending lengths.
0
0
10
20
X (mm) 30 40
50
60
0 0
10
20
X (mm) 30 40
Exp Opt
60
Exp Opt
–20
–20
–40
–40
–60
–60
–80
–80 Z (mm)
Z (mm)
50
–100
–100
–120
–120
–140
–140
–160
–160
–180
(a)
–180
(b)
12.22 Inverse optimisation. Comparison between experimental and optimised profiles with (a) full and (b) partial experimental profile optimisation.
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Figure 12.23 presents the experimental shapes for two bending lengths and calculated shapes with the optimised parameters. Discrepancies between the shapes confirm that Dahl’s relationship is not able to model the behaviour for low curvatures. For small bending lengths, the analysis of the moment and curvature along the profiles showed that the material could have an elastic behaviour. If Dahl’s model is not suitable for modelling elastic behaviour, a second identification was performed to verify whether it is able to model accurately non-elastic behaviour. This second inverse identification was performed using only the first-half points of the profile previously used for the first optimisation. This time, the optimisation gave Dahl’s parameters B = 2.162 N.mm and M0 = 0.077 N. For the simulated shape, the maximal curvature at the embedded edge remained equal to kL200 = 3.2 ¥ 10–2 mm–1. Both profiles are closer with this identification (Fig. 12.22b). The new average of the residues was r = 0.30 mm and the maximal value was 0.87 mm. These results confirm that Dahl’s relationship is capable of modelling non-elastic behaviour. Higher discrepancies can be observed obviously between profiles at small bending lengths with new optimised parameters.
0
0
20
40
60
X (mm) 80
100
120
140
160
–20
Z (mm)
–40
–60
–80 Exp Opt Exp Opt
–100
L150 L150 L100 L100
–120
12.23 Comparison between experimental and simulated profiles for lengths 100 and 150 mm.
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12.4
Conclusions
The bending behaviour of composite reinforcements has become significant in forming process simulations, especially when out-of-plane phenomena, like wrinkles, occur during the process. This behaviour is a specific complex multi-scale mechanical problem. On a macroscopic scale, bending behaviour is described by the constitutive moment–curvature relationship which is not linear and depends on the range of the curvature. Investigation of studies performed on clothing textiles showed that it is not possible to apply models and results coming from the clothing field directly to the composites field, as the loads are significantly higher in composite materials applications than in the clothing industry, the constitution and the rigidity are different, and consequently models for deformation of woven fabrics as developed for the clothing industry are often not applicable for composites. Whatever the scale used to approach the problem, experimental identification of the bending behaviour and modelling require a macroscopic bending test. Bending tests performed with the Kawabata Evaluation System on composite reinforcements showed that such textiles have non-elastic behaviour. As a result, the standard cantilever test designed for clothing textiles is not suitable because it allows evaluating only linear elastic properties. The KES bending test could be suitable but it has been designed also for clothing textiles and does not allow testing stiffer textiles such as composite reinforcements. Moreover, boundary conditions are not really handled. Therefore, a new bending test based on the cantilever test but with increasing bending length has been designed in order to identify non-elastic behaviour. The so-called flexometer enables one to test various reinforcements with varying thicknesses, different woven structures, and small or large bending rigidity. During the test, the specimen undergoes a series of cantilever tests depending on the measured bending length. For each bending length, an optical measure associated with image processing enables one to extract a mean line of the bent sample profile. The line is accurately defined by a set of Cartesian coordinates. The moment–curvature response is then deduced from the series of lines. This direct method is very interesting as it enables one to get the response and the user is free to find a model. However, curvature is not accurately evaluated in some cases because of the numerical double derivative computation. An inverse identification has therefore been developed. Based on a finite element analysis and an optimisation algorithm, this numerical method enables one to test a postulated bending behaviour model. This second approach allows the testing and identification of the bending behaviour of composite reinforcements. It is especially efficient in verifying a model. Tests performed on interlock carbon fabric showed that the repeatability of the deformation assessment was due mainly to the own repeatability of
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the material. A direct method was carried out and two series of tests were performed. The first were performed under the material’s own weight and a strip was glued to the free edge of the samples for the second. Gluing a strip allowed reaching a larger curvature and identifying the change from elastic to non-elastic behaviour. Tests performed on non-crimp fabric showed that inverse identification is a useful numerical tool for verifying a model. Dahl’s model is fairly good at reproducing bending behaviour at large curvatures for this material.
12.5
Acknowledgement
The author thanks Laurence Schacher of the Laboratoire de Physique et Mécanique Textiles de Mulhouse (ENSISA) for allowing him to perform the KES bending tests.
12.6
References
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Livesey, R. & Owen, J. 1964. Cloth stiffness and hysteresis in bending. Journal of the Textile Institute, 55, 516–530. Lomov, S. V., Truevtze, A. V. & Cassidy, C. 2000. A predictive model for the fabricto-yarn bending stiffness ratio of a plain-woven set fabric. Textile Research Journal, 70(12), 1088–1096. Lomov, S. V., Belov, E. B., Bischoff, T., Ghosh, S. B., Truong Chi, T. & Verpoest, I. 2002. Carbon composites based on multiaxial multiply stitched preforms. Part 1. Geometry of the preform. Composites Part A: Applied Science and Manufacturing, 33, 1171–1183. Lomov, S. V., Verpoest, I., Barbuski, M. & Laperre, J. 2003. Carbon composites based on multiaxial multiply stitched preforms. Part 2. KES-F characterisation of the deformability of the preforms at low load. Composites Part A : Applied Science and Manufacturing, 34, 359–370. Maison, S., Thibout, C., Garrigues, C., Garcin, J. L., Payen, H., Sibois, H., Coiffer, C. & Vautey, P. 1998. Technical developments in thermoplastic composites fuselages. SAMPE Journal, 34, 33–39. Marquardt, D. W. 1963. An algorithm for least-squares estimation of nonlinear inequalities. SIAM Journal on Applied Mathematics, 11, 431–441. Ngo Ngoc, C., Bruniaux, P. & Castelain, J. M. 2002. Modelling friction for yarn/fabric simulation. Application to bending hysteresis. Proceedings of the 14th European Simulation Symposium, Dresden, October. Parnas, R. S. 2000. Liquid Composite Molding. Hanser Gardner Publications, Cincinnati, OH. Peirce, F. T. 1930. The ‘handle’ of cloth as a measurable quantity. Journal of the Textile Institute, 21, 377–416. Peirce, F. T. 1937. The geometry of cloth structure. Journal of the Textile Institute, 28, 45–96. Platt, M. M., Klein, W. G. & Hamburger, W. J. 1959. Mechanics of elastic performance of textile materials: Part XIV: Some aspects of bending rigidity of singles yarns. Textile Research Journal, 29, 611–627. Potter, K. D. 1999. The early history of the resin transfer moulding process for aerospace applications. Composites Part A: Applied Science and Manufacturing, 30, 619–621. Prodromou, A. G. & Chen, J. 1997. On the relationship between shear angle and wrinkling of textile composite preforms. Composites Part A: Applied Science and Manufacturing, 28, 491–503. Rudd, C. D., Long, A. C., Kendall, K. N. & Mangin, C. G. E. 1997. Liquid Moulding Technologies, Woodhead Publishing, Cambridge, UK. Schnur, D. S. & Zabaras, N. 1992. An inverse method for determining elastic material properties and a material interface. International Journal for Numerical Methods in Engineering, 33, 2039–2057. Shanahan, W. J. & Hearle, J. W. S. 1978. An energy method for calculations in fabric mechanics: Part II: Examples of application of the method to woven fabrics. Journal of the Textile Institute, 69, 92–100. Soulat, D., Cheruet, A. & Boisse, P. 2006. Simulation of continuous fibre reinforced thermoplastic forming using a shell finite element with transverse stress. Computers and Structures, 84, 888–903. Sun, W. & Yuan, Y.-X. 2006. Optimization Theory and Methods. Springer, New York.
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Wang, J., Paton, R. & Page, J. R. 1999. The draping of woven fabric preforms and prepregs for production of polymer composite components. Composites Part A: Applied Science and Manufacturing, 30, 757–765. Yu, W. R., Zampaloni, M., Pourboghrat, F., Chung, K. & Kang, T. J. 2005. Analysis of flexible bending behavior of woven preform using non-orthogonal constitutive equation. Composites Part A: Applied Science and Manufacturing, 36(6), 839–850.
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13
Friction properties of reinforcements in composites
J. L. G o r c z y c a, K. A. F e t f a t s i d i s and J. A. S h e r w o o d, University of Massachusetts at Lowell, USA
Abstract: The use of binders during the forming of fabric-reinforced composites by thermostamping induces in-plane tensile forces as a result of friction between the fabric and the tooling and adjacent layers of fabric. These in-plane forces reduce the potential for the development of defects in the form of fabric wrinkling. However, if in-plane forces are too high, then fabric tearing can occur. Thus, it is critical to understand the relationship between the effective friction, the forming rate and other processing parameters to produce quality parts using this high-volume lowcost manufacturing process. This chapter summarizes recent research on the friction at the tool/fabric and fabric/fabric interfaces. Key words: thermostamping, thermoforming, friction, woven fabrics, composites.
13.1
Introduction
Composite materials are an attractive material choice for reducing vehicle weight and increasing occupant safety due to the ability to tailor their structural properties, e.g. stiffness, strength and energy absorption, and their high strength-to-weight ratios. However, before composites can be an attractive alternative for the auto industry to use in place of aluminum and steel, it is imperative that composite parts can be made at essentially the same cost and rate as their metal counterparts without sacrificing part quality. Thermostamping of woven fabrics is one manufacturing process that can potentially satisfy these requirements. Finite element modeling can be used to study how the various parameters associated with thermostamping influence the resulting part quality as a function of stamping rate and material choice. One important parameter in the process is friction. Too little friction will result in part wrinkling, while too much friction can result in part tearing. This chapter will present recent research efforts to understand the friction mechanisms associated with tool (metal)/fabric and fabric/fabric interactions during the thermostamping process. The study of the metal–fabric interaction is relatively new. Research into this area has increased as interest in forming processes such as thermostamping 397 © Woodhead Publishing Limited, 2011
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has intensified. Twintex®, which this chapter will use as an example of fabric reinforcement material, is commonly used in this process and is composed of fiberglass fibers commingled with polypropylene fibers. This material is very desirable for forming processes requiring rapid cycle times and is very easy to handle at room temperature. Although this chapter focuses on Twintex ®, the information contained in this chapter can be applied to numerous other woven and stitched fabrics that are combined with a formable matrix material. The polypropylene fibers in the Twintex® material are solid, but flexible, at room temperature and melt at approximately 160°C. During the thermostamping process, a fabric blank is aligned in a rigid steel frame (Fig. 13.1). To achieve the desired ply-stackup schedule, multiple layers of fabric are placed in the frame. The frame travels along the shuttle rails into an oven where the fabric is heated and the polypropylene melts. The frame is then shuttled from the oven to the forming press, which may or may not have heated tooling. At this point, metal binder rings are pressed down on the fabric, and subsequently the metal punch descends. As the punch pushes the fabric blank into the metal die, the friction between the fabric and the tooling (punch, die and binders) and the fabric induces an in-plane tensile force that assists in reducing the potential for wrinkling of the formed part. If the in-plane tensile force is too high, then the fabric may tear, and if the in-plane force is too low, then wrinkling may occur. Thus, it is critical to quantify the magnitude of the friction at the fabric/tool interfaces and fabric/fabric interfaces so the normal force induced by the binder is not too low or too high such that it compromises the quality of the formed part. Table 13.1 summarizes recent experimental work on friction at the metal–fabric interface and includes contributions from the current research. The research in Table 13.1 that is printed in italics is the main focus of this chapter, although the work of other researchers listed in Table 13.1 will also be discussed in this chapter. The experimental results generated by Chow (2002) and Gorczyca et al. (2004, 2005) focusing on the effect of processing parameters on the friction coefficient showed that velocity, normal force and tool temperature had Punch Fabric aligned in frame
Binder ring
Die Fabric heating
Alignment and binder pressure application
Punch application
13.1 Schematic example of the thermostamping process (Gamache, 2007).
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Polymers N/A3 Pre-consolidated glass/PP4 Twill
Pre-consolidated glass/PP Glass/PP preform
Clifford et al. (2001) Chow (2002)
Pre-consolidated glass/PP
Fabric preform (Murtagh et al., 1994)
Unidirectional or woven (Murtagh et al., 1994)
110–1033
Plain, twill
2
Isothermal, 180–332
Isothermal, 180 Fabric: 160–200 Tool: RT, 85–120 Fabric: 160–200 Tool: 21, 70–140 Fabric: 180 Tool: 25, 85–140 Isothermal, 175–205
Isothermal, 21 Isothermal, 180–220
N/A3 N/I
0, 90
N/I2
Fiber orientation ( o)
0/90
0–85 (Chow, 2002)
2.0–83.3
0/90, 45/45, 60/30, 90/0 Any/all orientations
0.33–8.33 0/90
0.2–0.66
£1.2 <9 N/I 8.33–41.6 0/90, 30/60, 45/45, 60/30, 90/0 1.67–83.3 0/90, 30/60, 45/45, 60/30, 90/0 1.67–83.3 0/90
2.5
£2
£0.83
RT1 Isothermal, 25–405
Velocity (mm/s)
Temperature (oC)
Fabric: 180–200 Tool: 85-140 ~50–1050 Fabric: ~200 (Chow, 2002) (Chow, 2002) Tool: 70–120 (Murtagh et al., 1994)
21.05–63.16
Twill, satin
16–63
218–4000
Plain Twill
88.6–1030
<100 387–1030
Plain
Twill Satin
<0.333 80–2500
20–150
0.63–0.470
Pressure (kPa)
RT: room temperature. N/I: not investigated. 3 N/A: not applicable. 4 PP: polypropylene. Italic type indicates research conducted at the University of Massachusetts Lowell.
1
Actual process parameters
ten Thije et al. (2008) Pre-consolidated glass/PP and glass/PPS Fetfatsidis (2009) Glass/PP preform
Vanclooster et al. (2008)
Gorczyca et al. (2004, Glass/PP preform 2005) Gamache (2007) Glass/PP preform
Carbon/PEEK
Murtagh et al. (1994, 1995) Maldonado (1998) Wilks (1999)
Plain, twill, nonwoven Unidirectional
Cotton and wool
Ajayi (1992a, b)
Structure
Material
Reference
Table 13.1 Comparison of process parameters for study of tool–fabric friction
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significant effects on the friction coefficient. The friction-test apparatus used in these studies was a constant-displacement apparatus. Thus, once nesting of the yarns occurred as the fabric was in motion, the applied load tended to decrease. This nesting phenomenon allowed for determining an effective static coefficient of friction, but not a dynamic coefficient of friction. The work of Gorczyca et al. (2005) also included finite element simulations of the thermostamping process using ABAQUS/Standard and showed the importance of incorporating a varying friction coefficient based on the variations in punch force calculated by the finite element simulations when a constant friction coefficient versus a varying friction coefficient was used in the models. To develop a fuller understanding of the friction phenomenon associated with thermostamping, Gamache (2007) and Fetfatsidis (2009) designed and built a constant-load friction-test apparatus. The constant-load apparatus could capture the static and dynamic friction coefficients, thereby allowing for a comprehensive examination of the dependence of friction on normal force, speed and temperature and for the inclusion of these processing parameters in finite element models of the thermostamping process. Gamache and Fetfatsidis expanded upon the work of Chow and Gorczyca et al. and a refined phenomenological friction model was developed. This refined model has been incorporated into ABAQUS/Explicit and LS‑DYNA as a user-defined friction subroutine (Fetfatsidis, 2009). This friction subroutine was used in finite element models of the friction test and the stamping process along with a user-defined material subroutine for composite fabrics developed by Jauffrès et al. (2009). The decision was made to switch from an implicit solver to an explicit solver to take advantage of the more robust contact algorithms available in explicit computational tools versus those currently used in the implicit solvers. The numerical results from ABAQUS/Explicit will be discussed in detail in Section 13.5. However, the reader should be aware that similar results can be obtained using other finite element software packages that have the ability to incorporate a user-defined friction model, such as LS-DYNA. The friction model has the ability to account for the variation in friction between the steel tool and the fabric that occurs as a result of changes in the processing parameters during the thermostamping operation, e.g. normal force on the fabric, relative velocity between contacting surfaces and temperature. The friction model is independent of the fabric material model. However, in this chapter, the friction model is used in conjunction with a user-defined material model that captures the change in material properties due to the reorientations of the tows that occur when the woven fabrics are deformed into geometries with compound curvatures. The results from the finite element simulations include a comparison of punch forces and yarn stresses between variable-friction and constant-friction models.
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401
Theory
Existing friction models include Coulomb (friction between dry surfaces) and hydrodynamic (friction between two surfaces completely separated by a thin layer of fluid). Often, hydrodynamic models are paired with rheological models, which consider the relationship between shear rate and shear stress of polymers at different pressures and temperatures. Currently, ASTM standards exist to determine the coefficient of friction of thin sheets. these standards consider the effects of normal load, temperature and pull-out speed on friction, but they do not account for other factors that are associated with thermostamping structural composite sheets, i.e., sheet viscosity and fiber orientation. For example, ASTM Standard D1894 is referenced in publications by a number of researchers (Ajayi, 1992a, b; Chow, 2002; Maldonado, 1998; Murtagh et al., 1995; Gorczyca et al., 2004; Gamache, 2007; Vanclooster et al., 2008; ten Thije et al., 2008; Fetfatsidis, 2009) whose works will be discussed in the following paragraphs. This astM standard is based on the theory of coulomb friction,
m=F N
13.1
where m is the friction coefficient, and F is the pull-out force required to overcome the normal force, N. this astM test method was designed for static and dynamic friction of thin plastic sheets and requires the pulling of a sheet between two horizontal platens at standard laboratory-temperature and constant-pressure conditions. the user prescribes the applied normal load and the pull-out speed. the pull-out force is recorded. thus, when there are two contacting surfaces, the resulting effective Coulomb friction coefficient can be calculated with the general equation F meff eff = 2N
13.2
Eqs 13.1 and 13.2 have been used by a number of researchers to determine the effective friction coefficient of fabric–metal and polymer–metal interfaces. Maldonado (1998) studied friction between thermoplastics, including polypropylene, and metal. he noted that normal force is not linearly related to the friction coefficient. The results of his study were to be used to understand the effect of friction on forming processes such as injection molding and extrusion and the ultimate relationship between part assembly and product evaluation. Ajayi (1992a, b) studied friction under an isothermal condition at room temperature for cotton and wool fabrics interacting with fabric, rubber and Perspex surfaces. From his research, he found that the friction coefficient decreased with an increasing number of tests on the same sample, the contacting surface affected the friction coefficient, and there was no consistent change © Woodhead Publishing Limited, 2011
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in friction coefficient with increasing velocity. In addition, he noted that the relationship between friction force and normal force was described by F = K · Mn
13.3
where F is the friction force, N is the normal force, K is the friction constant and n is the power index. The friction constant and power index are determined from a regression analysis performed on the experimental data. According to Eq. 13.3, the relationship between normal force and friction coefficient is nonlinear. Upon further research, Ajayi (1992b) noted that the friction coefficient of cotton plain-weave fabrics was affected by changes in yarn geometry such as crimp, thread spacing and balance. Murtagh et al. (1995) studied the effect of processing parameters on intraply slip and found that the viscosity behavior of thermoplastic longfiber reinforced composites used in their study (carbon/PEEK) followed a Bingham Power Law model as given by
t = t yield + K · g n
13.4
where t is the shear stress, g is the shear rate and K and n are constants determined from a regression analysis performed on the experimental data. Murtagh et al. (1995) researched the effects of temperature, normal pressure, fiber orientation (0° and 90° only), mold release agent and surface texture on the press-forming of continuous-fiber thermoplastic composites. From these results, it was found that the friction coefficient was not dependent on velocity at room temperature. However, the friction coefficient increased with increasing temperature and increasing velocity. this dependence of the coefficient of friction on velocity only at elevated temperatures suggested that a friction mechanism other than the coulomb friction mechanism was affecting the results, as velocity is not a variable in the coulomb friction model (Eq. 13.1). Also, as the normal force increased, the friction coefficient decreased. In addition, they found a dependence of the friction coefficient on orientation for unidirectional fabrics. one of the two methods they used to measure friction between two surfaces was ASTM Standard D1894. The other method consisted of pulling a material between two platens, which could be heated. this second test was used in the Murtagh et al. (1994) study for intraply slip. the boundary conditions of this test did not match the actual processing conditions, but this test allowed for maintaining an elevated steady-state temperature. Wilks (1999) designed a pull-out test similar to the one used by Murtagh et al. (1995), which used springs to apply normal pressure. He pulled a metal shim from between two platens covered with a preconsolidated fiberglass–polypropylene twill-weave fabric. He analyzed the effect of processing parameters on shear stress. his model accounted for the effects © Woodhead Publishing Limited, 2011
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of coulomb friction and hydrodynamic friction as given by
t = m · PN + h · g
13.5
where m is the Coulomb friction coefficient, PN is the normal pressure, h is the viscosity and g is the shear rate. Based on his experiments, Wilks (1999) found that pull-out velocity had the greatest effect on shear stress, followed by normal pressure and interface temperature, respectively. All of the works discussed from Ajayi (1992a, b), Maldonado (1998), Murtagh et al. (1994, 1995) and Wilks (1999) included results for the friction coefficient when the maximum sliding velocity was less than 9 mm/s, which is significantly slower than the maximum velocity typically reached in the actual forming process, e.g. 85 mm/s (Chow, 2002). in addition to studying intraply friction, clifford et al. (2001) studied the effects of temperature, normal pressure and velocity on the tool–ply friction interaction. They performed a pull-out experiment on a thermoplasticcomposite sheet between two steel platens and investigated the effect of those parameters on the resulting total shear stress. Because they noted a variation in shear stress with pull-out rate and temperature, their analytical model considered both Coulomb friction and experimentally obtained viscous resistance of the polymer film between the thermoplastic-composite sheet and the steel platens as
t = m · g + f · m · PN
13.6
where t is the shear stress, h is the viscosity of the matrix, g is the shear strain rate, f is the ratio of the area of fibers contacting the steel platen to the area of the steel platen, m is the Coulomb friction coefficient (assumed equal to 0.3) and PN is the normal pressure. clifford et al. (2001) found that this model gives reasonable agreement with experimental data for shear stresses greater than 0.02 MPa. In 2002, Chow proposed an analytical model for friction behavior of a woven fabric between the binder ring and the die from his test results for a commingled glass–polypropylene four-harness satin-weave fabric. his analytical model incorporated weighted effects of coulomb and hydrodynamic friction models to predict the effective friction coefficient for different testing parameter values that could be used in numerical simulations. he first theorized that the transition between these two friction mechanisms for various combinations of the testing parameters resulted in a relationship similar to the Stribeck curve (Hutchings, 1992), which plots the coefficient of friction versus the hersey number in the transition region between elasto-hydrodynamic lubrication and hydrodynamic lubrication (Figs 13.2 and 13.3). The Hersey number, H (Hutchings, 1992), sometimes referred to as the Stribeck number (Stachowiak and Batchelor, 2001) is a function of viscosity, h, speed, U, and normal load, N:
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H =
h·U N
13.7
Stribeck originally developed this theory in the early 1900s to study friction as it relates to both sliding and rolling bearings (Czichos, 1978; Hutchings, 1992; Stachowiak and Batchelor, 2001). As this theory is dependent on viscosity, relative velocity and normal load, it was a reasonable place to start for the study of friction at the interface of steel and a heated, commingled glass–polypropylene woven fabric. According to Stribeck’s theory, the friction coefficient is dependent upon resin viscosity (melted polypropylene), fabric velocity (related to the stamping rate and tool geometry), and normal pressure (binder and tool forces). chow (2002) proposed that this relationship was applicable to the thermostamping of a commingled glass–polypropylene four-harness satin-weave fabric after conducting a single series of experiments with one set of test parameters at six different Hersey numbers. He noted that his data as shown in Fig. 13.3 followed the trend indicated by the Stribeck curve (Fig. 13.2). However, his six values were not necessarily sufficient to explore the full range of the hersey number that would be applicable to the thermostamping process. Because the hersey number depends upon viscosity, a rheological model was necessary for the calculation of the Hersey number in Eq. 13.7. The viscosity term was determined through use of the Power Law of ostwald and de Waele (Fried, 1995):
h = m · g n –1
13.8
Coefficient of friction, µ
Boundary lubrication Hydrodynamic (full film) lubrication Elasto-hydrodynamic lubrication
Region of interest for Chow’s (2002) research H (m–1)
13.2 Theoretical Stribeck curve.
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0.25 6
Coefficient of friction
0.20
0.15 2
4
3
5
1 0.10
0.05
0.00 0.0000
0.0005
0.0010 0.0015 0.0020 Hersey number (m–1)
0.0025 0.0030
Test conditions Point
Tool temp. (°C)
Initial fabric temp. (°C)
Sliding velocity (mm/s)
Normal force (N)
1 2 3 4 5 6
85 85 85 85 85 85
200 200 200 200 180 200
16.67 16.67 8.30 16.67 16.67 41.67
4000 3000 1500 1500 1500 4000
13.3 Experimental Stribeck curve for a commingled glass– polypropylene four-harness satin-weave fabric (Chow, 2002).
Table 13.2 Power-law parameters for polypropylene Temperature, (°C)
g range, (s–1)
Consistency (Nsn/m2)
Power-Law index
180 190 200
100–400 100–3500 100–4000
6.79 ¥ 103 4.89 ¥ 103 4.35 ¥ 103
0.37 0.41 0.41
Source: Fried, 1995.
where m, the consistency, and n, the power-law index, are experimentally determined power-law parameters and g is the shear strain rate of the resin. Table 13.2 lists the power-law parameters for various shear strain ranges and temperatures of melted polypropylene. other rheological models considered were the Bingham and the herschel– Bulkley models. however, the power-law model was ultimately chosen as the rheological model to incorporate into the hersey number calculations
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because the results from the experimental data obtained by Chow (2002) matched the analytical results using that model best when compared to the results using the other two models. the shear strain rate, g , can be expressed as a function of relative velocity, U, and fluid film thickness, h:
g = U h
13.9
Substituting Eqs 13.8 and 13.9 into Eq. 13.7 gives ÊU ˆ H =m·Á ˜ Ë h¯
n –1
·U N
13.10
Chow (2002) assumed a constant film thickness of h = 0.07 mm based on research by clifford et al. (2001) who determined this value from optical microscopy for preconsolidated glass–polypropylene twill-weave samples. Chow (2002) began with an equation included in the work of Gelinck and Schipper (2000), for his research. Equation 13.11 combines the effects of coulomb friction and hydrodynamic friction to calculate an effective coefficient of friction, meff:
meff eff =
mc · N c + hr · g · Ar Nt
13.11
where mc is the Coulomb friction coefficient, Nc is the normal load pertaining to the coulomb friction, Nt is the total normal load, hr is the resin viscosity and Ar is the contact area of the fluid film. This model was appropriate for chow’s research because he theorized that his data fell within the elasto-hydrodynamic region of the stribeck curve where both coulomb and hydrodynamic friction mechanisms exist. The Coulomb portion of Eq. 13.11 is based on a single asperity from the fabric contacting the friction plate. The hydrodynamic portion of Eq. 13.11 is included to account for the fluid friction resulting from the melted resin layer contacting the friction plate. Applying the power-law model to Eq. 13.11 and assuming the power-law coefficients m and n are determined using the initial fabric temperature,
meff eff =
mc · N c + (m · g n –1 ) · g · Ar Nt
13.12
Finally, by accounting for the portion of coulomb friction affecting the effective friction coefficient with the symbol a, the master equation developed by Chow (2002) is derived:
meff eff =
mc · (a · N t ) + (m · g n –1) · g · Ar Nt
13.13
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The portion of Coulomb friction affecting the effective friction coefficient was not determined experimentally in Chow’s research. It was an empirical value based on the experimental results obtained from his tests on a fiberglass– polypropylene four‑harness satin‑weave fabric. Using experimental results from Gorczyca et al. (2005), Eq. 13.13 will be discussed further. Gorczyca et al. (2005) expanded upon the research of Chow (2002) by investigating the results from a larger range of Hersey numbers (Table 13.3 and Fig. 13.4). Table 13.3 Set of Hersey numbers explored by Gorczyca et al. (2005) for Twintex® Test
Hersey number (m–1)
A-1 A-2 B-1 B-2 C-1 C-2 D-1 D-2 E-1 E-2 F-1 F-2 G-1 G-2 H-1 H-2 I-1 I-2
7.45 7.45 9.00 9.00 1.86 1.86 2.40 2.40 3.37 3.37 3.60 3.60 4.31 4.31 1.19 1.19 1.87 1.87
¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥
10–4 10–4 10–4 10–4 10–3 10–3 10–3 10–3 10–3 10–3 10–3 10–3 10–3 10–3 10–2 10–2 10–2 10–2
Velocity (mm/s)
Normal force (N)
0.83 1.67 8.33 16.7 4.17 8.33 8.33 16.7 16.7 41.7 8.33 16.7 1.67 4.17 33.3 83.3 33.3 83.3
1129 1500 3095 4000 1161 1500 1161 1500 1069 1500 774 1000 356 500 392 550 249 350
Architecture: plain weave. Orientation: 0/90. Tool temperature: 85°C. Fabric temperature: 180°C.
Coefficient of friction
0.5 0.4 C1
D1 E2 D2 0.3 F1 G2 A1 B1 C2 E1 F2 G1 B2 A2 0.2
H2 H1
I1
I2
0.1
0 0.0E+00 2.0E-03 4.0E-03 6.0E-03 8.0E-03 1.0E-02 1.2E-02 1.4E-02 1.6E-02 1.8E-02 2.0E-02 Hersey number, m–1
13.4 Hersey number versus experimental friction coefficient.
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Then, an empirical friction model was determined from this expanded investigation by fitting a line to the results in Fig. 13.4: m = 6.12H + 0.27
13.14
where m is the friction coefficient and H is the hersey number. It was determined that a linear fit was appropriate for this research because the range of values studied in this research fall to the right of the trough region of the Stribeck curve, which is essentially linear (Fig. 13.5). Furthermore, this method was deemed appropriate because the results from additional tests conducted to examine the effect of different processing parameters on the friction coefficient fell within the trend predicted by that line in all cases except for the tool–temperature investigation. To account for the variation in tool temperature in the empirical friction model, a linear shift term, Stt, was added to the model: 13.15
m = (6.12H + 0.27) – Sn
where Stt is the shift term for the tool temperature effects. this shift term was determined from a line fit to the tool temperature versus friction coefficient data: mtt = –8.64 ¥ 10–4 Tn + 3.70 ¥ 10–1
13.16
where Ttt is the tool temperature in °C. The baseline value on this curve was taken as 85°C. The shift term, Stt, for Eq. 13.15 was the difference between mtt at the baseline temperature and mtt at the actual tool temperature determined: Stt = m ttB – m tta
13.17
Coefficient of friction, µ
Boundary lubrication Hydrodynamic (full film) lubrication Elasto-hydrodynamic lubrication
Region of interest for Gorczyca et al. (2005) research H (m–1)
13.5 Theoretical Stribeck curve.
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where m ttB is the coefficient of friction at the baseline temperature of 85°C (0.30) and m tta is the coefficient of friction determined at the actual tool temperature using Eq. 13.16. Figure 13.6 shows the effect of this shift term for the range of hersey numbers studied and for the complete range of tool temperatures investigated by Gorczyca et al. thus, an empirical friction model was developed that includes each of the processing factors studied that were found to affect the friction coefficient. This model is different from the theoretical model proposed by Chow (2002) for a four-harness satin-weave fabric. as discussed, his model contained a component related to Coulomb friction. Equation 13.12 is restated below so that it can be easily referenced in this discussion:
meff eff =
mc · (a · N t) + (m · g n –1) · g · Ar Nt
13.18
to review, mc is the coefficient of Coulomb friction, a is the theoretical component of the normal force associated with coulomb friction, and Nt is the normal load. the power-law viscosity parameters are m and n, g is the shear rate and Ar is the contact area of the fluid film. as previously stated, the test results obtained by Gorczyca et al. for the plain-weave fabric are in the region to the right of the trough in the stribeck curve. according to the stribeck theory, only hydrodynamic friction (i.e., no Coulomb friction) exists to the right of the trough in the Stribeck curve because a complete fluid boundary layer exists between the two surfaces (Fig. 13.2). Thus, a = 0 in Eq. 13.18 and the standard expression for hydrodynamic friction meff as a function of hydrodynamic friction is
meff eff =
(m · g n –1) · g · Ar Nt
13.19
Coefficient of friction
0.50 0.45
24°C 50°C 85°C 120°C 140°C
0.40 0.35 0.30 0.25 0.20 0.0E+00
5.0–03
1.0E–02 Hersey number (m–1)
1.5E–02
2.0E–02
13.6 Effect of tool-temperature shift term on coefficient of friction for the range of tool temperatures and Hersey numbers investigated by Gorczyca et al. (2005).
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At first glance, the relationship between the Chow model as depicted in Eqs 13.18 and 13.19 and the model developed by Gorczyca et al. (2005) expressed by Eq. 13.14 is not evident. However, similarities between the two models become apparent through a rearrangement of terms in Eq. 13.18. (As Chow’s model does not take into account the tool temperature, Eq. 13.14 is used for comparison as opposed to Eq. 13.15.) First, by expanding Eq. 13.18,
meff eff =
mc · (a · N t) (m · g n –1) · g · Ar + Nt Nt
13.20
Then, simplifying the first term on the right-hand side of Eq. 13.20,
meff = (mc · a ) · +
(m · g n –1) · g · Ar Nt
13.21
Now, substituting the Hersey number and viscosity (Eqs 13.6 and 13.8) into Eq. 13.21 and simplifying,
meff = (mc · a ) +
Ar · H h
13.22
Rewriting Eq. 13.22 in terms of constants and rearranging terms, meff = C1 · H + C2
13.33
Note that Eq. 13.23 is now in the same general form as Eq. 13.14. Additional refinements to the phenomenological friction model were made by Fetfatsidis (2009). His testing methods and results obtained from the loadcontrol friction-test apparatus will be discussed in more detail in sections 13.3 and 13.4, respectively. However, based on those results, he proposed a phenomenological friction model with a modified Hersey number. To start, he considered the physical definition of the Coulomb friction coefficient, m, as the ratio of the frictional (shear) stress across an interface, t, to the contact pressure, P, between contacting bodies:
m=t P
13.24
From experimental observation of an increasing coefficient of friction with increasing velocity and a decreasing coefficient of friction with increasing normal force, hydrodynamic lubrication can be assumed such that contacting surfaces are fully separated by the thermoplastic matrix. Considering the viscous resistance of the polypropylene film between the composite sample and the metal tooling, the coefficient of friction can be defined as
m=
h · g P
13.25
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where the viscosity, h, of the polypropylene has been defined by the Power Law of Ostwald and de Waele noted in Eq. 13.8 with the shear strain rate, g , defined by Eq. 13.9. Because power-law parameters for polypropylene are not available in the literature at a temperature of 170°C, the parameters at 180°C (Table 13.2) were used in the calculation of the resin viscosity. The consistency and power-law index at 180°C are only applicable to shear rates between 100 and 400 s–1. the velocities generally observed in the thermostamping process (Gorczyca et al., 2004) and studied in the current research yield shear rates within the applicable range assuming a fluid-film thickness of 0.07 mm as deduced from optical micrographs (Clifford et al., 2001). It has been shown that at shear rates between 100 and 400 s–1, the viscosity of polypropylene is nearly the same at temperatures of 170 and 180°C (Vanclooster et al., 2008), justifying the use of the power-law parameters at 180°C. Substituting Eqs 13.8 and 13.9 into Eq. 13.25 gives Êmˆ m = Á n˜ Ëh ¯
ÊU n ˆ ·Á ˜ Ë P¯
13.26
Equation 13.26 describes the influence of the film thickness, the velocity and the pressure on the hydrodynamic friction coefficient. Thus, n
H Mod = U P
13.27
where HMod is the modified Hersey number. Like the Hersey number, H, the modified Hersey number, HMod, takes the opposite effects of velocity and pressure on the friction coefficient and regroups them into a single parameter. Thus, equal modified Hersey numbers should theoretically correspond to equal coefficients of friction. Compared to the use of the conventional Hersey number, the modified Hersey number presents the advantage of not having to assume a value for the fluid-film thickness, h. the required assumptions are that the frictional behavior is hydrodynamic and that the fluid-film thickness is constant for a given modified Hersey number. Results obtained using the modified Hersey number are presented in Section 13.4.
13.3
Testing methodologies (static and dynamic friction coefficients)
ASTM Standard D1894 presents the standards for measuring the static and dynamic coefficients of friction of thin plastic sheets sliding over other thin plastic sheets or over other substances. The measurement of the coefficient of friction is taken from the ratio of a normal force applied on the plastic sheet, and the pulling force required to overcome the normal load to begin
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to slide. This ASTM Standard shows a variety of configurations for an apparatus to measure the coefficient of friction. These configurations have been modified by several researchers to be applicable to conditions seen during the thermostamping process for woven-fabric composite sheets, such as elevated temperatures.
13.3.1 Displacement control Initial designs of a friction-testing apparatus (Chow, 2002; Gorczyca et al., 2003) at the University of Massachusetts Lowell (UML) utilized a cam to position two steel plates and create a normal force on the fabric. The test device is shown in Fig. 13.7. As the fabric was pulled through the pressure plates, the normal force typically dropped due to the fabric thickness effectively decreasing. Thus, the test method was limited to measuring the static coefficient of friction. A typical normal-force vs time curve from this test setup is shown in Fig. 13.8. Note the fabric velocity was zero until ~6 s.
13.3.2 Load control To have a test method that can capture the static and dynamic coefficients of friction, a load-control apparatus was designed and built by Gamache
Cable to instron Motor speed reducer
Speed multiplier
Cam for normal pressure application
DC motor to drive cam
Pressure plates
Load cell attached to fabric holder
Fabric holder
Infrared oven
13.7 Displacement-control friction testing apparatus at UMass-Lowell.
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1600 Maximum tension occurs at this point
1500
Normal load, N
1400 1300 1200 1100 1000 900 800 0
2
4
6 Time, s
8
10
13.8 Normal load drops as fabric is pulled through the pressure plates.
(2007). The load-control apparatus uses an active control system to maintain a normal force on the test sample. The fabric is preheated to approximately 170°C using an infrared oven before being transferred in between the platens that are heated using silicone heating elements to a regulated temperature of 170°C. During the thermoforming process, the fabric temperature is believed to be between 160 and 170°C within the one second required to stamp the part (Fetfatsidis, 2009), i.e. above the melting temperature of polypropylene (~150°C (Lebrun et al., 2004)). The normal force is applied by a pneumatic spring. A servo-pneumatic system controls the air pressure in the spring and maintains a constant normal force. The system is used in a closed-loop control configuration (using LabVIEW) which continuously monitors the individual load output from each of the three normal-force load cells and the summed signal from these load cells and updates the command signal to obtain a desired force level which is prescribed by LabVIEW. In addition to prescribing the command signal for the air spring, the LabVIEW program also continuously acquires the signals from the normal-force load cells, tension-force load cell (used to measure the pulling force), and linear transducer (used to track fabric displacement). A DC motor drives a rack and pinion to pull the sample through the press. The use of a DC motor removes the need for the Instron tensile testing machine, which was used in the displacement-control device to pull the fabric. A close-up view of the components in the load-control test apparatus is shown in Fig. 13.9. Typically, when plotting the pull-out force, F, as a function of displacement, a peak force to initiate slipping is observed. This initial peak corresponds to the static coefficient of friction. Following the initial peak is a somewhat
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Servo-valve
Rack and pinion system
Air spring
Fabric frame
Fabric attachment point
Tension load cell
Track
Cable to rack and pinion
Load cell under bottom press plate Linear transducer
13.9 Close-up of the components of the load-control friction tester. 40 35
Load (N)
30 25 20 Sample 1
15
Sample 2 10
Sample 3
5 0
0
10
20 30 Displacement (mm)
40
50
13.10 Typical pull-out force vs. displacement curves resulting from friction experiments for load-control friction testing.
steady-state value of the pull-out force corresponding to the dynamic coefficient of friction. Using the constant-load test apparatus shown in Fig. 13.9, the effective friction coefficient is calculated using Eq. 13.2. Load–displacement curves for different Twintex® woven-fabric samples obtained from the loadcontrol friction tester are shown in Fig. 13.10. By maintaining a constant normal force, a steady-state value of the pullout force, and thus the dynamic coefficient of friction, can be obtained.
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However, the area of fabric being pulled through the clamped platens must also remain constant, because a drop in the fabric area can result in a drop in the effective pull-out force, thereby making it challenging to quantify a dynamic coefficient of friction. The load-control apparatus was designed such that the area of fabric being pulled through the clamped platens was constant during the test. This constant area requirement was satisfied by having the contact area of the tool shorter than the test fabric. Thus, as the fabric was pulled between the platens, the effective contact area between the tool and the fabric was unchanged. Initial tests showed significant deformation of the fabric, and the polypropylene resin was being squeezed toward the end of the sample as the aft portion of the fabric was drawn into the platens (Fig. 13.11). As a result, the two heated platens (62.2 ¥ 62.2 mm2) were replaced with larger steel heated platens (length 127.4 mm ¥ width 62.2 mm) and fabric samples were cut to the same size as the original platens (62.2 ¥ 62.2 mm2). With the modified platen dimensions, the fabric could be fully covered at the start of each test – preventing resin from being squeezed toward the end of the sample – and could be pulled several millimeters before the effective fabric area under the platens would begin to decrease as the sample left the platen area (Fig. 13.12). The fabric holder must be capable of clamping the fabric sufficiently such that the sample will not slip out of the holder or tear at high tensile loads. The original fabric holder (Fig. 13.13) provided a sufficient clamping force
Deformation of tows
Excess resin
(a)
(b)
13.11 Tow deformation and excess resin on (a) top and (b) bottom of a fabric sample.
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L
1/2 L
>1/2 L
>1/2 L
(a)
1/2 L (b)
13.12 (a) Original test conditions (fabric larger than platen); (b) modified test conditions (fabric smaller than platen).
Original holder
Modified holder
13.13 Fabric holder modification.
but it was time consuming to prepare the edges through consolidation (melt the polypropylene and allow it to harden, thereby stiffening the edges), punch holes along the edges, and then pass screws through the holes to mount the fabric in the holder. The redesign of the fabric holder reduced the sample preparation time (no consolidation step or hole punching) and featured a longer arm to rotate the fabric sample easily from the oven to the platens (Fig. 13.13). Large screws were used to provide the sufficient clamping force that is required to keep the fabric sample intact during each friction test. A series of pins lined the inner perimeter of the fabric holder, pinching into the edges of the fabric sample.
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In addition to measuring the static and dynamic coefficients of friction between the metal tooling and the fabric, the apparatus was modified to be capable of measuring the static and dynamic frictions between adjacent layers of fabric. With the enlarged (height and length) platens, initial attempts were made to wrap fabric around the platens such that the fabric sample inside the holder could be pulled between the platens and would slide against a stationary fabric to investigate the fabric/fabric friction. However, a sufficient method of keeping the stationary fabric wrapped around the platens could not be obtained, as the wrapped fabric frequently detached from the platens. In an alternate method, two pieces of fabric were clamped together on one end, and these two pieces of fabric sandwiched the fabric sample mounted in the fabric holder. As the platens close shut and the fabric inside the holder begins to pull, the ‘sandwich’ clamp is blocked by the closed platens, thus allowing only the middle layer of fabric to be pulled through the test device (Fig. 13.14). The load-control friction test apparatus at University of Massachusetts Lowell was validated by measuring the static and dynamic friction coefficients of a Teflon® sample against the steel tool surfaces and comparing them to values given in the literature. Other examples of load-control friction testing devices include the setup at Katholieke Universiteit Leuven (KUL) in Belgium and the setup at the University of Twente in the Netherlands. Figure 13.15 depicts the friction setup at KUL. For the KUL setup, two metal plates are attached to a frame, which is mounted onto a tensile testing machine. One plate is fixed on the frame while the other plate can be translated horizontally by a pneumatic cylinder. Electrical heaters inside the steel plates are used to warm the plates to the desired temperature. A fabric specimen is pressurized and heated between the metal plates and pulled by attaching it to the crosshead of the tensile testing machine. Interply friction is measured by gripping two stationary outer plies to the machine and pulling only the middle layer attached to the crosshead. The area of overlap between the middle ply and outer plies is 80 ¥ 80 mm². The middle ply is cut slightly larger than the outer plies, to prevent the yarns from deforming during the pull-out test (Vanclooster et al., 2008). Figure 13.16 shows a schematic representation of the load-control experimental N Top platen F Bottom platen
13.14 Schematic of fabric/fabric setup.
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13.15 Configuration for friction tests at KUL (Vanclooster et al., 2008).
setup used for measuring friction at the University of Twente. The fabric layers are heated between two steel pressure platens and pulled out while maintaining a constant normal load on the fabric. The temperature and the exerted normal force are controlled during the test. The total platen area is equal to 50 ¥ 50 mm2 with a thickness of 40 mm (ten Thije and Akkerman, 2009).
13.4
Experimental data
The test matrix shown in Table 13.4 was developed to observe how different combinations of pressures and velocities, i.e. different modified Hersey numbers, affect the friction between the tool and a Twintex® glass/ polypropylene balanced plain-weave fabric during the first second that it takes to form the sample to the shape of the die. Two sets of test parameters were chosen for three modified Hersey numbers that were selected for the
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13.16 Experimental pull-out setup at University of Twente in the Netherlands (Akkerman et al., 2007). Table 13.4 Test conditions studied for Hersey investigation Test
Modified Hersey number, Un/P [(m/s)n/Pa]
A-1 A-2 B-1 B-2 C-1 C-2
4.38 4.38 1.11 1.11 1.98 1.98
¥ ¥ ¥ ¥ ¥ ¥
10–7 10–7 10–6 10–6 10–6 10–6
Velocity (mm/s)
Pressure (kPa)
8.3 16.7 16.7 10.0 25.0 16.7
337 438 172 143 112 97
investigation. The same parameters were also tested to investigate the friction between adjacent layers of the plain-weave fabric. All samples were cut to 51 ¥ 76 mm2, and each test condition was done in triplicate. The results were plotted with error bars of one standard deviation. A symmetric two-layer sample was used for each test.
13.4.1 Tool/fabric friction – plain weave Figure 13.17 shows that equal modified Hersey numbers do produce similar coefficients of friction for the balanced plain-weave fabric passing over the tool surface, and that an upward trend exists between the modified Hersey
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Dynamic coefficient of friction
0.50
0.40
C-1 C-2 B-1 B-2
0.30
0.20 A-1 A-2
0.10
0.00 0.0E+00
5.0E-07
1.0E-06 1.5E-06 Un/P [(m/s)n/Pa]
2.0E-06
2.5E-06
13.17 Dynamic coefficient of friction as a function of modified Hersey number – balanced plain-weave fabric (tool/fabric).
number and the dynamic coefficient of friction. This upward trend is similar to the hydrodynamic region of the stribeck curve. these two observations validate the assumptions of hydrodynamic lubrication and constant fluid-film thickness, h, for a given modified Hersey number and the applicability of using the hersey number to the dynamic coefficient of friction. Equation 13.26 can be rearranged to solve for a varying fluid-film thickness, h, as a function of normal load and velocity from the experimentally obtained coefficients of friction: ÈÊ ˆ h = ÍÁ m˜ ÎË m ¯
1
Ê n ˆ ˘n · ÁU ˜ ˙ Ë P ¯˚
13.28
Using Eq. 13.28, the trend observed in the film thickness based on experimentally measured friction coefficients is consistent with the trend obtained by an analytical model developed by ten thije et al. (2008) that uses the Reynolds equation to determine the fluid-film thickness. For the same modified Hersey numbers, the order of magnitude of the fluid-film thickness compares well between the analytical model and the predictions based on experimental results (Fig. 13.18). It should be noted, however, that the analytical model used by ten thije et al. is for a similar fabric (same yarns) but a different weave (balanced twill-weave) heated to a temperature of 200°C. Nevertheless, the similarities in the trends and the order of magnitude of the fluid-film thickness between the analytical and experimental results confirm the relevance of the approach used.
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180
Fluid film thickness (µm)
160 140
C
120 100 80 60 40 20 0 0.0E+00
B
Experimental prediction Analytical prediction
A
1.0E-06
2.0E-06 Un/P [(m/s)n/Pa]
3.0E-06
4.0E-06
13.18 Comparison of predicted fluid-film thickness between experimental and analytical results (ten Thije et al., 2009).
Dynamic coefficient of friction
0.50
0.40 C-1 C-2
0.30
0.20
B-1 B-2 A-1 A-2
0.10
0.00 0.0E+00
5.0E-07
1.0E-06 1.5E-06 Un/P [(m/s)n/Pa]
2.0E-06
2.5E-06
13.19 Dynamic coefficient of friction as a function of modified Hersey number – balanced plain-weave fabric (fabric/fabric).
13.4.2 Fabric/fabric friction – plain weave When the balanced plain-weave fabric slides against another balanced plainweave fabric, the friction is shown to increase with increasing modified Hersey number (Fig. 13.19). The trend shown in Fig. 13.19 corresponds to the hydrodynamic region of the Stribeck curve. The fabric/fabric friction for the balanced plain-weave fabric was very similar to the tool/fabric coefficient of friction. This similarity indicates that the thickness of the resin film separating the contacting surfaces is comparable for these two cases, according to Eq. 13.28.
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13.5
Modeling of thermostamping
Thermostamping is a deep-drawing process that is often modeled with commercially available finite element codes. In this section, a hemispherical 3D geometry shown in Fig. 13.20 is used to demonstrate the modeling capabilities within the finite element code ABAQUS/Explicit and the importance of friction in the thermostamping process. Simulations using a single [0°/90°] layer are stamped into a hemispherical shape. The fabric model definition is based on a Twintex® commingled glass–polypropylene balanced plain-weave fabric at room temperature. A 200 kPa pressure applied by the binder to the fabric is within the range of pressures typically used in thermostamping. In the absence of any detailed friction data, researchers performing thermoforming simulations have assumed a constant friction coefficient of 0.3 at the fabric/tool and fabric/fabric interfaces. In experimental investigations by Gorczyca et al. (2004) and Fetfatsidis (2009), it was evident that the friction coefficient is affected by the stamping rate (fabric velocity), the binder pressure and punch force (normal force on the fabric) which may vary during the manufacturing process. Thus, a robust simulation will account for the variation in the friction coefficient over the fabric for the duration of the forming process. To study the importance of a varying friction coefficient, the punch reaction forces and yarn tensile stresses are Punch 76 mm Fabric Binder
Punch
Die
83.9 mm R 5 mm Binder Fabric R 20 mm 78 mm Die
13.20 Deep drawing of a hemisphere: geometry of the tools.
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postprocessed from analyses which use a constant coefficient of friction in the forming simulation and are compared to forming simulations that use a variable friction definition. An additional simulation with a zero coefficient of friction is also provided.
13.5.1 No friction During the thermostamping process, fabric layers slide relative to the metal binder(s), punch, and die as well as move relative to one another. During this rapid relative motion, it is clear that the in-plane forces that are induced by the state of friction can potentially affect the ultimate quality of the formed part. Therefore, neglecting friction in a simulation of the forming process will possibly lead to an unrealistic prediction of yarn stresses and punch forces. Figure 13.21 shows that for a displacement of 80 mm, the punch exerts a maximum force of 78 N. In the absence of friction, the predicted punch forces and yarn tensile stresses are expected to be significantly lower than actual values because there is no resistance to motion by frictional forces and the fabric layers are able to slide freely. If used as a predictive design tool, this simulation may provide unrealistic feedback to the design process.
13.5.2 Constant friction A constant coefficient of friction of 0.3 is often assumed in simulations of thermostamping to provide an average or global effect of the frictional forces 0
Punch force (N)
–100
–200
–300
–400 No friction –500 0
10
20
30 40 50 Punch displacement (mm)
60
70
80
13.21 Effect of no friction on punch force for [0°/90°] balanced plainweave fabric (stamping rate 45 mm/s).
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on the quality of the formed part. Using a constant friction coefficient of 0.3, the resulting punch forces as a function of punch displacement are compared to a frictionless simulation in Fig. 13.22. (Note that Figs 13.22–13.23 and 13.25–13.26 do not show the punch forces for the entire 80-mm punch displacement due to the large differences in punch forces among the various 0
Punch force (N)
–1000
–2000
–3000
–4000 Constant friction (0.3) No friction –5000
0
10
20
30 40 50 60 Punch displacement (mm)
70
80
13.22 Effect of constant friction (m = 0.3) versus no friction on punch force for [0°/90°] balanced plain-weave fabric (stamping rate 45 mm/s). 0 9.0 mm/s 22.5 mm/s
Punch force (N)
–1000
45.0 mm/s
–2000
–3000
–4000
–5000 0
10
20
30 40 50 Punch displacement (mm)
60
70
80
13.23 Effect of constant friction (m = 0.3) on punch force for different stamping rates (single-layer balanced plain-weave fabric).
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friction models. Therefore, the punch forces have been truncated to view differences between friction models more clearly.) Figure 13.22 indicates that the addition of friction into the simulation significantly increases the amount of force that the punch must exert to press the fabric layer into the die. While more realistic than a zero-friction scenario, assuming a constant friction coefficient limits the use of the simulation as a design tool that can optimize the manufacturing rate, i.e. make quality parts at the fastest rate possible, because ignoring the dependence of the friction coefficient on velocity and pressure does not allow for the consideration of how forming rate can affect part quality as friction varies with forming rate. For example, when using a constant tool/fabric friction coefficient of 0.3 in a single-layer simulation, varying the stamping rate from 9.0 to 22.5 to 45.0 mm/s showed no change in the punch reaction force as a function of stamping rate (Fig. 13.23). Variations in velocity and pressure over the surface of the fabric may significantly affect punch forces and yarn stresses globally. Thus, it is important to have the option to define a dynamically changing coefficient of friction from point to point across the fabric rather than an average or constant friction coefficient.
13.5.3 Variable friction ABAQUS/Explicit allows the user to implement frictional behavior via a subroutine (VFRIC). Experimental friction data can be linearized over the range of Hersey numbers that are of interest to thermostamping (Fig. 13.24) to relate a friction coefficient dependent on the modified Hersey number that accounts for variations in velocity and pressure.
Dynamic coefficient of friction
0.50
0.40
C-1
C-2
B-1 B-2
0.30
0.20 A-2 0.10
0.00 0.0E+00
A-1
5.0E-07
1.0E-06 1.5E-06 Un/P [(m/s)n/Pa]
2.0E-06
2.5E-06
13.24 Example of linear fit through experimental data – balanced plain-wave Twintex® fabric (tool/fabric).
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The user subroutine uses the nodal velocity, U, and force outputs, P = force/area, from the finite element model to calculate the modified Hersey number and find the corresponding coefficient of friction. This methodology was verified by first simulating the experimental friction test. Figure 13.25 compares the punch forces (stamping rate = 45 mm/s) from the variablefriction single-layer model with the punch forces from the frictionless and constant-friction models described previously. The variable-friction punch force is slightly less than that for a constant-friction assumption, but still significantly greater than for the zero-friction model. Varying the stamping rate from 9.0 to 22.5 to 45.0 to 180 mm/s using a variable-friction model shows that the punch reaction forces increase with increasing stamping rate for a fixed binder pressure of 200 kPa (Fig. 13.26). Assuming that the variable-friction models are accurate representations of the variation in the punch force as a function of position and rate, it is noted that at 45.0 mm/s the punch forces are greatly overpredicted by the constant-friction model (m = 0.3) at 9.0 mm/s but underpredicted relative to the 180 mm/s variable-friction model. As stated previously, as frictional forces increase, there may also be an increase in the yarn tensile stresses in addition to the punch force. Figure 13.27 shows that assuming a constant coefficient of friction (m = 0.3) greatly overpredicts the tensile stresses in the fabric yarns in the single-layer simulation. Similar to the punch force, the tensile stresses are also underpredicted at elevated velocity. Neglecting friction significantly underpredicts the tensile stresses in the yarns. Previous studies have shown that changes in friction 0
Punch force (N)
–1000
–2000
–3000 Constant friction (0.3) –4000
Variable friction No friction
–5000 0
10
20
30 40 50 60 Punch displacement (mm)
70
80
13.25 Comparison of punch forces between variable friction model, constant friction model (m = 0.3), and frictionless model for [0°/90°] balanced plain-weave fabric (stamping rate 45 mm/s).
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0
Punch force (N)
–1000
–2000
–3000
9.0 mm/s - variable friction 22.5 mm/s - variable friction 45.0 mm/s - variable friction
–4000
180.0 mm/s - variable friction 9.0 mm/s - constant friction
–5000 0
10
20
30 40 50 Punch displacement (mm)
60
70
80
13.26 Effect of variable friction on punch force for different stamping rates (balanced plain weave fabric). Maximum stress = 112 MPa
Maximum stress = 72 MPa
112.2 102.9 93.2 83.9 74.0 64.4 54.8 45.3 35.7 26.1 16.5 6.9 –2.6
(a)
(b)
Maximum stress = 3 MPa
(c)
13.27 Tensile stresses in fabric yarns (single [0°/90°] layer) with (a) constant friction coefficient (m = 0.3), (b) variable fabric friction, and (c) no friction. All models used stamping rates of 45 mm/s.
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that lead to higher or lower tensile stresses do not necessarily affect the manner in which the fabric shears. Instead, fabric shearing is controlled more by the kinematics of the fabric conforming to the shape of the mold (Fetfatsidis, 2009).
13.6
Conclusion
An overview of the thermostamping process for the forming of composite parts from woven fabrics was presented, and the importance of the fabric in-plane forces on the quality of such parts was discussed. These in-plane forces are a direct result of the coefficients of friction between the interacting surfaces, i.e. tool/fabric and fabric/fabric. Thus, the ability to account for any variation of the coefficients of friction during the forming process is critical to having a credible model of the forming process. It was shown that using Stribeck theory and a modified Hersey number leads to a friction model that can describe how the friction varies as a function of normal force and fabric velocity. A shift term can be introduced to account for the dependence of friction on temperature. A review of some friction test apparatuses was presented. To account for static and dynamic coefficients of friction, a load-control setup is required due to the potential compaction and nesting of the fabric during the test. The experimental data were implemented into the finite element code ABAQUS/Explicit via a user subroutine to capture the hydrodynamic frictional behavior of the fabrics during the thermoforming process. The user-defined friction subroutine was first validated using a finite element model of the experimental friction test, and accounts for dynamically changing coefficients as a function of variations in velocities and normal forces experienced locally by the fabric. Hemisphere stamping simulations were performed using constant, varying, and no-friction coefficient. The friction at the contacting interfaces significantly affected the force required by the punch to form the part and the resulting tensile stresses in the fabric yarns. Increasing the stamping rate led to an increase in the friction force, thus increasing the punch force and the tensile stresses in the yarns. Using a constant coefficient of friction showed that the punch force had no dependence on the stamping rate. Incorrectly predicting the fabric stresses could lead to unexpected manufacturing defects and not optimizing the processing speed, i.e. manufacturing rate.
13.7
References
Ajayi, J.O.: Fabric smoothness, friction, and handle. Textile Research Journal, 62, 52–59, 1992a. Ajayi, J.O.: Effect of fabric structure on frictional properties. Textile Research Journal, 62, 87–93, 1992b.
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Akkerman, R., Ubbink, M.P., de Rooij, M.B., and ten Thije, R.H.W.: Tool–ply friction in composite forming. Proceedings for the 10th ESAFORM Conference on Material Forming, Zaragoza, Spain, pp. 1080–1085, 2007. ASTM Standard D 1894-08: Standard Test Method for Static and Kinetic Coefficients of Friction of Plastic Film and Sheeting. ASTM International, Philadelphia, PA. Chow, S.: Frictional interaction between blank holder and fabric in stamping of woven thermoplastic composites. Lowell, MA: UMass, 2002. Clifford, M.J., Long, A.C., and de Luca, P.: Forming of engineered prepregs and reinforced thermoplastics. TMS Annual Meeting, Second Global Symposium on Innovations in Materials, Process and Manufacturing: Sheet Materials: Composite Processing, New Orleans, LA, 2001. Czichos, H: Tribology: a Systems Approach to the Science and Technology of Friction Lubrication and Wear. Elsevier Scientific, New York, pp. 130–156, 1978. Fetfatsidis, K.: Characterization of the tool/fabric and fabric/fabric friction for woven fabrics: Static and dynamic. Lowell, MA: UMass, 2009. Fried, J.R.: Polymer Science and Technology. Prentice-Hall, Englewood Cliffs, NJ, 1995. Gamache, L.: The design and implementation of a friction test apparatus based on the thermostamping process of woven-fabric composites. Lowell, MA: UMass, 2007. Gelinck, E.R.M., and Schipper D.J.: Calculation of Stribeck curves for line contacts. Tribology International, 33, 175–181, 2000. Gorczyca, J., Sherwood, J., and Chen, J.: Friction between the tool and the fabric during the thermostamping of woven co-mingled glass–polypropylene composite fabrics. 18th Annual American Society for Composites Conference, pp. 196–205, 2003. Gorczyca, J., Sherwood, J., Liu, L., and Chen, J.: Modeling of friction and shear in thermostamping process – Part I. Journal of Composite Materials, 38, 1911–1929, 2004. Gorczyca, J., Sherwood, J., and Chen, J.: A friction model for use with a commingled fiberglass–polypropylene plain-weave fabric and the metal tool during thermostamping. European Finite Element Revue, 14(6–7), 729–751, 2005. Hutchings, I.M.: Tribology: Friction and Wear of Engineering Materials. CRC Press, Ann Arbor, MI, 1992. Jauffrès, D., Sherwood, J.A., Morris, C.D., and Chen, J.: Simulation of the thermostamping of woven composites: Mesoscopic modelling using explicit FEA codes. Proceedings for the 12th ESAFORM Conference on Material Forming, Twente, Netherlands, 2009. Lebrun, G., Bureau, M.N., and Denault, J.: Thermoforming-stamping of continuous glass fiber/polypropylene composites: Interlaminar and tool–laminate shear properties. Journal of Thermoplastic Composite Materials, 17, 137–165, 2004. Maldonado, J.E.: Coefficient of friction measurements for plastics against metals as a function of normal force. Conference Proceedings Special Areas, Proceedings of the 1998 56th Annual Technical Conference, ANTEC, Part 3, p. 3431, 1998. Murtagh, A.M., Monaghan, M.R., and Mallon, P.J.: Investigation of the interply slip process in continuous fibre thermoplastic composites. Proceedings of the Ninth International Conference on Composite Materials, Madrid, Spain, pp. 311–318, 1994. Murtagh, A.M., Lennon, J.J., and Mallon, P.J.: Surface friction effects related to pressforming of continuous fibre thermoplastic composites. Composites Manufacturing, 6, 169–175, 1995. Stachowiak, G.W., and Batchelor, A.W.: Engineering Tribology, second edition. Butterworth Heinemann, Boston, MA, 2001.
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ten Thije, R.H.W., Akkerman, R., van der Meer, L., and Ubbink, M.P.: Tool–ply friction in thermoplastic composite forming. Proceedings for the 11th ESAFORM Conference on Material Forming, Lyon, France, 2008. ten Thije, R.H.W., and Akkerman, R.: Design of an experimental setup to measure tool–ply and ply–ply friction in thermoplastic Laminates. International Journal of Material Forming, 2, 197–200, 2009. Vanclooster, K., Lomov, S.V., and Verpoest, I.: Investigation of interply shear in composite forming. Proceedings for the 11th ESAFORM Conference on Material Forming, Lyon, France, pp. 957–960, 2008. Wilks, C.E.: Characterization of the tool/ply interface during forming. Nottingham: University of Nottingham, 1999.
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14
Permeability properties of reinforcements in composites
V. M i c h a u d, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland
Abstract: Permeability is defined from the equations of fluid flow through porous media. Modelling of the saturated permeability tensor is then presented using a historical perspective, going from the first geometrical models developed in soil science, to recent unit-cell models based on a precise description of composite reinforcements. Deviations from the saturated flow models are then presented, discussing in particular the effect of capillarity. Finally, experimental methods to measure permeability are described and discussed. Key words: permeability, saturation, Darcy’s law, liquid composite moulding, textiles.
14.1
Introduction
Reinforcements used in the production of polymer composite materials are in general thin filaments or fibres, assembled into yarns or tows, which are further assembled into a fabric. This initially dry fibre assembly thus constitutes a self-sustaining porous body or ‘preform’. During composite processing, a fluid precursor of the matrix phase (a thermoset resin, a thermoplastic polymer or pre-polymer) is made to infiltrate the open pore space within the preform. Upon subsequent chemical reaction or solidification of the matrix precursor, a composite material is produced. The drive to manufacture sound and homogeneous parts at the lowest cost has driven the need to predict the kinetics of the process, as well as the local void and fibre content distribution within the composite, and in some cases residual strain or stress fields that may have built up in the final part. All these final attributes of the process and the resulting part are influenced by the flow characteristics of the fluid matrix precursor into the preform, which are chiefly governed by the fluid viscosity and the resistance to flow brought by the reinforcement. This last point is generally expressed by the permeability of the preform, the object of the present chapter. We will first describe porous media constituted by the more common composite reinforcement preforms, and then define their permeability as it emerges from the equations of flow through porous media. Modelling of the saturated permeability tensor will be presented 431 © Woodhead Publishing Limited, 2011
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from a historical perspective, starting with the first geometrical models developed in soil science, and then moving to more recent unit-cell models that couple precise descriptions of the fluid flow path with realistic models of the reinforcement phase used in producing the composite. Deviations from such saturated models arise, however, when the fluid does not fill all open pore space within the preform: fluid flow is then said to be unsaturated, and the problem is then one of multiphase flow. Experimental methods used in the measurement of permeability are then described, and discussed by comparison with theoretical models.
14.2
The permeability tensor
14.2.1 Porous medium description Textile reinforcements used in composite processing have been described in detail in the previous chapters of this book, together with strategies used in modelling their internal geometry. The mechanical behaviour of dry textile reinforcement has also been described in several chapters: it is important to note that composite reinforcements tend to deform and shear rather easily, with a frequently hysteretic behaviour. For the purpose of the present chapter, the fabrics or preforms are considered as porous media to be invaded or ‘infiltrated’ by a fluid phase. Many methods to describe such preforms can be found in other branches of engineering dealing with porous media, such as soil science, reservoir engineering, textile engineering and membrane science (Bear, 1972; Dullien, 1979; Scheidegger, 1974). Like most porous media, reinforcements used in composite processing are statistical by nature, so a complete description of their pores would require mapping the internal geometry of the whole preform, a task that is still beyond reach in practical cases. A continuum mechanics approach, calling for average properties of the reinforcement, is thus used in most models. This rests on the definition of a representative volume element (REV), large enough to contain representative averages of all phases, solid, liquid and gas, yet small enough to be considered as a differential element on the scale of the preform, to which the model assigns only average values of relevant process parameters such as temperature or pressure. Figure 14.1a provides an example of REV for a reinforcement fabric, and Fig. 14.1b shows a typical micrograph, for a non-crimp fabric composite which was incompletely infiltrated, showing tows, matrix and porosity. A REV contains in the most general case what remains of the initial atmosphere, the fibres, and infiltrated liquid, in respective volume fractions Va, Vf and Vl, such that
Va + Vf + Vl = 1
14.1
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Probability of occurrence
(a)
Pore size (c)
(b)
14.1 (a) Representative volume element containing tows, fluid and pores; (b) micrograph of a carbon fibre non-crimp fabric, showing the tows, the matrix, and pores (part thickness is about 6 mm); (c) typical pore size distribution for a reinforcement fabric.
By similarity with soil mechanics, the fluid phase saturation S is defined as: S=
Vl 1 – Vf
14.2
where (1 – Vf) is the initial, dry preform, open porosity; S varies from 0 to 1 between a dry and a fully infiltrated (saturated) preform. The main descriptors of the dry porous medium are its porosity, its specific surface, and the pore distribution. The fibre volume fraction (complementary of porosity, and most often used in composite practice) is calculated as Vf = NS/(rf h), where N is the number of fabric layers, S the areal mass of the fabric, rf the density of the fibre material and h the height of the fabric stack. For a simple arrangement of aligned cylinders, packed on a square array, the maximum fibre volume fraction that can be attained is Vfmaxs = p/4. For a hexagonal array, Vfmaxh = p/2√3. The specific surface Sf of the porous medium per volume of material is rather difficult to measure as it is also linked to the surface roughness of the reinforcement, and thus its measure depends on the measurement method (Bear, 1972). For composite reinforcements, it can be measured using the BET
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(Brunauer, Emmet and Teller) technique based on the physical absorption of gases on a surface (Verrey, 2006), or alternatively it can be estimated using simple geometrical models. For example, it can be assumed that an assembly of parallel rods of radius r lies on a square array, Sfs = 2Vf/r, or for a hexagonal array, Sfh = 3Vf/r. Pore size distribution is a statistical descriptor of the porous medium that is also difficult to measure directly. It is often measured through gradual filling of the media with non-wetting fluids (Bear, 1972, p. 42). As textiles are most often formed of tows or yarns that are assembled into a preform, the porous medium is generally described by a bimodal pore size distribution, as sketched in Fig. 14.1c.
14.2.2 Fundamentals of flow in porous media The underlying physical phenomena for all composite processes include capillary or surface phenomena, transport of fluid, heat, and mass, the mechanics of preform deformation during infiltration, matrix solidification or chemical cross-linking, and also potential matrix/reinforcement chemical reaction during and after the process. Complete solutions of the flow equations will be described in detail in Chapter 19. In the following, we consider the general case of infiltration by a liquid of a compressible porous preform in which all initial porosity is interconnected (no closed pores), all attributes typical of textile reinforcements. We do not treat heat or mass transfer (as induced by a chemical reaction) since these are not directly used in defining permeability. To describe the flow of liquid in a porous medium, averaged values of relevant parameters, such as velocity or volume fraction, are used to derive equations for conservation of mass and momentum, all in a continuum mechanics approach. It is generally assumed that the densities of liquid and solid phases are constant; in most cases of non-compressible liquid fluids, this is a reasonable assumption; on the other hand it should be removed when infiltration by a gas is considered, which complicates the equations somewhat (Scheidegger, 1974). Mass conservation equations are written for the solid and the fluid phase, respectively, as ∂Vf + — (Vf us ) = 0 ∂t
14.3
∂((1 – Vf ) S ) + — ((1 – Vf ) S ul ) = 0 ∂t
14.4
and
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The momentum equation is generally written using Darcy’s law: (1 – Vf )S (u (ul – us ) = – K —P h
14.5
where K (a function of S and Vf) is the permeability of the porous medium in DV, h is the liquid viscosity, and P is the pressure in the liquid. K, the permeability, is thus defined from Eq. 14.5 as a tensorial quantity, with units m2. Equation 14.5 is written neglecting gravitational or other potential body forces, and is only valid provided the relevant Reynolds number, defined in relation to the average fluid velocity and the pore diameter, is less than about 1: this is most often the case for polymer composite processes because polymers have comparatively high viscosity. The left-hand side of Eq. 14.5 is called the superficial velocity, often also called the filtration velocity, which was initially defined by Darcy as the ratio of the volumetric flow rate Q out of a porous medium, over the cross-section of this porous medium, A (Darcy, 1856). For fully saturated flow in a rigid porous medium, the filtration velocity u0 is generally simply written as: Q = u0 = (1 – Vf )ul A
14.6
Finally, having neglected inertial and body forces in both solid and liquid, stress equilibrium is written using an extension of the effective stress principle developed for partially saturated soils (Wang, 2000): —s¢ – —(BS P) = 0
14.7
where s¢ is the effective stress acting in the solid, counted as positive in compression and averaged over a surface area comprising both solid and liquid. B is the Biot tensor; if the porous medium is isotropic, then B = bI, where I is the identity matrix, and b = 1 – C0/Cs, where C0 is the compression modulus of the fibre assembly, and Cs is the compression modulus of the fibre material itself. If the porous medium is not isotropic, as is often the case for composite reinforcements, the Biot tensor is still diagonal and all three diagonal terms depend on the compliance tensors of the fibre material, and of the fibre assembly (Tran, 2009). Since the fibre material is generally of very high modulus compared to the compressibility of the fibre bed, b = 1 in all directions is often used in composite processing, with very few exceptions (Tran, 2009). Initial and boundary conditions valid for each case complete the definition of the problem. Four main characteristics of the fibre preform and the fluid thus need to be known for a solution of the problem: the viscosity h, the dependence of the saturation S on the local pressure P, the stress–strain behaviour of the preform, and the permeability K. This last parameter is not only important; it is somewhat special in that it varies strongly with its underlying governing
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parameters: K increases with the square of the average pore diameter, and varies even more strongly with the pore volume fraction; it is thus a crucial parameter in modelling composite fabrication processes. In the most general case of multiphase flow, permeability as defined in Eq. 14.5 is a function of the preform volume fraction, fibre arrangement and stress state, as well as of the degree of fluid saturation in the preform, S. Following the approach developed in soil science, K is then generally separated into two terms, K = krKs. The saturated permeability, Ks, is the permeability tensor of the preform for fully saturated flow, a function of the internal geometry of pores in the reinforcement only. The relative permeability, kr, is a scalar ranging from 0 to 1, which is a function of S. In composite processing, most attention has so far been given to the determination of the saturated permeability, Ks. We will focus on this parameter in the next section, and will then briefly address the issue of relative permeability, kr.
14.3
Saturated permeability modelling for fibre preforms
14.3.1 Introduction and historical perspective By definition, the saturated permeability Ks is a characteristic of the preform only, which in principle does not vary with the nature of the infiltrant – provided of course that this infiltrant is an incompressible Newtonian fluid, which furthermore has fully infiltrated all open pores of the preform and flows in the Darcian regime of appropriately low Reynolds number. As indicated earlier, Ks is a tensor, so in principle nine values need to be identified for a three-dimensional case of infiltration: Ê K11 K12 K13 ˆ Á ˜ K s = Á K 21 K 22 K 23 ˜ ÁË K 31 K 32 K 33 ˜¯ For symmetry reasons, Kij = Kji, so only six values differ. Finally, if the principal axes of the tensor are found, it is possible to write the permeability tensor with only three distinct elements, Kx, Ky and Kz on the diagonal, all other terms becoming nil (Bear, 1972). For textile fabrics, it is often easy to determine that the axes of the coordinate system should be oriented with x and y in the plane of the fabric, and z corresponding to its thickness. As a result, in the composite community, three values of the saturated permeability tensor are considered, two in plane, and one through the thickness of the reinforcement, following the fabric geometry as shown in Fig. 14.2. It should be noted, however, that the warp and weft directions of the fabric do not necessarily correspond to the principal in-plane axes.
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x
z
y
14.2 Schematic of a reinforcement preform and principal axes.
14.3.2 Continuum mechanics models for dilute and concentrated fibre beds Capillary tube models The easiest and earliest model of permeability considers unidirectional fluid flow inside a cylindrical tube of radius R along a direction x parallel to the tube axis. The Hagen–Poiseuille solution of the Navier–Stokes equation in this simple case for the volumetric flow rate is (Bird, 2007): Q Ê –dpp /dxˆ Ê p R 4 ˆ R 2 dpp Q=Á , so s o = – ˜ A 8h dxx Ë 2h ¯ ÁË 4 ˜¯
14.8
Comparing Eq. 14.8 to Eq. 14.5, and considering the tube thickness as negligible, it is clear that Ks = R2/8. This simple relation intuitively shows that permeability is roughly in the order of magnitude of the square of the pore space dimension. In a composite reinforcement, the space between fibres has an order of magnitude of a few microns to a few hundred microns, hence permeability values are expected in the range from 10–9 to 10–13 m2. If there are N such tubes per unit area of cross-section normal to the direction of flow, then Vf = NpR2/1, and Ks = (1 – Vf)R2/8. These models are of course very limited as the description of the porous medium is far from realistic; in fact this view of flow through porous media is often misleading. This equation is at the basis of several ‘hydraulic radius’ models that were developed by ‘guessing’ an equivalent or average radius describing with sufficient accuracy the porous medium. The most accepted such model is that derived by Kozeny (1927), later modified by Carman (1937, 1956). In the various forms of this model, the porous medium is treated as a bundle of parallel capillary tubes that are not necessarily circular in cross-section. A constant is introduced that depends on the internal geometry of the porous
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medium. A well-known form of this equation, very often used in composite literature, is given below: 2 (1 – V ) f Ks = r 4ki,i Vf2
3
14.9
where r is the fibre radius, Vf the fibre volume fraction, and ki,i the Kozeny– carman constant (i = x, y, z) (Advani, 1994). The Kozeny–Carman equation is in general reasonably accurate for isotropic porous media (packed particle beds notably) or flow parallel to the axis of parallel fibres, a situation in which it was later re-derived by several authors in the composites literature aiming to arrive at a more precise estimate of kij. For example, Gebart (1992) derived a similar equation, with kxx = 1.78125 for a hexagonal fibre arrangement, and kxx = 1.65625 for a square arrangement. On the other hand this estimate does not work well for many other situations, notably for transverse flow through anisotropic fibrous porous media. This is due in particular to the fact that the Kozeny–Carman constant does not take into account the fact that there is a maximum packing volume fraction, at which the permeability drops to 0 for a fibre volume fraction Vf less than unity, at which touching fibres simply block transverse flow. Several authors have proposed to extend the validity of the Kozeny equation (Åström, 1992; Cai, 1993); however, the Kozeny–Carman constant is not known a priori and such extensions generally rely on an experimental fit valid only for a given fabric, which furthermore has to depend on flow direction, fibre volume fraction, fibre, fluid and pressure gradient, demonstrating the limits of the model. Resistance to flow models for a uniform distribution of cylindrical fibres Another approach to modelling flow in porous media is to consider a fluid, in which spheres or cylinders are suspended without the possibility to move. These suspended solid objects thus present a resistance to flow of the fluid, called a drag force, which can be calculated by solving the Stokes equations for flow of a fluid around a rigid isolated body. Such models are strictly valid only for diluted suspensions, as they do not include the interaction between neighbouring fibres. Happel and Brenner (Bear, 1972) extended this model to treat flow across a less dilute body of cylindrical fibres, introducing a coefficient l that takes into account the interaction between neighbouring fibres. Several other authors proposed models along these lines, proposing a ‘cell’ approach, in which the Stokes equations are solved and the geometry of the cell is defined according to a packing geometry, such as quadratic or hexagonal. For flow parallel to the fibre axis, only one fluid velocity component is present, and solutions are found, assuming zero velocity on the fibre surface,
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and zero velocity gradient at the cell surface, situating this mid-way through the liquid, across planes of symmetry. Most models fall along similar lines, with a general equation given as follows by Drummond and Tahir (Jackson, 1986): 2 Ks 1 ÊÁ –ln V + K + 2 V – Vf ˆ˜ = f f 2¯ r 2 4 Vf Ë
14.10
where K depends on the geometry of the array: K = –1.476 for a square array, and K = –1.354 for a hexagonal array. This curve is shown together with Eq. 14.9 with the coefficients proposed by Gebart, and the simple initial straight tube model, in Fig. 14.3. For flow perpendicular to the fibre axis, similar solutions have been proposed, which are reviewed by Jackson (1986). Solutions are again very similar, taking the following form: Ks = 1 (–ln Vf – 1.476 + 2Vf – 1.774 Vf2 + 4..076Vf3 + O (Vf4 )) r 2 8 Vf 14.11 for a square array, and: Ks = 1 (–ln Vf – 1.490 + 2Vf – 0.5V Vf2 + O (Vf4 )) 2 8 Vf r
14.12
1000
10
k/r2
0.1
0.001
10–5
Navier-Stokes Gebart, hexagonal Gebart, square Happel
10–7
0.2
0.4
Vf
0.6
0.8
1
14.3 Reduced permeability as a function of fibre volume fraction, for flow along the fibre axis.
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for a hexagonal array. These curves are shown in Fig. 14.4. For more concentrated fibre assemblies, nearly all resistance to transverse flow is imposed by the narrowest constriction between two fibres (the wellknown ‘bottleneck effect’). A solution for flow within this restricted space alone, based on fluid-flow patterns often derived from lubrication theory, coupled with an estimate of the spatial density of such constrictions, leads to a permeability value. Several relevant models based on lubrication theory are found in the literature, which give quite similar results, varying mostly by the assumptions taken to analytically solve the equations. These are found in references by Keller (1964), Sangani (1982), Jackson (1986), Gebart (1992), and Bruschke (1993). The most often used expression in the composite literature is that proposed by Gebart (1992), since it was the first to be presented in a composites journal: 5
2 Ks 16 Ê Vf maxs – 1ˆ = Á ˜ r 2 9p 2 Ë Vf ¯
14.13
for a quadratic arrangement of cylinders, and 5
2 Ks 16 Ê Vf maxh – 1ˆ = Á ˜ Vf r 2 9p 6 Ë ¯
14.14
1000
10
k/r2
0.1
0.001
10–5
10–7
Drummond, square Drummond, hexagonal Gebart, square Gebart, hexagonal Carman-Kozeny Bruschke, square Bruschke, hexagonal 0.2
0.4
Vf
0.6
0.8
1
14.4 Reduced permeability as a function of fibre volume fraction, for flow orthogonal to the fibre axis.
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for a hexagonal arrangement of cylinders. Another development was later proposed by Bruschke (1993) by taking a more accurate geometrical description to solve the flow equations for square arrays: ˆ K s 1 (1 – l 2 )2 Ê arc tan(( ((1 + l )/(1 – l )) l 2 = 3l + + 1˜ Á 2 3 2 3 2 r l Ë ¯ 1–l where l =
–1
14.15
4Vf /p , and for hexagonal arrays:
ˆ Ks (1 – lh2 )2 Ê arc tan( (1 + lh )/(1 – lh )))) lh2 = 1 + + 1˜ Á3 2 3 2 r 3 3 lh 1 – lh2 Ë ¯
–1
14.16
where lh = 2 3Vf /p . Equations 14.13 to 14.16 are also plotted in Fig. 14.4, together with the Carman–Kozeny model with the coefficient chosen so as to obtain the same value as with Eq. 14.14 for Vf = 0.4. Note that in fibre preforms, particularly in woven textiles, there are two ‘fibres’: the fibres themselves at the finer scale, but also, on a coarser scale at which much of the fluid flow will occur, the woven fibre tows. As a consequence, these models have been extended to elliptical cylinders, which correspond better to the shape of tows used in the composite reinforcements (Labrecque, 1968; Epstein, 1972; Phelan, 1996; Ranganathan, 1996; Papathanasiou, 2002; Markicevic, 2003; Merhi, 2007), or to non-Newtonian fluids (Bruschke, 1993). More recently, numerical solutions of fluid flow around bundles of cylinders removed the need for several of the above assumptions, and also gave efficient testbed data for evaluation of the validity of the analytical models (Cai, 1993; Berdichevsky, 1993; Gebart 1992; Bruschke, 1993; Nagelhout, 1995; Phelan, 1996). For flow perpendicular to the fibre axes, these showed that Eqs 14.13 to 14.16 are quite accurate in the high volume fraction range of general interest to composite processing. Numerical simulation results are, however, generally limited to simple periodic geometrical units.
14.3.3 Permeability prediction of multi-scale porous media using numerical models Since most reinforcements are not based on uniformly dispersed fibres, but rather comprise fibre bundles, which are often woven or knitted, there is a need to develop permeability predictions for multi-scale porous media. This has motivated the proposal of several models that rely on a more accurate description of the internal pore geometry in preforms of this type.
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The models vary by the solution method chosen to compute fluid flow through the fabric, but the strategy is often similar. In such models, the representative volume element contains fibre tows that are assumed to either constitute an impermeable solid or have themselves a porous structure. One example of such a unit cell is given in Fig. 14.5. To describe a fabric more accurately, it is also possible to produce a series of different representative unit cells that will then be assembled following a given pattern. Then, the relationship between the flow-rate across the unit cell and the pressure drop is calculated by a numerical method (computational fluid dynamics, or finite element method) or an analytical method (combining analytical solutions of flow inside the tow and in between tows) and a permeability value is thus computed for the cell. Flow between tows is generally modelled using the Stokes equation: —P = h—2ul
14.17
The boundary for flow at the tow surface is often treated as a no-slip boundary for impermeable tows, or by matching fluid velocities and/or pressure at the interface when the tow is permeable. In semi-analytic models, flow within the tow is described using Darcy’s law, Eq. 14.5; in some cases, an extension of Darcy’s law to high-velocity flow regimes, known as the Brinkman equation, is used instead, often because it presents the advantage of being more easily coupled, in simulations, with the Stokes equation: —P = –
h u + he — 2 u l Ks l
14.18
The brackets denote (REV) volume average properties, while he is an effective viscosity, which is used to match the shear stress values at the interface between d ul d ul the free flowing fluid and the porous medium: h = he , ddyy dy dy + – y =0
z
y =0
y x
14.5 Schematic of a unit cell used for numerical modelling; channels are dark grey and tows are light grey.
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where y = 0+ represents the boundary on the side of the free-flowing fluid, and y = 0– represents the boundary on the side of the fibre tow. This was used among others by Ranganathan (1996), Phelan (1996) and Song (2005) to model flow within fibre tows. Analytical models have also been developed; in this case the unit cell is greatly simplified so that a tractable solution of the equations can be written for flow within the tow and also between tows, the two being then assembled into a full permeability model. Recent examples are by Yu (2000), Lundström (2000b), Endruweit (2011) and Chen (2010). These models are useful for rapid evaluation of the influence of tow permeability, or intra-tow channel size on the permeability. For example, Chen (2010) showed that the in-plane permeability of a woven fabric scales with the third power of the channel width, while the through-thickness permeability scales with the fourth power of this parameter. Such models can also easily demonstrate that several fabrics having similar overall volume fractions of fibre do not have the same permeability. Many variations of the numerical models are found in the literature (Cai, 1993; Gebart, 1992; Markicevic, 2003; Papathanasiou, 1997, 2002; Verleye, 2008, 2010; Griebel, 2010; Nordlund, 2005, 2006), also extended to the case of non-Newtonian fluid flow (Loix, 2009). These are in general compared to analytical or experimental results, and the agreement is often, beyond the accuracy of the modelling method, related to the accuracy of the fabric description. Alternative models based on lattice Boltzmann methods or smooth particle hydrodynamics (SPH) methods have also been developed, initially at NIST (Spaid, 1997, 1998; Belov, 2004; Comas-Cardona, 2005). These are elegant but remain difficult to use in practice, as the necessary parameters lack a direct relation to physical characteristics of the preforms. The predictive power of these multi-scale models thus relies on the trade-off between a very accurate description of the porous medium and computation complexity. These models are best if a faithful description of the internal structure of the reinforcement is available. As a result, they are often coupled to unit cell models that describe the internal structure of the textiles, such as those described in Chapters 7 and 8. In addition, it is now possible to account for an evolving fibre distribution in the cell, caused by preform deformation when the fabric is sheared or compressed during the preforming operation. Recently, Verleye et al. (2010) compared the speed and accuracy of two methods, one based on a finite-difference solution of Stokes flow in a unit cell, the other using a 2D approximation of the fluid flow path. The Stokes flow solution was found to attain the desired accuracy of 10%, but for several hours of computing, making this solution time consuming. A look-up table with already computed values for a given fabric was proposed to give access to values for K in practical cases. Finally, the description of the porous medium can also be based on an
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experimental 3D reconstruction of a given textile; as mentioned, this is now feasible using X-ray microtomography, as the current resolution is somewhat below the micrometre using X-ray synchrotron radiation (Desplentere, 2005; Badel, 2008; Koivu, 2010).
14.4
Unsaturated permeability modelling
14.4.1 Relative permeability This parameter, inherited from soil science treatment of multi-phase flow, represents the additional resistance to fluid flow created when the pore space comprises a third phase, generally air or a gas. It thus depends on the fibre–matrix system, and is much more difficult to measure in the case of composite systems since model fluids cannot be used, or to predict theoretically given the far greater geometrical complexity of non-saturated flow. Therefore, models are mainly semi-empirical. Such models have been proposed and validated experimentally in soil science, and take the general form: kr = Sn
14.19
where n is an exponent typically between 1 and 3 for particle-based soils (Spitz, 1996) while S is considered here as the saturation in the non-wetting phase (this assumes that the resin is considered in this chapter as the nonwetting phase, but it is possible to consider the inverse case). Alternative models introduce a parameter l (Brooks, 1964), named the pore size distribution index: 2+ l ˆ Ê kr = S 2 Á1 – (1 – S ) l ˜ Ë ¯
14.20
or additional parameters L, M and b (Van Genuchten, 1980) that are related to the shape of the S(P) curve : 1ˆ Ê kr = S L Á1 – (1 – S ) M ˜ Ë ¯
2M
14.21
The approach also implies, as indicated in Section 14.2.2, that the saturation curve be measured for the given reinforcement/matrix system (Patel, 1996a, b; Nordlund, 2008). Although this approach is standard in multi-phase flow modelling for other branches of engineering, it has seldom been used toward composite process modelling. There are examples however, that either were based on formulae derived from soil science (Markicevic, 2006; Patel, 1996a,b; Michaud, 1994) or used inverse determination of Eqs 14.19 and 14.21 from experiment (Dopler, 2000; Nordlund, 2008). This is due to the difficulty in evaluating the necessary parameters with the matrix phase, and
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to the fact that most models developed in soil science implicitly assume that capillary forces are independent of fluid velocity, which is not always true when high-viscosity matrices are used (Verrey, 2006).
14.4.2 Alternative methods: use of sink terms Another approach taken by researchers in the composite field is to consider saturated fluid flow, but introduce pores that will gradually get filled during impregnation, typically representing a tow. In general, flow is assumed to first take place in between the fabric tows, with the resin only penetrating later and more slowly the tows themselves, gradually behind the infiltration front. This implicitly assumes that the fluid does not wet the fabric. In that case, the permeability values are evaluated using saturated models, with one value for the tow permeability, and another value for the overall permeability of the fabric. A sink term is then introduced in the models, providing fairly good agreement when adjusting the saturation rate for tow infiltration (Binetruy, 1997, 1998, 2006; Acheson, 2004; Zhou, 2006; Kuentzer, 2006; Lawrence, 2009; Bréard, 2003; Wolfrath, 2006; Bayldon, 2009). As a result, the overall saturation of the fabric varies with time, but no specific relative permeability model is needed. These models have proven to help solve the issue of permeability evolving during flow progression, but are also heavily reliant on empirical or simplified models, or on the use of experimental values to back-calculate the saturation kinetics.
14.5
Permeability measurement methods
As described in the previous sections, several models have been developed to predict the permeability of an assembly of fibres or tows; however, these are based either on analytical solutions for very simple geometrical descriptions of the pore structure, or on numerical simulations that describe a specific (and often still idealized) pore structure but require one to build a dedicated computer model implying sometimes long computation times. In all cases, validation of the models is crucial; hence it is necessary to confront the models to experimentally measure values of permeability. Also, for practical purposes of composite manufacturing, it is sometimes sufficient to directly measure preform permeability, without the need for a model. Many experimental methods have been proposed to measure permeability of fibre preforms; since there is no established procedure or norm for permeability testing of reinforcements for composite processing (whereas norms exist for permeability measurement to air of textiles (ISO, 1995), for clothing, industrial textiles, air bags, and of course soils), all researchers have developed inhouse methods depending on the fabric type, flow geometry and available funding. As a result, the literature reports many measurement methods and
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results for permeability of fabrics, which are not easily comparable (Sharma, 2010). Parnas et al. (1995, 1997) at NIST proposed to use a reference fabric for standardization of permeability measurement methods, and a benchmark exercise was later reported (Lundström, 2000a), where scatter was observed and attributed to differences in specimen preparation. A benchmark exercise is currently running between several laboratories (Laine, 2010; Abter, 2011) with the aim of reaching practical guidelines for permeability measurement as applied to composite reinforcements. In principle, permeability measurement is rather straightforward: following Darcy’s law, Eq. 14.5, if a fluid of known viscosity flows under a known pressure differential through a known thickness and area of porous medium, measurement of the volumetric flow rate leads to the permeability. This is the method proposed by Darcy in 1856. With fibre composites, reinforcements are often highly anisotropic; as a consequence, in measuring the permeability of fibre preforms, flow of the fluid must be directed along selected orientations to arrive at each of the relevant component(s) of the permeability tensor. In practice, this measurement leads to a large variability, because the porous medium itself is statistical in nature (and this is also true for reinforcements used in composites), or because the measurement method leads to many potential variations. The choice of test fluid is also important: different fluids have been used that ideally have a viscosity close to that of the pre-polymer used in practice. For many cases of liquid composite moulding, thermoset resins are in the viscosity range between 0.3 and 1 Pa.s. As a result, water is never used; rather, several vegetal or mineral oils, corn syrup, dilute polymer solutions (such as PEG solutions), or the resin itself have been used (Sharma, 2010). If the flow is saturated and no capillary effects are present (meaning flow is fully or analogously saturated), all fluids should give the same results, provided each has a stable Newtonian behaviour and does not degrade during flow or react with the reinforcement or its sizing. Temperature control of the fluid during the test is also a crucial point, unfortunately often neglected.
14.5.1 Through-thickness permeability measurement For through-thickness permeability measurement, techniques used for composite reinforcements are all somewhat similar, being based on unidirectional saturated flow across a stack of reinforcement layers, as shown schematically in Fig. 14.6. This is similar to the methods used in soil science for permeability measurement of granular porous media. The critical point is to prevent edge flow between the porous medium and the container; this is in general addressed by using tight-fitting samples, a side membrane that can be fitted to the preform, or by blocking lateral flow (Merhi, 2007; Wu, 1994; Endruweit, 2002; Ouagne, 2010; Drapier, 2002, 2005; Buntain, 2003;
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Gas pressure inlet
Flow rate measurement
Preform maintained inside a tube structure Valve and fluid pressure manometer
14.6 Schematic of through-thickness permeability measurement.
Comas-Cardona, 2007; Song, 2009). Some authors included the possibility of simultaneously compressing the reinforcement stack in the permeameter device, with a goal of evaluating the change of permeability with volume fraction in a direct measurement (Comas-Cardona, 2007; Buntain, 2003; Ouagne, 2010).
14.5.2 In-plane permeability measurement In-plane permeability measurement leads to the identification of the two in-plane values of the permeability tensor, Kx and Ky, that are necessary for almost all LCM processes where in-plane flow predominantly takes place. As a result, diverse methods have been proposed, which can be classified into several measurement types (Sharma, 2010; Verleye, 2010): (1) based on the flow geometry, radial versus unidirectional; (2) based on the inlet boundary condition: constant flow rate or constant inlet pressure; and (3) based on the type of flow: saturated continuous flow, or transient filling of the reinforcement, generally called ‘unsaturated’ measurement. In saturated unidirectional measurements, the permeability is calculated directly from Eq. 14.5 as: Ks =
–Qh Ltot A DP
14.22
where Ltot is the length of the preform in the direction of flow, A its crosssection, and DP the pressure difference in the fluid between outlet and inlet (hence negative), generally measured by a pressure transducer at the flow inlet.
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In unsaturated unidirectional measurements, monitoring of the flow front position with time, L(t) is used to compute permeability, based on integration of Eq. 14.5 with the mass conservation equation (Eq. 14.4), assuming saturated flow. The flow front position is most often measured by visual monitoring through a transparent mould as shown in Fig. 14.7 (Ferland, 1996; Verrey, 2006). Alternative methods have also been proposed, using fibre optic sensors (Ahn, 1995), thermistors (Weitzenböck, 1998), pressure transducers (Endruweit, 2006), or electrical resistance measurements (Luthy, 2001). If flow is under constant applied pressure at the inlet, it is shown that L2(t)/t = y2 is a constant, which is experimentally measured as the slope of the L2(t) versus t curve, and the overall permeability K is measured as: K=
– (1 – Vf )hy 2 2 DP P
14.23
where DP is the (constant) pressure difference in the fluid between the flow front and the inlet (Michaud, 2001). Q t If flow is forced to proceed at a constant flow rate, then L (t ) = A (1 – Vf ) (the flow front position varies linearly with time), and the inlet pressure increases with time, so that: K=
– Q 2h t A 2 (1 – Vf ) DP P
14.24
where DP is the pressure difference between inlet and outlet at instant t. To evaluate DP, an assumption that is often made is to neglect the capillary pressure at the flow front, and set the front pressure to the gas pressure in Mould
Stopwatch
Outlet
Video camera
Injection unit
Valve Pressure sensor
TV + recorder
14.7 Schematic of in-plane permeability measurements, allowing both measurement methods, through flow front monitoring, for unsaturated measurements, and through flow rate measurement after complete filling, for saturated measurements.
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the preform ahead of the infiltration front. This has led to the observation that the overall permeability K measured this way is often different from Ks measured in a saturated flow experiment and depends on the model fluid used (De Parseval, 1997; Pillai, 1998, 2004; Slade, 2001; Tan, 2007; Bréard, 2003; Verleye, 2010). The discrepancy appears greater for fabrics with wider pore size distributions, for example woven fabrics versus random fibre mats (Pillai, 2004). This is related to the wetting characteristics of the fluid as compared to air: according to Eq. 14.19, the permeability value K is reduced by a factor kr < 1 when the fluid is not wetting, and increased when the fluid is wetting (as we considered saturation in non-wetting fluid in Eq. 14.19). Moreover, an overall measurement of K thus depends on the potentially changing saturation level during the experiment. As a result, it is advised to prefer saturated flow experiments if one seeks to measure the ‘true’ saturated permeability. Unsaturated flow measurements are nonetheless often used for practical assessment of the flow kinetics, which is fairly accurate provided the test fluid is close in properties (viscosity, surface tension) to the resin used in the composite process. If the principal directions of the fabric are not known a priori, three measurements are needed in three different in-plane directions to obtain the two in-plane principal permeabilities and their orientation axes. To gain time, with fabric preforms, the in-plane permeability is often measured by infiltrating the model fluid through a central hole in a thin fibre preform made of the fibre lay-up of interest clamped within a flat transparent mould. The progression of the resulting elliptical flow front, directly showing the principal directions of the fabric, is measured using a CCD camera, and the two principal values of the permeability tensor in this plane are deduced from Darcy’s law and the mass conservation equations solved in cylindrical coordinates (Adams, 1987, 1988; Wang, 1994; Weitzenböck, 1999a, b; Lekakou, 1996; Parnas, 2000; Liu, 2007). This relatively easy experiment may, however, again lead to erroneous measurement because of the potential influence of capillarity. Alternative methods for permeability measurement with planar flow use an array of pressure sensors, distributed at several locations close to the surface of the mould, to monitor the fluid pressure at these locations during the test, instead of the flow front. This a priori alleviates the capillary issues (Liu, 2007; Wietgrefe, 2010). Advantages of the radial flow experiments are the lack of race-tracking flow at the edges between the fabric and the mould, and the rapid identification of the principal directions of the permeability tensor. Comparison of results for in-plane permeability measurements shows that the measurement method is not consistent yet, and leads to very scattered results (Laine, 2010; Gommer, 2009; Abter, 2011). Several reasons have been identified, in addition to potential issues of test fluid viscosity change during the experiments. A major reason is the intrinsic variability of the
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fabrics, which has been demonstrated in several studies to be a dominant factor (Hoes, 2004; Lundström, 2000a). As a result, permeability values should be considered as a statistical measurement, and a large number of experiments are required to precisely measure the statistical parameters. Another major reason is a high probability of experimental errors, due for example to manual fabric cutting and placement, to mould deflection during flow, to uncontrolled fabric compression during flow if pressure is high, to race tracking, or to errors in the flow position or fluid pressure measurements. Permeability measurement thus still remains a difficult task, with errors within a factor of 2 often encountered. A potentially valuable method to calibrate measurements is to use a stiff reference porous medium of well-known permeability (Morren, 2009; Vechart, 2010).
14.5.3 Gas permeability measurement The methods described above apply when the test fluid is a viscous liquid. If the porous medium is sensitive to fluids, for example as may be the case with natural cellulose-based fibre mats (Pettersson, 2006) or biological tissues, or if the fabric already contains a fluid phase as for prepregs (Sequeira-Tavares, 2009), it may become necessary to use gas as the test fluid. Equations for flow and mass conservation of the gas phase are similar, but the gas phase is compressible. Solutions have been developed again in soil science (Van Groenewoud, 1968; Stonestrom, 1989; Shan, 1992). The permeability is quite readily measured by placing the porous medium between two gas chambers, one kept at a given pressure Pa, the other one left to equilibrate from an initial pressure Pi, different from Pa, until Pa is reached. The permeability k is then obtained as (Li, 2004; Sequeira-Tavares, 2009): ))(Pa – Pi )˘ kAP Pa È(P + P(t ))( ln Í a = t ˙ ( P – P ( t ) ))( )( P + P ) m ZV a i ˚ a Î a
14.25
where A is the-cross section of the porous medium, Z its thickness, V the volume of the gas chamber of initial pressure Pi, and ma the viscosity of the gas phase. The measured value of permeability is the slip-enhanced permeability, as there may be slip effects at the reinforcement surface, also known as Klinkenberg or Knudsen effects, which alter the value. By conducting several experiments at various levels of pressure, the saturated bˆ Ê permeability value can be obtained as k = K s Á1 – ˜ , where b is a parameter P¯ Ë of the porous medium, deduced from experiments (Wu, 1998; Tanikawa, 2009; Bear, 1972).
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451
Conclusion and future trends
This chapter was intended to provide insight into the theoretical and experimental methods previously and currently used for assessing the permeability of textile reinforcements. Permeability is a key parameter in liquid composite moulding processes; it is a well-defined concept, but still a difficult parameter to quantify, in large part because textile reinforcements are not constant in their internal geometry. As a result, statistical variations should be taken into account, both in the models and in the experimental methods. In addition, the fibre distribution inside a unit cell of fabric is not uniform, and depends on the stress-state of the fabric, so that permeability should also be defined as a function of the stress-state of the textile preform (compression, shear, etc.). Surface tension effects also play a role when unsaturated flow is used to measure permeability, and an adequate method to extract the saturated value from unsaturated experiments is still lacking. Progress is nonetheless observed, on the modelling side thanks to the improvement of computational fluid dynamics methods using real descriptions of the fabric unit cell, and on the experimental side as best practice methods are progressively introduced to standardize the set-ups and analysis. As a result, a large palette of possible modelling methods is now available, depending on the level of precision one seeks to attain: simple analytical models provide quick first estimates, more advanced numerical models based on a precise unit cell description provide accurate data, and complete statistical models, based on a statistical description of the fabric unit cell coupled with simulations, should provide the most realistic picture. Finally, these models should eventually be always coupled to those simulating the unit cell geometrical change with the local stress-state and/or temperature in the mould, so as to map more precisely the local permeability variations during processing of a part.
14.7
References and further reading
(Abter, 2011) Abter, R. et al., Experimental determination of the permeability of textiles, a benchmark exercise. Composites Part A, 42: 1157. (Acheson, 2004) Acheson, J.A., P. Simacek, and S.G. Advani, The implications of fiber compaction and saturation on fully coupled VARTM simulation. Composites Part A, 35: 159. (Adams, 1987) Adams, K.L., and L. Rebenfeld, In-plane flow of fluids in fabrics: structure/ flow characterization. Text Res J 57(11): 647–654. (Adams, 1988) Adams, K.L., W.B. Russel, and L. Rebenfeld, Radial penetration of a viscous liquid into a planar anisotropic porous medium. Int J Multiphase Flow 14(2): 203–215. (Advani, 1994) Advani, S.G., ln Flow and Rheology in Polymer Composites Manufacturing, ed. R.B. Pipes, Elsevier, Vol. 10, p. 591. (Ahn, 1995) Ahn, S.H., W.I. Lee, and G.S. Springer, Measurement of the three-dimensional
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permeability of fiber preforms using embedded fiber optic sensors. J Comp Mater 29(6): 714–733. (Åström, 1992) Åström, B.T., R.B. Pipes, and S.G. Advani, On flow through aligned fiber beds and its application to composites processing. J Comp Mater 26(9): 1351–1373. (Badel, 2008) Badel, P., E. Vidal-Sallé, E. Maire, and P. Boisse, Simulation and tomography analysis of textile composite reinforcement deformation at the mesoscopic scale. Comp Sci Technol 68: 2433–2440. (Bayldon, 2009) Bayldon, J.M., and I.M. Daniel, Flow modeling of the VARTM process including progressive saturation effects. Composites Part A 40: 1044. (Bear, 1972) Bear, J., Dynamics of Fluids in Porous Media, Dover Publications, New York. (Belov, 2004) Belov, E.B., S.V. Lomov, I. Verpoest, T. Peters, D. Roose, R.S. Parnas, K. Hoes and H. Sol, Modelling of permeability of textile reinforcements: lattice Boltzmann method. Comp Sci Technol 64: 1069. (Berdichevsky, 1993) Berdichevsky, A.L., and Z. Cai, Preform permeability predictions by self-consistent method and finite element simulation. Polym Comp, 14(2): 132–143. (Binetruy, 1997) Binetruy, C., B. Hilaire, and J. Pabiot, The interactions between flows ocurring inside and outside fabric tows during RTM. Comp Sci and Technol 57: 587–596. (Binetruy, 1998) Binetruy, C., B. Hilaire, and J. Pabiot, Tow impregnation model and void formation mechanisms during RTM. J Comp Mater, 32(3): 223–245. (Binetruy, 2006) Binetruy, C., B. Gourichon, and P. Krawczak, Experimental investigation of high fiber tow count fabric unsaturation during RTM. Comp Sci Technol 66: 976. (Bird, 2007) Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, revised 2nd edition, Wiley, Chichester, UK. (Bréard, 2003) Bréard, J., Y. Henzel, F. Trochu, and R. Gauvin, Analysis of dynamic flows through porous media. Part I: Comparison between saturated and unsaturated flows in fibrous reinforcements. Polym Comp, 24: 391. (Brooks, 1964) Brooks, R.H., and A.T. Corey, Hydraulic properties of porous media. Colorado State University Hydrology Papers, p. 27. (Bruschke, 1993) Bruschke, M.V., and S.G. Advani, Flow of generalized Newtonian fluids across a periodic array of cylinders. J Rheology, 37(3): 479–498. (Buntain, 2003) Buntain, M.J., and S. Bickerton, Compression flow permeability measurement: a continuous technique. Composites Part A 34: 445. (Cai, 1993) Cai, Z., and A.L. Berdichevsky, An improved self-consistent method for estimating the permeability of a fiber assembly. Polym Comp, 14(4): 314–323. (Carman, 1937) Carman, P.C., Fluid flow through granular beds. Trans Inst Chem Eng 15: 150. (Carman, 1956) Carman, P.C., Flow of Gases through Porous Media, Butterworths Scientific Publications, London. (Chen, 2010) Chen, Z.R., L. Ye, and M. Lu, Permeability predictions for woven fabric preforms. J Comp Mater, 44(13): 1569–1586. (Comas-Cardona, 2005) Comas-Cardona, S., P. Groenenboom, C. Binetruy, A generic mixed FE-SPH method to address hydro-mechanical coupling in liquid composite moulding processes. Composites Part A 36(7): 1004–1010. (Comas-Cardona, 2007) Comas-Cardona, S., C. Binetruy, and P. Krawczak, Unidirectional compression of fibre reinforcements. Part 2: A continuous permeability tensor measurement. Comp Sci Technol 67: 638.
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(Darcy, 1856) Darcy, H., Les Fontaines Publiques de la Ville de Dijon. Victor Dalmont, Paris. (De Parseval, 1997) De Parseval, Y., K.M. Pillai, and S.G. Advani, Simple model for the variation of permeability due to partial saturation in dual scale porous media. Transport in Porous Media 27: 243. (Desplentere, 2005) Desplentere, F., S.V. Lomov, D.L. Woerdeman, I. Verpoest, M. Wevers, and A. Bogdanovich, Micro-CT characterization of variability in 3D textile architecture. Comp Sci Technol 65: 1920–1930. (Dopler, 2000) Dopler, T., A. Modaressi, and V.J. Michaud, Simulation of metal composite isothermal infiltration processing. Metall Mater Trans 31B: 225–233. (Drapier, 2002) Drapier, S., A. Pagot, A. Vautrin, and P. Henrat, Influence of the stitching density on the transverse permeability of non-crimped new concept (NC2) multiaxial reinforcements: measurements and predictions. Comp Sci Technol, 62: 1979. (Drapier, 2005) Drapier, S., J. Monatte, O. Elbouazzaoui, and P. Henrat, Characterization of transient through-thickness permeabilities of Non Crimp New Concept (NC2) multiaxial fabrics. Composites Part A 36: 877. (Dullien, 1979) Dullien, F.A.L., Porous Media, Fluid Transport and Pore Structure, Academic Press, New York. (Endruweit, 2002) Endruweit, A., T. Luthy, and P. Ermanni, Investigation of the influence of textile compression on the out-of-plane permeability of a bidirectional glass fiber fabric. Polym Comp 23(4): 538–554. (Endruweit, 2006) Endruweit, A., P. McGregor, A.C. Long, and M.S. Johnson, Influence of the fabric architecture on the variations in experimentally determined in-plane permeability values. Comp Sci Technol 66(11–12): 1778–1792. (Endruweit, 2011) Endruweit, A., and A.C. Long, A model for the in-plane permeability of triaxially braided reinforcements. Composites Part A 42: 165–172. (Epstein, 1972) Epstein, N., and J. Masliyah, Creeping flow through clusters of spheroids and elliptical cylinders. Chem Eng J 3: 169–175. (Ferland, 1996) Ferland, P., D. Guittard, and F. Trochu, Concurrent methods for permeability measurement in resin transfer molding. Polym Comp, 17(1): 149–158. (Gebart, 1992) Gebart, B.R., Permeability of unidirectional reinforcement for RTM. J Comp Mater, 26(8): 1100–1133. (Gommer, 2009) Gommer, F., S. Lomov, K. Vandenbosche, et al., Error assessment in permeability measurement using radial flow method. Adv Comp Lett, 18(4): 123–130. (Griebel, 2010) Griebel, M., and M. Klitz, Homogenization and numerical simulation of flow in geometries with textile microstructures. Multiscale Modelling and Simulation 8(4): 1439–1460. (Gutowski, 1987) Gutowski, T.G., et al., Consolidation experiments for laminate composites. J Comp Mater 21: 650–669. (Hoes, 2004) Hoes, K., D. Dinescu, H. Sol, et al., Study of nesting induced scatter of permeability values in layered reinforcement fabrics. Composites Part A 35(12): 1407–1418. (ISO, 1995) ISO 9237, Textiles, determination of the permeability of fabrics to air. (Jackson, 1986) Jackson, G.W., and D.F. James, The permeability of fibrous porous media. Can J Chem Eng, 64: 364–374. (Keller, 1964) Keller, J.B., Viscous flow through a grating or lattice of cylinders. J Fluid Mech, 18: 94–96. (Koivu, 2010) Koivu, V., M. Decain, C. Geindreau, et al., Transport properties of
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heterogeneous materials. Combining computerised X-ray micro-tomography and direct numerical simulations. Int J Comp Fluid Dyn 23(10): 713–721. (Kozeny, 1927) Kozeny, J., Über Kapillare Leitung des Wassers in Boden. Sitzungsberichte Akad Wiss Wien Math Naturwiss Kl., Abt 2a, 136: 271–306 (in German). (Kuentzer, 2006) Kuentzer, N., P. Simacek, S.G. Advani, and S.Walsh, Permeability characterization of dual scale fibrous porous media. Composites Part A 37: 2057– 2068. (Labrecque, 1968) Labrecque, R., The effects of fiber cross-sectional shape on the resistance to the flow of fluids through fiber mats. TAPPI 51: 8–15. (Laine, 2010) Laine, B. et al., Experimental determination of textile permeability, a benchmark exercise. Paper 87, 10th Int Conf on Flow Processes in Composite Materials (FPCM10), Switzerland, 11–15 July 2010. (Lawrence, 2009) Lawrence, J.M., V. Neacsu, and S.G. Advani, Modeling the impact of capillary pressure and air entrapment on fiber tow saturation during resin infusion in LCM. Composites Part A 40: 1053–1064. (Lekakou, 1996) Lekakou, C., et al., Measurement techniques and effects on in-plane permeability of woven cloths in resin transfer molding. Composites Part A 27: 401–408. (Li, 2004) Li, H.L., J.J. Jiao, and M. Luk, A falling-pressure method for measuring air permeability of asphalt in laboratory. J Hydrol 286(1–4): 69–77. (Liu, 2007) Liu, Q., R. Parnas, and H. Giffard, New set-up for in-plane permeability measurement. Composites Part A 38: 954–962. (Loix, 2009) Loix, F., L. Orgeas, C. Geindreau, et al., Flow of non-Newtonian liquid polymers through deformed composites reinforcements. Comp Sci Technol, 69(5): 612–619. (Lundström, 2000a) Lundström, T.S., R. Stenberg, R. Bergström, H. Partanen, and P.A. Birkeland, In-plane permeability measurements: a Nordic round-robin study. Composites Part A 31(1): 29–43. (Lundström, 2000b) Lundström, T.S., The permeability of non-crimped stitched fabrics. Composites Part A 31: 1345–1353. (Luthy, 2001) Luthy, T., M. Landert, and P. Ermanni, 1D-permeability measurements based on ultrasound and linear direct current resistance monitoring techniques. J Mater Process Manu 10(1): 25–43. (Markicevic, 2003) Markicevic, B, and T.D. Papathanasiou, An explicit physics-based model for the transverse permeability of multi-material dual porosity fibrous media. Transport in Porous Media 53(3): 265–280. (Markicevic, 2006) Markicevic, B., and N. Djilali, Two-scale modeling in porous media: relative permeability prediction. Phys Fluids 18: 033101. (Merhi, 2007) Merhi, D., V. Michaud, L. Kampfer, P. Vuilliomenet, and J.-A.E. Månson, Transverse permeability of chopped fiber bundle beds: effect of an elliptical bundle cross-section. Composites Part A 38: 739–746. (Michaud, 1994) Michaud, V.J., L. Compton, and A. Mortensen, Capillarity in isothermal infiltration of alumina fiber preforms with aluminum. Metall Trans 25A(10): 2145– 2152. (Michaud, 2001) Michaud, V., and A. Mortensen, Infiltration processing of fiber reinforced composites: governing phenomena. Composites Part A, 32: 981–996. (Morren, 2009) Morren, G., M. Bottiglieri, S. Bossuyt, et al., A reference specimen for permeability measurements of fibrous reinforcements for RTM. Composites Part A 40(3): 244–250.
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(Shan, 1992) Shan, C., R.W. Falta, and I. Javandel, Analytical solutions for steady-state gas-flow to a soil vapor extraction well. Water Resour Res 28(4): 1105–1120. (Sharma, 2010) S. Sharma, D. A. Siginer, Permeability measurement methods in porous media of fiber reinforced composites. Appl Mech Rev 63: 020802. (Slade, 2001) Slade, J., K.M. Pillai, and S.G. Advani, Investigation of unsaturated flow in woven, braided and stitched fiber mats during mold-filling in resin transfer molding. Polym Comp 22: 491. (Song, 2005) Song, Y.S., K. Chung, T.J. Kang, and J.R. Youn, Numerical prediction of permeability tensor for three dimensional circular braided preform, by considering intra-tow flow. Polymers and Polymer Composites, 13: 323–334. (Song, 2009) Song, Y.S., D. Heider, and J.R. Youn, Statistical characteristics of out-ofplane permeability for plain-woven structure. Polym Comp 30(10): 1465–1472 (Spaid, 1997) Spaid, M.A.A., and F.R. Phelan, Lattice Boltzmann methods for modeling microscale flow in fibrous porous media. Phys Fluids 9(9): 2468–2474. (Spaid, 1998) Spaid, M.A.A., and F.R. Phelan, Modeling void formation dynamics in fibrous porous media with the lattice Boltzmann method. Composites Part A 29(7): 749–755. (Spitz, 1996) Spitz, K., and J. Moreno, A Practical Guide to Groundwater and Solute Transport Modeling. John Wiley & Sons, New York. (Stonestrom, 1989) Stonestrom, D.A., and J. Rubin, Air permeability and trapped-air content in two soils. Water Resour Res 25(9): 1959–1969. (Tan, 2007) Tan, H., T. Roy, and K.M. Pillai, Variations in unsaturated flow with flow direction in resin transfer molding: an experimental investigation. Composites Part A 38: 1872. (Tanikawa, 2009) Tanikawa, W., and T. Shimamoto, Comparison of Klinkenberg-corrected gas permeability and water permeability in sedimentary rocks. Int J Rock Mech Min 46(2): 229–238. (Tran, 2009) Tran, T., C. Binetruy, S. Comas-Cardona, and N.-E. Abriak, Microporomechanical behavior of perfectly straight unidirectional fiber assembly: Theoretical and experimental. Comp Sci Technol, 69(2): 199–206. (Van Genuchten, 1980) Van Genuchten, M.T., A closed form for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44: 892–898. (Van Groenewoud, 1968) Van Groenewoud, H., Methods and apparatus for measuring air permeability of soil. Soil Sci 106(4): 275–279. (Vechart, 2010) Vechart, A.P., R. Masoodi, and K.M. Pillai, Design and evaluation of an idealized porous medium for calibration of permeability measuring device. Adv Comp Lett, 19(1): 35–49. (Verleye, 2008) Verleye, B., R. Croce, M. Griebel, M. Klitz, S. Lomov, G. Morren, H. Sol, I. Verpoest, and D. Roose, Permeability of textile reinforcements: Simulation, influence of shear and validation. Comp Sci Technol 68: 2804–2810. (Verleye, 2010) Verleye, B., S. Lomov, A. Long, I. Verpoest, and D. Roose, Permeability prediction for the meso–macro coupling in the simulation of the impregnation stage of resin transfer moulding. Composites Part A 41: 29–35. (Verrey, 2006) Verrey, V., V. Michaud and J.-A.E. Månson, Dynamic capillary effects in liquid composite molding with non-crimped fabrics. Composites Part A 37(1): 92–102. (Wang, 1994) Wang, T.J., C.-H. Wu, and L.J. Lee, In-plane permeability measurement and analysis in liquid composite molding. Polym Comp; 15(4): 278–288. (Wang, 2000) Wang, H.F., Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology. Princeton University Press, Princeton, NJ. © Woodhead Publishing Limited, 2011
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Microscopic approaches for understanding the mechanical behaviour of reinforcement in composites
D. D u r v i l l e, Ecole Centrale Paris/CNRS UMR 8579, France
Abstract: An approach to the mechanical behaviour of textile composites at the scale of their constituting fibres, using an implicit finite element simulation code, is proposed in this chapter. The approach is based on efficient methods and algorithms to detect and take into account contactfriction interactions between elementary fibres. It allows one to model samples of woven textile composites made of a few hundreds of fibres, with an elastic matrix. The approach is employed first to determine the initial configuration of woven fabric samples, before applying to them various loadings in order to identify their mechanical properties under various solicitations. Key words: finite element, contact-friction between fibres, mechanical properties of woven fabrics, modelling of fibrous materials at microscopic scale.
15.1
Introduction
Textile composites manufactured from woven fabrics are formed of arrangements of fibres and matrix. The organization of fibres into two levels – the tows and the woven fabric – gives a multiscale aspect to these materials, characterized by a complex mechanical behaviour whose nonlinearities are mainly due to the interactions between components. Various modellings of this mechanical behaviour are possible depending on the scale at which phenomena are addressed. Macro-models at the scale of the fabric require a homogenized approach to the fabric behaviour. At a lower scale, having identified meso-models representing the behaviour of elementary tows, it is possible to model the woven structure as an assembly of tows, taking into account interactions between them. Models at intermediate scales cannot yet account for phenomena occurring at the scale of fibres, such as local compaction or rearrangement of fibres within tows, and which can be responsible for nonlinear effects at the global scale. The approach presented here aims at modelling samples of textile composites at the scale of fibres, using a simulation code based on the finite element 461 © Woodhead Publishing Limited, 2011
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method, which was specially developed for the study of the mechanical behaviour of entangled materials, in order to identify their mechanical response when subject to various types of loadings (Durville 2005, 2009, 2010). The key issue in this approach at microscopic scale, which considers all fibres constituting the studied samples, is the taking into account of contact-friction interactions developed between these fibres. The solving of the mechanical problem involving a large number of contacts in a reasonable computation time, and using an implicit solver, is made possible by the development of methods and algorithms to detect contacts and to handle the nonlinear interactions they generate. The development of such methods allows one to handle assemblies made of a few hundreds of fibres, and subject to large displacements and strains, and to simulate the behaviour of small samples of woven fabrics directly at the scale of fibres. Two main results are derived from this approach. The first one is the computation of the initial configuration, using a special process which gradually moves fibres until satisfying the chosen weaving pattern. The trajectory of each fibre is thus obtained through the solution of a global mechanical equilibrium, which offers an accurate geometrical description of the arrangement of fibres and tows within the fabric. Then this woven structure can be submitted to different loading cases, to identify its mechanical behaviour. This way of tackling the problem at the scale of fibres has two main advantages. First of all it eliminates the need for the definition of the initial configuration, which can be very complex. Secondly, no intermediate model at meso-scale is required, and the only parameters to be defined are the characteristics of fibres (radius and elastic parameters) and the parameters governing contact-friction reactions. As the fabric is joined to an elastic matrix to form the composite, a particular modelling is made to account for the presence of this matrix and for its interactions with fibres. A solid mesh, not conforming with the meshes of fibres but with an overlapping region of controlled thickness with the tows, is automatically generated on both sides of the fabric. Connection elements are then introduced in the overlapping region to ensure the link between the fibres and the matrix. This modelling allows one to simulate the application of loading cases inducing large displacements on composite samples, coupling the approach at the mesoscopic scale of fibres with the consideration of an elastic matrix. To present the proposed approach, the chapter is organized as follows. The next section describes the interests of the approach at microscopic scale. This is followed by a discussion on the modelling approach to textile composites. The chapter concludes with a section on application examples.
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Interests and goals of the approach at microscopic scale
15.2.1 Complexity of textile composite reinforcements Because they are made of arrangements of fibres, tows and matrix, textile composite reinforcements appear as heterogeneous materials with a multi-scale structure. If they can be considered as continuous structures at a macroscopic scale, at lower levels, they must be viewed as discrete assemblies of tows and fibres. Their global behaviour thus involves both continuous aspects, essentially related to the behaviour of fibres and tows in longitudinal directions, and discrete aspects related to interactions between tows or fibres, and influencing mostly the behaviour of these components in transverse directions. Due to their low transverse and bending stiffnesses, these materials may undergo large deformations which can change the geometry and the arrangement of tows and fibres, thus inducing couplings and nonlinearities at different levels.
15.2.2 Determination of the initial geometry An accurate definition of the geometry of woven fabrics used as reinforcement may be essential for two reasons. First, a good knowledge of porosities between tows and fibres is required to assess the permeability of such materials with respect to the injection of resin during the impregnation process. Second, running finite element simulations at the meso-scale in order to assess the mechanical response of textile composites requires one to be able to describe the geometry of tows in order to mesh the different components corresponding to the tows and the matrix. However, the arrangement of fibres and tows in a woven assembly is hard to determine a priori, since it results from complex interactions between these components during the weaving process. The way fibres rearrange within tows, and the way tows deform accordingly in different directions, both play a predominant role in the determination of the trajectories of fibres and tows, and of the shapes of tow cross-sections. Geometrical approaches have been developed to model the initial geometry of textile materials. Robitaille et al. (2003) developed an approach to determine volumes occupied by matrix and tows in an unit-cell, by identifying elementary volumes based on geometrical constructions. Verpoest and Lomov (2005) approximated the trajectories of yarns by geometries minimizing the yarn deformation energy, and generated the volume of yarns by sweeping these trajectories with cross-sections of constant shape but with varying sizes. Hivet and Boisse (2008) tried to identify precisely the varying contours of crosssections along the yarn, depending on their relative positions with respect to the other yarns, to provide an accurate geometrical description of the © Woodhead Publishing Limited, 2011
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woven structure, avoiding in particular penetration between tows. All these approaches are designed to supply mechanical models, and particularly finite element models, with a relevant geometrical description of components of a textile composite. Geometrical parameters on which these approaches are based need to be identified from observations on manufactured fabrics. An alternative way to approach the geometry of textile reinforcements is to simulate the manufacturing of these structures so that their components (yarns or fibres) take naturally their equilibrium positions depending on the interactions they develop with other components. In this way, only very few geometrical assumptions are needed. Dynamic explicit FE simulation codes have been used in particular for this purpose. Finckh (2004) simulated the weaving of fabrics, taking into account several fibres within each yarn. Pickett et al. (2009) simulated the braiding of a preform made of 96 yarns, representing each yarn by bar elements, using also an explicit FE simulation code. Simulation by means of so-called digital elements was proposed by Miao et al. (2008) and Wang et al. (2010) to calculate the geometry of woven and braided structures, considering several fibres within each yarn. Similarly, the approach we propose here aims at determining the initial geometry of samples of woven fabric, by making tows made of several fibres fulfil gradually the weaving pattern chosen for the fabric. This approach, characterized by an implicit method to solve the mechanical equilibrium, is based on methods and tools specially developed for the mechanical modelling of entangled materials.
15.2.3 Approach to the mechanical behaviour of textile reinforcements Beside geometrical issues, textile composites are characterized by complex mechanical behaviours at different scales. Tows display a very specific mechanical behaviour due to the fact that they are formed of an assembly of discrete fibres. Their behaviour in the longitudinal direction mainly depends on the elasticity of fibres, whereas their behaviour in transverse directions is essentially ruled by contact-friction interactions between fibres. Friction furthermore introduces couplings between loadings in different directions, since frictional forces between fibres depend on normal reactions, which depend themselves both on the curvature of fibres and on their tensile stress. Because of these friction effects, the twist of tows and their tensile stress have a large influence on their transverse behaviour. The scale of tows can be qualified as meso-scale. Identifying nonlinear and coupled behaviours at this mesoscale is a difficult task which requires a sound understanding of phenomena taking place between fibres. Although many attempts have been made to propose such mesomodels (see, for example, Lomov et al., 2007, Boisse et al., 2010), this remains an open issue.
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Taking into account the individual fibres constituting the tows, the formulation of models at the meso-scale is no longer necessary. Only the mechanical behaviour of fibres, the geometrical arrangement of these fibres and the friction interactions between them need to be defined to represent the global behaviour of the tow.
15.3
Modelling approach to textile composites at microscopic scale
To identify the mechanical behaviour of textile composites, small samples made of fibres and matrix, and subject to various loadings on their edges, are studied. The global problem is set in the form of the seek of the mechanical equilibrium of an assembly of fibres, under quasistatic assumptions and considering finite strains and large displacements. To achieve such a modelling, the mechanical behaviour of each constituent component, namely the fibres and the matrix, have first to be accounted for through appropriate models. Then interactions between these components (contact-friction interactions between fibres and connections between the fibres and the matrix) need to be represented. Finally, as the edges of the considered samples consist of assemblies of fibres, a complex driving of boundary conditions must be implemented in order to apply the desired loadings without interfering too much with the changes of the local arrangement of fibres within tows.
15.3.1 Finite strain beam model for fibres An enriched kinematic beam model has been adopted to represent the behaviour of fibres. According to Antman’s theory (Antman 2004), this model describes the kinematics of any beam cross-section by the means of three vector fields, one for the position of the centre of the cross-section, and two directors to represent first-order deformations of the cross-section. According to this model, the position of any particle x of the beam, identified by its three components (x1, x2, x3) in a material configuration, can be expressed as follows:
x(x1, x2, x3) = x0(x3) + x1g1(x3) + x2g2(x3)
15.1
where x0(x3) is the position of the centre of the cross-section, and g1(x3) and g2(x3) are two directors of the section (Fig. 15.1), these three vectors depending only on the curvilinear abscissa x3. This expression can be interpreted as a first-order Taylor expansion of the position vector with respect to the transverse coordinates (x1, x2) of the particle. In accordance with the expression of the position, the displacement of any particle is written in the following way:
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2
Material particle 1
x x3
3
Reference configuration
15.1 Beam kinematic model.
u(x1, x2, x3) = u0(x3) + x1h1(x3) + x2h2(x3)
15.2
where u0(x3) is the displacement of the centre of the cross-section, and h1(x3) and h2(x3) are the variations of the section directors. No assumption is made on the norm of the section directors, nor on the angle between them. This means that the cross-sections of the beam can deform depending on the variations of the section directors. However, as the surfaces of cross-sections are generated from the two directors, they remain plane when deformed. If their initial shape is circular, they can deform to any elliptical shape. This kinematic model considers nine degrees of freedom per crosssection, and is therefore a little more expensive than models assuming rigid cross-sections and using only six degrees of freedom per cross-section. This enriched kinematic model, however, offers some advantages. First, the use of directors to define the kinematics of cross-sections avoids the handling of finite rotations and greatly simplifies the formulation of the model. Second, thanks to the deformations of cross-sections introduced by these directors, no component of the Green–Lagrange strain tensor is a priori zero. This allows one to use standard constitutive laws involving all components of the strain tensor, and to reproduce in particular the Poisson’s effect. Adaptation of stiffness to model macro-fibres The actual number of fibres in each tow may be too high for the problem to be solved in reasonable time. To circumvent this limit, a reduced number
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of so-called macro-fibres, with a larger diameter, can be considered. If Nf is the actual number of fibres in a tow, with diameter Df, and Nm is the number of macro-fibres chosen to represent these actual fibres, the diameter Dm of macro-fibres is calculated to keep the same cross-sectional area:
Dm = (Nf/Nm)1/2 Df
15.3
Since the axial stiffness of a fibre is proportional to its cross-sectional area, a macro-fibre will have an axial stiffness equivalent to that of the number of actual fibres it represents. However, as the bending and torsional stiffnesses depend on the second moment of area, If = pr4/4, which is proportional to the radius to the power 4, the bending and torsional stiffnesses of the macrofibres would be overestimated. To correct this effect, the second moment of area of the macro-fibre, denoted Im, is calculated as
Im = Nf/Nm If
15.4
15.3.2 Contact-friction interactions between beams: a central issue The taking into account of contact-friction interactions is the main issue in the problem considered here. The high number of fibres leads to a high density of contacts between fibres, and the modelling of contact and friction interactions introduces additional nonlinearities. The consideration of these nonlinearities is what makes the approach relevant to follow phenomena at the scale of fibres, but complicates the solving of the problem. Effective methods are required to detect contact, and appropriate models for contact and friction, leading to efficient numerical algorithms, must be developed to reduce the number of iterations needed per loading step using an implicit solver. Geometrical aspects: detection of contacts within an assembly of fibres Due to the large deformations undergone by the tows, contacts between fibres change continuously during the loading. Not only can their relative locations with respect to the fibres change, but some of them can appear and disappear. As contacts are not fixed, their detection must be regularly updated, and the method employed to detect them needs to be effective. The goal of the detection of contact is to associate entities on the surface of fibres to which non interpenetration conditions should be applied. Because of the one-dimensional geometry of fibres, we will assume that a contact zone between two fibres is represented by a line. Depending on the relative orientations between the two fibres, the contact line formed between them can have very different extensions: for two perpendicular fibres, it reduces to
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a point, whereas it may have the same length as the fibres when they follow parallel trajectories. In order to account for these different configurations, and consistently with the approximation of the geometry of fibres by finite element shape functions, we propose to check contact between fibres by means of punctual contact elements constituted by pairs of material particles that are predicted to enter into contact. Generation process of contact elements The generation of contact elements often follows a master–slave strategy. Determining one point on the surface of the slave structure, a corresponding target is searched on the opposite master surface, using a given contact search direction, which is usually taken as the normal vector to one of the two surfaces. This method can be criticized for not providing a symmetrical treatment for both structures, due to the fact that the contact search direction is determined from the geometry of only one of the two structures. Instead of determining contact from one structure with respect to the other, the approach taken here consists in considering both structures with respect to an intermediate geometry defined as an approximation of the actual contact line. In this way, the contact search direction is determined from this intermediate geometry, and both structures are considered symmetrically. The method is implemented according to the following steps. First, proximity zones, defined as pairs of parts of fibres that are close enough to each other, are determined within the whole assembly of fibres. In order not to be too time consuming, this search of proximity zones is performed evaluating distance criteria at test points distributed on the fibres with a coarse discretization. The kth proximity zone between fibres i and j, denoted Zijk, is defined as follows by a pair of intervals of curvilinear abscissae on both fibres (see Fig. 15.2):
Zijk = {[ai, bi],[aj, bj]}
15.5
For each proximity zone, the intermediate geometry is defined as the average of both parts of the line constituting the proximity zone. Each point on this geometry, identified by its relative curvilinear abscissa s, is defined by xint (s) = 1 [x0i (ai + s(bi – ai)) + x0j (aj + s(bj – aj))] 15.6 2 This average geometry, viewed as a means to approximate the unknown actual geometry of the contact line, is then employed as geometrical support for the discretization of contact. Contact is checked at some discrete points xc defined by their abscissa sc on the intermediate geometry:
xc = xint (sc)
15.7
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Proximity zone ai bi
bj aj Intermediate geometry
15.2 Determination of a proximity zone between two fibres. Plane orthogonal to the intermediate geometry
xi xint (s) xj Intermediate geometry
15.3 Determination of material particles constituting a contact element.
defined as the pair of material particles located on the surface of fibres that are predicted to enter into contact at this location:
Ec (xc) = (xi, xj) such that xi, xj enter into contact at xc
15.8
The determination of the two material particles that are candidate to contact is performed in two steps (see Fig. 15.3). First we determine the position of the centres of cross-sections that are candidate to contact by the intersections between the plane orthogonal to the intermediate geometry going through xc and the centrelines of the fibres. Next, for each cross-section, the particle
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that is candidate to contact is localized on the outer contour of the section using the directions between both centres of cross-sections. Linearized contact conditions Once pairs of material particles constituting contact elements have been determined, kinematic conditions designed to prevent interpenetration between fibres must be established for these contact elements. A gap distance, gap(Ec), is defined between these particles according to a normal contact direction N(Ec): gap(Ec) = (xi(xi) – xj(xj), N(Ec))
15.9
In order to fulfil non-interpenetration conditions, the gap distance defined for each contact element must remain positive. The determination of the normal direction N(Ec) is essential in the formulation of this condition. This direction needs to be properly oriented in order to prevent interacting fibres from crossing through each other. Depending on the angle between fibres, it is calculated in different ways using the positions of the centres of crosssections and the tangent vectors to the centroidal lines. Mechanical models for contact and friction The convergence of algorithms used to solve contact problems depends to a large extent on the mechanical models expressing contact-friction interactions at contact elements. Mechanical model for normal contact To enforce kinematic contact conditions, a penalty method is employed, yet with two main refinements. The first is a regularization of the standard penalty method, by a quadratic function for very small penetrations. Instead of defining the normal reaction as simply proportional to the gap, with a penalty parameter kN, it is assumed to be quadratic with respect to the penetration for penetrations lower than a given regularization threshold preg, and linear for larger ones, according to the following expression:
RN(Ec) = 0,
if gap(Ec) ≥ 0
RN(Ec) = (kN/(2preg)) (gap(Ec))2,
if –preg ≤ gap(Ec) ≤ 0 15.10
RN(Ec) = – kN gap(Ec) – (kN/2)preg, if gap(Ec) ≤ preg
The regularization ensures a continuity of the derivative of the normal reaction with respect to the gap at the origin. This improvement is very useful to stabilize the contact algorithm, particularly for contacts with very
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low reaction force, the status of which may be oscillating from one iteration to the next. The second improvement of the penalty method is provided by a local adaptation of the penalty parameter for each proximity zone. The determination of this parameter is a delicate issue since it controls the penetration allowed by the penalty method at contact elements. The quadratic regularization is efficient provided a small amount of contact elements is concerned by the regularization, and thus have penetrations under the regularization threshold. However, since contact loads may vary widely from one contact zone to the other, and during the loading, a constant and uniform penalty parameter cannot ensure penetrations of the same order of magnitude in different contact zones, as required to guarantee the efficiency of the quadratic regularization. This is the reason why the penalty parameter is adapted for each proximity zone, so as to control the maximum allowed penetration within each contact zone. With this adaptation, penetrations registered at different contact elements within a given proximity zone are different, depending on the load exerted at each contact element, but the maximum penetration is approximately limited by the maximum allowed penetration. Mechanical model for friction A regularized friction model, based on the Coulomb law and allowing a reversible relative displacement before gross sliding occurs, is used to account for tangential interactions between fibres. The fact that contact elements have no continuity in time requires an appropriate procedure to transfer history variables related to the reversible relative displacement from one step of computation to the following, between different contact elements. For this purpose information related to this quantity for each particle of the contact element is attached to the material configuration of fibres, and interpolated at the location of the contact element each time these elements are updated. Algorithmic aspects The modelling of contact-friction introduces various nonlinearities in the global problem that need to be solved by appropriate algorithms. The first nonlinearity is related to the generation of the contact elements which depends on the solution. As the solution can change significantly during a loading increment, the positions of contact elements should be updated. The relations defining the positions of particles of contact elements cannot be differentiated, which prevents the use of a Newton-like algorithm. A fixedpoint algorithm is therefore dedicated, at a first level, to iterations on the determination of contact elements, within each loading increment. Contact elements being fixed, another fixed-point algorithm is used to iterate on the
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normal contact directions involved in the expression of the gap function. All other nonlinearities (contact status, sliding, finite strains) are treated by a Newton–Raphson algorithm. The nesting of loops of three levels tends to increase the number of total iterations, but is necessary to consider large loading increments. Efficient algorithms are all the more needed to solve nonlinearities related to mechanical models for contact and friction.
15.3.3 Modelling of the matrix and its interactions with textile reinforcements Composite parts are made of a combination of textile reinforcements and matrix, and both components must be considered to model the behaviour of a textile composite sample. Composite samples considered here are made of the juxtaposition of two matrix layers on both sides of the fabric. The exact determination of the volume occupied by the matrix is a difficult issue, since it depends on the manufacturing process employed and on the way the matrix impregnates the tows. An accurate geometrical description of this matrix volume may contain many details on a very small scale, and meshing such a geometry can turn out to be a difficult task, yielding a large number of small finite elements that could significantly increase the computational cost. To overcome this difficulty, we choose to approximate the actual geometry of the matrix in order not to take into account small details. The matrix is assumed to penetrate textile tows with a given penetration depth, and a solid structured mesh is generated to approximate the geometry it occupies. This meshing is performed independently of the finite element discretization of fibres, and the resulting meshes for the matrix and the fibres are therefore non-conforming. The continuity of displacements on the interface between these components is therefore no longer ensured by the sharing of common nodes, and additional developments are required to model the connection between these structures. Special connection elements are introduced for this purpose. This global modelling of the matrix in interaction with the fabric tries to reproduce first-order effects of the coupling between textile reinforcements and the matrix. Meshing of the matrix The meshing of the matrix is performed once the initial configuration of the woven fabric has been computed, providing an accurate description of the volume occupied by fibres. In order to mesh the volume of each matrix layer, a regular meshing is first carried out on the external face of the layer. Each node of this face is then projected vertically on the first fibre encountered in the fabric. This projection point is then moved to penetrate to a given depth inside the fabric.
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A structured solid mesh for the layer is finally generated between these two faces (see Fig. 15.4). This way of meshing the matrix is pretty simple and does not require any particular geometrical tool. The penetration of the internal surface of the matrix layer inside the fabric provides an overlapping region between the two components within which the connection between both structures will be modelled. Introduction of connection elements between the fibres and the matrix Because the nonconforming meshes for the fibres and the matrix no longer ensure the continuity of displacements at the interface, a particular modelling of the coupling between these two structures is required. Particular connection elements are introduced for this purpose at nodes of fibres located inside solid elements of the matrix. For each fibre node If, located inside a finite element Em of the matrix, we seek the corresponding material particle x, defined by its relative coordinates (x1, x2, x3) within the matrix finite element, whose position x(x) is characterized by:
x(x) = SJ,m wJ,m(x1, x2, x3) XJ,m = XI,f
15.11
where Jm are the nodes of the solid finite element of the matrix, wJ,m are the shape functions associated with these nodes, and XI,f and XJ,m are respectively the position of the considered fibre node, and the positions of the nodes of the solid element of the matrix. For each fibre node located inside a matrix finite element, a connection element is generated between this node and the corresponding material particle in the matrix volume, by introducing a connection stiffness kc between the
15.4 Mesh of the volume occupied by the matrix generated on both sides of the fabric.
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two points. The resulting interaction force between the two points is written as
Rc = kc (SJ,m wJ,m(x1, x2, x3) XJ,m – XI,f)
15.12
The introduction of the virtual work of this force induces couplings between degrees of freedom associated with the fibre node and with the matrix nodes of the corresponding matrix element. The connection stiffness kc must be chosen carefully. As the displacement fields in the matrix finite elements are interpolated with a coarser discretization than they are at the level of fibres, and as several fibres can be involved within the same matrix finite element, taking a too strong connection stiffness would amount to prescribing a coarse kinematics to the set of fibres involved in the matrix finite element, and to lock somehow the possible relative motions between these fibres. In order to give back more flexibility to the connection, by allowing it some extension, the connection stiffness is computed for each connection element as a function of the Young’s modulus of the matrix and of the size of the finite elements of the fibre and the matrix.
15.3.4 Modelling of boundary conditions Problems with boundary conditions Dealing with boundary conditions to apply at the ends of the tows and of the fibres on the edges of the studied sample, we are confronted with a contradiction. On the one hand, we would like to apply displacements at the ends of the fibrous components on the edges of the sample to maintain them, whereas on the other hand, fibre ends and tow ends should be allowed to become rearranged according to the loading to which they are subjected. Hierarchical organization of the composite The hierarchical organization of the composite sample in different levels related to the fibres, the tows, the woven fabric and the matrix, must be respected by the application of boundary conditions, to allow one to apply simultaneously different conditions to the ends of components of different levels. For example, one may wish to fix the end of a tow while allowing the ends of its constituting fibres to become rearranged. Introduction of rigid bodies at ends of components To apply different conditions according to the different hierarchical levels of components, we introduce rigid bodies at their ends. Boundary conditions can then be applied to these rigid bodies. Ends of subcomponents can then be
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attached with proper conditions to the rigid body associated with the upper level. rigid bodies are generated for each edge of the sample, and for each end of the tows. By this way, ends of tows on one edge of the sample can be attached with appropriate connection conditions to the rigid body associated with this edge, and ends of fibres constituting a tow can be attached to the rigid body associated with its end. The connection conditions established between a set of ends and a rigid body must enable one to prescribe globally a given displacement or a given force to this set of ends, while allowing them to be rearranged. Average connection conditions have been developed to meet this need. The employed rigid bodies are defined by four nodes (Nm, ND1, ND2, ND3) (see Fig. 15.5), the first node being the master node of the rigid body, and the other three forming an orthonormal frame. These four nodes are rigidly connected to maintain the orthonormality of the frame expressed by the following conditions: (x(NDi) – x(M), x(NDj) – x(M)) = dij
15.13
To prescribe an average displacement di in the ith direction of a moving rigid body to a set of ne ends, represented by the nodes (Nek), we express the following condition: (1/ne) S (x(Nek) – x(M), x(NDi) – x(M)) = di k
15.14
Since the rigid body to which end nodes are attached is moving, this average condition is nonlinear. Nodes involved in such a conditions are constrained only on average, and each particular node can move differently, provided the mean displacement corresponds to the prescribed value. The use of the average connection conditions together with the introduction of moving rigid bodies at the ends of tows and on the edges of the sample Fibre end node
ND2
N ek
Nm ND3
ND1
Rigid body associated with the tow end
15.5 Rigid body used to define boundary conditions at the end of a tow.
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is essential to apply appropriate boundary conditions on the edges of the sample.
15.3.5 Determination of the initial configuration of the woven fabric The knowledge of the initial configuration is a key issue for woven fabrics. As the arrangement of fibres and tows within the fabric results from the different stages of the manufacturing process, trajectories of fibres and tows making up the fabric can hardly be known a priori. One way to access these geometries by calculation could be to simulate the whole manufacturing process, and in particular the weaving process. Such a simulation would require one to consider significant lengths of yarns, which would be too expensive using an implicit solver. Simulations of weaving and braiding processes have been proposed (Finckh 2004; Pickett et al. 2009), yet using explicit dynamic simulation codes. To avoid too expensive a simulation of the weaving process, we propose an alternative method consisting of progressively moving tows forming the fabric until their arrangement fulfils the chosen weaving pattern. To do this, we start from an arbitrary initial configuration, where all tows, formed of a compact arrangement of fibres, are in the same plane, interpenetrating each other (Fig. 15.6). From the weaving pattern, at each crossing between two tows, it is possible to state which tow should be above or below the other. Normal contact direction at crossings between tows
15.6 Starting configuration for the determination of the initial geometry of the woven fabric, and normal contact directions transiently chosen for fibres at crossings between warp and weft tows for a plain weave.
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During the stage of determination of the initial configuration, contact conditions between fibres are modified so that fibres of initially intersecting tows are gradually moved in the right direction until there is no more penetration between tows. Two adjustments to the contact conditions are made for this purpose. The first one is the choice of the normal contact condition. As we know, from the weaving pattern, which fibre should be above the other at each crossing between tows, this normal contact direction is oriented to satisfy this condition. The second adjustment is related to the evaluation of the gap between particles of contact elements. The observed penetrations during this initial stage have no real physical meaning and can be very important. As the treatment of contact conditions tends to reduce the penetrations to zero, this would lead to too important relative motions between fibres, preventing the convergence of the global solution algorithm. To make the fibres move gradually, the gap considered between two particles is limited to a given value. The relative displacement between the two concerned fibres thus does not exceed this value. The global increment of displacement for each loading step during this initial stage can then be controlled to guarantee the convergence of the solution algorithm.
15.4
Application examples
Different results obtained on the same intial set of tows, arranged according two different weaving patterns and subject to various loading cases, are presented in this section. The main characteristics of this initial set of tows, geometrical characteristics of the fabric and mechanical characteristics of fibres are respectively summarized in Tables 15.1, 15.2 and 15.3.
Table 15.1 Characteristics of the model Number of tows in weft direction Number of tows in warp direction Number of fibres per tow Total number of fibres Number of finite elements per fibre Total number of nodes Total number of degrees of freedom Estimated number of contact elements
4 4 76 608 32 39,520 355,680 100,000
Table 15.2 Geometrical characteristics of the fabric Radius of fibres Tow cross-sectional area Initial distance between tows (before crimp)
0.0201 mm 0.0967 mm2 0.88 mm
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7300 MPa 0.3 0.20 0.15
Axial stress (MPa) 2500 2000
1000
0
–1000
–2000 –2500
15.7 Computed initial configuration for the plain weave.
15.4.1 Determination of the initial configuration of the woven structures The first task assigned to the proposed approach is the determination of the a priori unknown initial configuration of the woven structures. Starting from the same initial arrangement of eight tows, each made of 76 fibres, two different patterns of plain weave and twill weave are chosen to compute this initial geometry. For this first stage of simulation, 21 increments are needed, each of them requiring about 30 nonlinear iterations. Tension forces are applied in the directions of the tows, before being reduced to very low values during the final increments. The obtained initial configurations are shown in Figs 15.7 and 15.8. Useful geometrical information can be derived from these results, in particular the crimp of the fabric and the local curvatures in all fibres. The determination of the trajectory of each individual fibre within the fabric allows one to describe the trajectories and cross-sectional shapes of tows. To illustrate the method of determining the initial configuration, Fig. 15.9 shows the evolution of the shapes of the tows during this first stage. Figure 15.10 compares slices of the computed initial configuration for the two different weaves and shows the different cross-sectional shapes. These
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Axial stress (MPa) 1300 1000
0
–1000 –1300
15.8 Computed initial configuration for the twill weave.
n=0
n=3
n=6
n=9
n = 21
15.9 Details of the evolution of tow cross-sections for different increments during the computation of the initial configuration for a plain weave (left) and a twill weave (right).
descriptions can be of great interest for the construction of meso-models at the scale of tows.
15.4.2 Application of test loading cases on the dry fabric Test loading cases can be applied to samples of dry fabric once their initial configurations have been determined.
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Twill weave
15.10 Slices of the computed intial configuration for the plain and twill weaves. 200 Plain weave Twill weave
180 160
Axial force (N)
140 120 100 80 60 40 20 0
0
0.002
0.004 0.006 Axial strain
0.008
0.01
15.11 Loading curves for the equibiaxial extension.
Equibiaxial extension To simulate an equibiaxial extension on both samples, incremental displacements are applied on the edges of the samples until a 1% strain is reached in the warp and weft directions. The loading curves (Fig. 15.11) show a nonlinear effect at the beginning of the loading, that may be attributed to changes induced in the arrangement of fibres. Figures 15.12 and 15.13 show a compaction of fibres within tows, and a reduction of the undulations of tows, mainly in the twill weave. Shear test A shear test is simulated on the plane weave sample by applying an incremental lateral displacement on one edge of the sample, while exerting
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Axial stress (MPa) 2800 2000 1000 0 –1000 –1700
15.12 Slice of the plain fabric, before and after applying a 1% equibiaxial traction. Axial stress (MPa) 2000
1000
0 –500
15.13 Slice of the twill fabric, before and after applying a 1% equibiaxial traction.
a small tensile force in the warp and weft directions. The global loading is divided into 45 increments, and about 15 nonlinear iterations are needed to solve the problem for each increment. The shear loading curve (Fig. 15.14) displays a nonlinear behaviour similar to those observed in benchmark tests on different woven fabrics (Cao et al. 2008). The shear force increases until approaching a locking angle for which there is almost no more space between tows, as can be seen in Figs 15.15 and 15.16.
15.4.3 Application of loading cases to the composite sample Various loading cases can be applied to the composite sample, considering an elastic matrix on both sides of the fabric. The use of rigid bodies to control boundary conditions enables one to apply global conditions on the edge of the sample. Applying, for example, rotations around different directions while keeping free certain displacements, it is possible to simulate either a global bending (Figs 15.17 and 15.18) or a twisting (Fig. 15.19). These results demonstrate the ability of the approach to consider large displacements and strains, even in the presence of a matrix. Interesting coupling effects between the matrix and the fabric should be studied by this means.
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Shear force (N)
7 6 5 4 3 2 1 0
0
5
10
15 20 Shear angle (degree)
25
30
15.14 Loading curve for the shear test for the plain weave sample. Axial stress (MPa) 3000 2000
0
–2000 –3000
15.15 Mesh of modelled plain weave sample before and after the application of the shear deformation.
15.5
Conclusions
An approach to the finite element simulation of the mechanical behaviour of textile composite samples at the scale of constituting fibres has been presented in this chapter. Based on an implicit solver, the approach considers each individual fibre with a finite strain beam model and focuses on the detection and modelling of contact-friction interactions developed between these fibres. The coupling between fibres and an elastic matrix discretized with a coarse mesh is taken into account by the means of connection elements generated between the two structures. Appropriate boundary conditions are established
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Axial stress (MPa) 2500 2000 1000 0 –1000 –2000 –2500
15.16 Mesh of two particular tows of the plain weave sample before and after the application of the shear deformation.
15.17 Deformed mesh of the plain weave composite sample subject to a global bending.
on the edges of the studied samples using rigid bodies defined at ends of different components. The efficiency of the proposed models and algorithms allows one to handle samples made of a few hundreds of fibres. The approach is first employed to determine the unknown intial configuration of the woven structure, providing geometrical descriptions at the level of fibres and tows in terms of trajectories and cross-sectional shapes. Once their initial geometry has been computed,
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15.18 Slice of deformed mesh of the plain weave composite sample subject to a global bending.
15.19 Deformed mesh of the plain weave composite sample subject to a global twisting.
various loading cases can then be applied to the studied samples in order to identify their mechanical properties. This approach at microscopic scale is characterized by the fact that it requires the identification of very few parameters (mechanical characteristics of fibres, geometrical description of the weaving pattern), and provides a wide
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range of results, from the determination of the initial configuration to the characterization of the mechanical state at the scale of fibres during loading. For this reason, it should be helpful to identify behaviours of components at upper scales, from both a geometrical and a mechanical point of view.
15.6
References
Antman S S (2004), Nonlinear Problems of Elasticity, Springer, New York. Boisse P, Aimene Y, Dogui A, Dridi S, Gatouillat S, Hamila N, Khan M A, Mabrouki T, Morestin F, Vidal-Sallé E (2010), ‘Hypoelastic, hyperelastic, discrete and semidiscrete approaches for textile composite reinforcement forming’, Int J Mater Form, 3 (Suppl 2), 1229–1240. Cao J, Akkerman R, Boisse P, Chen J, Cheng H S, de Graaf E F, Gorczyca J L, Harrison P, Hivet G, Launay J, Lee W, Liu L, Lomov S V, Long A, de Luycker E, Morestin F, Padvoiskis J, Peng X Q, Sherwood J, Stoilova T, Tao X M, Verpoest I, Willems A, Wiggers J, Yu T X, Zhu B (2008), ‘Characterization of mechanical behavior of woven fabrics: Experimental methods and benchmark results’, Composites: Part A, 39, 1037–1053. Durville D (2005), ‘Numerical simulation of entangled materials mechanical properties’, J Mater Sci, 40, 5941–5948. Durville D (2009), ‘A finite element approach of the behaviour of woven materials at microscopic scale’, in J.-F. Ganghoffer, F. Pastrone (eds): Mechanics of Microstructured Solids, Lecture Notes in Applied and Computational Mechanics, 46, 39–46. Durville D (2010), ‘Simulation of the mechanical behaviour of woven fabrics at the scale of fibers’, Int J Mater Form, 3 (Suppl 2), 1241–1251. Finckh H (2004), ‘Numerical simulation of the mechanical properties of fabrics’, LSDYNA Users Conference, Bamberg, Germany. Hivet G and Boisse P (2008), ‘Consistent mesoscopic mechanical behaviour model for woven composite reinforcements in biaxial tension’, Composites: Part B, 39, 345–361. Lomov S V, Ivanov D S, Verpoest I, Zako M, Kurashiki T, Nakai H, Hirosawa S (2007), ‘Meso-FE modelling of textile composites: Road map, data flow and algorithms’, Comp Sci Technol, 67, 1870–1891. Miao Y, Zhou E, Youqi Wang Y, Cheeseman B A (2008), ‘Mechanics of textile composites: Micro-geometry’, Comp Sci Technol, 68, 1671–1678. Pickett A K, Sirtautas J, Erber A (2009), ‘Braiding simulation and prediction of mechanical properties’, Appl Compos Mater, 16, 345–364. Robitaille F, Long A C, Jones I A, Rudd C D (2003), ‘Automatically generated geometric descriptions of textile and composite unit cells’, Composites: Part A, 34, 303–312. Verpoest I and Lomov S V (2005), ‘Virtual textile composites software WiseTex: Integration with micro-mechanical, permeability and structural analysis’, Comp Sci Technol, 65, 2563–2574. Wang Y, Miao Y, Swenson D, Cheeseman B A, Yen C F, LaMattina B (2010), ‘Digital element approach for simulating impact and penetration of textiles’, Int J Impact Eng, 37, 552–560.
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Mesoscopic approaches for understanding the mechanical behaviour of reinforcements in composites
E. V i d a l - S a l l é, INSA Lyon, France and G. H i ve t, Polytech Orléans, France
Abstract: Composite reinforcements can be modelled at various scales. The mesoscopic one is an intermediate between the macroscopic scale for which the whole woven fabric is considered, and the microscopic one in which each fibre is taken into account. The mesoscopic scale considers the woven fabric as an assembly of yarns. Various approaches are used depending on the considered characteristic length of the yarn. Dedicated constitutive equations have to be built which capture correctly the specific behaviour due to the underlying microstructure. Corresponding behaviour identification is generally realized using macroscopic tests. Each step of a modelling is described and illustrated for dry fabrics. Key words: yarn modelling, contact, geometrical modelling, finite element, material identification.
16.1
Introduction
Composite reinforcements are made by the assembly of several tows or yarns, themselves including thousands of continuous fibres. Consequently, they can be seen at various scales. The largest scale that can be considered is the macroscopic one where the whole mechanical part is considered, and the smallest is the microscopic level, i.e. the scale of a single fibre. An intermediate scale is the mesoscopic one, that is to say the scale of a yarn (Fig. 16.1). When the reinforcement is observed at the macroscopic level, the fabric is considered as an anisotropic continuous material exhibiting mechanical properties inherited from its meso- and microstructures. The fabric is modelled using membranes, or shells if bending is taken into account (Spencer 2000; Dong et al. 2001; King et al. 2005; Shahkarami and Vaziri 2007). The modelling must take into account the behaviour specificities of the fabric (Carvelli and Poggi 2001), especially the necessary property updating due to large strains (especially large in-plane shear) (Yu et al. 2002, 2005; Peng and Cao 2005; Xue et al. 2005). The main drawback of that approach is the fact that it does not include crimp and interlacement effects, which are 486 © Woodhead Publishing Limited, 2011
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Microscopic scale
Macroscopic scale
Mesoscopic scale
16.1 The three scales of composite reinforcements.
important features of fabric reinforcement behaviour. One way to identify macroscopic properties of fabrics is to use homogenization of results coming from lower-scale observations. The best approach would consist in realizing simulations at a scale from which the material is really continuous, i.e. the scale of the fibre. Some authors (Zhou et al. 2004; Durville 2005, 2007, 2008; Miao et al. 2008) adopted this approach with applications to metallic braids (Wang and Sun 2001; Zhou et al. 2004; Miao et al. 2008) and knitted fabrics (Duhovic and Bhattacharyya 2006). All of those simulations use a reduced number of fibres in each yarn for computational time reasons. The consistency of those approaches is then questionable when the number of fibres is greater than 5000. An intermediate way is to build a model representative of the yarn behaviour and able to capture its main specificities in terms of forces and geometry. This constitutes a good compromise between realism and complexity. When modelling a woven fabric at the yarn scale, several points have to be addressed: • • •
What information is being looked for? What is the geometrical precision level required? What is the characteristic length of interest?
Depending on the answers to these questions, the mechanical modelling will be more or less precise with a possible choice between 3D and shell modelling. Whatever the answer is, several steps have to be achieved: geometrical modelling of the yarn or yarn set; choice of a constitutive equation with its identification; experimental validation of the proposed modelling. Each of those points is dealt with in the present chapter. Section 16.2 briefly reviews the main specificities of woven fabric mechanical behaviour. Then, Section 16.3 points out the mechanical behaviour of a yarn, and Section 16.4 deals with mesoscopic geometrical modelling of the yarn in its environment, i.e. in the yarn assembly which constitutes the woven fabric. Section 16.5 presents various theoretical approaches which model the mechanical behaviour of a yarn and the identification of the material parameters using finite element
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analyses. Section 16.6 gives some examples of the use of mesoscopic modelling and Section 16.7 gives a non-exhaustive list of possible improvements of the modelling.
16.2
Mechanical behaviour of the reinforcement
As far as mechanical properties are concerned, composite reinforcements are specific materials because they can be considered either as a material or as a structure. Their mechanical behaviour then results from this lower-scale structural constitution but has to be identified at the ply scale for applications such as manufacturing. This particularity is undoubtedly responsible for the absence of a universally accepted constitutive law. Experimental determination of the mechanical behaviour of dry fabrics accordingly passes through a series of classical tests representative of the different loadings. Up to now the great majority of fabrics used for industrial applications are 2D or interlock fabrics constituted of yarns oriented in two directions. This section is focused only on those fabrics.
16.2.1 Biaxial tensile behaviour Looking at the strain energy brought into play under the possible loadings enables one to identify the first particularity of these materials: the tensile energy is significantly higher compared to all the others (shear, bending, etc.). Measuring the tensile properties of woven fabrics is therefore a crucial issue. Due to the yarn interlacement (woven fabrics) or stitching (non-crimp fabrics), the tensile behaviour of fabrics is biaxial. Biaxial tensile behaviour is defined by the coupling between the tensions and longitudinal strains in both yarn directions. Since the tension in one direction depends on the strains in both directions, two surfaces represent the biaxial tensile behaviour of a fabric, each defining a tension as a function of both axial strains (Fig. 16.2). Identifying the biaxial tensile behaviour of a fabric therefore consists of determining those two tension surfaces for an unbalanced fabric (one for a balanced fabric). The latter can be measured experimentally via the use of a biaxial tensile device. Different principles exist, among which are systems using two deformable lozenges (ASTM 2002) implemented on a standard traction device or two autonomous axes moved by electric engines (Fig. 16.3), both devices acting on cross-shaped specimens. Biaxial tensile results can also be depicted with curve networks for better readability. An example of experimental results is presented in Fig. 16.2 in the case of a Twintex® plain weave. Submitted to a tensile load, the undulated yarns tend to become straight, inducing the transverse crushing in the second yarn direction. This first step is responsible for the first non-linear part of the biaxial behaviour, often
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16.2 Tension surface for a 2 ¥ 2 carbon twill (warp tension T11 as a function of warp strain E11 and weft strain E22) and corresponding curve families.
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16.3 Biaxial tensile device.
encountered for standard fabrics and illustrated in Fig. 16.2. When the yarns are straight or when the transverse stiffness of the second network becomes too high, crushing of the transverse yarns is no longer possible and single-yarn tensile behaviour is therefore reached, the latter being almost linear. Since the yarns are woven and undulated in both warp and weft directions, the phenomenon is biaxial and an equilibrium in the yarn undulation changes is reached under a biaxial tensile stretching. The physical phenomenon involved in biaxial deformation of fabrics is consequently well known. Nevertheless, as often when dealing with dry composite reinforcements, data and scientific
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studies are not so commonly found; obtaining a standard repeatable accurate characterization method is therefore still to be achieved (Buet-Gautier and Boisse 2001).
16.2.2 In-plane shear As previously mentioned, fabric strain energy is mainly due to tensile stiffness. Nonetheless, one of the most interesting mechanical properties of woven fabrics is their ability to shear, which makes them particularly suitable for the forming of complex non-developable shapes. This explains why in-plane shear is the most studied type of loading for dry fabrics. The experimental identification of the in-plane shear response is classically made using two principles (Kawabata et al. 1973b; Prodromou and Chen 1997; McGuinness and ÓBrádaigh 1998; Wang et al. 1998; Potter 2002; Lebrun et al. 2003; Sharma and Sutcliffe 2004; Peng et al. 2004; Cao et al. 2008; Potluri et al. 1996; Launay et al. 2008; Lomov and Verpoest 2006; Harrison et al. 2004): the picture frame test, which is a deformable frame (Fig. 16.4a), and the bias extension test, which is a tensile test with the fabric yarns oriented ±45° from the loading direction (Fig. 16.4b). To the first order, these two tests can be considered as pure in-plane shear tests and enable one to obtain the classical non-linear shear curve (Fig. 16.5). Here again, the structural aspect of the fabric is responsible for this behaviour, an explanation supported by mesoscopic optical measurements of the displacement field (Fig. 16.5). The shear curve can be subdivided into three parts: for small shear angles (up to 30° in Fig. 16.5) the warp and weft yarns rotate, each yarn having a solid body motion. Figure 16.5 confirms this analysis, showing that the relative displacement field inside a yarn is a rotation field and strains in the yarn are negligible. During this first phase, the shear stiffness is very low, because it is due only to small frictional forces between yarns. The subsequent increase of the shear angle leads to closing of the space between yarns of the same network (Boisse et al. 2005), so the second part of the curve is a transition step during which side contacts between yarns start to exist and yarns continue to slightly rotate. This second step is characterized by the round part of the in-plane shear curve that links the first soft quasi-linear part and the third much stiffer one. When the so-called ‘locking angle’ is reached, the weaving does not allow this rotation any more. Pores are closed and yarns are subjected to side contacts with their neighbours. The relative displacement field inside a yarn shows that the yarn is in lateral compression, the latter getting more important as the shear angle increases (Fig. 16.5). During this last phase, the shear load is due mainly to this lateral compression, so the shear stiffness thus increases quickly. For some weavings and especially for complex ‘interlock’ weavings, the stiffness increase is more progressive so that it can be more or less difficult
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16.4 Principles of (a) the picture frame and (b) the bias extension test.
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16.5 Optical observation of the yarn kinematics during an in-plane shear test on a glass plain weave with mesoscopic displacement fields for various shear angles before and after the locking angle.
to identify a clear locking angle. Anyway, whatever the fabric considered, the increase of the shear stiffness induces another deformation mode: bending, which becomes less energetic than the shear mode. Wrinkles are therefore classically observed (Fig 16.6) as soon as the locking angle is reached. The mechanical mechanisms involved in the in-plane shear of a dry fabric are clearly identified; nevertheless, in this case also, performing pure shear tests remains a difficult task because of the perturbations resulting from the high tensile stiffness (Launay et al. 2008). Discrepancies in the results tend to be mastered better (Hivet and Duong 2010) but are still high, as has been demonstrated with the results of the benchmark study driven by Cao et al. (2008).
16.2.3 Bending The wrinkling phenomenon previously observed unquestionably results from the competition between the shear and the bending stiffness of the fabric. Furthermore, position, shape and number of wrinkles are strongly related to the bending stiffness of the layer. The fabric bending behaviour has thus to be investigated. It has been widely studied for garment fabrics because it is an important issue as far as cloth comfort is concerned. Two standard tests are classically used: the standard cantilever test (ASTM 2002; ISO 1978) and the Kawabata bending test (KES-FB2) (Kawabata 1980). The first test assumes an elastic linear behaviour and enables the determination of only one parameter: the bending stiffness. However, the
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16.6 Appearance of wrinkles at high shear angles during a picture frame test.
bending behaviour has been proven not to be linear elastic and the standard cantilever test is therefore not really suitable. The second test was designed for textile fabric and enables one to record the bending moment versus the curvature during a bending cycle. It enables a non-linear and hysteretic behaviour to be shown but, since it has been designed with the aim of testing clothing textiles, it is not really suitable for composite reinforcements, which are often thicker and stiffer. An extension of the cantilever test has recently been proposed. It is based on the principle of the cantilever test but coupled with an optical measurement acquisition system in order to obtain consistent data for composite reinforcements (Fig. 16.7a). This test can deal with any type of composite reinforcement and, even if much work remains to be done concerning the understanding of the bending behaviour of fabrics, it has already permitted the conclusion that a hysteretic elastic–plastic law expressing the bending moment as a function of the curvature seems to be a good approach for the real bending behaviour (de Bilbao et al. 2010). One curve used to identify the model is presented in Fig. 16.7b. The work concerning both the bending behaviour of composite reinforcements and the determination of a consistent bending behaviour law is just beginning; significant progress can be expected on this topic in the next few years.
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16.7 (a) Bending device with CCD camera, and (b) bending curve of the Hexcel G1151® fabric.
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16.2.4 Compaction Many results can be found in the literature concerning compaction of one or more layers of fabrics. The use of a standard load cell or any compression device coupled with adapted plates indeed enables one to identify the compaction behaviour of fabrics (Saunders et al. 1998). One difficulty remains the definition of the intrinsic thickness of the fabric and therefore the identification of the initial very low stiffness, because of the heterogeneity of the upper surface. As for the compaction behaviour of the yarn itself, the compaction behaviour of a stack of fabric layers is non-linear. A typical pressure–thickness curve of a woven fabric presented in Chen and Chou (1999) is depicted in Fig. 16.8; its exponential or hyperbolic shape resulting from fibre stacking modification through the compaction loads. The first soft part is primarily dominated by fibre reorganization inside the yarn (see Section 16.3) leading to the reduction of pores and gaps among the fibres and yarns. The progressive closing of these pores leads to a progressive increase of the compressive stiffness. When fibre displacements are no longer possible the stiffness becomes very high and reaches the order of magnitude of that of the fibre. When dealing with multiple layers the same type of behaviour is observed but quantitative values, especially the initial thickness and the stiffness increase, are strongly dependent on the relative positioning of the layers in terms of nesting (inplane displacement between the layers) and orientation (orientation of the yarns of the different layers) (Chen and Chou 2000). Figure 16.8 corresponds to a multi-layer compaction test. Such a test remains difficult to perform and analyse because of this sensitivity to nesting and orientation, these two parameters being difficult to master accurately in practice.
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16.8 Typical pressure–thickness curve of woven fabrics under compaction for several layers of plain weave.
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16.2.5 About coupling Interesting and consistent results have been obtained through experimental approaches on the fabric scale concerning elementary membranous solicitations, nevertheless, industrial loadings (such as forming, for instance) often consist of coupling between those solicitations. Yet, approaching coupling phenomena is shown to be difficult because of the very high ratio between tensile stiffness and the other stiffnesses. The first experimental results have been proposed for the coupling between shear and tension in Willems et al. (2008) and Hivet and Duong (2010), for example, but a huge amount of work remains to be done in this field and the experimental approach may not be the best one. Simulation at the meso-scale can be a very helpful tool in reaching this goal (Badel et al. 2007; Lin et al. 2008), as will be developed in the next sections. Concerning compaction, as has been explained above, a strong coupling exists also between membrane loading and compaction behaviour of fabrics. This coupling has been illustrated by experimental observations (Potluri et al. 2004) and deduced from biaxial tensile tests (Boisse et al. 2001), but here again, direct experimental coupling results are still missing and are difficult to obtain; much work therefore remains to be done before an acceptable level of knowledge is reached on this topic.
16.2.6 Friction identification Friction occurs at several scales. At the meso-scale, friction occurs between yarns when the deformation of the fabric imposes relative displacement between them, for example during in-plane shear (Dumont et al. 2003; Gorczyca et al. 2004). Some authors have realized experimental studies of friction on impregnated composites (Suresha et al. 2010) and dry fabrics (Martinez et al. 1993; Rebouillat 1998; Allaoui et al. 2009). Those studies point out the difficulty in determining values of the Coulomb-Amonton friction coefficient with confidence (Nadler and Steigmann 2003; Howell and Mazur, 1953) and the very specific friction behaviour of dry fabrics, characterized in particular by very high variations of the friction forces at the steady state. All the above-mentioned experiments have been carried out on ply-toply friction. Generally, this coefficient is chosen to be between 0.2 and 0.4 with a constant isotropic distribution. If this is questionable, up to now no extensive experimental evidence has been published to confirm or refute this assumption. In conclusion, much interesting experimental work has been done on the experimental characterization of the mechanical behaviour of dry fabrics. Nevertheless, the results obtained are subjected to quite a high level of © Woodhead Publishing Limited, 2011
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discrepancy. Much further research is therefore needed to achieve an accurate and repeatable characterization of dry composite reinforcements. Complementarities between numerical and experimental approaches are undoubtedly a promising way and mesoscopic modelling would be an interesting tool. The following section deals with the mechanical behaviour of the yarn considered as a continuous material.
16.3
Mechanical behaviour of the yarn
As mentioned in the introduction, a mesoscopic modelling of woven reinforcements considers the material at the yarn level. This means that the meso-structure of the fabric is explicitly modelled. Consequently, the mechanical behaviour of the yarn material needs to be studied. As the microstructure of the fibre bundle, i.e. the fibre arrangement, is not explicitly modelled, the constitutive behaviour of the yarn material must exhibit specificities linked to the fact that the material is not really continuous in the same manner as fabric material. Depending on the application it is built for, the yarn arrangement is not always the same. Some yarns are built with parallel fibres while others are twisted. In order to guarantee a high tensile stiffness for high-performance composites, the twist angle is generally weak. Anyway, high-resolution x-ray tomography imaging shows that the yarn material is strongly oriented (Badel et al. 2008) (see fig. 16.9). The consequence, from a mechanical point of view, is the strong tensile stiffness in relation to the other rigidities of the material, as mentioned for fabric materials. The possible relative movement between fibres inside the bundle makes the bending stiffness of a yarn particularly weaker than the classical beam bending stiffness, even though the slenderness of the yarn is similar to that of a beam. In the same manner, the transverse behaviour of the yarn strongly depends on the actual fibre volume fraction: the denser the tow, the stiffer it is.
16.9 3D reconstruction of X-ray tomography imaging of a 2 ¥ 2 carbon twill.
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The present section deals with the main characteristics of a yarn that must be captured by a constitutive law representative of the yarn material.
16.3.1 Tension A yarn is made up of thousands of fibres joined together. When subjected to longitudinal tension, the fibres are reorganized in order to better resist the tensile load. Consequently, depending on the micro-structure of the yarn (namely the twist angle and fibre density), the tensile response would present various initial non-linearities (Rao and Farris 2000). The tensile loading induces an untwisting movement. When this straightening movement is completed, the tensile behaviour of the yarn becomes linear (see Fig. 16.10). Consequently, the yarn tensile behaviour depends mainly on three parameters: the number of fibres, the nature of the fibres and the twist angle. For the nominal tensile stiffness only the first two parameters are relevant. Such a stiffness is generally expressed in N (implying newtons per unit strain) and not via a Young’s modulus as for classical continuum mechanics analyses. In the case of a yarn made of parallel fibres, if it were possible to realize a tensile test ensuring that the applied load is exactly longitudinal, the relative positions of the fibres would not change. As this is not possible, boundary conditions cause the cross-section of the yarn to change. Moreover, for uncoated fibre bundles, the cohesion between fibres is only ensured by the environment of the tow: if the yarn is extracted from the fabric, it does not maintain its cohesion. Consequently, it is easier to realize tensile tests on fabrics than on single yarns (Buet-Gautier and Boisse 2001). In that case, X-ray tomography shows that, because of the boundary conditions, when subjected to tension, the density of the fibre bundle increases (Badel et al. 2008).
Tensile stress
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16.10 Non-linear tensile curve for an initially twisted fibre bundle.
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16.3.2 Transverse behaviour Observing the tow in the direction perpendicular to the fibre direction, an unloaded yarn presents a certain density of fibres which can be evaluated with the ratio between the fibre cross-sectional area and the apparent area of the yarn. Although this is theoretically possible, it is not really easy as the initial yarn section strongly depends on the boundary conditions applied to the yarn. But if we consider the yarn in its natural environment, i.e. in the fabric, its initial cross-section is better known and its evolution can be observed. Here again, X-ray tomography is a good tool as it allows one to observe the internal structure of the yarn in situ (Badel et al. 2008). Transverse behaviour covers various types of macroscopic loading: transverse compression of fabric; in-plane loading of fabric, and biaxial tension (Dumont et al. 2003; Badel et al. 2008). Two phenomena can be distinguished: densification and distortion. During the biaxial tensile test, the warp network imposes compression on the weft yarns, which implies that the fibres get closer to each other in the bundle. Experiments on a single yarn are very difficult to implement for the same reason as previously mentioned: the cohesion of the yarn is only ensured by its environment. Consequently, the major part of the experiments deals with fabrics. Moreover, as the fibre bundle interacts with its neighbours, friction plays a major role in the global results of the experiments. Some main points can be highlighted using several fabric tests. As has been shown in the previous section, transverse compaction of single- and multi-layer (Saunders et al. 1998) fabrics is strongly non-linear. That non-linearity is mainly due to the non-linearity of the yarn compaction itself. The compaction phenomenon is the fact that fibres get closer inside the bundle; consequently, when the voids become smaller than the fibre diameter, it becomes more difficult to continue the compaction movement. The high stiffness of carbon or glass fibres makes it difficult to deform the fibres themselves. Generally, it is admitted that the transverse compressive stiffness of the fibre bundle is dependent on longitudinal tension. Direct experimental evidence is rare but this can be illustrated by biaxial tensile tests (BuetGautier and Boisse 2001; Gasser et al. 2000). It can be explained by the fact that fibre bundles are not granular materials: they have a third dimension and the organization of the fibres in the longitudinal direction is not perfect. When fibres move against one another, their global twist changes (even if we consider an untwisted yarn) and they can be tightened by those small movements. As the tensile stiffness is very high, small tension can produce a large effect. In-plane shear of fabric also activates the deformation of the fibre bundle (see Fig. 16.11). Several phenomena occur: the crimp changes, inducing a weak tension of the yarn and a small apparent density change; the angle © Woodhead Publishing Limited, 2011
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16.11 (a) Initial and (b) sheared geometries of a 2 ¥ 2 carbon twill for in-plane shear: comparison between (c) initial and (d) sheared cross-section of the yarns. Black lines correspond to simulation results.
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between warp and weft yarns changes, inducing a change in the boundary conditions and a change of shape; and lateral contacts occur for large shear angles, inducing a fabric thickness increase and a stronger apparent density change. This means that the ratio between width and thickness changes. The global external shape of the yarn also changes (see Fig. 16.11). These evolutions can be evaluated using x-ray tomography (see Fig. 16.11). The cross-sectional area changes as a result of the compaction of the yarn and the shape changes and evolves along the yarn. As for the fibre density, when the global shape of the fibre bundle section changes, the fibres are reorganized in the tow and can be slightly tensed. This tunes the stiffness behaviour. Consequently, this mechanism is also non-linear.
16.3.3 Bending Since yarn is made up of thousands of fibres, those fibres can move in relation to their neighbours and this makes the yarn very soft in response to certain types of loading. In particular, even though a yarn has the geometry of a beam, its bending stiffness is smaller because of the relative motion that can occur between the fibres constituting the tow (Lahey and Heppler 2004; de Bilbao et al. 2010). Consequently, it is very easy to bend a yarn as it is quite like bending each fibre. Nevertheless, it has been shown that it is necessary to take the lower bending stiffness into account in a mesoscopic analysis of a yarn (Gatouillat et al. 2010). Although the bending stiffness of each fibre plays a major role in yarn bending behaviour, the cohesion forces that exist in the tow are also important. For a free, uncoated yarn, this second parameter is negligible but, when the yarn is observed inside the fabric, some forces exist between warp and weft networks and the cohesion of the yarn is also ensured by the inter-yarn friction. This is the reason why fabric bending is a not a fully reversible phenomenon (Ghosh et al. 1990).
16.3.4 Friction identification Friction occurs at several scales: at the meso-scale, friction occurs between yarns when the deformation of the fabric implies relative displacement between them, for example during in-plane shear (Dumont et al. 2003; Gorczyca et al. 2004). A mesoscopic-scale approach has to take these slidings into account explicitly. But friction also occurs at a lower scale as mentioned above: relative displacement between fibres inside the tow activates friction which has to be traduced by the constitutive equations of the yarn material. A friction parameter is not easy to identify but for ballistic applications it is a key point, since friction is responsible for the energy dissipation. The main way to evaluate friction is by use of the pull-out test. But to our knowledge,
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links between inter-yarn friction and results of the pull-out test are not well established. Models and experimental approaches have been developed, but the difficulty of tackling contact loads between yarns during pull-out tests leads one to consider many simplifications. Results of applying this approach to Kevlar yarns can be found in, for example, Duan et al. (2005) and Kirkwood et al. (2004). A constant friction coefficient with no dependency on the contact pressure is assumed, which seems to be a coarse assumption according to what is classically encountered in fabric/fabric or fabric/steel friction (Ajayi and Elder 1997). The major part of frictional studies has dealt with fabrics and not with yarns. Nevertheless, for consistent use of the yarn material in finite element analysis, an improvement of that parameter is an important point to address. The mesoscopic modelling of woven fabrics involves both geometrical and mechanical assumptions. The next section presents the geometrical modelling of yarns, and the following one gives some examples of mechanical modelling.
16.4
Geometric modelling
In the present section, two kinds of woven reinforcements are considered, namely 2D and 3D reinforcements. 2D reinforcements represent a large part of the woven reinforcements used. They are made by periodic arrangement of warp and weft yarns, constituting generally an orthogonal network. An extension of that definition concerns braided reinforcements where the two networks are not necessarily orthogonal. Nevertheless, as they are periodic, it is always possible to define the smallest geometrical pattern which defines the periodicity of the fabric. This is named the representative unit cell (RUC). Its definition depends on the weave. Three main weaves are used: plain, twill and satin. Those 2D fabrics can be modelled using 3D, shell or even beam approaches. The first is the most complex. It is extensively described in the present section. The second one, giving rise to smaller problems, allows one to consider greater domains with fewer details. Its main application is in impact simulation (Barauskas and Abraitiene 2007; Shahkarami and Vaziri 2007; Nilakantan et al. 2010; Sapozhnikov et al. 2007). The beam approach is generally used for braided (Wang and Sun 2001; Zhou et al. 2004; Miao et al. 2008) or knitted fabrics (Duhovic and Bhattacharyya 2006).
16.4.1 Beam and shell or membrane modelling for 2D fabrics Geometrically, a fibre bundle is a very slender structure with a strongly oriented material. It seems natural to model it using a beam structure, but
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generally, standard beam theory does not account for cross-sectional changes and it has been shown (Badel et al. 2008) that this is an important issue to manage. Nevertheless, for some applications for which the real yarn cross-section is not of primary importance, like ballistic simulations, some authors have used a beam approach with viscoplastic yarn behaviour (Tan and Ching 2006). A shell approach partially avoids the problem of beam theory as it allows compaction of the yarn in the in-plane direction. Moreover, those elements can capture correctly the bending behaviour of thin structures. Two types of shell finite elements are used in the literature: solid and classical shell elements. A single element is necessary in the width direction of the yarn. Variable nodal thickness allows one to account for the yarn shape (see Fig. 16.12a). This solution has been used by Nilakantan et al. (2010) for ballistic applications. Such modelling allows the use of few degrees of freedom in a finite element simulation and makes possible the modelling of a large domain of an armour. Other authors use classical Kirchhoff shell elements
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(Barauskas and Abraitiene 2007; Shahkarami and Vaziri 2007) with one or more elements in the width direction (see Fig. 16.12b). Some authors consider that the very weak bending stiffness of the yarn material enables the use of membrane elements instead of shell elements (Sapozhnikov et al. 2007). The unit cell for a plain weave requires only 216 degrees of freedom in such a case. Here again, the main application is ballistic modelling. Unfortunately it has been shown by Gatouillat et al. (2010) that even if it is weak, the bending stiffness plays a major role in the behaviour of a woven fabric, especially when the forming process is considered: without bending stiffness, there is no effort between warp and weft yarns during a draping and the yarns slip without resistance. This leads to rapid decohesion of the fabric (see Fig. 16.13). That modelling of the fabric gives information about the fabric resistance and the fabric stiffness when used for ballistic simulations. The example of Fig. 16.13 shows a forming simulation. Such an approach allows the capture of wrinkle onset, straightening but also sliding between warp and weft networks, which cannot be achieved with the macroscopic continuous models.
16.4.2 Representative unit cell for 2D fabrics For some applications, like resin injection modelling within the framework of liquid composite moulding (LCM) forming processes, evaluation of the
16.13 Deep drawing modelling using shell elements for the yarn modelling.
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permeability tensor needs as precise as possible an identification of the pore space. Beam or shell modelling of the yarn is not precise enough and 3D modelling of the yarn is then necessary. But up to now, the available computational capabilities have not allowed one to model precisely a large domain. Consequently, it is necessary to limit the studied domain to the smallest one, i.e. the RUC. The present section explains the road map of an efficient 3D modelling of the yarn inside the RUC. The geometrical description of the fabric has therefore a unique purpose: giving to finite element codes a consistent geometrical description of the yarn geometry in order to realize mechanical or permeability simulations. A microscopic model representing all the fibres inside each yarn could be a solution to get accurate mechanical and permeability models; this approach is developed by Durville (2007), but computational capabilities only allow the modelling of a few tens of fibres where a few thousands are necessary. This section will therefore be focused on models that today ensure the best compromise between accuracy and complexity: mesoscopic models. Considering the simplest weave architecture, the plain weave, various unit cells can be chosen (Fig. 16.14). All of them allow a complete building of the whole fabric but, for a problem to be periodic, the periodicity of the geometry is not enough: the load and boundary conditions must also be periodic. Consequently, depending on the calculations that are planned, one or the other choice would be more relevant. In the following, the unit cell will be the smallest geometrical pattern which allows complete building of the fabric by orthogonal translations, because the main issues to address are the same whatever the dimensions of the unit cell.
(a) RUC type 1
(b) RUC type 2
16.14 Two different RUCs for plain weave: both are geometrically correct but only type 1 allows a correct description of the macroscopic uniform strain field corresponding to in-plane shear of the fabric.
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16.4.3 Structure of 3D mesoscopic models Yarns are composed of an assembly of fibres. There are many types of fibres, differing by their constitutive material (carbon, glass, etc.) and geometry (diameter, length). They can be obtained by different processes (Scardino 1989). As has been developed in previous sections, the reorganization of fibres inside the fibre bundle is a key point to understanding the evolution of yarn geometry. If the cross-section of a yarn is defined as the envelope of the fibres, it strongly depends on the yarn structure. A consistent geometrical model should therefore be able to represent any types of sections and any type of evolution. The second crucial point is to fulfil consistency between the two yarn networks. The interlacement between these two networks leads to a contact surface which has to be modelled accurately for mechanical simulation and for permeability simulation, otherwise significant errors are obtained in the simulations (Hivet and Boisse 2005; Hivet et al. 2006). Two families of mesoscopic models can be considered. In the first one, yarns are assumed to be an assembly of rods jointed by articulations. Some models have been developed using rigid rods, all the rigidities being introduced by springs (Ben Boubaker et al. 2003). For some models, including the well-known Kawabata model, hinged rods are elastic in tension. Transverse stiffness (crushing of yarns) is introduced using springs (Kawabata et al. 1973a,b; Kawabata 1989). Those models are simple and fairly efficient, but if a 3D model is extrapolated there are some interpenetrations between warp and weft yarns (Fig. 16.15). Moreover, when using the Kawabata model, coefficients identified for the transverse crushing behaviour, in order to obtain a good correlation between model and experiment for tension surfaces, are not physically consistent (Buet-Gautier and Boisse 2001). X3 Interpenetration zone
X2
X1
16.15 Interpenetration in the 3D articulated segment model.
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16.4.4 Consistent mesoscopic models Models of the second family are much more sophisticated. These have been developed in order to give a better description of the fabric geometry at the initial state. Yarns are supposed to be continuous solids and the fabric is built up by the interlacement of those solids. Most models assume that the yarn section remains constant along a curvilinear trajectory. The yarn 3D geometry is then obtained through the sweep of the constant cross-section along the trajectory. The curvilinear trajectory, i.e. the mean line of the yarn, can be composed of sinusoids, splines, circles, or polynomials with elliptic crosssections (Lomov et al. 2000). Some of these models have been applied for the simulation of biaxial tension. Results can be found in the literature and the consistency between results obtained from different models is presented in Lomov et al. (2003). In these models, since the yarn section remains constant, it is not related to the transverse trajectory; the contact zone is therefore not very well described. The contact surface between yarns is indeed brought back to a point or does not exist; consistency (represented by the common surface) is therefore not accurately ensured. Some interpenetration or, on the contrary, spurious voids due to the model can be noticed at the contact zone between warp and weft yarns. Above all, experimental observations (Hivet and Boisse 2005; Badel et al. 2008) have shown that the assumption of a constant section along the yarn is not relevant. More precise models have been developed so as to deal with this section variation (Hivet and Boisse 2005; Kuhn and Charalambides 1999). The yarn geometrical model is still defined by the volume generated by the sweep of sections along a trajectory. The section is still defined by the envelope of the yarn cross-section (Fig. 16.16a). But the section varies along the trajectory, taking into account the reorganization of fibres near the contact zone. The trajectory is still defined by the longitudinal section of the yarn (Fig. 16.16b). In fact, these models are intrinsically consistent because the trajectory is constrained by this necessary 3D consistency and the contact part of the trajectory will consist of the same conic surface as the cross-sectional one (Fig. 16.16c). In the contact-free zone, no lateral load is applied to the yarn; consequently, since the bending stiffness of yarns is very weak, the contact-free part of the trajectory is assumed to be straight (Fig. 16.16d). Hence, the trajectory is composed of conics and straight segments linked by tangency conditions so that the model fulfils accurately the fabric description. The full 3D model of the yarn is obtained through a smooth interpolation between the control sections along the imposed trajectory. The interpolation is obtained using CAD software, such as PROEngineer® or Catia V5, which include a ‘swept blend’ feature that is able to build volumes using control sections and trajectories. The final 3D model of the fabric is built with the
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Contact free part
Contact part (a) Trajectory
(b) Contact zone
(c) Free zone
(d)
16.16 Different zones to be considered for a consistent modelling of mesoscopic 3D geometry of the yarn: (a) section; (b) trajectory; (c) contact zone; (d) free zone.
assembly of these yarns and is fully consistent (no interpenetration or spurious voids). Figure 16.17 presents an example for a 4 ¥ 3 twill weave.
16.4.5 Identification of the consistent 3D model The complete 3D model of a 2D fabric may be identified by measuring from three parameters for balanced fabrics to seven for unbalanced fabrics. These seven parameters can be yarn width, yarn density, crimp in each direction, and thickness of the fabric. These measurements are quite easy to get for fabrics, so the model is easy to identify. Three CAD models can be created for each pattern (plain, twill and satin weave) and be completely parameterized to become the masters (generic models) of a CAD family. So a new fabric
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(a)
A
B
B
A
(b)
16.17 3D consistent geometrical model of 4 ¥ 3 twill: (a) 3D view of 4 ¥ 3 twill weave; (b) cut in different planes.
model is obtained by changing the values of the calculated 3D parameters without any new modelling on the CAD software.
16.4.6 Hexahedral mesh These geometrical models have to be meshed in order to perform solid or fluid mechanics simulations. For mechanical simulations, the geometrical CAD model has to be transformed so as to obtain a hexahedral mesh of the elementary pattern. The 3D geometry of yarns is complex. Therefore, a powerful automatic hexahedral-meshing algorithm is needed in order to be able to mesh the fabric. The complex patterns are meshed using specific features of meshing software such as Patran® because each yarn is not a simple glide of a section along
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a trajectory. Top and bottom surfaces of each yarn are meshed identically with quadrangles and hexahedral elements are built by the subdivision of the above-mentioned hexahedra extremities. The mesh is automated using the PCL language, modified so that it is parameterized and applicable to any type of 2D fabric. For permeability evaluation, the complementary model has to be built. Using the advanced tools of standard CAD software, it is easy to get the complementary geometry (Fig. 16.18) that can be discretized with voxels to perform permeability simulations.
16.5
Behaviour identification and finite element modelling
As mentioned in Section 16.3, the identification of the yarn constitutive behaviour is a difficult task as the yarn cohesion is only ensured by the boundary conditions. As a consequence, the best way to proceed is to use inverse identification using macroscopic (i.e. whole fabric) experimental tests in conjunction with finite element modelling. Depending on the modelling strategy chosen, the identification procedure and parameters vary. The first part of the present section presents two ways for modelling the yarn behaviour using hypo- or hyper-elastic approaches. For both, and based on experimental observations (Potluri et al. 2006; Badel et al. 2008), the fibre bundles can be considered to be transversely isotropic. X-ray tomography observation of the yarn cross-section justifies this assumption as the fibre External surfaces for extracting the complementary volume
Meshing of solid skeleton
Meshing of the fluid RUC
16.18 Different steps to be fulfilled for permeability calculations.
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distribution is quasi-isotropic (Badel et al. 2008). The main specificity of such material lies in its strong stiffness in the fibre direction, which can differ from transverse stiffness in a ratio greater than several hundreds (Gu 2007). In the same manner, in-plane and out-of-plane shear stiffness are very small (Potter 2002; Sun and Pan 2005). Strong anisotropy means that the mechanical model strictly follows the material directions in order to calculate stresses, or stress increments in the material frame.
16.5.1 Hypo-elastic approach for yarn modelling Hypo-elastic models have been proposed for material at large strain (truesdell 1955; Xiao et al. 1997; Belytschko et al. 2000): — = C :D
16.1
where D and C are the strain rate tensor and the constitutive tensor, respectively. — , called the objective derivative of the stress tensor, is the derivative for an observer who is fixed with respect to the material. This is the more popular approach for mainly two reasons: it is easy to implement for large strains or rotations; and it is easy to introduce non-reversible behaviour like viscous or plastic behaviour. Using such an approach, two points have to be addressed: determining the constitutive tensor C and the objective derivative of the Cauchy stress tensor — . When considering the constitutive behaviour of a single yarn, it is necessary to realize finite displacement analyses. Although longitudinal stretches are generally small (due to the strong tensile stiffness of the yarn), the other strains and displacements can be large. Considering a bending load, the corresponding strains are small but the rotations are large, and it is then necessary to implement constitutive laws available for large displacements. In the same manner, the shear stiffness of a yarn is very weak, which makes the shear behaviour particularly soft. Moreover, because part of the yarn behaviour is due to friction between fibres inside the tow, it is obvious that its mechanical behaviour is not fully reversible. The difficulty, for strong anisotropic materials, lies in the consistent choice of the objective derivative. Let’s consider the following equation: Êd ˆ — = Q . Á (Qt ..Q)˜ .Qt Ë dt ¯
16.2
Q is the rotation from the initial orthogonal frame to the so-called rotating frame where the objective derivative is made. the most common objective derivatives are the Green–Naghdi (Dienes 1979) and Jaumann (Dafalias 1983) ones. they use the rotation of the polar decomposition of the deformation
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gradient tensor f = R.U (the default in aBaQUS/Explicit®), and the corotational frame (the default in aBaQUS/Standard®), respectively. they are routinely used for analyses of metals at large strains. It has been shown that, in the case of a material made of fibres oriented by the vector f1, the proper objective rotational derivative is based on the rotation of f1, i.e. (Hagège et al. 2005; Boisse et al. 2006; ten Thije et al. 2007; Badel et al. 2009): = f i ƒei0
16.3
where f1 =
f.e10 f.e10
f2 =
,
f.e 02 – (f.e 02 . f 1 ) f 1
(
)
f.e 02 - f.e 02 . f 1 f 1
,
f 3 = f1 ¥ f 2
16.4
Equation (16.1) is integrated over a time increment Dt = tn+1 – tn using the formula of Hughes and Winget (1980), widely used in finite element codes at finite strains: [ n+1] f n+1 = [ n ] f n + [C n+1/2 ] f n+1/2 [D] f n+11//2 i
i
i
i
16.5
where [D] f n+1/2 = ([D] f n+1/2 Dt ) and [S] f n stands for the matrix of the components i
i
i
of any tensor S expressed on the basis fi ƒ fj ƒ . . . ƒ fm at time tn. In eq. 16.5 the constitutive matrix [C]fi is expressed on the basis {fi} oriented by the fibres at the current time. This is a major advantage because its shape and its components are known or can be determined on this basis where longitudinal and transverse behaviour are distinguished. The first is defined by the stiffness of the fibres in the direction f1 (Fig. 16.19), whereas the second characterizes the behaviour of the fibre bundle in the plane (f2, f3). Considering the transverse isotropy assumption allows splitting the strain tensor applied to the transverse section into ‘volumetric’ and ‘deviatoric’ parts. As the plane strain state is considered, the term ‘surface’ will be used instead of ‘volumetric’. The spherical part of the strain tensor is representative of the cross-sectional area changes, i.e. the fibre density changes; and the deviatoric part of the strain tensor is representative of the shape changes of the yarn cross-section. these two modes of deformation are observed by X-ray tomography (Badel et al. 2008): Fig. 16.11 exhibits both compaction of the fibre network (cross-sectional area changes), and rearrangement or change of shape of the fibre bundle. The transverse isotropy assumption allows one to uncouple those two phenomena in the isotropy plane (f2, f3). Finally, two successive decompositions are realized: separation between longitudinal and transverse behaviour; and separation between the ‘surface’ and ‘deviatoric’ parts for the transverse behaviour itself:
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f2
f3 f1
16.19 Local frame of the fibre bundle with f1 longitudinal and f2 and f3 transverse directions.
È e11 Í [] fi = Í Í sym. Î
e12 0
e13 0 0
˘ È 0 ˙ Í ˙+Í ˙ Í sym. ˚ Î
0 e 22
0 e 23 e 33
˘ ˙ ˙ ˙ ˚
16.6
= [L ] fi + [T ] fi where stands for the tensorial accumulation of DDt in the rotating frame. the following developments only concern the plane (f2, f3) and the restriction of [ T ] fi to (f2, f3) is considered: È e 22 [ T ] fi = Í ÍÎ sym.
e 23 ˘ È e s ˙=Í e 33 ˙ Í 0 ˚ Î
0 ˘ È ed ˙+Í e s ˙˚ ÍÎ e 23
e 23 –ed
˘ ˙ ˙˚
16.7
where es = 1 (e22 + e33) is the ‘surface’ strain component and ed = 1 (e22 2 2 – e33) and e23 are the ‘deviatoric’ ones. This formalism is widely used in plasticity. It has also been used in fibre bundle micromechanics in Simacek and Karbhari (1996). as integration is a linear operation, the same separation is valid for strain increments [DT ] fi and then for stress increments [Dst]fi. thus the decomposition leads to: Dss = ADes Dsd = BDed
16.8
Ds23 = CDe23 where Dss, Dsd, Des and Ded are respectively ‘surface’ and deviatoric stress and strain increments (stress components being defined in the same way as strain components by eq. 16.7), and A, B and C are elastic coefficients. It can
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be shown that B = C. Eventually the transverse constitutive tensor (used in eq. 16.5) contains two independent elastic coefficients, which corresponds to an isotropic two-dimensional medium. Using Voigt notation: È ( A + B)//2 Í [C t] fi = Í ( A – B)/2 Í 0 Î
( A – B)/2 ( A + B)//2 0
0 0 B
˘ ˙ ˙ ˙ ˚
16.9
To complete the description of this model, the form of the coefficients A and B must be specified for which basic physical assumptions are used. Under compaction the yarn becomes stiffer for both surface and deviatoric behaviours due to the densification of the fibre network (up to the maximum achievable density). Note also that under longitudinal tension, the surface behaviour should be stiffer. The influence of longitudinal tension on the deviatoric behaviour is not as easy to anticipate but can be assumed to be weak. For the sake of simplicity, it is considered in the present model that tension has no influence on deviatoric behaviour. From these assumptions, it is suggested to give the following form to coefficients A and B: A = A 0e
–pes ne11
B = B 0e
–pes
e
16.10
the transverse constitutive model requires four parameters. the determination of these parameters is explained in Badel et al. (2008). For longitudinal behaviour, and considering the transverse isotropy assumption, the constitutive tensor will require two more independent parameters: the longitudinal Young modulus E1 and the shear modulus G12. Finally, six parameters are necessary to characterize the yarn material. Several other laws have been proposed for fabric transverse behaviour (Cai and Gutowski 1992; Chen and Chou 1999; Comas-Cardona et al. 2007; Kelly 2008), but most of them concern compaction of the fabric and they are often one-dimensional models.
16.5.2 Hyper-elastic approach When using a hyper-elastic approach, it is assumed that the second PiolaKirchoff stress tensor S derives from an energy potential W: S = 2 ∂W ∂C
16.11
where C is the right Cauchy–Green strain tensor. Using the representation theorems, it has been shown (truesdell and Noll 1992) that for transverse isotropic materials the energy potential can be expressed using five different strain invariants:
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W(C) = W(I1, I2, I3, I4, I5)
16.12
with I1, I2 and I3 the classical invariants of C:
I1 = tr(C); I 2 = 1 (tr 2 (C) – tr (C 2 )); I 3 = det(C) 2
16.13
I4 and I5 are structural invariants. If f1 is the anisotropy direction, the structural tensor f1 can be built with
F 1 = f 1 ƒ f1
16.14
Then I4 and I5 are built with
I4 = f1·C·f1; I5 = f1·C2·f1
16.15
Building a hyper-elastic constitutive model consists then in the choice of consistent strain energy potential. The basic assumption is a fully reversible assumption. Even though the yarn behaviour is not fully reversible (Ghosh et al. 1990; Lahey and Heppler 2004), some authors (Guo et al. 2006; Menzel and Steinmann 2001; Bonet and Burton 1998; Weiss et al. 1996; Holzapfel et al. 1996, 2000; Itskov and Askel 2004; Schröder and Neff 2003; Balzani et al. 2006; Wysocki et al. 2008) have proposed hyper-elastic models essentially for pre-impregnated yarns. Their formulations are based on the superimposition of an orthotropic potential representative of the fibres and an isotropic potential representative of the matrix behaviour. The main applications are for biological tissues. An efficient formulation for dry fabrics is already an issue to be addressed.
16.5.3 An example of parameter identification for a hypo-elastic model The present section, illustrating the parameter identification on a glass plain weave, will sum up the whole chapter as it requires all the steps mentioned in the previous sections. Considering the hypo-elastic model described in this section, six parameters have to be determined, namely coefficients E1, G12, A0, B0, n and p. Longitudinal Young modulus The initial longitudinal Young modulus E10 is determined directly using a tensile test on the fabric as shown in Fig. 16.20. As experimental data are given in terms of yarn stiffness, the Young modulus has to be updated during the simulation in order to ensure a constant stiffness of the yarn even when its cross-sectional area changes, using the simple relation:
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Tensile force per yarn (N)
300 Single yarn
250
Uniaxial tension for the fabric
200 150 100
Numerical fitting
50 0 0
0.002
0.004 0.006 Total longitudinal strain
0.008
0.01
16.20 Experimental determination of Young modulus of yarn from tensile testing.
E1 = E10
S0 S
16.16
where E10 is an initial Young modulus related to the initial cross-sectional area S0. the actual cross-sectional area S is updated during the tensile straining using the transformation gradient (see eq. 16.4). All the other parameters need inverse method identification. Consequently, finite element calculations are necessary for the following stages of identification. The five remaining parameters are linked and all of them have an influence on the transverse behaviour. Three kinds of tests can be used: the biaxial tensile test, the in-plane shear test, and the compression tests. in the present example, biaxial tensile and picture frame tests are used. For both test, a constant macroscopic strain field is assumed. Boundary conditions the way to obtain a precise geometrical description of the fabric has been extensively described in Section 16.4. Boundary conditions on the cell must ensure both periodicity conditions and macroscopic strain field. The periodicity conditions depend on both geometry and loading (Miehe and Dettmar 2004; Badel et al. 2007). From the geometrical point of view, the pattern must present symmetry relative to the cutting plane, and from the load point of view, it has to be possible to ensure the macroscopic strain field, leaving free small movements around the symmetry plane. if the periodicity is planar, translation vectors P can be expressed by linear combination of two elementary vectors P1 and P2:
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P = S ma Pa , ma Πa =1
16.17
the boundary ∂V of the RUC can be split into two pairs {∂V–a, ∂V+a}a = 1,2. let us consider two points of the boundary, images of each other by an elementary translation Pa (a = 1, 2). Let them be called paired points. Due to periodicity: Xa- Œ ∂Va-; Xa+ Œ ∂Va+ Xa+ – Xa- = Pa
16.18
the transformation (X) of the structure can be decomposed into a macroscopic (or average) part m(X) and a periodic (or local) fluctuation w(X). to ensure the periodicity of the deformed structure, we need to verify: w(Xa- ) = w(Xa+ ) forr pair paired ed points Xa- Œ ∂Va- and Xa+ Œ ∂Va+, a = 1, 2
16.19
As a consequence, in the deformed configuration: xa+ – xa- = m (Xa+ ) – m (Xa- )
16.20
Since m is known, eq. 16.20 is a kinematic condition on each point of the boundary. A difficulty exists when the geometrical boundary of the RUC is not made of material points, because in this case it cannot be applied. in the plain weave example shown in Fig. 16.14, the first type of RUC (called type 1) makes ensuring the periodicity and boundary conditions simpler because no contact change can occur on the frontier. this is particularly critical for in-plane shear analysis (Badel et al. 2007). Inverse method and optimization procedure Some authors have successfully used the inverse method with an optimization procedure to characterize the elastic properties of a material (Schnur and Zabaras 1992; Gasser et al. 2000). For the method to be efficient, it is necessary to choose the correct mechanical test. the biaxial tensile test has been chosen here with a strain ratio between warp and weft stretches equal. Moreover, for such a test, the RUC can be reduced as there are more symmetry planes on the macroscopic strain field. Results from Buet-Gautier and Boisse (2001) are used. This allows optimizing the values of the five remaining coefficients. The friction coefficient between yarns has to be chosen. Here, a constant value of the Coulomb-Amonton coefficient of 0.24 has been chosen from the literature. When using the biaxial tensile test, this parameter is not crucial,
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as the relative displacements between yarns for such kinematics are not very large. The optimization procedure uses a classical Levenberg–Marquardt algorithm (Schnur and Zabaras 1992). Simultaneous optimization of the five parameters is not easy to realize because not all the parameters have the same weight. The best optimization sequence has been found to be to first optimize exponents n and p, and then coefficients A0 and B0. The shear modulus g12 is arbitrarily chosen. In this way it is possible to get a good approximation with only three iterations. Although the values of A0 and n are obtained with reasonable confidence, this is not the case for coefficients B0 and p. These two depend strongly on the initial values because the test is not sufficiently sensitive to shape change. The use of a second test–the shear test, for example – could be an alternative, with kinematics more sensitive to the shape change of the cross-section. With those limitations the following values are obtained (Badel et al. 2008). Longitudinal Young modulus El (MPa) Transverse parameter A0 (MPa) Transverse parameter B0 (MPa) Transverse parameter n Transverse parameter p Longitudinal shear modulus Gl (MPa)
52,500 1.86E–3 6.25E–3 4060 39.9 20
The use of a compression test could also be an alternative. But in such a case, the relative displacements between yarns are large and a good evaluation of the friction coefficient is necessary. Without precise values, this test remains difficult to use for the identification of mechanical parameters. Local validation The optimization procedure has been realized using a macroscopic force– displacement curve. Five parameters are necessary to achieve this fitting. Consequently, the uniqueness of the solution is not ensured. In order to validate the results, various observations have been realized: comparing the numerical and experimental force–displacement curves for other tests such as biaxial tensile tests with other strain ratios or picture-frame tests; and comparing the local geometries of the yarn for biaxial tension or in-plane shear using x-ray tomography.
16.6
Finite element simulations, use and results
As briefly described in the introduction, 3D mesoscopic-scale simulations have mainly two direct applications: giving a physically based interpretation of macroscopic constitutive behaviour of woven fabrics; and giving the solid
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skeleton of a deformed fabric for flow simulations (the injection phase of the LCM processes).
16.6.1 Constitutive behaviour at greater scale The basic assumption is that the mechanical characteristics of the yarn material depend on the yarn constitution, namely the nature and diameter of the fibre, the number of fibres and the fibre organization (twisted or untwisted yarn). Consequently, it is theoretically possible to find the mechanical properties of a fabric knowing the yarn parameters and the fabric architecture. For 3D tailored fabrics, it could be interesting to predict the mechanical behaviour of the fabric before the weaving operation. In this way, it could be possible to optimize the weaving in order to get a better compromise between formability and in-use performance. This has been done successfully by Grujicic et al. (2009) for ballistic application.
16.6.2 Permeability computations A very important application of mesoscopic simulations is the calculation of permeability tensors for computing resin flow in a preform using Darcy’s law. The position and density of the solid reinforcement at the end of the draping phase constitutes the input data for the Stokes (or Stokes and Brinkman) flow calculation. While some methods have been proposed to leapfrog that step using approximate geometries (Belov et al. 2004; Verleye et al. 2006; Demaria et al. 2007), calculation of the flow based on the geometry obtained by mesoscopic analysis is important because it takes into account the current deformed shape of the fabric, as shown in Fig. 16.21 (Nguyen et al. 2010). The whole procedure is summarized in Fig. 16.18. The first step consists of meshing the solid skeleton, i.e. the yarns in the initial configuration; then a macroscopic uniform strain field is applied and the complementary volume is extracted in order to realize flow simulations. Depending on the nature of the permeability code (finite element or finite differences), the complementary volume is more or less difficult to identify. The example of Fig. 16.18 corresponds to a finite element mesh, but if a finite difference code like “CELPER2” is used (Boust et al. 2002) the geometrical discretization is easier and can be realized using an image processor like AMIRA.
16.6.3 Forming defects The previous section was mainly concerned with 3D modelling of the yarn. It is obvious that the same kind of work has to be done with shell modelling of the yarn material. In such a case, the mechanical behaviour is a simplified version of the 3D one. The possibilities offered by the smallest number of
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Permeability (m2)
2 ¥ 10–9
1 ¥ 10–9 Kxx 5 ¥ 10–10
Kyy 2 ¥ 10
–10
0
10
20 30 Shear angle (°)
40
50
16.21 Evolution of the permeability tensor of the 2 ¥ 2 carbon twill as a function of the shear angle. 90 Theoretical angle Mean angle from simulation Measured angle
80
Shear angle (°)
70 60 50 40 30 20 10 0 0
10
20
30 40 Displacement (mm)
50
60
70
16.22 Shear angle during a bias extension test using the mesoscopic analysis.
degrees of freedom per unit cell allow simulating a larger domain and then simulating large-scale defects like wrinkles or network sliding phenomena. Figure 16.22 shows that the mesoscopic scale analysis is able to capture the right shear angle between warp and weft yarns of a bias extension test (Gatouillat et al. 2010). Network sliding is shown in Fig. 16.13.
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16.6.4 Ballistic impact For ballistic applications an important issue is the energy absorption by the armour during the impact. It has been shown that friction is as important as the mechanical behaviour of the yarn itself for energy consumption (Duan et al. 2005). This is the reason why mesoscopic-scale simulations allowing yarn sliding are of great interest for such applications (Rao et al. 2009; Nilakantan et al. 2010; Sapozhnikov et al. 2007).
16.7
Conclusions and future trends
The present chapter has successively pointed out the usefulness and key points of a consistent modelling of woven fabrics at meso-scale. Under a single term, several realities are depicted: 3D modelling of the yarn geometry with precise representation, or shell modelling for the whole fabric. The corresponding applications are also different: mechanical behaviour estimation and permeability estimation for the precise 3D modelling; and forming defects and ballistic impact simulations for the shell modelling. For both approaches, several aspects need to be improved. For the precise 3D modelling, both the geometrical description and the mechanical characterization need improvement. As mentioned in Section 16.4, for 3D weaves the building of a consistent model without spurious voids or interpenetrations has to be achieved. The section on variability for high fibre volume fraction fabrics proposes to take into account mechanical constraints, not only geometrical constraints. The mechanical characterization covers two main aspects, The first is friction between yarns, which needs to be explored not only in terms of the friction factor but mostly in terms of interface constitutive behaviour. It is obvious that yarn-to-yarn friction cannot be an isotropic phenomenon. This point is important also because the inverse method used to characterize the yarn itself requires finite element simulations for tests for which the relative displacements between yarns are not negligible. The second aspect is mechanical testing, for which several possibilities need to be explored: compression, in-plane shear and bending. Up to now, the shear stiffness that is representative of the bending has been evaluated using the biaxial tensile test, but this is not a good option: bending needs a specific study. When all these aspects have been improved with a reasonable degree of confidence, it should be possible to simulate the weaving operation from the row yarn. Then, it should be possible to evaluate other important aspects such as yarn damage and residual stresses with reasonable accuracy.
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Continuous models for analyzing the mechanical behavior of reinforcements in composites X.Q. P e n g, Shanghai Jiao Tong University, P.R. China and J. C a o, Northwestern University, USA
Abstract: In this chapter two continuous models based on a geometric transformation approach and an energy approach, respectively, for analyzing mechanical behavior of woven fabric composites are proposed. In the geometric transformation approach, a convected coordinate system is taken to describe contravariant stresses and covariant strains. The transformations between the contravariant/covariant components and Cartesian components of the stress and strain tensors provide an approach for deriving a non-orthogonal constitutive relation. In the energy approach, a simple hyperelastic constitutive model is developed. The strain energy function is decomposed into tensile energy from fiber yarn stretches and shearing energy from yarn angular change. Key words: woven fabric, material constitutive law, anisotropy, nonorthogonal model, hyperelastic model.
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Introduction
Woven composite fabrics usually undergo small membrane extension along the yarn directions while experiencing large angular variation between weft and warp yarns during deformations. The reorientation and redistribution of fiber yarns result in a significantly anisotropic material behavior. The effective elastic properties are very sensitive to fiber orientation and usually have a pronounced material nonlinearity. Despite an enormous amount of work in the field, there is no widely accepted model that describes accurately all the main aspects of fabric mechanical behavior.[1] The multiscale nature of woven composites allows both discrete as well as continuous approaches. In discrete approaches,[2–4] analysis is carried out at unit cell level in which each yarn and fabric is modeled. Nevertheless, because of very large number of yarns or fibers, the computational effort is significant so these approaches are limited to small domain analysis. Conversely to these discrete approaches is to consider woven fabrics as a continuum. The main benefit of this approach is that it can be used in a standard finite element method. Some orthotropic material models were developed via the continuous 529 © Woodhead Publishing Limited, 2011
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approach for woven composites and fabrics.[5–10] However, the anisotropic material behavior caused by the complex fiber reorientation during forming should be captured in a constitutive model. By using the stress energy density referring to the deformed volume, Luo and Chou[11] derived a nonlinear constitutive model for flexible composites composed of aligned continuous fibers in an elastomeric matrix under finite deformation. The theoretical derivation was based upon the Eulerian description to account for the material nonlinearity including stretchshear coupling. The fiber orientation was obtained from strains in their model through an iterative calculation. Vu-Khanh and Liu[12] predicted the forming-induced fiber rearrangement by using the well-known pin-joint net idealization model (Mark and Taylor[13]). The deformed woven fabrics (non-orthogonal architecture) were then characterized by a laminate composed of four fictional unidirectional plies. The equivalent thermomechanical properties of the deformed woven fabric were predicted based on the classical laminated plate theory. This approach works well when the shear deformation is relatively small. Taking into account the effect of fiber orientation on material anisotropy, Yu et al.[14] developed a non-orthogonal constitutive model for woven composites based on force analysis along weft and warp fiber yarns under a continuum mechanics framework. A unit cell structure was used to extract the macroscopic material properties of woven composites. An analytical form for the material properties relating stresses and strains was then derived. However, the derivation was based on small deformation assumption and the paper did not present the stress distributions and characteristic force–displacement responses in their numerical examples. Based on the stress and strain transformations in the orthogonal and nonorthogonal coordinate systems and the rigid body rotation matrix, Xue et al. [15] developed a constitutive model for characterizing the non-orthogonal material behavior of woven composite fabrics under large deformation. The geometric and material nonlinearity was taken into account. The model was validated by comparing numerical shearing results of a plain weave composite sheet with experimental data. Nevertheless, some questions can be still asked about the physical meaning of the stress transformation and stress definition in the non-orthogonal coordinate system. Numerically, Hsiao and Kikuchi[16] applied the homogenization method to analyze and optimize the thermoforming process of woven composites. A fiber orientation model was used in their approach to trace the woven-fabric microstructure evolution (fiber rearrangement) during forming. Based on the assumption of instantaneously rigid solid fiber suspended in a viscous nonNewtonian polymer melt, the material properties of thermoplastic composites under various fiber orientations were obtained by the homogenization method and then tabulated as a database. The instantaneous homogenized properties of each global finite element were then obtained by interpolation
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in the database. Though this approach is able to capture the evolution of non-orthogonality of woven composites during forming, it is computationally costly for complex fiber architectures and forming processes. Cherouat and Billoët[17] simulated the deep-drawing and laying-up processes of woven fabrics by super-elements combining truss and membrane elements. In their model, each warp and weft yarn composing the fabric was modeled by truss elements with elastic properties and the connecting points of truss elements were assumed to be hinged. The viscous resin was modeled by membrane elements and was kinematically coupled to the fabric at those connecting points. The fiber reorientation and the thus induced non-orthogonality on the material behavior of woven composites can be efficiently delineated by the application of truss elements. However, the lack of bending stiffness in the truss elements would be a possible problem in the forming simulation. A number of researchers have developed hypoelastic constitutive models for characterizing the behavior of woven composite fabrics at large strain. [18, 19] Based on an objective derivative, these show that using the fiber rotation allows one to keep the orthotropic direction coinciding with the fiber direction. This approach is carried out for a single direction. Khan et al. [20] extended this approach to the macro-scale to perform a forming simulation that had two directions. In this model the warp and weft fibers rotation tensor can correctly trace the specific behavior of woven materials. The results obtained by this model show good accordance with experimental data of the double dome benchmark.[21] Aimene et al.[22] proposed a hyperelastic model for composite reinforcement forming simulation. In this chapter two continuous models, based on a geometric transformation approach and an energy approach, respectively, for analyzing the mechanical behavior of woven fabric reinforcements in composites are proposed. In the geometric transformation approach, a non-orthogonal constitutive model is developed under the same material characterization framework as in Peng and Cao [10] to characterize the anisotropic material behavior of woven composite fabrics under large deformation. [23] In this non-orthogonal model, a convected coordinate system, whose in-plane axes are coincident with the weft and warp yarns of woven composites, is embedded into shell elements modeling composite fabrics. A model is also developed to trace fiber reorientation during deformation. Contravariant components of the stress tensor and covariant components of the strain tensor are introduced into the convected coordinate system. The transformations between the contravariant/covariant components and the Cartesian components of the stress and strain tensors provide an approach for deriving the global nonorthogonal constitutive relation for woven composites. In the energy approach, based on fiber-reinforced continuum mechanics theory, a simple hyperelastic constitutive model is developed to characterize
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the anisotropic nonlinear material behavior of woven composite fabrics under large deformation. The strain energy function for the hyperelastic model is additively decomposed into two parts nominally representing the tensile energy from weft and warp yarn fiber stretches and the shearing energy from fiber–fiber interaction between weft and warp yarns, respectively. The two proposed material characterization approaches are demonstrated on a balanced plain weave composite fabric. The equivalent material properties are obtained by matching with experimental data of tensile and shearing tests on the woven composite fabric. Assumptions used in our presented constitutive models are as follows: 1. The tensile-shear behavior is assumed to be decoupled in the local convected coordinate system. This tensile-shear decoupling was well verified in the experimental studies on the biaxial tensile tests of woven composite fabrics.[24, 25] This observation makes the modeling work much easier. 2. The tensile behavior of fiber yarns is assumed not to be affected by gap closure due to shearing. These first two assumptions are related and both are reasonable when the shear angle is low. However, when the shear angle is close to the locking angle, these two assumptions might be invalid; more research work is needed in this area. 3. Yarn slippage is negligible. An outline of the remainder of the chapter is given as follows. In Section 17.2, the general non-orthogonal constitutive model is derived based on tensor analysis. The specific format for woven composite fabrics is provided in Section 17.3. The detailed application to a plain weave composite fabric is presented in Section 17.4. Experimental tensile and bias extension tests are introduced to determine the equivalent material properties (tensile modulus and shear modulus) of the plain weave composite fabric. A model for tracing the fiber orientation is also developed in this section. By using the developed non-orthogonal constitutive model and fiber orientation model, a user-defined material subroutine is coded for a commercial FEM package, ABAQUS/Standard. Numerical simulations for the bias extension and shearing tests of plain weave composite fabrics are then implemented in Section 17.5. The results are compared with experimental data to validate the proposed non-orthogonal model. In Section 17.6, a general fiber-reinforced hyperelastic constitutive model is provided. The specific format and its detailed application to a plain weave composite fabric are provided in Section 17.7. The equivalent material parameters of the plain weave composite fabric for the hyperelastic model are determined by using experimental data of uniaxial tensile and shearing tests. Section 17.8 gives a summary on the continuous constitutive models.
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17.2
533
Continuum mechanics-based non-orthogonal model
For a body shown in Fig. 17.1, we first set up a stationary Cartesian coordinate system with an orthonormal basis {ei} for a Euclidean vector space, to define stress and strain tensors in the body. alternatively, we can choose another arbitrary vector basis {gi}, as shown in Fig. 17.1. For simplicity, e3 and g3 are not shown in Fig. 17.1. Both of them are assumed to be perpendicular to the 1–2 plane under plane-stress condition for shell elements. nevertheless, it must be noted that the following derivations are suitable for general 3D cases. The vector basis {gi} can be neither orthogonal nor normal and can be considered as a covariant basis for the euclidean vector space. The relation between the two bases {ei} and {gi} can be defined as gi = Pij ej
17.1
in which summation over an index (in this case j) is implied by its repetition. This convention is followed throughout this chapter without further comment except where an explicit statement is made to the contrary. We can write the inverse of eq. 17.1 in the form ei = Qij gi
17.2
where the coefficients
Qij
satisfy
Pi j Q kj = d ik = Pjk Qij
17.3
where d ik is the Kronecker delta. e2
g2
g2
g1
e1
g1
17.1 Coordinate systems in the non-orthogonal model.
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The reciprocal basis of {gi}, denoted by {gi}, is a contravariant basis for the Euclidean vector space and is defined uniquely by[26] ÏÔ 1, gi ◊g j = d ij = Ì ÔÓ 0,
i=j i ≠j
17.4
By using relation 17.4 in conjunction with eqs 17.1, 17.2 and 17.3, we can show that the contravariant basis vectors {gi} are related to {ei} according to gi = Q ij e j
17.5
Suppose that the strain tensor at a material point in the body is expressed in the contravariant coordinate system {gi} and the Cartesian coordinate system {ei}, respectively, as = eij gi ƒg j ,
= eijei ƒ ej
17.6
where the covariant components eij of the strain tensor can be tranformed from the Cartesian components eij according to
eij = Pim Pjn e mn
17.7
The stress components corresponding to the covariant components eij of the strain tensor in a constitutive relation are the contravariant components (denoted as t ij ) of the stress tensor expressed in the covariant coordinate system {gi}, = t ij gi ƒg j
17.8
The contravariant stress components t ij are related to the Cartesian components tij (the same as tij since {ei} are self-reciprocal) according to
t ij = t ij = Pmi Pnjt mn
17.9
The constitutive relations under the covariant coordinate system can be given as
t ij = C ijkl e kl
17.10
where C ijkl are the contravariant components of the fourth-order constitutive tensor C in the covariant coordinate system {gi}: C = C ijkl gi ƒ g j ƒ g k ƒ gl
17.11
The contravariant components C ijkl are related to the Cartesian components Cijkl (the same as Cijkl since {ei} are self-reciprocal) of the constitutive tensor C according to © Woodhead Publishing Limited, 2011
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C ijkl = Qmi Qnj Qok Q lp C mnop = Qmi Qnj Qok Q lp Cmnop
535
17.12
The inverse of eq. 17.12 is Cijkl = Pmi Pnj Pok Ppl C mnop
17.13
equtions 17.12 and 17.13 provide a way for transforming the elastic matrix in the non-orthogonal coordinate system to its counterpart in the orthogonal Cartesian coordinate system or vice versa. In this chapter, we will first determine the elastic matrix in the non-orthogonal coordinate system, and then transform it to the orthogonal system by using eq. 17.13.
17.3
Non-orthogonal constitutive model for woven fabrics
To trace the rotation of fiber yarns in woven fabrics during deformation, we choose the two vectors, g1 and g2, to coincide with the current weft and warp directions of the fabrics, as shown in Fig. 17.2. For simplicity, g1 and g2 are chosen to be unit vectors. The vectors {gi} construct a convected coordinate system which reflects fiber reorientation during deformation. The coefficients Pij in eq. 17.1, which relate {gi} to {ei}, can be obtained from Fig. 17.2 by a simple geometric analysis as È cosa P = Pi j = Í ÍÎ cos(a + q )
˘ sina ˙ sin(a + q ) ˙˚
17.14
To consider the non-orthogonality of woven fabrics caused by fiber reorientation during shear deformation, we need to express the stress tensor and strain tensor in the convected coordinate system. By combining eq. 17.7 with 17.14 and representing it in a rate form for the convenience of g2 g2
g1
q a
O O
g1 Before deformation
e1
e1 After deformation
17.2 Schematic of a deformed plain weave structure with shear deformation.
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FEM implementation, the covariant components deij of the strain increment d can be transformed from the Cartesian components deij explicitly as È de11 Í ÍÎ de12
È de11 de12 ˘ ˙ = PÍ de 22 ˙˚ ÍÎ de12
de12 de 22
˘ ˙P T ˙˚
17.15
Similarly, by combining eq. 17.9 with 17.14, the contravariant stress components dt ij are related to the Cartesian components dtij explicitly as È dt 11 Í ÍÎ dt 12
dt 12 dt 22
˘ È dt11 ˙ = PT Í ˙˚ ÍÎ dt12
dt12 ˘ ˙P dt 22 ˙˚
17.16
now what we need is an appropriate constitutive law relating the stress and strain increments. experimental studies on biaxial tensile tests of woven composite fabrics have verified that the shear stress and the direct stresses can be treated as uncoupled. [23, 24] Hence, it should be reasonable to assume that the contravariant elastic matrix, which relates the contravariant stresses to the covariant strains in the convected coordinate system, has an orthotropic format. Consequently, we can rewrite the constitutive relations eq. 17.10 in a rate form according to the contravariant elastic matrix as Ï dt11 Ô Ì dt 22 Ô 12 Ó dt
¸ È D 11 () D 12 () 0 Ô Í 12 ˝ = Í D () D 22 () 0 Ô Í 0 0 D 33 () ˛ ÍÎ
˘Ï ˙ Ô de11 ˙ Ì de 22 ˙Ô ˙˚ Ó dg12
¸ Ô ˝ Ô ˛
17.17
or {dt} = [D ()]]{d {de}
17.18
where dg12 = 2de12 . Here the contravariant elastic matrix is essentially the tangent modulus and depends on the current strains in the fabric. From Eq. 17.15, we have for {de} Ï de11 Ô Ì de 22 Ô dg 12 Ó
¸ È (P11 )2 Ô Í 1 2 ˝ = Í (P2 ) Í Ô 1 1 ˛ ÍÎ 2 P1 P2
(P12 )2 (P22 )2 2 P22 P12
˘Ï ˙ Ô de11 2 1 ˙ Ì de 22 P2 P2 ˙ 2 1 1 2 ( P1 P2 + P1 P2 ) ˙˚ ÔÓ dg 12 P11P12
¸ Ô ˝ Ô ˛
È ˘ Ï de ¸ coss 2a sin 2 a ssina cos cosa Í ˙ Ô 11 Ô 2 2 = Í coss (a +q ) sinn (a + q) q) ssin( sin in((a +q )ccos( os(a +q ) ˙ Ì de 22 ˝ Í ˙ Ô dg Ô sina ssin sin in(a +q ) in( ssin in(2a +q ) in(2 ÍÎ 2cosa cos(a +q ) 22si ˙˚ Ó 12 ˛ 17.19
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or {de} = [T1 ]{de }
17.20
where dg12 = 2de12. Similarly, we can obtain the relation between {dt} and {dt} from eq. 17.16 in a matrix format as Ï dt 11 Ô Ì dt 22 Ô dt Ó 12
¸ Ô ˝ Ô ˛
È coss 2a cos 2 (a +q ) 2cosa cos((a +q ) ˘ Ï dt11 Í ˙Ô = Í sinn 2a sin 2 (a +q ) sin 2ssin ina sin(a +q ) ˙ Ì dt 22 Í ˙ Ô 12 ÍÎ sinna cosa sin(a +q )cos(a +q ) sin(2a +q ) ˙˚ Ó dt
¸ Ô ˝ 17.21 Ô ˛
or {dt } = [T1]T {dt}
17.22
Substituting eqs 17.17 and 17.20 into 17.22 yields {dt } = [T1 ]T [ D ()][T1 ]{d ]{de } = [D ]]{de }
17.23 Consequently, the elastic matrix [D] in the orthogonal Cartesian coordinate system can be transformed from the contravariant elastic matrix [D ] in the convected coordinate system by [ D ] = [T1 ]T [[D D ][T1 ]
17.24
To be consistent with aBaQUS, we will use the logarithmic strain tensor for large deformation analysis.
17.4
Specific application for a plain weave composite fabric
a numerical approach for the material characterization (orthogonal model) of woven composite fabrics was developed in Peng and Cao.[10] In that approach, a unit cell was first built to represent the periodic pattern in a balanced plain weave fabric. The characteristic force–displacement curves of the composite fabric were then obtained by simulating tensile and shear tests on the unit cell. a shell element with the same outer size as the unit cell was then used to represent the unit cell. The equivalent material properties for the shell element were obtained by reproducing the force–displacement curves. In this section, we will modify this approach by including the non-
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orthogonal constitutive relations derived in the previous sections and using experimental force–displacement curves to obtain the contravariant elastic matrix [D ] in eq. 17.17. The balanced plain weave fabric used is a fiber-reinforced thermoplastic composite (glass fiber and polypropylene resin). The material properties of the constitutive phases of the composite can be found in Peng and Cao.[10] The geometric characteristic parameters for the plain weave fabric are yarn width 3.72 mm yarn spacing 5.14 mm and fabric thickness 0.39 mm. now, the task is to determine the equivalent contravariant elastic moduli 11 22 D D , D D, D 12 and D 33 in the non-orthogonal constitutive model. as a natural way, we are expected to express these contravariant elastic moduli as functions of the covariant strain components in the convected coordinate system. However, the covariant strains cannot reflect the real fiber yarn stretch and angle change between yarns; neither can the logarithmic strain tensor. Here we choose the Green–Lagrange strains to define the effective material properties for the plain weave fabric. This is due to the fact that the Green–Lagrange strain can reflect the real deformation status along the fiber yarn directions and the angle change between the weft and warp yarns. Figure 17.3 shows a single square shell element under trellising deformation. In Fig. 17.3, the undeformed square element has an edge length of L = 5.14 mm and a total displacement of D = 2.12 mm. Figure 17.4 shows the green–Lagrange and logarithmic direct strains versus displacement in the single shell. In Fig. 17.4, the solid line represents the logarithmic direct strain P
D L
q/2 q/2
17.3 Shear angle in trellising test.
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0 0
0.5
1
1.5
2
2.5
–0.02
Direct strain
–0.04 –0.06 –0.08 –0.1 –0.12 –0.14 –0.16
Logarithmic direct strain Green direct strain Displacement (mm)
17.4 Direct strains in trellising test of a single shell element obtained from two different strain measurements.
and the filled squares denote the Green–Lagrange direct strain. Under the pin-joint assumption for the trellising deformation mode, the edge length of the shell element should remain unchanged during the deformation, which means that the direct strain should be zero. This is faithfully captured by the Green–Lagrange strain, as shown in Fig. 17.4. Consequently, it is chosen in the definition of the tensile modulus, which depends mainly on the stretch status in the fiber yarn. Figure 17.5 shows the Green–Lagrange (filled squares) and logarithmic (solid line) shear strains as well as the sinusoidal values of the shear angles (dotted line) in the single element under trellising. The shear angle is the angle change of the shell element during trellising and can be calculated as follows (see Fig. 17.3), Ê ˆ q = 90 o – 2cos –1Á 2 L + D ˜ Ë 2L ¯
17.25
As shown in Fig. 17.5, the Green–Lagrange shear strain is identical to the sinusoidal value of the shear angle and can reflect the real shear deformation in the woven composite. Hence, based on continuum mechanics concepts, we will choose the Green–Lagrange strain tensor to define the equivalent contravariant elastic moduli.
17.4.1 Tensile moduli for the plain weave composite fabric The tensile moduli D 11 and D 22 were determined in Peng and Cao[10] by tensile tests on the unit cell. Since for uniaxial or biaxial tensile deformation modes, © Woodhead Publishing Limited, 2011
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0.8 0.7
Shear strain
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.5
1 1.5 Displacement (mm)
2
2.5
17.5 Shear strains in trellising test of a single shell element obtained from two different strain measurements.
the non-orthogonal coordinate system coincides with the orthogonal one, it should be appropriate to directly take over the tensile modulus formulation defined in Peng and Cao.[10] However, by studying the experimental results of uniaxial tensile tests on the plain weave fabric, it is found that some modifications are necessary to reflect the realistic tensile behavior of the composite fabrics, i.e., the gap between fiber yarns should be taken into account and an appropriate compaction model should be developed to model the compression between fiber yarns. The uniaxial tensile tests of the plain weave fabric were carried out on a Sintech-20/G tensile testing machine under room temperature. The pulling speed was 0.25 mm/min. Figure 17.6 shows the setup for a single fiber yarn and a composite fabric. The tensile load versus strain curves are shown in Figure 17.7. Four fabrics were prepared for the tensile test on the plain weave composite fabric. Each fabric included eight fiber yarns in the width direction. As can be seen from Fig. 17.7, the testing results for the composite fabrics are quite scattered. This can be attributed mainly to the various initial stretching statuses in the samples. Generally, the load–strain curve of the woven composite fabric can be roughly divided into three stages, taking sample number 3 as an example. First, the fabric has a very small initial tensile modulus (the slope of the load–strain curve) until reaching a strain level of about 0.8%, which is due to the undulation of fiber yarns in the fabric. Secondly, from the strain level of 0.8% to about 2.3%, the load–strain curve is approximately linear with a high tensile modulus. Then due to the
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541
(a)
(b)
17.6 Uniaxial tensile tests on plain weave composite fabrics.
breakage of some fibers in the fabric, the load-strain curve becomes very nonlinear and has a much smaller tensile modulus than in the second stage. The tensile behavior for a single yarn, on the contrary, is quite different
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Peng & Cao[10] Shell element Fabric #4
Load (N/yarn)
200
Yarn
Fabric #3 100 Fabric #2 Fabric #1 0 0
0.01
0.02
0.03 Strain
0.04
0.05
0.06
17.7 Uniaxial tensile test results on plain weave composite fabric.
from that of a fabric. As shown in Fig. 17.6, there is no initially soft stage in the load–strain curve of the yarn. The tensile modulus of the yarn is much larger than that of the fabric. The fiber breakage at about 0.6% strain level immediately results in the loss of loading capacity in the yarn. As a comparison, we also present the numerical uniaxial tensile load–strain curve of the unit cell in Peng and Cao[10] by a dashed line in Fig. 17.7. It can be clearly seen that the unit cell actually possesses the same uniaxial tensile behavior as a single yarn instead of a fabric. This can be attributed to the gaps in the fiber yarns. Fiber yarns in a dry fabric are a collection of thousands of tiny fibers with a diameter at micrometer level, and there are unavoidably gaps between fibers. However, the volume fraction of the voids between fibers was not taken into account in the calculation of the homogenized elastic constants for fiber yarns. Consequently, the unit cell, as modeled with solid continuum elements, treats all gaps between fibers having the same material properties as the fibers. This results in the unrealistic high tensile modulus in the uniaxial test of the unit cell. Consequently, we cannot directly take over the tensile modulus formulation defined in Peng and Cao.[10] As an alternative, we take directly the experimental tensile load–strain curve of the fabric instead of the numerical results from the unit cell to determine the tensile modulus for the fabric. While efforts are currently underway to reduce the variation in the tests, the testing result of fabric number 3 is taken in the determination of the tensile modulus, as it roughly represents the averaged tensile response of the plain weave
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composite under uniaxial tensile test. The tensile modulus so obtained has the following formulation: Ï 0.1 Ô Ô 2000 D ii (MPa) = Ì 1 + exp(– 600(e i – 0.01 01)) Ô 1200 Ô Ó
ei < 0 0 £ e i < 0.022
17.26
e i ≥ 0.022
where i = 1 or 2 and no summation for i. ei are the green–Lagrange direct strain components. Since fiber yarns buckle immediately under a compression load, it is expected that the equivalent tensile elastic moduli D 11 and D 22 should have a negligible compression stiffness (here we assume a constant as 0.1 MPa). Figure 17.8 shows the equivalent tensile modulus versus the direct strain. As can be seen from Fig. 17.8, the equivalent tensile modulus has three stages, which correspond to the tendency in the experimental tensile load–strain curve of the fabric. Considering the weak interaction between the weft and warp yarns under tension, the term D 12 is assumed to have a very small value: D 12 (MPa) = 0.02 ¥ min(D 11, D 22 )
17.27
The numerical uniaxial tensile load–strain curve from shell elements by using the equivalent tensile modulus formulation in eqs 17.26 and 17.27 is shown in Fig. 17.7 by a solid line with squares.
2500
Tensile modulus (MPa)
2000
1500
1000
500
0 0
0.01
0.02 0.03 Tensile strain
0.04
0.05
17.8 Equivalent tensile modulus for the plain weave composite fabric.
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17.4.2 Shear modulus for the plain weave composite fabric The experimental bias extension test of the plain weave composite fabrics is used to determine the shear modulus D 33 . The bias extension test is essentially a uniaxial extension test with the dimension of the test sample in the loading direction relatively greater than the width, and the yarns initially oriented at ±45° to the loading direction. The bias extension test of the plain weave composite fabrics was carried on a Sintech-20/g tensile testing machine under room temperature. The pulling speed was 1 mm/min. Figure 17.9 shows the setup for the bias extension test. Figure 17.10 shows a bias extension sample. The samples used in the bias extension test were chosen to have an aspect ratio of 2. The size of the bias extension sample is determined by the square ABDE in Fig. 17.10. We will call it an N-yarn sample if the line aB contains N fiber yarns. Two kinds of samples were prepared for the bias extension test: 16-yarns and 24-yarns, and each kind had four samples. The 16-yarn samples correspond to a sample size of 111.3 mm ¥ 222.6 mm and the 24-yarn samples correspond to a sample size of 167 mm ¥ 334 mm. Figure 17.11 shows a deformed sample under the bias extension test. The raw experimental data for the 16-yarn samples are shown in Fig. 17.12a. As can be seen from Figure 17.12a, the experimental repeatability is not good. However, if we shift the curve of fabric number 2 horizontally to the right by about 2 mm and the curves of fabrics 3 and 4 horizontally
17.9 Experimental setup for bias extension test.
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D
H
E
L
F
545 C
I
B
A
G
J
17.10 Bias extension sample.
17.11 Deformed bias extension sample.
to the left by about 1 mm, we can find the adjusted loading curves of the four samples overlap with each other very well, as shown in Figure 17.12b. This shift is reasonable since the bias extension load is very small at the initial stage. What is important here is the slope of the loading curve that represents the characteristic material behavior of the plain weave composite
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30
Load (N)
Fabric 2 Fabric 1
20
Fabric 3 10
Fabric 4 0
0
5
10
15 20 25 Displacement (mm) (a)
30
35
30
35
40
30 16 16 16 16
25
yarns yarns yarns yarns
– – – –
#1 #2 #3 #4
Load (N)
20 15 10 5 0
0
5
10
15 20 25 Displacement (mm) (b)
40
17.12 Bias experimental load-displacement curves of 16-yarn fabrics: (a) raw data; (b) adjusted data.
fabric under bias extension. The raw data of the 24-yarn samples are shown in Figure 17.13a. Good repeatability of the experimental data is obtained by shifting the curves of fabrics 2 and 4 horizontally to the right by about 0.5 mm and the curves of fabrics 1 and 3 horizontally to the left by about 0.5 mm (see Figure 17.13b). As can be seen from Figs 17.12 and 17.13, the load–displacement curves for the bias extension tests have a very flat stage
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40 Fabric 2
35
Fabric 4
30
Load (N)
25 20 Fabric 3
15 10
Fabric 1
5 0
0
10
20
30 40 Displacement (mm) (a)
50
60
30 24 24 24 24
25
yams yams yams yams
– – – –
#1 #2 #3 #4
Load (N)
20 15 10 5 0
0
10
20 30 Displacement (mm) (b)
40
50
17.13 Bias experimental load-displacement curves of 24-yarn fabrics: (a) raw data; (b) adjusted data.
followed by a steep increase tendency. The flat stage is mainly due to the undulation in the fiber yarns. After certain deformation, the yarn undulation is squeezed out and compaction happens between the weft and warp yarns, which results in shear locking. The load increases tremendously with the displacement after shear locking. The adjusted experimental results of the 16-yarn samples will be used to determine the equivalent shear modulus
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G 12 , and those of the 24-yarn samples will be used later to validate our non-orthogonal material model. a multi-element model is built to simulate the bias extension test of the 16-yarn samples first. Figure 17.14 shows the FEM model. The grippers were not modeled in the numerical simulation. Instead, a fixed boundary condition is imposed to the bottom edge of the mesh, and the nodes along the top edge have the same displacement, as shown in Fig. 17.14. The deformed mesh with the contour of the Green–Lagrange shear strain is shown in Fig. 17.15, which shows three discernible deformation zones of the fabric under the bias extension mode. No significant deformation occurs in Zone I. In Zone II, the deformation is a combination of shearing and extension. The main mode of deformation in Zone III is shearing. Most of the deformation of the fabric occurs in this zone. The contour plots of the green–Lagrange direct strains (which reflect the stretching status of fiber yarns during the deformation) are shown in Fig. 17.16. As can be seen from Fig. 17.16, the fiber yarns from the corners of the clamped end to the center of the free edges have the biggest stretching deformation. These simulation results conform very well with our experimental observations. The averaged experimental results of the 16-yarn samples with error bars are shown in Fig. 17.17 by a solid line. By adjusting the contravariant shear modulus D 33 imposed on the shell elements to match the experimental load–displacement curve of the 16-yarn samples under the bias extension test, we obtain the equivalent shear modulus D 33 as a function of the engineering (green–Lagrange) shear strain g12 (see Fig. 17.18), 3 2 D 33 (MPa) = 5.027 g 12 – 1.21 21g 12 + 0.194 g 12 + 0..075 075
17.28
17.14 FE mesh for bias extension simulation.
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SDV13 SNEG, (fraction = –1.0) (Ave. Crit.: 75%) +6.995e–01 +6.299e–01 +5.603e–01 +4.908e–01 +4.212e–01 +3.516e–01 +2.820e–01 +2.124e–01 +1.429e–01 +7.328e–01 +3.698e–01
Zone III
Zone I
Zone II
17.15 Deformed mesh with contour of Green–Lagrange shear strain in bias extension simulation.
SDV14 SNEG, (fraction = –1.0) (Ave. Crit.: 75%) +1.500e–02 +3.500e–03 –8.000e–03 –1.950e–02 –3.100e–02 –4.250e–02 –5.400e–02 –6.550e–02 –7.700e–02 –8.850e–02 –1.000e–01
(a) ex
(b) ey
17.16 Contour of Green–Lagrange direct strains in the fabric (x: the weft fiber direction; y: the warp fiber direction).
As can be seen from Fig. 17.18, overall, the equivalent shear modulus is highly nonlinear and very small in comparison to the tensile modulus. There is an initial plateau stage followed by a steep increase stage, which conforms with the shear–locking phenomena in woven composite fabrics
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25
Load F (N)
20
15
10
5
0
0
10
20 Displacement (mm)
30
40
17.17 Match the 16-yarn results to obtain equivalent shear modulus.
Contravariant shear modulus (MPa)
3.5 3 2.5 2 1.5 1 0.5 0 0
0.2
0.4 0.6 Engineering shear strain g12
0.8
1
17.18 Contravariant shear modulus D 33 as a function of the engineering shear strain g12 in the convected coordinate system.
under bias extension mode. The simulated load-displacement curve for the 16-yarn samples is shown in Fig. 17.17 by a solid line with filled squares. A good match is expected because we obtain the shear modulus by fitting the experimental data.
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now we have determined all the equivalent elastic constants through eqs 17.26–17.28. With the combination of the transformation relations 17.24 as well as the constitutive equation 17.17, we can design a user material subroutine in aBaQUS/Standard for shell elements to represent the macroscopic material behavior of the plain weave composite.
17.4.3 Fiber orientation model Fiber yarns experience pronounced reorientation during the forming process. The fiber orientation has a significant leverage on the effective material properties of woven composite fabrics. The implementation of the proposed non-orthogonal constitutive model makes it essential to keep track of the fiber orientation during deformation, as stated in Section 17.3. Here we develop a simple way for tracing the fiber orientation based on continuum mechanics concepts. For the purpose of generality, we consider an arbitrary four-node shell element used in the modeling of a woven composite fabric sheet, as shown in Fig. 17.19. To reflect the initial fiber orientation, we embed a Cartesian material coordinate system {ei} in the shell element with e1 and e2 coinciding with the original weft and warp directions of the fabrics. The convected coordinate system {gi} conforms with the material coordinate system {ei} in the initial configuration. After a deformation, the material coordinate system will experience a rigid-body rotation according to the rotation matrix R and the updated material coordinate system can be expressed as ÏÔ e1¢ = R◊e1 Ì ÔÓ e¢2 = R◊e 2
17.29
The two vectors g1 and g2, which construct the convected coordinate system, can be regarded as two infinitesimal direction vectors in the shell element. e¢2
g¢2
e2 g2
g¢1
e¢1 e1 g1
F = RU
17.19 Schematic for fiber reorientation model.
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According to the definition of the deformation gradient tensor F, we directly have the following relations for the updated {gi} as, ÏÔ g1¢ = F◊g1 17.30 Ì g ¢2 = F◊g 2 Ô Ó which represents the new orientation of the fiber yarns after deformation. Figure 17.19 shows the schematic of this fiber orientation model. This model is simple and can be easily integrated into our user material subroutine by setting {ei}and {gi} as state variables to trace the fiber reorientation. The vectors {ei} and {gi} are updated in the user material subroutine according to the rigid body rotation matrix R and the deformation gradient tensor F, respectively, in each time increment.
17.5
Validation of the non-orthogonal model
The developed non-orthogonal constitutive model is validated in this section by comparing our numerical simulation results to their corresponding experimental data. The experimental tests used in the validation include the 24-yarn bias extension test and trellising tests.
17.5.1 Comparison with the 24-yarn bias extension test data The first validation is done with the bias extension test of the 24-yarn samples. The experimental results were shown in Fig. 17.13. The averaged experimental results with error bars are shown in Fig. 17.20 by a solid line. The numerical load–displacement curve obtained from the FEM simulation is shown in Fig. 17.20 by a solid line with filled squares. As can be seen from Fig.17.20, very good agreement between the numerical bias extension load and the experimental results of a 24-yarn fabric is obtained.
17.5.2 Comparing with trellising test data Chen et al.[27] presented a detailed experimental setup for the trellising test of the plain weave composite fabric, as shown in Fig. 17.21. The shear frame has a size of 216 mm ¥ 216 mm. The effective size of the trellising test samples is outlined by the black lines shown in Fig. 17.21. The region outside the black lines is defined as the arm parts. In each arm part of the composite fabric, the fiber yarns parallel to the shear frame were taken out. Hence there was no crossover in the four arm parts, thus ensuring a pure shearing of the test sample.
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20 24 yarns – FEM 24 yarns – test
Load F (N)
15
10
5
0
0
10
20 30 Displacement (mm)
40
50
17.20 Load–displacement curves for 24-yarn bias extension.
17.21 Trellising shear frame with a testing sample which has fiber yarns in the arm parts being pulled out.[27]
Four samples with an effective size of 30 in2 were prepared for the trellising tests.[27] The tests were run at room temperature and at a crosshead rate of 50 mm/min. Figure 17.22 shows the experimental trellising loads versus shear
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25
Load (N)
20
in2 in2 in2 in2
–#1 –#2 –#3 –#4
15 10 5 0 0
10
20 30 40 Shear angle (deg)
50
60
17.22 Trellising results of 30 in2 samples.[27]
17.23 FEM model for trellising composite fabrics.
angle of the four samples. As can be seen from Fig. 17.22, the trellising test data are also quite scattered and a shifting adjustment similar to that in the bias extension test was tried to improve the repeatability. Little improvement can be obtained. A multi-element FEM model, as shown in Fig. 17.23, was used to simulate the shear frame (trellising) tests of the plain weave composite fabrics. The fabric patches were modeled by four-node shell elements (S4R). The shear
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frame was modeled by truss elements instead of beam elements to eliminate the effect of the frame on the trellising test results, as truss elements have no bending stiffness. Each edge of the frame can only rotate along the joints that connect to another frame edge. The shell elements were connected with the truss elements without slippage and separation by sharing the same nodes at the interfaces. One corner of the shear frame was clamped and the diagonally opposite corner was pulled to simulate the experimental shear frame tests. The user material subroutine incorporating the fiber orientation model is applied to the four-node shell elements. FEM simulations were implemented by prescribing displacements along the diagonal line at the pulling corner. The loading force is obtained by recording the reaction force at the clamped corner. Since the arm parts of the composite fabrics did not have any yarn crossovers in the experiments, it is reasonable to assume that the experimental trellising loads are purely due to the shearing of fiber yarns in the central region. In the numerical simulations, however, the shell elements in the arm parts have the same equivalent material properties as those in the central region. As a result, the deformation of the arm parts will definitely contribute to the total trellising load. To compare with the experimental data, we have to isolate out the force contributed from the deformation of the central region. This is implemented by the following approach. The strain energy of each shell element in the central region and the arm part region is recorded and added up to give the total strain energy in each region. We can then obtain the strain energy ratio between the two regions. We can assume that the work done by the pulling load in the trellising deformation is totally converted to the strain energy of the fabric; the total pulling load in the numerical simulation should then be divided into two parts, Pcenter and Parm, proportionally according to the strain energy ratio between the central square region and the arm parts of the composite fabric. We can then consider Pcenter as the numerical trellising load and will compare it with the experimental trellising loads. Figure 17.24 shows the numerical trellising load versus shear angle for the composite fabrics with central sizes of 30 in2, by filled squares. As a comparison, the averaged experimental trellising load with error bars is also shown in Fig. 17.24, by a solid line. As can be seen from Fig. 17.24, generally, the numerical results predicted from the non-orthogonal constitutive model are in good agreement with experimental data. The validation of the proposed non-orthogonal constitutive model is thus completed. By including coupling between tension and shearing in the local convected coordinate system, Lee et al.[28] extended this non-orthogonal model and applied it in the numerical simulation of composite fabric thermostamping. In the following sections, we will present another continuous model based on an energy approach.
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Load F (N)
20
15
10
5
0
0
10
20 30 Shear angle (deg)
40
50
17.24 Numerical results compared with experimental data for the 30 in2 test.
17.6
General fiber-reinforced hyperelastic model
In the continuum mechanics framework, the mechanical behavior of a composite with hyperelastic ground material reinforced with a family of unidirectional aligned fibers can be represented by a strain energy function W , which can be expressed as a scalar function of the right Cauchy–Green deformation tensor C = FTF and the original fiber directional unit vector a0 (Spencer[29]), i.e.,
W = W(C, a0)
17.31
where F = ∂x/∂X is the deformation gradient tensor. X represents the position of a material particle in the original (undeformed) configuration, while x is the position of the corresponding particle in the current (deformed) configuration. The strain energy function may be written in terms of invariants Ii as
W(C, a0) = W(I1, I2, I3, I4, I5)
17.32
where the invariants are given by I3 = detC, I1 = trC, I 2 = 1 [(trC)2 – trC 2 ], 2
I 4 = a 0 ◊C◊a 0 = la2, I5 = a0·C2·a0
17.33
where la is the stretch of fiber a. If the hyperelastic body is reinforced by an extra family of fibers with original fiber directional unit vector b0, the strain-energy function W is an
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isotropic invariant of C, a0 ƒ a0, b0 ƒ b0 and can be expressed as a function of the following invariants:
I1, I2, I3, I 4 = la2 , I5
I 6 = b0 ◊C◊b0 = lb2 , I5 = a0 · C2 · a0
17.34
where lb is the stretch of fiber b, and cos2f = a·b/(||a||·||b||)is the cosine of the angle between the two families of fibers in the deformed configuration. The second Piola–Kirchhoff stress tensor is obtained directly from the hyperelastic strain energy function as S = 2∂W/∂C
17.35
The Cauchy stress tensor, s, is given by s = I3–1FSFT, or 8 ∂I ∂I s ij = I 3–1Fik Fjl S Wm ÊÁ m + m ˆ˜ Ë ∂Ckl ∂Clk ¯ m=1
where Wm denotes ∂W/∂Im. From Eq. 17.34 we have
∂I1 = d kl , ∂Ckl
∂I 2 = I1d kl – Ckl , ∂Ckl
∂I 4 = ak0 al0, ∂Ckl
∂I 5 = 2 ak0 a 0p C pl , ∂Ckl
∂I 6 = bk0 bl0, ∂Ckl
∂I 7 = 2bk0 b p0 C pl , ∂Ckl
17.36
∂I 3 = I 2d kl – I1Ckl + Ckp C pl , ∂Ckl
∂I 8 –1 = –(I 4-11ak0 al0 + I 6-1bk0 bl0 )I 8 + 2(I 4 I 6 ) 2 cos2fak0 bl0 ∂Ckl 17.37
Substituting Eq. 17.37 into 17.36 and applying the Cayley-Hamilton theorem, we obtain the following expression for the Cauchy stress tensor s: s = 2 I 3-1F ∂W F T ∂C
È( I 2W2 + I 3W3 )I + W1B – I 3W2 B–1 + (I 4W4 – I 4-1 I 8W8 )a ƒ a ˘ ˙ Í ˙ = 2I 3–1 Í+ I 4W5 (a ƒ Ba + aB ƒ a ) + (I 6W6 – I 6-1 I 8W8 )b ƒ b ˙ Í Í+ I W (b ƒ Bb + bB ƒ b) + (I I )– 21 W cos2f (a ƒ b + b ƒ a )˙ 4 6 8 ˚ Î 6 7 17.38
where I is the second order unit tensor and B is the left Cauchy–Green tensor B = FFT.
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17.7
Specific fiber-reinforced hyperelastic model for woven composite fabrics
Due to friction between fiber yarns, there is some energy dissipation during forming. Nevertheless, the energy required to deform woven composite fabrics can be approximately treated as the strain energy of the corresponding deformation status in our hyperelastic approach during the forming process, provided that the unloading process is not considered. Similar treatment is implied in the development of hypoelastic models for woven composite fabrics.[18, 19] Therefore, the fiber-reinforced constitutive framework in Section 17.6 is applicable to woven composite fabrics. Compared with sheet metal forming, woven composite fabrics usually undergo small membrane extensions along the yarn directions while experiencing large angular variations between weft and warp yarns during forming. Woven fabrics can be regarded as campsites reinforced with two families of fibers only and the ground substance is ignored in the development of the hyperelastic constitutive model, as illustrated in Fig. 17.25. The strain energy can be decomposed into two parts: one is from fiber stretch; the other is from fiber–fiber interaction due to angular rotation between weft and warp yarns. Consequently, the strain energy function can be written as W = W (C, a 0, b 0 ) = WaF (I 4 ) + WbF (I 6 ) + W FF (I 8 ) 17.39 where WF is the contribution from fiber stretch and WFF is the shearing energy resulting from fiber–fiber interaction. Then from Eq. 17.38, the Cauchy stress tensor s is given by s=
È(I 4W4 – I 4-1 I 8W8 )a ƒ a + (I 6W6 – I 6-1 I 8W8 )b ƒ b˘ ˙ ˙ Í+ (I I )- 21 W cos2f (a ƒ b + b ƒ a ) 4 6 8 ˚ Î
2 I 3-1 Í
b0
17.40
b
a
2j a a0
O Before deformation
O After deformation
17.25 Geometric structure and deformation modes of plain weave composite.
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With the decomposition of the strain energy function in Eq. 17.39, the material characterization of woven composite fabrics can be greatly facilitated: 1. By matching experimental data from uniaxial tensile tests along one fiber yarn direction, the specific format for WF can be obtained. 2. By matching experimental data from picture frame tests (pure shearing), the specific format for WFF can be obtained. The plain woven thermoplastic composite fabric used in the development of the non-orthogonal model (Section 17.4) is taken as an example to demonstrate the proposed fiber-reinforced hyperelastic model for composite fabrics. The characteristic load–displacement curves of the composite fabric were obtained from experimental uniaxial tensile (Fig. 17.6) and shearing (picture frame) tests (Fig. 17.21). The experimental uniaxial tensile load–displacement curve is transformed to a stress–strain curve and is shown in Fig. 17.26. The curve is smoothed by curve-fitting, and then by integrating the stress over strain and noting that e1 = la – 1 = I 4 – 1 , we obtain the tensile strain energy density versus (I4 – 1), as shown in Fig. 17.27. The tensile strain energy density can be approximated by a function of (I4 – 1) as
WFb(I4) = k1(I4 – 1)3 + k2(I4 – 1)2, for I4 ≥ 1.0
17.41
where k1 = 1340.81, k2 = 20.68 with a unit of MPa. For balanced plain weave composites, WFb(I6) has the same functional format as WFb(I4). It is assumed that composite fabrics have no compressive stiffness. Hence WF is assigned a small constant when fiber is under contraction (l < 1). 40 Experiment
35
Curve-fitting
Yarn stress (MPa)
30 25 20 15 10 5 0
0
0.005
0.01
0.015 0.02 Direct strain
0.025
0.03
0.035
17.26 Uniaxial tensile stress–strain curve on plain weave composite.
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0.6
Curve-fitting
0.5 0.4 0.3 0.2 0.1 0
0
0.01
0.02
0.03 0.04 I4 – 1
0.05
0.06
0.07
17.27 Strain energy density vs (I4 – 1) in uniaxial tensile test. F
u L
q /2
q /2 2j
17.28 Schematic for trellising test.
Figure 17.28 shows the schematic of the picture frame (trellising) test setup for woven composite fabrics. A simple geometric analysis on Fig. 17.28 gives the relation between the angle j and displacement u:
cosj =
2L + u 2L
17.42
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where L is the frame length. The testing frame size is 300 mm ¥ 300 mm and the effective size of the tested composite fabric is 210 mm ¥ 210 mm. The smoothed average load–displacement curve is shown in Fig. 17.29. Integrating the load curve over the displacement gives the work needed to deform the composite fabric, which is assumed to be transformed to strain energy. Assuming incompressibility for the woven fabric, the strain energy density is obtained by dividing the strain energy over the fabric volume (210 mm ¥ 210 mm ¥ 210 mm). The obtained strain energy density versus the invariant I8 is shown in Fig. 17.30. By curve-fitting with a polynomial function, we obtain the shearing strain energy function as
WFF(I8) = k3(I8)2 + k4I8
17.43
where k3 = 0.426, k4 = 0.246 with a unit of MPa. Adding Eqs 17.41 and 17.43 gives the total strain energy for a composite fabric under general deformation. According to the definition of the deformation gradient tensor F, we directly have the following relations for the updated fiber orientation after deformation: ÏÔ a = F ◊ a 0 17.44 Ì b = F ◊b0 Ô Ó Then the Cauchy stress tensor s can be computed according to Eq. 17.40. This completes the development of the hyperelastic fiber-reinforced constitutive model for plain woven composite fabric. 300
250
Force (N)
200
150
100
50
0 0
0.05 0.1 Displacement (m)
0.15
17.29 Trellising load curve for plain weave fabric.
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Strain energy density (Mpa)
0.25 Experiment Curve fitting
0.2
0.15
0.1
0.05
0
0
0.1
0.2
I8
0.3
0.4
0.5
17.30 Shear strain energy density vs I8 in trellising test.
17.8
Conclusions
In this chapter two continuous models, based on a geometric transformation approach and an energy approach, respectively, for characterizing the nonlinear anisotropic mechanical behavior of woven fabric are proposed. In the geometric transformation approach, the stress and strain transformation relations between global orthogonal coordinates and the local non-orthogonal coordinate system bring out the non-orthogonal constitutive relation for woven composite fabrics. The combination of the non-orthogonal constitutive model and the fiber orientation model makes it feasible to capture the anisotropic material behavior of woven composite fabrics under large shear deformation. The major assumptions in developing the non-orthogonal constitutive model are the decoupling between the shearing and tensile response, and the decoupling of the tensile responses between the weft and warp directions. The model is validated by comparing numerical results with experimental data of bias extension tests and shear frame tests of a plain weave composite fabric. In the energy approach, a simple hyperelastic fiber-reinforced constitutive model is developed to represent the anisotropic nonlinear mechanical behavior of woven composite fabrics during forming. The strain energy function is decomposed into two parts as tensile and shearing energies to facilitate the material characterization process. The proposed constitutive modeling approach is demonstrated on balanced plain weave composites. The specific format for the strain energy function is determined from load–displacement curves of uniaxial tests for the tensile part and of picture frame (trellising) tests for the shearing part. The development of the continuous models builds up a
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solid foundation for the numerical simulation and processing optimization for composite fabric forming.
17.9
Acknowledgment
The support from the National Science Foundation of China (Grant no. 50975236) is gratefully acknowledged.
17.10 References 1. King M J, Jearanaisilawong P, ‘A continuum constitutive model for the mechanical behavior of woven fabrics’, International Journal of Solids and Structures, 2005, 42, 3867–3896. 2. Cherouat A, Billoët J L, ‘Mechanical and numerical modelling of composite manufacturing processes deep-drawing and laying-up of thin pre-impregnated woven fabrics’, Journal of Materials Processing Technology, 2001, 118, 460–471. 3. Sharma S B, Sutcliffe M P F, ‘A simplified finite element model for draping of woven material’, Composites Part A: Applied Science and Manufacturing, 2004, 35, 637–643. 4. Duhovic M, Bhattacharyya D, ‘Simulating the deformation mechanisms of knitted fabric composites’, Composites Part A: Applied Science and Manufacturing, 2006, 37, 1897–1915. 5. Weissenbach G, Limmer L, Brown D, ‘Representation of local stiffness variation in textile composites’, Polymers & Polymer Composites, 1997, 5, 95–101. 6. Gowayed Y, Yi L, ‘Mechanical behavior of textile composite materials using a hybrid finite element approach’, Polymer Composites, 1997, 18, 313–319. 7. Wang C, Sun C T, Gates T, ‘Elastic/viscoplastic behavior of fiber-reinforced thermoplastic composites’, Journal of Reinforced Plastics and Composites, 1996, 15, 360–377. 8. Wang C, Sun C T, ‘Experimental characterization of constitutive models for PEEK thermoplastic composite at heating stage during forming’, Journal of Composite Materials, 1997, 31, 1480–1506. 9. Chen J, Lussier D S, Sherwood J A, Cao J, Peng X Q, ‘The relationship between materials characterization and material models for stamping of woven fabric/ thermoplastic composites’, in: Proceedings of the 4th International ESAFORM Conference on Material Forming, Liège, Belgium, 127–130, 2001. 10. Peng X Q, Cao J, ‘A dual homogenization and finite element approach for material characterization of textile composites’, Composites Part B: Engineering, 2002, 33, 45–56. 11. Luo S Y, Chou T W, ‘Finite deformation and nonlinear elastic behavior of flexible composites’, Journal of Applied Mechanics, 1988, 55, 149–155. 12. Vu-Khanh T, Liu B, ‘Prediction of fiber rearrangements and thermal expansion behavior of deformed woven-fabric laminates’, Composites Science and Technology, 1995, 53, 183–191. 13. Mark C, Taylor H M, ‘The fitting of woven cloth to surfaces’, Journal of the Textile Institute, 1965, 47, T477–T488. 14. Yu W R, Pourboghrat F, Chung K, Zampaloni M, Kang T J, ‘Non-orthogonal
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constitutive equation for woven fabric reinforced thermoplastic composites’, Composites Part A: Applied Science and Manufacturing, 2000, 33, 1095–1105. 15. Xue P, Peng X Q, Cao J, ‘A non-orthogonal constitutive model for characterizing woven composite’, Composites Part A: Applied Science and Manufacturing, 2003, 34, 183–193. 16. Hsiao S W, Kikuchi N, ‘Numerical analysis and optimal design of composite thermoforming process’, Computational Methods in Applied Mechanics and Engineering, 1999, 177, 314–318. 17. Cherouat A, Billoët J L, ‘Finite element model for the simulation of preimpregnated woven fabric by deep-drawing and lay-up process’, Journal of Advanced Materials, 2000, 32, 42–53. 18. Boisse P, Gasser A, Hagège B, Billoët J L, ‘Analysis of the mechanical behaviour of woven fibrous material using virtual tests at the unit cell level’, Journal of Materials Science, 2005, 40, 5955–5962. 19. Badel P, Vidal-Sallé E, ‘Large deformation analysis of fibrous materials using rate constitutive equations’, Computers & Structures, 2008, 86, 1164–1175. 20. Khan M A, Mabrouki T, Vidal-Sallé E, Boisse P, ‘Numerical and experimental analyses of woven composite reinforcement forming using a hypoelastic behaviour. Application to the double dome benchmark’, Journal of Materials Processing Technology, 2010, 210, 378–388. 21. Cao J, Akkerman R, Boisse P, et al., ‘Characterization of mechanical behavior of woven fabrics: experimental methods and benchmark results’, Composites Part A: Applied Science and Manufacturing, 2008, 39, 1037–1053. 22. Aimene Y, Hagège B, Sidoroff F, Vidal-Sallé E, Boisse P, Dridi S, ‘Hyperelastic approach for composite reinforcement forming simulations’, International Journal of Material Forming, 2008, 1, 811–814. 23. Peng X Q, Cao J, ‘A continuum mechanics based non-orthogonal constitutive model for woven composite fabrics’, Composites Part A: Applied Science and Manufacturing,, 2005, 36, 859–874. 24. Boisse P, Gasser A, Hivet G, ‘Analyses of fabric tensile behavior: determination of the biaxial tension-strain surfaces and their use in forming simulations’, Composites Part A: Applied Science and Manufacturing, 2001, 32, 1395–1414. 25. Buet-Gautier K, Boisse P, ‘Experimental analysis and modeling of biaxial mechanical behavior of woven composite reinforcements’, Experimental Mechanics, 2001, 41, 1–10. 26. Ogden R W, Non-linear elastic deformations. Ellis Horwood, Chichester, UK, 1984. 27. Chen J, Lussier D S, Liu L, Experimental trellising test of plain woven composite fabrics. Internal report, University of Massachusetts in Lowell, 2003. 28. Lee W, Um M K, Byun J H, Boisse P, Cao J, ‘Numerical study on thermo-stamping of woven fabric composites based on double-dome stretch forming’, International Journal of Material Forming, 2010, 3, S1217–S1227. 29. Spencer A J M, Continuum Theory of the Mechanics of Fiber-Reinforced Composites, Springer, New York, 1984.
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18
X-ray tomography analysis of the mechanical behaviour of reinforcements in composites
P. B a d e l, Ecole des Mines de Saint Etienne, France and E. M a i r e, INSA-Lyon, France
Abstract: X-ray tomography is a non-destructive technique which allows reconstruction of the internal geometry of opaque and massive materials in three dimensions (3D). This chapter first discusses the capabilities of this technique in the field of the mechanics of composite reinforcements. It then describes methods of analysing reconstructed data at different characteristic length scales. The chapter also includes an example of the use of such analyses to develop a numerical model of the mechanical behaviour of a woven composite reinforcement. Key words: x-ray tomography, textile composite reinforcements, multi-scale analysis, mechanical properties, finite element analysis.
18.1
Introduction
X-ray tomography is a non-destructive technique which allows reconstruction of the internal geometry of opaque and massive materials in 3D. Currently, this technique is widely used in materials science (see Baruchel et al., 2000 and Stock, 2008 for review articles) and its potential contribution to the study of composite reinforcements is very promising. Composite reinforcements are multi-scale materials. In these materials, one can distinguish the scale of the fibres (referred to as the microscopic scale in this chapter), the scale of the yarns, made of thousands of fibres (referred to as the mesoscopic scale), and the scale of the final composite part itself (referred to as the macroscopic scale). The current performances offered by laboratory tomography equipments, including in situ setups (see Buffiere et al., 2010), make it a suitable and easy to use technique for the analysis of such reinforcements. Indeed, the resolutions which can be reached facilitate the study of any of the three above-mentioned length scales. The advantage of tomography is that it gives access to local 3D observations inside the sample that are not possible with standard microscopy techniques, because these are restricted to surface observations. In addition, the more conventional 2D techniques face several technical difficulties when dealing with composite reinforcements. 565 © Woodhead Publishing Limited, 2011
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For instance, cross-sectioning samples can be quite difficult, and almost impossible in the case of dry reinforcements (non-infiltrated preforms). In the first section of this chapter, the principles of the technique will be presented, as well as its use in the field of composites and dry reinforcements. Recent developments related to the mechanics of woven reinforcements and their deformation mechanisms will also be addressed. Tomographic data usually require post-processing and the second section of this chapter will describe two specific methods of analysis, which can be used to analyse the mesoscopic and microscopic properties of the reinforcements. Finally, an example of an application encompassing all the aspects of this type of problem will be dicussed in the third section, where a finite element simulation model of the mechanical behaviour of textile reinforcements is developed and validated based on in situ tomographic observation of the deformation of a preform.
18.2
X-ray tomography of composite reinforcements
The underlying principle of X-ray tomography is based on X-ray attenuation by matter and on the recording of transmitted X-ray beams through an object. This basic principle is used in the well-known X-ray radiography method. Tomography requires not only one radiograph to be collected but a series of different radiographs of the sample in a given state, obtained by rotating it through 360°. This set of data is then treated by mathematical algorithms in order to reconstruct a three-dimensional map of the local attenuation coefficient of the object. This first section briefly describes the principles of this technique and introduces the ways in which it can be used for the purpose of analysing composite reinforcement geometry under mechanical loadings.
18.2.1 Principle of X-ray tomography X-rays have many physical properties and are used in multiple applications. The property of interest for tomography is their ability to penetrate through dense materials. X-rays interact with matter in two main ways: diffusion and attenuation (change of phase can also be an issue but not in the present study). In classical X-ray tomography, only attenuation is considered. The phenomenon of X-ray attenuation induces an intensity decrease of the beam passing through the matter, due to photons being absorbed (via electron excitation or ejection). The intensity decrease dI of a parallel beam through an infinitesimal thickness dx is proportional to the beam intensity I, the attenuation coefficient m of the matter, and dx:
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dI = Imdx
567
18.1
In the general case of a heterogeneous attenuation coefficient, the integration of this law yields the following relationship of Beer–Lambert between the incident and transmitted beam intensities, or equivalently between the number of incident photons N0 and the number of transmitted photons N along a path x: I = N = exppÊ ÁË I0 N0
Ú
x
ˆ m ((xx )dx˜ ¯
18.2
Thus N will depend on the thickness only if the material is homogeneous. if not, it will depend on the local attenuation at any point of the specimen scanned. These variations create the contrasts typically observed in X-ray attenuation radiographs. This principle is also used in tomography, which consists of performing a series of radiographs at multiple viewing angles of the specimen to obtain a full set of radiographs corresponding to a 360° rotation of the object (see the schematic setup in Fig. 18.1). From this data set, a map of the local attenuation coefficient can be reconstructed at any point. Schematically, an equation like Eq. 18.2 above can be written for each pixel of each radiograph. This gives a linear function of the local attenuation coefficient in each voxel of the 3D volume to be reconstructed (note the vocabulary: a voxel is a volume element, an extension of the term pixel which designates a picture element): ÊN ˆ lnÁ 0 ˜ µ S mik k Ë N ij ¯
18.3
where N0 is the number of incident photons, Nij is the number of transmitted photons recorded by the detector j in radiograph i, and mik is the total value of local attenuation coefficient of each voxel k summed along the beam
X-ray source
Incident beam
Transmitted beam
Rotation axis
Detector
18.1 Schematic of the X-ray tomography equipment.
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which reaches the detector j. The resolution of the system of linear equations obtained yields the unknown mik. In practice, dedicated algorithms have been developed to perform this reconstruction step. The usual reconstruction method involves a projection of the radiographs into the Fourier space followed by an inverse transform, hence the name ‘back-projection method’ (Feldkamp et al., 1984). More details about the principle of X-ray tomography can be found in Baruchel et al. (2000) and Stock (2008). The equipment required for tomography consists mainly of two elements: an X-ray source and an X-ray detector. Usually, small laboratory X-ray sources use vacuum tubes in which a cathode filament is heated up in order to emit electrons towards an anode target. Due to the high electrical field between the anode and the cathode, these electrons are accelerated and collide with the target, causing X-rays to be emitted, as well as high energy loss in the form of heat. Another method of X-ray production is that of synchrotron facilities. It is widely used in material sciences because the intensity and quality of the radiation allows for more diversified applications. Various types of detectors exist, using different technologies depending on the application. Detectors play a crucial role in the quality of the final reconstruction. Their size partly conditions the best resolution reachable and their response time directly influences the scanning time. Note that, in this chapter, resolution refers to the voxel edge size.
18.2.2 Tomography of woven textile composite reinforcements X-ray tomography has multiple applications in material science (Baruchel et al., 2000) and has already been used in the field of composite materials. Indeed, with other common observation techniques, major difficulties may arise when dealing with composite materials and especially with dry preforms. Although adding resin can keep the preforms in position, it may also distort their original geometry (Chang et al., 2004; Potluri et al., 2006). Tomography offers the major advantage of not having to touch or modify the specimen to internally observe it. Tomography has been used mainly for two types of observation: firstly, to observe composite parts and study phenomena such as rupture and decohesion between matrix and reinforcement (Raz-Ben Aroush et al., 2006), and secondly, to observe dry reinforcements in their initial state. For example, the geometric variability of woven reinforcements has been addressed by Desplentere et al. (2005). Another direct application of tomographic reconstructions consists of generating geometric models for possible use in numerical simulations (Pandita and Verpoest, 2003; Schell et al., 2006). In this section, another possible use of tomography for the observation of deformed dry reinforcement will be
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presented. This type of application is useful in analyses of the mechanical behaviour of dry reinforcements. It can provide support to constitutive assumptions, or it can be used to validate simulation results. Note also that in common laboratory tomographs the use of a conic beam allows a wide range of resolutions, though a trade-off between resolution and the size of the specimen has to be made. Due to this flexibility, woven composite fabrics can be analysed at the two important scales in terms of microstructure, namely the mesoscopic and the microscopic scales. Unloaded reinforcement tomography scans Basic imaging of composite reinforcements consists of positioning a sample on the rotating object holder. Two examples of woven reinforcements are described in this chapter. The first one is a complex interlock reinforcement hereafter referred to as G1151. The purpose of such an interlock structure is to tie several layers together. This fabric features a large unit cell size (the unit cell being the smallest volume that allows reconstructing the whole fabric by translation repetitions only), therefore a resolution of 20 mm was used to make the acquisition depicted in Fig. 18.2. In addition, a glass plain weave fabric was scanned in its unloaded state at a resolution of 3 mm, i.e. at the
(a1)
1 cm
(a3) (a2)
(b)
1 mm
18.2 Imaging of two different woven reinforcements in their unloaded state; G1151 was imaged at a voxel size of 20 mm, the glass plain weave at 3 mm: (a1) G1151: three successive slices in warp planes; (a2) G1151: three successive slices in weft planes; (a3) G1151: 3D view of the unit cell; (b) glass plain weave.
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microscopic scale. Figure 18.2 shows several reconstructed slices of these reinforcements. These slices could be obtained because of the 3D nature of the data which allows performing virtual cuts. The 3D view of G1151 was obtained after removing the voxels belonging to the air phase. This 3D view is close to what can be observed visually. In situ mechanical loading of reinforcements Loaded state geometries are of great interest for many reasons. They can help in understanding the underlying mechanisms of deformation of the reinforcements, and also provide an ideal tool to validate the output of models in terms of geometry. Nevertheless, under mechanical loading the acquisition is much more challenging because a dry reinforcement never keeps a frozen state, and fixing it in position with resin does not ensure that it is absolutely representative of the resin-free geometry. For these reasons, the strategy used in such material science problem is to perform mechanical testing in situ, i.e. directly inside the tomograph. To this aim, a specifically dedicated setup has to be developed. This equipment, recently developed by Badel (2008) for woven composite reinforcements, is presented in the following. A schematic view of this device is provided in Fig. 18.3. The guideline for this experimental device was to enable biaxial tension and/or shear deformation of textile reinforcements in situ inside the tomograph. These requirements lead to constraints, namely: • • •
Using non-absorbing materials or materials with low absorbance in the field of view Adapting the dimensions to the size of both the tomograph itself and that allowed for a given resolution Providing enough strength for a biaxial tension loading of about 20 N per yarn.
For biaxial loading, a cross-shaped specimen is used. The structure of the loading device is also in the shape of a cross positioned vertically on the rotation stage, with arms at ±45° to the rotation axis of the tomograph. The observation zone is thus small enough to provide a reasonable resolution. This device is made of hardwood with metal/polymer grips providing a firm hold on the yarns, which can be loaded (with a given load applied to each yarn) prior to gripping. In addition, the arms of the cross-shaped device are hinged, which enables shear deformation to be applied in the central part of the specimen (i.e. the observation zone). It is therefore possible to combine shear and biaxial tension loadings. High-resolution images (i.e. at the microscopic scale) of the deformed reinforcements were also obtained. In this particular case, it was necessary to freeze the specimen with cyanoacrylate glue applied only outside the
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Grips Cross-shaped specimen
X-ray source
X-ray beam
Shear angle positioning Holding axle (a)
P
P (b)
(c)
18.3 In situ mechanical loading device: (a) global schematic view; (b) shear loading position of the device; (c) biaxial tension loading.
region of interest, in order not to affect the yarns to be observed with glue, which could lead to possible effects on the geometry. Examples of loaded reinforcement imaging at the mesoscopic and microscopic scales, as well as their use for mechanical analyses, are given in the following section.
18.3
Analyses of the structure of a textile reinforcement
The versatility of tomography in terms of resolution (even for small laboratory systems) makes it a powerful tool for analysing the structure of composite
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reinforcements such as textile reinforcements. Resolutions of about 10–20 mm are sufficient to perform mesoscopic analyses and permit the scanning of large-sized specimens in most tomographs. Because of this, it is possible to observe several unit cells (over a lateral distance of several yarns) in the same specimen, for instance to analyse the shape and the spatial trajectory of the yarns. At resolutions below 3–4 mm, one can also consider microscopic analyses, the fibres being individually distinguishable. This kind of observation is helpful in providing additional information on the microstructure of the yarns and the spatial fibre distribution (fibre-rich vs fibre-poor regions).
18.3.1 Mesoscopic-scale analysis The observations made at the mesoscopic scale during deformation are used mainly to perform shape analyses of the yarn cross-sections. This allows us not only to evaluate the initial geometry of undeformed reinforcements, but also to characterise their deformed geometries. Qualitative analyses of such images provide interesting results. The images of the unloaded G1151 reinforcement emphasise the complexity of this woven structure characterised by non-trivial initial geometries. It can be observed, for instance, that superimposed weft yarns generally do not lie in the same plane even though the fabric is unloaded (see Fig. 18.2(a1)). Furthermore, a single yarn can feature several different cross-sectional shapes depending on its location inside the fabric (especially for warp yarns, see Fig. 18.2(a2)). These cross-sections can be elliptic, almost flat, or quasi-rectangular in some cases. This type of observation can be used for the purpose of developing geometric models of composite reinforcements, as exemplified in Hivet and Boisse (2008). Regarding the deformed state, only qualitative analyses have been performed at the present time for such a complex reinforcement. When studying the geometry of simpler reinforcements such as the glass plain weave introduced earlier, quantitative analyses are more feasible and enable us to go beyond qualitative comments. It is interesting to quantify characteristic geometric measurements between undeformed and deformed states. The parameters that are to be measured and that provide an insight into the geometric evolution of cross-sectional shapes can be, for instance, the height (or thickness), the width, or the cross-section’s surface of the yarns. This type of information is related to the ability of the yarns to deform (i.e. to change their shape) and to compact (i.e. to change their cross-sectional area). A systematic analysis of cross-sectional shapes acquired by X-ray is not straightforward. The geometry of textile reinforcements in their unloaded state is highly variable due to the unconstrained nature of this state. In practice, significant geometric differences may exist between two cross-sections
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observed in two different unit cells of the same fabric. Under mechanical loading this variability is much reduced because the mechanical constraint induces a geometrical constraint. However, the initial geometry often leaves some marks of the variability, which are more or less pronounced depending on the type of mechanical loading. This is especially true in cases where biaxial loading is applied, whereas shear loading provides a much more constrained geometry due to the contact configurations, hence a much less variable geometry. To overcome this problem, it is possible to compute average cross-sectional shapes over the whole specimen observed. In order to be as representative as possible, a large number of cross-sections are required. To maximise the number of cross-sections taken into account, it is necessary to identify all the cross-sections that are supposed to be the same from a geometric point of view, i.e. the sections in the reconstructed sample which are supposed to be in the same location in the elementary representative cell (ERC). With this aim, the notion of geometric configuration is introduced. The cross-sections offering the same geometric configuration will be averaged together. Geometric configurations are defined using the symmetries and periodicity of the fabrics. For instance, the example in Fig. 18.4(a) shows five different but ‘equivalent’ cross-sections (labelled S1 to S5) for the geometric configuration, corresponding to the crossover between the warp and weft directions. These five sections can therefore be used to compute the average cross-sectional shape for this geometric configuration, because they are situated in equivalent positions of the ERC. In the following, only this position will be considered for non-sheared fabric (Fig. 18.4). When the fabric is sheared, it is relevant to consider additional geometric positions as the yarn undergoes more complex deformation patterns along its trajectory. Here, the positions of the four configurations which have been chosen are marked in Fig. 18.4(b). The image analysis procedure for computing an average cross-sectional shape at a given geometric configuration consists of five steps: manually identifying the set of n cross-sections, transforming them into binary images, computing the geometric centre of each section, summing the whole set of n binary images after alignment of their geometric centres, and thresholding the resulting image at the grey level n/2. This procedure is illustrated by the schematic chart in Fig. 18.5. A typical result is given in Fig. 18.6. The figure shows the evolution of the cross-sectional shape during an equi-biaxial tension test performed on the glass plain weave (the tension force is 11 N per yarn). While the width of the cross-section is almost constant during this test, the cross-section is significantly reduced as evidenced by the ratio of area (86%) between the unloaded and loaded states. The figure also shows that this reduction of the area is mainly due to a reduction in the height of the yarns. Additional
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Symmetry s4
s2 s1
s3
s5
Periodicity (a)
Symmetry
1
2
3
4
Periodicity (b)
18.4 Example of geometric configurations in a glass plain weave: (a) unsheared state: one configuration is observed; (b) sheared state: four configurations are considered.
results are presented in section 18.4.2 and are used for the validation of FE simulations.
18.3.2 Microscopic-scale analysis When tuning the resolution at values below 4 microns, fibres can be distinguished individually within the yarn. Observations at this resolution provide a powerful means of analysing the microstructure of the yarn, more precisely the spatial distribution of fibres. The examples and results presented in this section are based on the study of the glass plain weave introduced in Section 18.2.2. Figure 18.7 shows reconstructed slices of the reinforcement under three different mechanical loadings: unloaded, biaxial tension loading (arbitrary tension for this acquisition) and shear loading. From a qualitative point of view, a clear trend can be observed. Whatever the mechanical loading, the
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Manual identification of the cross-sections
575
Binarisation + Computation of geometric centres 1
2
Superimposition of the sections
3
4 Thresholding + Edge detection
18.5 Scheme of the procedure used to determine the average crosssection at a given geometric configuration. Profile of the transverse yarn Average crosssection shape h0 Unloaded section
w0
Area: S0
S/S0 = 0.86 h/h0 = 0.89 Loaded section
w/w0 = 1.005
18.6 Evolution of the cross-sectional shape of a glass plain weave during an equi-biaxial tension test (tension is 11 N per yarn).
fibre bundle tends to be denser in its loaded state, although from a visual point of view, the fibre distribution appears to remain isotropic in each of the three slices. A quantitative analysis of the morphology of these images is proposed in the following. The proposed analysis consists of computing the two-point covariance of binary images of the microstructure. This type of procedure is described in Doumalin et al. (2003) and was also used for other types of microstructures, in Delarue and Jeulin (2003), for instance. The covariance C(x) is the probability of two points, separated by the distance x, belonging to the same
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1 mm (a)
1 mm (b)
1 mm (c)
18.7 Glass plain weave observed at the microscopic scale; resolution range: 2.85–5 mm: (a) unloaded state; (b) biaxial tension state (arbitrary tension); (c) sheared state (46°).
phase. In the study of composite reinforcements performed here, the phase considered is that of the fibres. Although three-dimensional by nature, the microstructure of the bundles is well known in the fibre direction. Therefore, only transverse cross-sectional 2D images are used for the analysis. In principle, if the spatial distribution is homogeneous, the covariance C(x) is equal to the volume fraction of the phase when x tends to 0 and to the square of this fraction when x increases. Unfortunately, this kind of result is very sensitive to the quality of the binary images. Those acquired here on the glass plain weave display a slightly too low resolution and the accuracy of the phase separation (obtained by a homogeneous thresholding) does not enable reliable results for volume fractions. Nevertheless, when calculated in two perpendicular directions (in this case along the width and along the thickness of the yarn), the covariograms provide information about the spatial distribution of fibres within the yarn. If the covariograms are similar in both directions (i.e. if the position of the peak is situated at the same value of x), the sample can be assumed to be isotropic. In the present study, the covariograms are calculated from a series of relatively fast operations: binarisation of the image, Fourier transform, computation of the square of the result and finally, inverse Fourier transform from which histograms are plotted along the directions chosen (Doumalin et al., 2003). Figure 18.8 presents the covariance curves in two perpendicular directions as averaged over several square-shaped measurement zones. Each
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577
3
Grey level (0–255)
200 2 150
100
50 Covariance in direction 2 Covariance in direction 3 0
0
10
20 30 Distance (µm)
40
50
18.8 Covariance analysis on the sheared glass plain weave: average covariograms obtained along directions 2 and 3 indicated on the image. Averaging operation is performed over more than 20 square zones depicted in black.
zone contains about 70 fibres. This result has been obtained from a sheared reinforcement, but note that the results are very similar when using a specimen under biaxial tension. The analysis was not performed using an unloaded fabric because the significant variability in the unloaded state and the initial dispersion of the fibre distribution make it difficult to obtain results. The curves obtained can be interpreted as follows. The first minimum corresponds to the average distance of the air gap between two fibres. Likewise the first maximum corresponds to the average distance of the first neighbouring fibre in the direction considered. It can be noted that for both directions the first minimum and the first maximum are located at the same distance x, which indicates an isotropic distribution of fibres. The slight differences of the covariograms are attributed to grey-level inhomogeneity and reconstruction artefacts due to the strong anisotropy of the objects. Indeed, fluctuations of the average grey level (visible in Fig. 18.3(c), for instance) induce thresholding differences and modify the covariograms at long distances. Apart from this, the set of observations and measurements made in this section provides a strong support to the hypothesis of a quasi-isotropic distribution of the fibres in the transverse plane during the deformation of a bundle. This hypothesis is of great importance for the development of yarn constitutive models. This point is addressed in the following section where direct applications of tomography analyses introduced in this section are presented.
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18.4
Application of the mechanical behaviour of woven reinforcements to finite element simulations
Previous sections have demonstrated the versatility of X-ray tomography for the analysis of composite reinforcements. Appropriate equipment and dedicated methods make it a powerful tool to study the multi-scale geometric and microstructural properties of composite reinforcements such as woven reinforcements. A direct application of these analyses can be carried out with the intention, for instance, of developing numerical models of the mechanical behaviour of textile reinforcements at the mesoscopic scale. In the following section, a FE model aimed at determining deformed geometries of textile reinforcements at the mesoscopic scale is presented. Deformed geometries are required in many applications such as permeability assessment and composite damage prediction, where they are of primary importance (Loix et al., 2008; Ladevèze et al., 2006). In this section, X-ray tomography is used both to support constitutive hypotheses and to validate the model from the point of view of deformed geometries.
18.4.1 Constitutive modelling of fibrous yarns In order to obtain results that are as realistic as possible, the constitutive model of the composite reinforcement during the forming process is the centrepiece. Hence, when the analysis is performed at the mesoscopic scale, a constitutive model is developed for the yarn itself. Yarns used in woven reinforcements are made of thousands of quasi-parallel fibres, which make the yarn mechanical behaviour very specific. Relative sliding is possible between fibres. It is in general not possible to model each of them, therefore the material inside the yarn will be considered as a continuum here. The equivalent continuum behaviour must take into account the fibrous nature by recreating its specificities. Two of them are of the highest importance. The first of these is the high longitudinal stiffness of the fibres compared to the other rigidities, which requires a lot of attention due to the large associated strain energy. The second is the transverse behaviour of the yarns, which obviously has a great influence on the deformed shape. In this section, these two aspects of the yarn constitutive model are briefly addressed. Additional details can be found in Badel et al. (2008a, 2008b). Based on the microscopic analysis developed in Section 18.3.2, the mechanical behaviour of the yarn is assumed to be transversely isotropic where the anisotropy direction is that of the fibres, hereafter denoted as f1. This hypothesis is hard to verify experimentally. Nevertheless, from the previous section it is clear that the transverse microstructure is rather isotropic from the point of view of the fibre spatial distribution. It can be noted that
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some work on particle-reinforced composites has already shown that if the spatial distribution of the reinforcement is initially at random and thus isotropic, the isotropy tends to be preserved after deformation. Although a slight anisotropy could be measured by precise image processing, the effect on the mechanical properties was found to be undetectable (Lebail, 2001). Therefore, the assumption of transverse mechanical isotropy for the yarn behaviour is reasonable. Continuum constitutive models dealing with the mechanics of composite reinforcement have been developed within the frameworks of hyper-elasticity and hypo-elasticity. The former was recently shown to have an interesting potential application to macroscopic simulations of composite forming (aimène et al., 2010) and could be also used for mesoscopic purposes. The mesoscopic continuum constitutive model presented here is written within the framework of a rate constitutive equation (or hypo-elasticity) (Truesdell, 1955; Xiao et al., 1997; Hughes and Winget, 1980) and mainly based on the initial work of Hagège (2004). rate constitutive equations are often used in Fe analyses at large strains (Hughes and Winget, 1980; Belytschko et al., 2000). in such equations, the Cauchy stress rate — is related to the strain rate D by a constitutive tensor C : — = C :D
18.4
To avoid rigid body rotations that can affect the stress state, the derivative —, called the objective derivative, is the derivative for an observer (or equivalently a basis) who is fixed with respect to the matter. Because this requirement is not uniquely defined, there are several objective derivatives. let {ei} = {e1, e2, e3} denote an orthonormal material basis, i.e. a rotating basis which is fixed with respect to the matter. In this basis, Eq. 18.4 can be integrated over a time increment ∆t = tn+1 – tn and yields the widely used formula of Hughes and Winget (1980) for the stress update: [ n +1]en +1 = [ n ]en + [C n +1/2 ]en +1/2 [D]en +11//2 i
i
i
i
18.5
where [D]en +1/2 = Dt[D]en +1/2 is the strain increment over ∆t and the notation i i [S]en stands for the matrix of any tensor S expressed in the basis {ei}. Note i that this equation is only valid when it is written in the material basis used for the objective derivative. When dealing with highly anisotropic materials such as yarn in a composite reinforcement, it is crucial to make a good choice for the material basis. The rationale behind this statement is that minor errors in following the material anisotropy direction would lead to significant stress update errors. This aspect and possible consequences were addressed in detail by Badel et al. (2008a). The issue of the high longitudinal stiffness of the yarn can be treated by
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using a material basis that accurately follows the anisotropy direction. This is achieved with a basis linked to the fibre direction f1 (see Fig. 18.9) and also means that an objective derivative based on the fibre direction rotation is used. equation 18.5 is then written as: [ n +1]f n +1 = [ n ]f n + [C n +1/2 ]f n +1/2 [D]f n +11//2 i
i
i
i
18.6
Introducing this specific constitutive equation in a FE code may require user-defined coding. Questions regarding this implementation in commercial FE codes have been addressed in Badel et al. (2008a) where algorithms and numerical results are presented. Note also that in this equation, the constitutive matrix [C]fi is expressed in the basis {fi} oriented by the fibres at the time. This is a major advantage because the components are known or can be easily determined in this basis where longitudinal and transverse behaviours are properly distinguished. In some applications, these components can be assumed to be constant. In others, it may be necessary to update them in order to take into account the strain state of the material. For yarns of composite reinforcements, the longitudinal behaviour is thus defined by the stiffness of the fibre bundle in direction f1 (which can be determined by simple mechanical tests on a single yarn), whereas the transverse behaviour characterises the behaviour of the fibre bundle in the plane (f2, f3). The second constitutive specificity of a yarn, its transverse behaviour, must then be defined. Transverse behaviour is also very important because it accounts for shape changes in the yarn cross-section. in this section, we consider only the transverse plane (f2, f3). as shown in Section 18.3.2, the yarns tend to be transversely crushed in any type of solicitation (Fig. 18.7). The presented approach (Badel et al., 2008b) suggests that spherical and deviatoric phenomena within the transverse plane should be uncoupled (in two dimensions, the term ‘spherical’ should be replaced by ‘circular’ but Fibre bundle
Single fibre
f3 f2 f1
18.9 Schematic of a fibre bundle illustrating the material basis linked to the fibre direction.
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we use it for comprehension and convenience). The reason for this choice arises from observations of tomography scans at the microscopic scale, where two modes of deformation of the cross-section are clearly distinguished: compaction of the network of fibres (for instance in biaxial tension, Fig. 18.7(b)) and change of shape of the fibre bundle (noticeable in pure shear, Fig. 18.7(c)). These two modes correspond respectively to spherical and deviatoric transformations of the cross-section. in the following, it is further assumed that these two modes are uncoupled, which remains valid under the assumption of isotropy in the plane (f2, f3). From these assumptions, the transverse strain state can be written as: È es [](f2,f3 ) = Í ÍÎ 0
0 es
˘ È ed ˙+Í ˙˚ ÍÎ e 23
e 223 –e d
˘ ˙ ˙˚
18.7
where e s = 12 (e 22 + e 33) is the spherical strain component while e d = 1 2 (e 22
– e 33) and e23 are the deviatoric strain components. This formalism where spherical and deviatoric contributions are uncoupled is often used in plasticity models. It was also previously used in fibre bundle micromechanics (Cai and Gutowski, 1992; Simacek and Karbhari, 1996). In addition, since the constitutive model is assumed to be elastic, though non-linear, the same decomposition stands for stresses. it also holds for strain and stress increments. Based on the assumption of uncoupling introduced in the previous paragraph, the following constitutive equation is then proposed for the transverse behaviour (note that incremental form is necessary to comply with eq. 18.6): Ds s = A0 e – pes e ne11 De s Ds d = B0 e – pes De d
18.8
Ds 23 = B0 e – pes De 23 The reasons for these choices and analytical proofs for some parameters to be equal are detailed in Badel (2008) and Badel et al. (2008b). Briefly, this form is obviously non-linear and ensures that the material becomes transversely stiffer when the fibre network is denser. It also makes the spherical behaviour stiffer under longitudinal tension, which recreates the increase of effort required to compact a pre-tensioned yarn. For the sake of simplicity, it is considered in the present model that tension has no influence on the deviatoric behaviour. Both specificities of the yarn behaviour have been addressed. This kind of model can be implemented in commercial codes. Identification of the related constitutive parameters is not straightforward, especially for transverse
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behaviour. Due to the complexity of transversely testing yarns and the complex non-linear relations between the parameters, an inverse approach is necessary. Experimental data for equi-biaxial tension were used to this aim (Buet-Gautier and Boisse, 2001). The identified model was then introduced into a mesoscopic FE model to determine the mechanical behaviour of composite reinforcements.
18.4.2 Validation of the FE simulations Besides the constitutive equation presented in the previous section, the mesoscopic FE model of woven composite reinforcements was developed using the geometric model of Hivet and Boisse (2008). A detailed description of this FE model and its specifications can be found in Badel et al. (2007) and a view of the deformed unit cell of the model is pictured in Fig. 18.10. The aim of this section is to show how tomography can be helpful in validating such a numerical model from the point of view of the deformed geometries. The glass plain weave presented earlier is used, again, for the purpose of illustrating this section. Two types of mechanical tests were modelled: a pure shear test and a test of biaxial tension followed by pure shear. The former is very important because shear is the preferred mode of deformation of woven fabrics, whose complex geometries are to be formed. The latter is also relevant in the sense that the influence of tension on the shear response is known to be significant. However, not much is currently known about this phenomenon. Reliable mechanical experiments are very challenging to set up, but have already emphasised the influence of tension (Launay et al., 2008). As shown in Section 18.3.1, the in situ loading device presented in Section 18.2.2 enables the study of deformed geometries at the mesoscopic scale. The procedure described in 18.3.1 can be applied to extract average crosssectional shapes of the yarns. This kind of analysis is used here to evaluate the model for both the test of pure in-plane shear and that of biaxial tension followed by shear. Figure 18.10 shows comparative results obtained from X-ray tomography analyses and from the FE model for the pure shear test. The shear angle is about 46°. Four geometric configurations are chosen (see Fig. 18.4), for which the cross-sectional shape is averaged over the whole specimen. Several interesting trends can be noted. From a qualitative point of view, note that the model captures the main geometric trends observed in tomography: section asymmetry, significant decrease of the width w, flat sides, rounded sides and overall decrease of the area. The most significant differences occur at the tips of the sections, due to the initial mesh that does not enable pointed tips. From a quantitative point of view, it is proposed to focus on the evolution of area and width of section four (fourth geometric configuration
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Initial section Area S0
Width w0
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Average cross-section from X-ray tomography Contour line of the FE model
Shear deformed sections #1 S
Mean area ratio S/S0 Simulation: 0.71 Tomography: 0.77
#2 Mean width ratio w/w0 Simulation: 0.74 Tomography: 0.77
#3
#4 w
18.10 Pure in-plane shear of the glass plain weave fabric (46°), comparison of the FE model and X-ray tomography analysis. On the left, 3D global views. On the right, comparison of the four crosssections, with quantitative results.
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considered). The ratios of area between initial and deformed states are 0.77 for the measure by tomography versus 0.71 for the simulation, while those of width are 0.77 and 0.74 respectively. The model tends to deform more than the actual reinforcement does. Though the difference is minor, it could be attributed either to the identification of constitutive parameters, or to the model itself. The illustrations in Fig. 18.11 correspond to the comparison of the model with tomography analyses for a test of equi-biaxial tension (20 N per yarn) followed by 38° pure in-plane shear. The same global trends as appeared in the shear test are also observed here. In addition, the area ratio after the phase of tension is 0.82 for the experimental measure versus 0.80 for the model, whereas after shearing it displays a slightly more significant difference with 0.92 versus 0.86. It is possible that the model reaches its limits of validity under this type of solicitation. Another possible explanation is related to the Initial section Area S0
Average cross-section from X-ray tomography Contour line of the FE model
After biaxial tension phase T = 20N S1 Area ratio S1/S0 Simulation: 0.80 Tomography: 0.82 After shear phase (38°) S2 #1
#2
Mean area ratio S2/S1 Simulation: 0.86 Tomography: 0.92
#3
#4
18.11 Equi-biaxial tension (20 N per yarn) followed by pure in-plane shear. Comparison of cross-sections from the FE model and from X-ray tomography analysis.
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experimental setup: the yarns may have slipped slightly in the grips during the required operations before the last tomography acquisition. It is also very likely that the yarn undergoes minor ‘viscous’ reorganisations of the fibre distribution during that time, hence minimising the yarn tension and affecting its deformed state.
18.5
Conclusion
The purpose of this chapter was to introduce the potential of X-ray tomography for the mechanical analysis of composite reinforcements. This non-destructive technique has been presented through a brief description of the principle and examples of images obtained for composite reinforcements, showing thus its ability to address multi-scale properties of such materials. Due to a dedicated in situ loading device, tomography was shown also to be a powerful tool for investigating the relationship between the microstructure and the mesoscopicand macroscopic-scale mechanics of these materials. Two approaches were developed to analyse tomographic data. At the mesoscopic scale, the approach was purely geometric, whereas morphologic considerations were investigated at the microscopic scale. It was demonstrated how these analyses are useful in developing numerical models of the mechanical behaviour of woven composite reinforcements. Through this chapter, some trends and improvements have been identified. The results presented here are encouraging regarding the mechanical analysis of composite reinforcements. Devices allowing both mechanical loading and measurements in situ are currently being developed. Although they are widely used in materials science, applying these techniques to composite reinforcements remains a challenge. As well as making the comparison with models much more reliable, they will also improve our understanding of the complex relations between microstructure, geometry and mechanical loading. The most promising contribution of X-ray tomography in this field, from our point of view, is concerned with microstructural analysis, as the micromechanics of the fibre bundle are still partially unknown. To address this huge potential, an improvement in the quality of imaging will be necessary so as to make morphological analyses more versatile and accurate. This is quite challenging due to the planar shape of composite reinforcements which makes unexpected artefacts more likely. An increase in resolution would make these analyses easier, but it will require more efficient and larger facilities. Together, these possible improvements would enable us to obtain more quantitative information about microstructural characteristics and especially about the evolution of microstructure under mechanical loading. This would shed light on the micromechanics of fibre bundles and composite reinforcements.
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18.6
References
Aimène Y, Vidal-Sallé E, Hagège B, Sidoroff F, Boisse P (2010), ‘A hyperelastic approach for composite reinforcement large deformation analysis’, J Compos Mater, 44, 5–26. Badel P (2008), Mesoscopic analysis of the mechanical behavior of textile composite reinforcements – Validation with X-ray tomography, PhD thesis, Lyon: INSALyon. Badel P, Vidal-Sallé E, Boisse P (2007), ‘Computational determination of in-plane shear mechanical behaviour of textile composite reinforcements’, Comput Mater Sci, 40, 439–448. Badel P, Vidal-Sallé E, Boisse P (2008a), ‘Large deformation analysis of fibrous materials using rate constitutive equations’, Comput Struct, 86, 1164–1175. Badel P, Vidal-Sallé E, Maire E, Boisse P (2008b), ‘Simulation and tomography analysis of textile composite reinforcement deformation at the mesoscopic scale’, Compos Sci Technol, 68, 2433–2440. Baruchel J, Buffiere JY, Maire E, Merle P, Peix G, editors (2000), X-ray tomography in material science, Paris, Hermès. Belytschko T, Wing KL, Moran B (2000), Nonlinear finite elements for continua and structures, Chichester, UK, John Wiley & Sons. Buet-Gautier K, Boisse P (2001), ‘Experimental analysis and modeling of biaxial mechanical behavior of woven composite reinforcements’, Exp Mech, 41, 260–269. Buffiere JY, Maire E, Adrien J, Masse JP, Boller E (2010), ‘In situ experiments with X ray tomography: an attractive tool for experimental mechanics’, Exp Mech, 50, 289–305. Cai Z, Gutowski T (1992), ‘The 3-D deformation behavior of a lubricated fiber bundle’, J Compos Mater, 26, 1207–1237. Chang SH, Sutcliffe MPF, Sharma SB (2004), ‘Microscopic investigation of tow geometry changes in a woven prepreg material during draping and consolidation’, Compos Sci Technol, 64, 1701–1707. Delarue A, Jeulin D (2003), ‘3D morphological analysis of composite materials with aggregates of spherical inclusions’, Image Anal Stereol, 22, 153–161. Desplentere F, Lomov SV, Woerdeman DL, Verpoest I, Wevers M, Bogdanovich A (2005), ‘Micro-CT characterization of variability in 3D textile architecture’, Compos Sci Technol, 65, 1920–1930. Doumalin P, Bornert M, Crépin J (2003), ‘Characterisation of the strain distribution in heterogeneous materials’, Méc Ind, 4, 607–617. Feldkamp LA, Davis LC, Kress JW (1984), Practical cone-beam algorithm, J Opt Soc Am, 1, 612–619. Hagège B (2004) Simulation du comportement mécanique des milieux fibreux en grandes transformations, PhD thesis, Paris: ENSAM. Hivet G, Boisse P (2008), ‘Consistent mesoscopic mechanical behaviour model for woven composite reinforcements in biaxial tension’, Compos Part B, 39, 345–361. Hughes TJR, Winget J (1980), ‘Finite rotation effects in numerical integration of rate constitutive equations arising in large deformation analysis’, Int J Num Meth Eng, 15, 1862–1867. Ladevèze P, Lubineau G, Marsal D (2006), ‘Towards a bridge between the micro- and meso-mechanics of delamination for laminated composites’, Compos Sci Technol, 66, 698–712.
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Launay J, Hivet G, Duong Av, Boisse P (2008), ‘Experimental analysis of the influence of tensions on in plane shear behaviour of woven composite reinforcements’, Compos Sci Technol, 68, 506–515. Lebail H (2001), Caractérisations microscopiques et mécaniques de matériaux composites et étude des relations aux propriétés élastiques associées, PhD thesis, Lyon: INSALyon. Loix F, Badel P, Orgeas L, Geindreau C, Boisse P (2008), ‘Woven fabric permeability: From textile deformation to fluid flow mesoscale simulations’, Compos Sci Technol, 68, 1624–1630. Pandita SD, Verpoest I (2003), ‘Prediction of the tensile stiffness of weft knitted fabric composites based on X-ray tomography images’, Compos Sci Technol, 63, 311–325. Potluri P, Parlak I, Ramgulam R, Sagar TV (2006), ‘Analysis of tow deformations in textile preforms subjected to forming forces’, Compos Sci Technol, 66, 297–305. Raz-Ben Aroush D, Maire E, Gauthier C, Youssef S, Cloetens P, Wagner HD (2006), ‘A study of fracture of unidirectional composites using in situ high-resolution synchrotron X-ray microtomography’, Compos Sci Technol, 66, 1348–1353. Schell JSU, Renggli M, Van Lenthe GH, Muller R, Ermanni P (2006), ‘Micro-computed tomography determination of glass fibre reinforced polymer meso-structure’, Compos Sci Technol, 66, 2016–2022. Simacek P, Karbhari VM (1996), ‘Notes on the modeling of preform compaction: I – Micromechanics at the fiber bundle level’, J Reinf Plast Compos, 15, 86–122. Stock SR (2008), ‘Recent advances in X-ray microtomography applied to materials’, Int Mater Rev, 58, 129–181. Truesdell C (1955), ‘Hypo-elasticity’, J Ration Mech Anal, 4, 83–133. Xiao H, Bruhns OT, Meyers A (1997), ‘Hypo-elasticity model based upon the logarithmic stress rate’, J Elasticity, 47, 51–68.
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19
Flow modeling in composite reinforcements
E. R u i z and F. T r o c h u, École Polytechnique de l’Université de Montréal, Canada
Abstract: In composite manufacturing by resin injection through a fibrous reinforcement, several phenomena occur involving the flow of resin driven by capillarity, gravitational or pressure forces. In order to study the manufacturing of pieces of complex shape made by liquid composite molding (LCM), a numerical simulation of the mold-filling process is required. Numerical process simulation can be done by combining classical finite element techniques to a flow-tracking approach which allows the movement of the resin flow front through the finite element mesh of the part. In this chapter, the techniques for computing the Darcy’s flow presented in Chapter 14 are described. The flow tracking technique is also mathematically described and exampled in one dimension. A numerical solution of the manufacturing of a truck fender is also presented to illustrate the practical application of these numerical approaches. Key words: liquid composite molding (LCM), composites processing modeling, Darcy’s flow, resin transfer molding (RTM), permeability.
19.1
Introduction
In the last decades, diverse composite processing techniques have been investigated to improve part quality and reduce manufacturing costs. In the processing of thermosetting polymer matrices, liquid composite molding (LCM) techniques such as resin transfer molding (RTM) and vacuum assisted resin infusion (VARI) have demonstrated potential advantages over traditional methods. In the aerospace field, the RTM process has now become mature technology to manufacture high-performance composites by resin injection. In some industrial sectors, these technologies are still expensive and the present research aims mainly at reducing cycle time. Appropriate tool design is crucial to ensure proper impregnation of the fibrous reinforcement and curing of composite parts. Air bubbles, dry spots or resin-rich areas are common problems that must be avoided during the filling stage. Residual stresses, warpage and spring-in are another category of problems that occur during LCM processing related to thermal or resin cure gradients. Numerical process simulation is critical to successfully design the mold, reduce the risk of defective parts and optimize the production cycle. The advantage of computational techniques to prevent filling problems and 588 © Woodhead Publishing Limited, 2011
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get better molding results has been the goal of many investigations in the past [1–11] and has been widely used in industry for process design and optimization [12–14]. Analysis of mold deformation during resin injection, comparison of various injection strategies and optimization of the injection flow rate [10, 12] illustrate the potential advantages brought by numerical simulation. This chapter focuses on the analytical and numerical solutions of the resin flow and cure in LCM processes. First, the governing equations of fluid flow and resin cure will be presented, and then analytical solutions are derived in simple geometries. The numerical approach is described to solve the flow, heat transfer and cure problems in the general case. Finally, the solutions of a series of isothermal problems are exposed.
19.2
Governing flow equations
A typical mold-filling process to manufacture composite parts involves the injection of a liquid resin into the mold cavity. As illustrated in Fig. 19.1, the central issue in the analysis of mold filling consists of tracking the free surface between the filling material (i.e., the liquid resin) and the escaping gas (usually air) present in the mold cavity before the injection. The flow of fluid through a fibrous network has been analyzed at both the microscopic and macroscopic levels. In the microscopic analysis, the flow through the fibrous reinforcement is governed by the Navier–Stokes equation. However, Vent Gp ∂W = Gq » Gp » Gd Ê ∂p ˆ Gq Á = q ˜ G p (p = p) Ë ∂nˆ ¯ Fluid flow front Œ Gd
Impregnated domain W
Injection Gq , Gp ∂p =0 ∂nˆ
19.1 Schematic representation of the domain W representing the mold cavity. The boundary is defined by ∂W = Gq » Gp » Gd. (Note that Gd was deliberately removed for clarity.)
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this approach may be misleading because of the complex architecture of the reinforcement and the difficulty of finding appropriate boundary conditions. In macroscopic investigations, the flow of resin through a fiber bed is modeled by Darcy’s law, which governs fluid flows through porous media as introduced in chapter 14. In order to simulate the filling process, several assumptions are required to simplify the problem. in general, the fabric reinforcement placed in the mold cavity is assumed to be rigid during filling, and inertia effects can be neglected because of the low Reynolds number, usually below 1. Furthermore, at the pressure level generated in LCM molds to drive the flow, surface tension may be neglected compared to the dominant viscous force. Based on these assumptions, the mass conservation of the fluid phase can then be written as: div(r·v) = 0
19.1
where r is the density of the resin injected and v is the average fluid velocity, also called Darcy’s velocity, namely the velocity at which the fluid actually travels in the porous medium, rather than the observed macroscopic velocity of the flow front. Note that we consider here that the flow takes place in rigid molds, otherwise the equation of mass conservation (19.1) should consider also the deformation of the fabric structure. The flow of an incompressible fluid through a porous medium can be calculated from Darcy’s law as follows: v = – 1 [K ] · —pp m
19.2
where [K] is a 3 ¥ 3 permeability tensor, m is the resin viscosity and —p is the pressure gradient. Equation 19.2 relates the three components of the superficial fluid velocity vector to the fluid pressure. This relationship is only valid for Newtonian fluids and ignores gravity effects and mold deformation. Note that these effects can also be considered using a more elaborated equation [1]. Combining equations 19.1 and 19.2, the partial differential equation that governs the fluid flow in porous media is finally written as follows: Ê 1 ˆ divv Á – [K ] · —p˜ = QS Ë m ¯
19.3
where p is the scalar potential on W (see Fig. 19.1). The term [K]/m represents the dual scalar field and QS is a volume source term representing fiber saturation [10, 11] (see equation 14.5 in Chapter 14 for details). For fully saturated flows, QS can be neglected and equation 19.3 equates to zero. Darcy’s velocity v is related to the resin superficial velocity vs, namely the observed velocity of the flow front, via the porosity of the porous medium f as follows: © Woodhead Publishing Limited, 2011
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vs = v f
591
19.4
Since the porosity f is smaller than 1, the superficial velocity is always larger than Darcy’s velocity. The solution of the resin flow inside a rigid mold can then be obtained by solving equations 19.3 and 19.4.
19.3
Analytical solution
Equations 19.3 and 19.4 can be solved analytically for simple geometries such as rectangular or circular molds. In the simplest case, we consider a rectilinear mold as shown in Fig. 19.2. In this case the resin is injected at constant pressure Pinj and impregnates the fibers through the length a of the mold. The superficial velocity of the resin flow can be calculated in one dimension along the x-axis from equations 19.3 and 19.4, neglecting capillary effects, as follows: Pin dxx injj – Pvacuum vs = v = f = – K dP = K f dt f m dx dx f m xf
19.5
where Pvacuum is the relative vacuum pressure inside the mold cavity prior to and during resin injection, K is the permeability of the fibrous reinforcement along the x-direction, and xf is the distance from the injection gate to the fluid front position. Equation 19.5 can be rearranged and integrated in time and space as follows: xf
Ú0
x dx =
tf
Ú0
K (P – P inj vacuum ) dt f m inj
19.6
Integrating both sides of equation 19.6 leads to the solution of the resin flow front position xf in time. Rearranging this equation allows calculation of the time to fill the mold tfill at a distance xf as follows: xf2f fm m xf2 = 2K (P Pin injj – Pvacuum )t or t fill f = fm 2K (Pin P injj vacuum )
19.7
a
Injection port Pinj
b
vs xf
19.2 Schematic representation of a rectangular mold containing a fibrous reinforcement being impregnated by a liquid resin injected at constant pressure.
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Figure 19.3 shows the solution of a longitudinal injection in RTM using both forms of equation 19.7. The quadratic relationship between the flow position and time can be observed, as well as the inverse quadratic relationship of the resin flow rate at the inlet port with time. This implies that the resin velocity will be very high during the first seconds of the injection and decrease quickly after a few centimeters of impregnation. This important fluid velocity variation may be the cause of improper impregnation of the fibers due to the dual-scale nature of the fibrous reinforcement as presented in Chapter 14. In high-end applications such as in the aeronautical or aerospace industries, the resin is injected at a controlled speed in order to improve the impregnation of the fibers. Darcy’s law can also be solved for a longitudinal injection with a constant flow rate specified at the inlet port. In this case, equation 19.4 can be treated as follows: vs =
Qinj Qinj dxxf v in in = = fi xf = t dt f f Afront f A fr front fr
19.8
where Qinj is the resin flow rate at the inlet port and Afront is the area of the fluid flow front (i.e., the area of the mold transversal to the fluid flow). Since the flow rate is constant, the position of the fluid flow in the mold is proportional to the injection time. In this particular case, even if the fluid flow rate is constant at the inlet port, the fluid pressure will not be constant due to the pressure drop and the constant flow velocity along the flow path. To calculate the evolution of the inlet fluid pressure, equations 19.3 and 19.4 can be solved as follows: 2 Qinj mQinj in in = – K dP fi Pin = P + t injj vacuum 2 f Afront fm dxx f Afront K front fr
1.0E-03
1 Flow front Flow rate
Flow front position (m)
0.9 0.8 0.7
Pinj: 9 ¥ 105 Pa Pvacuum: 0 Pa (atmospheric) µ: 1 Pa.s Vf: 50% K: 10–9 m2 Mold length: 1 m Mold width: 0.3 m Mold thickness: 3 mm
0.6 0.5 0.4 0.3 0.2 0.1 0
19.9
0
50
100
150 Time (s)
200
250
Injection flow rate (m3/s)
vs =
1.0E-04
1.0E-05
1.0E-06
19.3 Analytical solution of the resin flow through a longitudinal mold as described in Fig. 19.2 Resin is injected at constant pressure (see equation 19.7).
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1
Flow front position (m)
0.9 0.8
Flow front Pressure at injection
2.0E+06
0.7 0.6 0.5
xµt P inj µ t
0.4
1.0E+06
0.3 0.2
Pressure at injection Pinj
where Pinj is the fluid pressure at the inlet port and Pvacuum is the relative vacuum pressure inside the mold cavity. Equation 19.9 implies that a linear relationship exists between the fluid flow position and the pressure at the inlet port during a longitudinal injection at constant flow rate. This can be observed in the example of Fig. 19.4. Note that both the resin flow front position and pressure at the inlet gate exhibit a linear dependence with time. Comparing the solution at constant injection flow rate (Fig. 19.4) with the one at constant pressure of Fig. 19.3, the first case leads to a fluid pressure at the end of filling which is twice the value at constant pressure. This indicates that a more rigid mold is required for injections at constant flow rate. Other analytical solutions of the Darcy’s flow in RTM can be obtained for simple geometries. Table 19.1 presents a list of analytical solution for three kinds of flow paths: rectangular, radial divergent and radial convergent. Also, constant injection pressure and constant flow rate boundary conditions can be considered. Figure 19.5 shows a comparison of three injection strategies for the same part of 1 m2 injected at constant pressure of 105 Pa. If part is injected longitudinally, the expected filling time would be 125 seconds. If a hole is made in the center of the mold and the part is injected in a radial divergent flow, it will take 165 seconds to completely fill the mold. Finally, if a peripheral runner is machined in the mold and the part is injected in a radial convergent manner, it will take only 18 seconds to fill the mold. It is then clear that the radial convergent flow is the fastest solution for any part while the radial divergent flow is the slowest approach. However, because of the complexity of controlling converging fluid paths and the uniformity of the peripheral runner, this approach is only used after practical knowhow has been gained on these techniques.
Qinj: 2.8 ¥ 106 m3/s Pvacuum: 0 Pa (atmospheric) µ: 1 Pa.s Vf: 50% K: 10–9 m2 Mold length: 1 m Mold width: 0.3 m Mold thickness: 3 mm
0.1 0 0
100 Time (s)
200
0.0E+00
19.4 Analytical solution of the resin flow through a longitudinal mold as described in Fig. 19.2 Resin is injected at constant flow rate (see equation 19.8).
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Table 19.1 Analytical solutions of the resin flow in RTM Pressure distribution
Flow rate
Filling time
2 phK P0 m r ln m rff
Ê rt ˆ rm2 – rt2 ˆ fm Ê 2 Árt ln Á ˜ + ˜ 2 KP0 Ë 2 ¯ Ërm ¯
Constant pressure ln r rff rm ln rff
Radial convergent
P0
Rectilinear
Ê ˆ P0 Á1 – x ˜ x Ë ff ¯
y mhK P0 m x ff
fm 2 x 2 KP0 m
Radial divergent
ln r rff P0 r ln t rff
2 phK P0 m r ln ff ri
fm 2 KP0
Radial convergent
mQ0 ln r 2 phK rff
Q0
fph (rm2 – ri2 ) Q0
Rectilinear
mQ0 (x – x ) y mhK ff
Q0
fhx my m Q0
Radial divergent
mQ0 r ln ff 2 phK r
Q0
fph (rm2 – ri2 ) Q0
Ê Êrm ˆ ri2 – rm2 ˆ 2 Árm ln Á ˜ + ˜ 2 ¯ Ë ri ¯ Ë
Constant flow rate
Xm Xff
Xm Xff
Xff
Xm
Radial convergent
19.4
Rectilinear
Radial divergent
Numerical solution
Although analytical solutions are simple to compute, they are limited to simple geometries. For more complicated two- and three-dimensional part geometries, numerical methods have to be implemented to compute Darcy’s law and predict the resin flow patterns during mold filling. In the past two decades, several numerical techniques have been developed to solve the fluid flow in LCM. One of the first techniques consisted of solving Darcy’s flow by finite differences (FD) with boundary fitted coordinates [1]. In this approach, the mesh used to describe the impregnated region in the mold changes at every computational time step as the saturated domain evolves. This requires significant computational effort to recalculate the positions of
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180 160
Filling time (s)
140 120 100 80 60 40
Longitudinal Radial convergent Radial divergent Pinj: 106 Pa Pvacuum: 0 Pa (atmospheric) µ: 0.5 Pa.s Vf: 50% K: 10–9 m2 Mold length: 1 m Mold width: 1 m Mold thickness: 3 mm Injection diameter: 10 mm
20 0 0
0.2
0.4 0.6 Mold dimension
0.8
1
19.5 Comparison of three injections at constant inlet pressure (see Table 19.1 for analytical solutions).
the nodes at every time step, thus making this solution slow and unstable. In addition, this technique is mostly limited to 2D geometries due to the difficulty of moving a three-dimensional mesh. Another numerical technique used in the past is the boundary element method (BEM) [15, 16]. This method is more accurate than FD. However, it also requires remeshing of the flow front. Hence it suffers from similar limitations as finite differences. Meshless techniques such as smooth particle hydrodynamics (SPH) have also been tested for solving isothermal Darcy’s flow [17]. This approach uses particles moving on a limited domain instead of a moving mesh. In this approach, the local properties of the fluid at any point in space can be estimated by taking a weighted average of those properties over a surrounding volume. SPH is well suited for computing fluid–structure interactions such as reinforcement deformation or core displacement during injection. However, these investigations have shown that uncertainties are generated by this approach, thus seriously limiting its applications to LCM. Finally, the most relevant and widely used numerical technique to solve Darcy’s flow in composites manufacturing is based on the finite element method (FEM). This approach was proven during the past two decades to be accurate and robust even for three-dimensional non-isothermal injection conditions where the resin viscosity varies with temperature and cure. This technique uses a fixed mesh, which describes the geometry of the part to be filled by the liquid resin. In order to follow the positions of the fluid flow front in time on the fixed mesh, the control volume/finite element method (CV/FEM) was used by most investigators [2–5, 7, 9]. The CV approaches a simple numerical scheme to track the moving boundary by associating a
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fill factor to each node of the mesh. One drawback of this method is that it requires constructing the control volumes from the elements of the mesh that are connected to a given node. Although in two dimensions this is quite straightforward, in three dimensions it becomes a bit cumbersome. The method followed by Trochu et al. [3] is slightly different: the fill factors being associated with the elements of the mesh, the motion of the flow front can be easily followed by recording the filling of each element. Note that the accuracy of the flow front tracking, and hence of the mold filling prediction, depends in both approaches not only on the size of the elements of the mesh, but also as described in the sequel on the type of finite element approximation selected to solve Darcy’s boundary value problem at each calculation step. Note that different mesh refinement techniques have been applied to improve the accuracy of the flow front prediction [18, 19]. By following more accurately the motion of the flow front, these approaches have reduced the overall error of the numerical solution. However, it was found that remeshing requires more computational effort than the actual solution of Darcy’s problem at one time step, thus limiting the applications of this technique. in most commercial software, the standard CV/FEM technique on a fixed mesh has been adopted as the most relevant approach to solve mold-filling problems based on Darcy’s flow in porous media.
19.4.1 Potential formulation To solve equations 19.1 and 19.2, a weak formulation can be written in terms of weighted residuals [8]. In the case of Darcy’s flow, the unknown scalar field p to be computed is the relative fluid pressure drop through the flow direction. Defining a test function equal to the scalar potential p and introducing a space F(W) of shape functions w, the elliptic partial differential equation 19.3 is replaced by an equivalent variational or weak formulation obtained as the integral of the scalar potential p on W multiplied by w. The weak formulation can then be expressed as follows: Ê 1
ˆ
ÚW w · div ÁË – m [K ] · —p˜¯ dW + ÚW (w · f ) dW = 0
19.10
for any test function w belonging to the space F(W). if Green’s theorem is applied for a Newtonian and incompressible fluid without any source term, the integration by parts gives (see Fig. 19.1 for more clarity): Ê 1
ˆ
ÚW —w · ÁË – m [K ] · —p˜¯ ∂W – ÚG
d
(w · (nˆ · v )) ∂G ∂G d = 0, " w ŒF (W W) 19.11
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The variational formulation of equation 19.11 characterizes the mathematical solution of the continuous Darcy’s flow through domain W of boundary Gd .
19.4.2 Finite element discretization The discrete finite element method (FEM) consists of finding a set of shape functions w that properly approximate the scalar potential p on the whole domain W. The scalar potential is unknown in W, but defined on the boundaries Gq, Gp and on the flow front. To compute the scalar potential p, the domain W is decomposed into small elements of simple geometrical shape called finite elements (FE). The scalar p is then approximated into the FE by a linear combination of shape functions specifically defined from the shape of the element. Knowing the scalar p on each FE allows calculating the integral solution of equation 19.11 by summing up all scalar functions p on the elements of the mesh. For a general shape function sn and test function sj associated with the group of nodes in the finite volume Ve, the Galerkin FEM formulation becomes: Ne
È
S pn
ÚV
n =1 Í Î
e
˘ Ê[K ] ˆ –—s j · Á —sn ˜ dVe ˙ = Ë m ¯ ˚
ÚG
e
(s j ·(nˆ · v )) d G e , "s j ŒVe 19.12
where Ne are the element nodes and Gd the boundary of the finite volume Ve. The scalar potential field p can be computed as: N
p(x ) = S pn sn (x ) n =1
19.13
where sn(x) is a piecewise linear shape function and x is the position vector in the domain W. In matrix notation, equation 19.12 can be rewritten as: [M]{p} = {R}
19.14
where [M] represents the N ¥ N stiffness matrix of the scalar potential field p. An element of matrix M is defined as: a jn =
ÚV
e
Ê[K ] ˆ —s j · Á —sn ˜ dVe , ffor or j, n, n …, N Ë m ¯
19.15
and {R} is a vector {rj} containing the boundary conditions: rj =
ÚG
e
(s j · (n (nˆ · v )) ∂G e , j = 1, …, N
19.16
The approximate solution of the scalar potential p is then obtained by solving
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the linear system (19.14) of N equations with N unknowns, where N is the number of nodes in the mesh.
19.4.3 Mass conservation and div-conform approximation In the finite element discretization of incompressible flows through porous media, the fluid mass is typically assumed to be constant within a defined control volume (cV). considering the geometry of the elements to represent the control volume where the mass balance equation is solved [3], the potential scalar field p representing the fluid pressure will be obtained at each node of the FE (or control volume). Since the fluid pressure is a scalar field, its derivative (i.e., the fluid velocity according to equation 19.2) is constant on the element. The fluid mass transported between adjacent elements can then be computed as the scalar product of the normal vector at the interface between two adjacent elements by the mass velocity r · v (see Fig. 19.6a). Two kinds of finite element approximation are commonly used to solve the potential scalar field p: conforming or non-conforming elements. The conservation of the fluid mass at the interface between two adjacent elements is not ensured with a conforming finite element approximation. As matter of fact, the continuous shape function obtained by Lagrange interpolation at the element nodes (see Fig. 19.6b) results in a piecewise linear scalar approximation of the pressure field on each finite element of the mesh. The gradient of the scalar potential, i.e., the dual scalar field v, is constant on each element, but the normal fluid velocity is not continuous across the element interface (edge E in Fig. 19.6b). As a result of this discontinuity across the interfaces between neighboring elements, the approximate solution of the flow equation (19.2) does not satisfy exactly the mass balance equation (19.1). A discontinuous shape function is constructed by interpolating the pressure field on the element edges as depicted in Fig. 19.6c. This non-conforming interpolation leads to a functional space of shape functions that ensures a continuous mass flow, i.e., nˆ · r v is constant across the interfaces between neighboring elements. On the other hand, equation 19.3 with a source term nˆ 3 ·(rv )
+1
N3
N1
0
(rv ) nˆ 1·(rv )
nˆ 2 ·(rv ) (a)
E
N3 –1
+1
E3 E2
N2 0 (b)
E1 N2 (c)
19.6 Triangular finite element: (a) incoming and outgoing flows in the FE; (b) conforming shape function at node N1; (c) non-conforming shape function associated with edge E1.
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implies that the integral of the divergence in an element is equal to the balance of resin mass flowing across its boundary BC: div(r v ) dV = Ú nˆ · r v dA ÚcV div( Bc
19.17
In numerical analysis, this kind of approximation is called div-conform. Therefore the fluid mass is conserved across the interfaces between elements, and hence in the flow during the simulation of mold filling. As demonstrated in [9], the use of non-conforming finite elements ensures mass conservation and results in a more accurate numerical simulation of injection molding.
19.4.4 Finite elements for LCM flow simulation Usually, mold-filling simulations are carried out in thin shell geometries for the following reasons: 1. Composite parts have usually a small thickness compared to their surfaces so that usually the through-thickness flow can be neglected. 2. The generation of a three-dimensional mesh for a part of small thickness is a complex operation and the setup of boundary conditions is not as simple as for 2D parts. 3. The number of finite elements required to obtain accurate solutions makes three-dimensional simulations tedious in terms of computer time. For these reasons, 2D finite elements are more commonly used for solving Darcy’s flow in composite materials. The simplest FE geometry consists of a triangle with three degrees of freedom. For the triangular non-conforming finite element of Fig. 19.7, the three degrees of freedom are assigned at the middle points of the element edges. considering a classical reference triangle with a local linear approximation space of dimension 3, the element shape functions se are written as follows [20]: Ïs1 = 1 – 2h Ô se = Ìs2 = 2 (z + h ) – 1 0 £ z , h £ 1 Ôs = 1 – 2z Ó3
19.18
where z, h and t are the local coordinates in a local reference system. The gradients of these shape functions can be written as a 3 ¥ 3 matrix of the form: È 0 –2 0 ˘ se = Í 2 2 0 ˙ Í ˙ ÍÎ –2 0 0 ˙˚
19.19
The triangular non-conforming shape function (eq. 19.18) is used to solve the
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2
3 t
y 1 x
5
h
6
4 0.5
1 z
h
1
1
0 3
2 3
2 –0.5
0
1
1
1
z
Reference triangular non-conforming finite element
Reference prismatic finite element (6 nodes)
19.7 Triangular and prismatic (six-node) non-conforming finite elements. The degrees of freedom are assigned at the middle edges; z, h and t denote the local coordinates in the reference element.
FE approximation of equations 19.14–19.16, while the gradient of the shape function (eq 19.19) allows computing directly the fluid flow in the FE. in many cases, three-dimensional simulations are required to accurately predict the injection time and detect local filling problems, mainly for structural parts a few inches thick. Because composite parts are made out of thin laminates, parallel layers of three-dimensional finite elements can easily be generated automatically to model the properties of each ply. A prismatic finite element containing six nodes is then of practical use for solving three-dimensional flows in shell-like parts. As shown in Fig. 19.7, a prismatic non-conforming element can be constructed with degrees of freedom on each edge of the triangular faces. The linear function space will have in this case six degrees of freedom. The element shape functions s e of the Prism6 element will be: (1 – t ) · 0.5 Ïs1 = (1 – 2h ) · (1 Ôs = (2z + 2h – 1) · (1 – t ) · 0.5 Ô2 Ôs3 = (1 – 2z ) · (1 (1 – t ) · 00.5 .5 0 £ z, h £ 1 se = Ì (1 + t ) · 0.5 –1 £ t £ 1 Ôs4 = (1 – 2h ) · (1 Ôs5 = (22z + 2h – 1) · (1 + t ) · 0.5 Ô (1 + t ) · 00.5 .5 Ós6 = (1 – 2z ) · (1
19.20
where the third dimension variable t is referenced from the element midplane. The gradient of the shape function is the following 6 ¥ 3 matrix:
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È Í Í Í se = Í Í Í Í Í Î
– (1 (1 – 2h ) · 0.5 ˘ ˙ (1 – t ) – (2z + 2h – 1) · 0.5 ˙ 0 – (1 – 2z ) · 0.5 ˙ ˙ – (1 + t ) (1 – 2h ) · 0.5 ˙ (1 ˙ (1 + t ) (22z + 2h – 1) · 0..5 ˙ 0 (1 – 2z ) · 0.5 ˙ (1 ˚
601
– (1 – t )
0 (1 – t ) – (1 – t ) 0 (1 + t ) – (1 + t )
19.21
Note that the Prism6 element is non-conforming with div-conformity in the bottom and top planes of the triangular faces, but it is not div-non conform through the element thickness. In other words, the resin mass will be conserved in the element plane, but mass may be lost through the element height (or thickness). Because the element thickness is usually small compared to the part surface, the accuracy of through-thickness flow calculations remains acceptable with this simple approach.
19.4.5 Flow front advancement The impregnation of the fiber bed by the fluid resin is a dynamic process where the resin flow front is in continuous displacement towards the exit of the mold. This unsteady flow can be solved by considering a succession of quasi steady-state approximations, dividing the problem into a sequence of spatial and transient analyses. At each computed time step, the free surface Gd representing the flow front position moves inside the mold cavity W (see Fig. 19.1). The new location of the free surface Gd is defined by its previous position and the actual velocity field. The methodology to track the displacement of the flow front consists of defining a scalar field S(x, t) in W, called the saturation coefficient (or fill factor). In a fully saturated porous medium (fully impregnated preform) S is equal to 1, while in the unsaturated region (dry fabric) its value is zero. In the partially saturated medium (in the vicinity of the flow front position), S(x,t) lies between these two values. The fill factor is transported in the partially and fully saturated regions until it finally reaches its fully saturated value S = 1 everywhere in the cavity. The pure transport equation for a scalar field S is: ∂S + v · —S = 0 ∂t
19.22
As previously described, the control volumes used in the non-conforming finite element approximation are the elements of the mesh [3]. The flow front is then advanced by transporting the fill factor across the inter-element boundaries (the element edges in 2D meshes or the faces in the 3D case). In order to follow the displacement of a moving flow front on a fixed
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mesh, a fill parameter fe, 0 ≤ fe ≤ 1, is associated to each element of the mesh. As illustrated in Fig. 19.8, this parameter represents, at any given time, the amount of fluid contained in the element. For a porous medium of porosity f, we have fe =
We , 1 ≤ e ≤ Ne fVe
19.23
where We is the volume of fluid inside element e and Ve represents the geometrical volume of the element. At the beginning of mold filling, all fill factors are zero, and at the end they all become equal to 1. During mold filling, each element is associated with a filling time, which represents the instant of complete filling of the element with resin. The knowledge of the filling time for each element of the mesh describes completely the filling of the mold. This approach is less precise to keep track of the flow front than a complete remeshing at each time step of the fluid saturated domain. However, the accuracy can be improved as the mesh is refined. This method provides stable and satisfactory results to follow the motion of the resin front during mold filling. It has none of the drawbacks of the first family of methods based on systematic remeshing at each calculation step. To track the evolution of the flow front, a simple algorithm based on fluid mass conservation can be devised, which provides a fairly accurate first approximation of mold filling. This approach is schematized in the example of Fig. 19.9. Let us assume that the mold is decomposed in a mesh of Ne elements (Ne = 8 in Fig. 19.9) and that the injection is performed through an edge of element 1 at a constant flow rate Q = 1 cm3/s. For sake of simplicity, it is assumed that all elements have the same volume Ve = 2 cm3, 1 ≤ e ≤ Ne and that the porosity of the reinforcement is f = 0.5. It takes Dt0 = 1 s to fill
fe > 1
Overfill region
fe = 1 –
Ve: total volume of element e f · Ve: pore volume of element e
0 ≤ fe ≤ 1 We: fluid volume content inside element e fe = 0 –
19.8 Definition of the fill factor associated with each finite element in the mesh.
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Q = 1 cm3/s t0 = 0
1 2
4 3
t3 = 5 s
6
8
8
5
7
7
t1 = 1 s
4 2
6
3
6
5
8
Ve = 2 cm3 f = 0.5
t4 = 7 s 8
7 4
t2 = 3 s
6 5
t5 = 8 s
8 7
19.9 Example of mold filling simulation at constant flow rate on a fixed FE mesh.
the first triangular element. Then Dt1 = 2 s are necessary to fill elements 2 and 3. Since the flow front is still connected with two empty elements, again we choose Dt2 = 2 s to fill elements 4 and 6, and Dt3 = 2 s to fill elements 5 and 7. The last element, element 8, will be filled in 1 s, so Dt4 = 1 s. The whole mold is filled in 8 s as expected. Figure 19.9 shows the sequence of filled elements and the corresponding filling times.
19.5
Application examples
A simple one-dimensional case with a known mathematical solution can be used to validate the numerical approximation of Darcy’s flow by the FEM. As shown in Fig. 19.10, a radial divergent flow has been used to evaluate the accuracy of the FE modeling. The velocity of the radial flow can be computed using the equation proposed in Table 19.1 for a constant pressure boundary condition. The error of the FEM is estimated as the difference between the mathematical and the numerically computed flow velocity. The mesh in Fig. 19.10 shows the finite element numerical solution. The color gradient indicates the variation in the pressure field from the inlet in the center (bottom left) towards the exit of the part in the outside boundary (upper right). The error of the FE prediction can be decreased by increasing the number of finite elements in the mesh. As shown in Fig. 19.10, the number of finite elements along the flow length Ne has been increased from 10 to 180 triangular elements. For only 10 elements along the length, the solution gives an error below 10% when using non-conforming finite elements. However,
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Number of finite elements (Ne) 20 60
5
1.4
180 Conforming Non-conforming
1.2
18% 10%
Log (error)
1 0.8
6%
0.6
3.5%
0.4 2%
0.2
1%
0 –0.2
Percentage error
604
h 0
0.5
Ne 1 1.5 Log of nodal distance (h)
2
2.5
19.10 Convergence analysis on a radial divergent flow using conforming and non-conforming finite elements.
with conforming elements, the error increases to more than 20%. This error decreases for a larger number of elements and becomes less than 1% for 180 elements along the flow path. However, note that non-conforming finite elements give a better accuracy than conforming elements. If the number of finite elements is large enough, both approximations converge towards the exact solution.
19.5.1 Application to three-dimensional flows Typical composite parts manufactured by LCM are very often thin shells consisting of planar facets of uniform thickness, where the fabric preform is made of homogeneous layers of constant thickness. This implies that the permeability tensor is assumed to have everywhere a symmetry plane parallel to the part mid-surface with the in-plane tensor of the component layers not differing by more than one order of magnitude. This approach allows the fluid flow to be calculated in two dimensions, in fact in the midplane of the part. When important variations in the permeability tensor appear between fabric layers, the resin will flow at different velocities in each ply. If the planar velocity strongly differs from one layer to the other, a through-thickness flow is generated across the plies. In this case, the averaged 2D flow solution no longer matches the real three-dimensional flow, and the mid-plane approach is no longer acceptable. In addition, through-thickness phenomena such as dry spots may appear in the low permeability layers, which remain hidden in
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the averaged flow analysis. In the above cases, three-dimensional solutions must be implemented for accurate prediction of the filling process. To demonstrate the accuracy of the three-dimensional FEM in predicting the resin flow through the thickness of the part, a laboratory test was carried out in a rectangular RTM mold. The experiment performed by Diallo et al. [21] involves the injection of pressurized oil into a rectangular cavity filled with a stack of fibrous reinforcements. Figure 19.11 shows a schematic of the experimental set-up used to record the flow front position in time. The mold assembly is made of two tempered glass plates of 930 mm ¥ 130 mm ¥ 19 mm. The mold cavity thickness was set with a 12 mm thick spacer. A constant injection flow rate was performed with a hydraulic cylinder mounted on a tensile testing machine. The reinforcement was a thermo-formable glass fabric EB-315-E01-120 from Brochier, which has an anisotropic permeability and surface density of 315 g/m2. For this fabric with a fiber volume fraction of 45%, the measured permeability in the weft direction is Kweft = 6.5 ¥ 10–10 m2, in the warp direction Kwarp = 5 ¥ 10–11 m2 and through-thickness Kz = 2.0 ¥ 10–11 m2. The oil viscosity measured at room temperature was 0.103 High permeability layer – 6.5 ¥ 10–10 m2
Injection press
Low permeability layer – 5.5 ¥ 10–11 m2 Impregnated region Flow front location
Data acquisition system Electrical wires Mold The PC speakers are not included in the Laboratory! Wires grid template High permeability
6 mm 3 mm z x 0
Low permeability
–3 mm 100 mm 100 mm 100 mm
–6 mm
19.11 Schematic diagram of the through-thickness flow front shape measurement system used by Diallo et al. [21].
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Pa.s. As illustrated in Fig. 19.11, 20 small wires of 0.1005 circular inches (~0.35 mm diameter) were inserted between the plies to measure the shape of the through-thickness flow front. One wire was placed at the injection port and used as ground. As the oil impregnates the wires, an electrical contact is established and the position of the flow front can be recorded in time. In order to create a through-thickness flow, the preform was constructed from the superposition of 20 plies of fabric aligned in the weft direction, and 20 plies aligned in the warp direction. The upper plies form a highpermeability layer that speeds up the flow, while the low permeability of the lower plies delays the flow. In this experiment, the injection flow rate was maintained constant at 3.04 cm3/s. To simulate this laboratory injection, the three-dimensional mesh of Fig. 19.12 was generated using prismatic (sixnode) non-conforming finite elements. As shown in Fig. 19.12, the 3D model is a brick of size 500 mm ¥ 12 mm ¥ 20 mm with four prism6 elements per layer (a total of eight finite elements through the thickness). The injection boundary condition is set on the faces of the finite elements on the left side of the plate. A comparison of the experimental flow with simulation results appears in Fig. 19.13. At the start of filling, a distortion in the flow front develops quickly (for injection times less than 20 s). Afterwards the two in-plane flows reach an equilibrium state and the distortion is stabilized. The progression of the flow front on the bottom and top mold surfaces is in good agreement with experimental results. The caption of Fig. 19.13 shows the through-thickness profile of the flow at 88 seconds, which corresponds very well to the measures of the electrical wires. This experimental validation demonstrates the use of three-dimensional finite element simulations to predict complex flows in thick composite parts manufactured by LCM. As a further validation exercise, the FEM method was compared with experiments in the case of a spherical three-dimensional flow. Bréard [22, 23] has performed central injections in a thick mold containing an anisotropic reinforcing material. The position of the spherical 3D flow front was detected by X-ray radioscopy. As drawn in Fig. 19.14, in this example the rectangular Injection rate 3.04 cm3/s
High permeability layer
Kweft = 6.5 ¥ 10–10 m2 Kwarp = 5.5 ¥ 10–11 m2
12 mm
Low permeability layer 500 mm
19.12 Three-dimensional model used to simulate the experiments of Diallo et al. [21]. In each layer of the laminate, the mesh contains eight prism6 finite elements in the through-thickness direction.
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0.5 Experimental, diallo et al. [21] Numerical simulaton (prism 6) Flow front location (m)
0.4 High permeability layer Kweft 6.5 ¥ 10–10 m2
0.3
Low permeability layer Kwarp 5.5 ¥ 10–11 m2
0.2
2.8 cm
0.1
Experimental
88s
Numerical 0.0 0
20
40
60 80 Filling time (s)
100
120
140
19.13 Flow front progression in the top and bottom mold surfaces. The flow front distortion is stabilized after 20 s. The numerical through-thickness flow front at 88 seconds is shown in the bottom right caption and is very close to the experimental profile. Silicon oil viscosity = 0.1 Pa.s Fiber volume fraction = 22% KX = KY = 3.95 ¥ 10–10 m2 KZ = 9.3 ¥ 10–11 m2
Injection pressure 1.89 ¥ 105 Pa 300 mm
Kx
20 mm Kz
19.14 Schematic drawing of a spherical central injection. The ratio of the fibrous preform between the planar and transverse permeability is 4.25. X-ray radioscopy was implemented to detect the progression of the three-dimensional fluid flow (Bréard et al. [23]).
mold has an internal cavity volume of 300 ¥ 300 ¥ 20 mm3. Silicon oil was injected through the cylindrical inlet (5 mm diameter) from the center of the mold. The injection pressure was 1.89 ¥ 105 Pa, and the silicon oil viscosity 0.1 Pa.s at room temperature. The in-plane permeability of the porous material used was KX = KY = 3.95 ¥ 10–10 m2, and the transverse permeability KZ =
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9.3 ¥ 10–11 m2. The ratio between the in-plane and transverse permeability is KX/KZ = 4.25. A three-dimensional mesh was constructed on a quarter of the rectangular mold with Prism6 elements extruded in the thickness direction. The model contains around 30 elements along the x and y axes, and 50 elements in the transverse direction (z axis). Elements are refined around the injection gate to ensure a good reproduction of the boundary condition. Figure 19.15 shows a comparison of predicted and measured flow fronts in spherical injection. Simulated flow fronts correspond well to experimental observations, although a small divergence is observed at the beginning of the injection (prior to 2 s). The reason for these differences between experimental flow front locations and numerical calculations may be explained by a slight compaction of the preform surrounding the injection gate at the beginning of the injection. To quantify the error on the flow front prediction, a comparison with an elliptic three-dimensional flow was carried out. Considering a cavity similar to the previous experimental case, but filled with an in-plane isotropic material, but with a different transverse permeability, the flow front position at time tfill can be obtained by solving Darcy’s equation for an elliptic flow [20]: t fill f =
˘ fm È 2 Ê 2 Rf ˆ R – 3˜ + R02 ˙ 1 6 K ÍÎ f ÁË R0 ¯ ˚ p0 – pf
19.24
where K is the isotropic permeability and R0 and Rf denote respectively 50 Experimental, Bréard et al. [23] Numerical (prism6) Flow along X
Flow front location (mm)
40
30
Flow along Z
20
10
0
0
2
4
6
8 10 Filling time (s)
12
14
16
19.15 Comparison of predicted and measured flow fronts in spherical injection. When permeability is measured correctly, numerical results are in good agreement with experiments.
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the injection gate radius and the position of the flow front. The properties considered here are a permeability K = 10–10 m2, a porosity of 0.5, a fluid viscosity of 0.1 Pa.s and an injection radius of 1.12 mm, while the relative injection pressure p0 – pf was set to 105 Pa. Meshes were constructed with different numbers of elements in the radial direction. Prism6 and tetrahedron elements were used to run a series of filling simulations. Figure 19.16 shows a comparison of the numerical results with the three different meshes. Note that the mesh with 30 prism6 elements along the radial direction shows a better agreement with theoretical values than the mesh with 50 tetrahedra. The error on the flow front prediction at the end of filling can be seen in Fig. 19.17. The convergence of the prism6 mesh is higher than for tetrahedron elements. For the same number of elements along the radius, the prism6 mesh gives a smaller error on the flow front estimation than tetrahedra. Note that, in this analysis, the fluid flow occurs in a three-dimensional space. The pressure gradient in the x, y and z directions will be of the same order of magnitude. Although this comparison is very convincing about the ability of the proposed model to predict three-dimensional flows, a loss of fluid mass appears due to the div-non conformity in the through-thickness direction. Therefore, even if the flow front is well predicted, some fluid mass may be lost during the filling simulation due to the inaccuracy of the throughthickness solution of prism6 elements.
0.10 0.09
Flow front position (m)
0.08 0.07 0.06 0.05 0.04 0.03
Analytical
0.02
Tetrahedra, 50 elements on radius
0.01
prism6, 30 elements on radius
prism6, 15 elements on radius
0.00 0
200
400
600 800 Filling time (s)
1000
1200
1400
19.16 Comparison of flow front positions calculated for different mesh sizes with the theoretical solution of an elliptic flow through an isotropic material.
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100
Error in flow front prediction (%)
prism6 elements Tetrahedra
10
1
0.1 10
100 Number of elements in radius
1000
19.17 Convergence of mesh refinement for the elliptic flow through an isotropic material. For the same number of elements on radius, the prism6 solution yields better results than the tetrahedron one.
19.5.2 Application to complex parts The truck fender of Fig. 19.18 is an example to demonstrate the practical use of numerical flow simulation of composites manufacturing by resin injection. This part has a complex three-dimensional shape of variable thickness, and contains different fibrous materials and stacking sequences. The finite element mesh generated from the CAD geometric model posses 17,000 triangular elements and 8500 nodes. To properly represent the variable properties, the part is decomposed into 13 zones with different laminate structures. In LCM, because of the cutting of the fibers and the free space created along the mold edges, the fluid (resin) may flow through preferential paths along the edges of the part. This phenomenon, called the edge effect or race tracking, can be numerically modeled using an enhanced permeability in the elements located along the edges [24–26]. During mold closure, preform interference often appears in surfaces with double curvature, in straight chamfers or in fillets with different internal and external radii. These draping problems create fiber or void-rich zones that strongly influence the flow during mold filling. In some cases, the fluid flow is faster in these regions (resin-rich zone) than in the adjacent preform. In other cases, the flow slows down in regions of high fiber volume content and surrounds the low permeability zones. In the fender model, diverse draping problems are considered, which translate into zones of higher or lower permeability and porosity. The injection strategy consists of four inlet gates around the part and
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611
Vent positions 1.6 m
1 Zones
2
3
2 4
19.18 Truck fender used to demonstrate the capability of the finite element process simulation software. The mesh contains 17,000 triangles and 8500 nodes. The injection strategy consists of four inlet gates at the upper part corners, at the stiffener ends, and in the laminated zone. An impermeable foam core is wrapped by fabric skins in zone 10. Draping effects are represented as regions of high or low permeability.
two vents placed in the central region of the fender. The two inlet gates at the upper corners (cones 1 and 2 in Fig. 19.18) are connected to an open channel free of any reinforcement. This channel allows a quick evolution along the part length (x-axis) and then distributes the resin vertically. This filling strategy is usually called a peripheral injection. Inlet 3 is also connected to an open channel at the end of the stiffener. This strategy is used to ensure a proper impregnation of the longitudinal rib. Injection gate 4 is positioned at the interior of the light support, where the laminate is made of different materials. This central injection gate permits the filling up of the low permeability zone containing the above-mentioned laminate. For all injection gates, the injection pressure is maintained constant at 3.0 ¥ 105 Pa. An isothermal resin viscosity of 0.1 Pa.s was used in this analysis. To expel the air from the cavity, two vents are set in the mold: one at the center of the elliptic shape (vent 1), and the other in the fender lower corner (vent 2). The vent in the elliptic shape of the part is closed when the resin arrives at it. Predicted filling times for this injection strategy are given in Fig. 19.19.
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The flow develops longitudinally through free edges. Possible dry spots appear when flows merge. 1 Flow front position 8 seconds after injection.
1.2 4 sec. D
13
4
1 2
1.2
D
4
3
1.2
se
13
4
8 sec.
13
21
17
17 sec .
c.
8 4
24.2 2 4
Flow front location prior to reaching vent point 2
19.19 Result of the numerical RTM flow simulation of the truck fender. A total filling time of 24.6 s was calculated. ‘D’ indicates possible dry spots.
An injection time of 24.5 s was obtained when the resin arrived at the second vent. The flow front profiles clearly show that resin runs along the upper open channel (between gates 1 and 2) and through the vertical right panel (below gate 2). Possible dry spots have been detected at the interference between the upper descending and the central ascending flows (denoted as ‘D’ in Fig. 19.19). The calculated flow front distribution in the light support is uniform and does not show any impregnation problem. About 8 seconds after the beginning of injection, the resin fronts merge and create an air bubble entrapped in the center of the part. The air bubble exits through the vent gate 1. Approximately at 20 seconds, the resin arrives at vent 1, which
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is automatically closed. This generates a dry spot. At the end of filling (after 21 s), the resin flow takes the shape of a bubble, although no dry spots were numerically detected. This example of a real composite part demonstrates the application of numerical simulation to resin transfer molding in order to simulate the impregnation of the fiber bed. The resulting flow path brings key information on the process and on the quality of the final component. It also helps the user to better understand the complex behavior of the resin flow during manufacturing.
19.6
Conclusions
This chapter sums up the theory of numerical simulation of composites processing by liquid resin injection through fibrous reinforcements. Firstly, the equations governing fluid flows in molds containing the dry fibers are presented together with mathematical solutions for a series of onedimensional cases. Secondly, Darcy’s boundary value problem is solved at each time step in the resin-saturated domain by the finite element method. The equations of the discrete problem are written for various kinds of finite element approximations on triangles, tetrahedra and prisms. As illustrated by simulation results, non-conforming finite elements are shown to possess the remarkable property of conserving the resin mass during mold filling. After describing the methodology followed to track the moving flow front on a fixed mesh, examples of numerical solutions for two- and three-dimensional flows are presented and numerical results are validated against experimental data. The numerical techniques presented in this chapter are currently used by process engineers to predict, control and optimize the manufacturing of high-performance composites by resin injection through dry fiber beds. Finally, the fabrication of a composite truck fender by resin transfer molding is simulated in order to demonstrate the application of process flow simulation on a typical industrial example.
19.7
References
1. Trochu F., Gauvin R., Gao D. M. Limitations of boundary-fitted finite differences method for the simulation of the resin transfer molding. Journal of Reinforced Plastics and Composites, 1992, 11(7): 772–786. 2. Bruschke M. V., Advani S. G. A finite-element/control volume approach to mold filling in anisotropic porous media. Polymer Composites, 1990, 11(6): 398–405. 3. Trochu F., Gauvin R., Gao D. M. Numerical analysis of the resin transfer molding process by the finite element method. Advances in Polymer Technology, 1993, 12(4): 329–342. 4. Trochu F, Boudreault J. F., Gao D. M., Gauvin R. Three-dimensional flow simulations for the resin transfer molding process. Materials and Manufacturing Processes, 1995, 10(1): 21–26. © Woodhead Publishing Limited, 2011
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5. Maier R., Rohaly T., Advani S. G., Fickie K. A fast numerical method for isothermal resin transfer molding filling. International Journal for Numerical Methods in Engineering, 1996, 39: 1405–1417. 6. Voller V., Peng S. An algorithm for the analysis of polymer filling molds. Polymer Engineering and Science, 1995, 35(22): 1758–1765. 7. Maier R., Rohaly T., Advani S. G., Fickie K. A fast numerical method for isothermal resin transfer molding filling. International Journal for Numerical Methods in Engineering, 1996, 39: 1405–1417. 8. Achim V., Ruiz E., Soukane S., Trochu F. Optimization of flow rate in resin transfer molding (RTM). 8th Japan International SAMPE Symposium and Exhibition (JISSE-8), Tokyo, 18–21 November 2003. 9. Remacle J.-F., Bréard J., Trochu F. A natural way to simulate flow driven injections in liquid composite molding. Proc. CADCOMP 98, Computer Methods in Composite Materials, 1998, 6: 97–107. 10. Ruiz E., Achim V., Soukane S., Trochu F., Bréard J. Optimization of injection flow rate to minimize micro/macro-voids formation in resin transfer molded composites. Composites Science and Technology, 2006, 66(3–4): 475–486. 11. Wolfrath J., Michaud V., Modaressi A., Månson J.-A. E. Unsaturated flow in compressible fiber preforms. Composites Part A, 2006, 37: 881–889. 12. Trochu F., Demaria C., Ruiz E., Dambrine B., Godon T. RTM process simulation and optimization of composite fan blades reinforced by 3D woven fabrics. Joint American Society for Composites/ Association for Composite Structures and Materials Conference (ASC/CACSMA), Newark, DE, 15–17 September 2009. 13. Lawrence J., Holmes S. T., Louderback M., Simacek P., Advani S. G. The use of flow simulations of large complex composite components using the VARTM process. 9th International Conference on Flow Processes in Composite Materials (FPCM-9), Montréal, Canada, 8–10 July 2008. 14. Octeau M.-A., Cloutier F., Feuvrier J., Soukane S., Trochu F. Numerical simulation of mold filling and thermal behavior in the manufacturing of an aeronautical composite part. 9th International Conference on Flow Processes in Composite Materials (FPCM-9), Montréal, Canada, 8–10 July 2008. 15. Soukane S., Trochu F. Application of the level set method to the simulation of resin transfer molding. Composites Science and Technology, 2006, 66: 1067–1080. 16. Yoo Y.-E., Lee W. I. Numerical simulation of the resin transfer mold filling process using the boundary element method. Polymer Composites, 1996, 17(3): 368–374. 17. Comas-Cardona S., Groenenboom P., Binetruy C., Krawczak P. A generic mixed FE-SPH method to address hydro-mechanical coupling in liquid composite moulding processes. Composites Part A, 2005, 36: 1004–1010. 18. Béchet E., Ruiz E., Trochu F., Cuillière J.-C. Re-meshing algorithms applied to mould filling simulations in resin transfer moulding. Journal of Reinforced Plastics and Composites, 2004, 23(1): 17–36. 19. Soukane S., Trochu F. New remeshing applications in resin transfer molding. Journal of Reinforced Plastics and Composites, 2005, 24(15): 1629–1653. 20. Ruiz E., Achim V., Trochu F. Coupled non-conforming finite element and finite difference approximation based on laminate extrapolation to simulate liquid composite molding processes. Part I: Isothermal flow. Science and Engineering of Composite Materials, 2007, 14(2): 85–112. 21. Diallo M. L., Gauvin R., Trochu F. Experimental analysis and simulation of flow through multi-layer fiber reinforcements in liquid composite molding. Polymer Composites, 1998, 19(3): 246–256. © Woodhead Publishing Limited, 2011
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22. Bréard J. Matériaux Composites à Matrice Polymère. PhD thesis, Université du Havre, France, 1997. 23. Bréard J., Trochu F. Modélisation de la dynamique des écouplements d’un fluid réactif à travers un milieu poreux déformable en condition anisotherme. Application: procédés LCM. Internal report, Ecole Polytechnique de Montréal, 1999. 24. Hammami A., Gauvin R., Trochu F., Touret O., Ferland P. Analysis of the edge effect on flow patterns in liquid composite molding. Applied Composite Materials, 1998, 5: 161–173. 25. Hammami A., Gauvin R., Trochu F. Modeling the edge effect in liquid composites molding. Composites Part A, 1998, 29(5–6): 603–609. 26. Junying Yang, Yu Xi Jiaa, Sheng Suna, Dong Jun Mac, Tong Fei Shi, Li Jia An. Enhancements of the simulation method on the edge effect in resin transfer molding processes. Materials Science and Engineering A, 2008, 478: 384–389.
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20
Modelling short fibre polymer reinforcements for composites
P. L a u r e, Université de Nice-Sophia Antipolis, France and L. S i l v a and M. V i n c e n t, Mines-ParisTech, France
Abstract: The chapter is focused on fibre-reinforced thermoplastics. The fibre reinforcement improves their mechanical properties but the fibre orientation which results from the injection moulding process has a major influence. The chapter contains results of experimental observations of fibre length distribution, concentration and orientation for complex enough situations in order to describe problems encountered in real industrial processes. Then, the models which allow describing the evolution of fibre orientation in a flow motion are exposed. Finally, computations are made with an injection moulding simulation tool (Rem3D) in which these models are available. The comparison with experimental data concerning fibre orientation prediction gives information on the validity and the influence of various parameters associated with these models. Key words: thermoplastic composite, reinforced polymers, fibre orientation, finite element computation.
20.1
Introduction
The chapter is focused on fibre-reinforced thermoplastics. There are two types of materials. Short fibre-reinforced pellets are obtained by mixing glass fibres and a thermoplastic matrix, polypropylene or polyamide for instance. The fibre content is usually between 30 and 50 wt%. The fibre length is distributed around a mean value of 500 mm. Long fibre-reinforced pellets are obtained by continuous impregnation of a fibre, by a pultrusion-like process. After solidification, the impregnated fibres are cut at a length around 10 mm. The fibre length is the same as the pellet length. Both materials are semi-product, adapted for fast production processes such as injection moulding or extrusion. As with any composite, their properties are a function of the properties of the matrix, the fibres and the interface between matrix and fibres, but also of the fibre microstructure. This notion covers first the fibre orientation. Indeed, fibres get oriented by the flow, and their properties are anisotropic. It also covers the fibre length: it depends on the initial fibre length in the pellet, but also on the plasticising and injection steps of the process. Lastly, there is the fibre concentration, which is usually around 30–50 wt%, that is, about 616 © Woodhead Publishing Limited, 2011
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half in volume for glass fibres, according to the glass and polymer densities. But the transport of the fibres by the polymer may be heterogeneous and fibre-rich or fibre-poor regions may be found. The second section of this chapter reviews the experimental observations, and the variations of fibre length distribution, fibre concentration and orientation are discussed. The experimental procedures giving these data are also briefly described. The third section presents the models describing the evolution of fibre orientation in the flow motion. The most recent models which deal with concentrated suspensions are also discussed, as well as the coupling with the rheological behaviour. Finally, numerical computations are performed on three typical examples and the influence of various parameters encountered in the models is analysed. Then, the limitations of these models for real parts are discussed.
20.2
Observations
20.2.1 Fibre length distribution As mentioned before, fibre length in the moulding depends on the initial fibre length in the pellet, and on the fibre degradation in the screw-barrel system of the plasticising unit and in the mould cavity. Most methods for measuring fibre length are destructive (Kamal et al., 1986; Chin et al., 1988; Franzen et al., 1989; Gupta et al., 1989; Denault et al., 1989; Akay et al., 1995; Avérous et al., 1997; Davidson and Clarke, 1999). The composite matrix is dissolved or burnt out. The fibre lengths are measured by image analysis of microscope pictures. The number of fibres is large (nearly 3000 per mm3 for 30 wt% and fibres 300 mm long and 15 mm in diameter!) and a careful representative selection must be made. The image analysis technique must be able to take into account a large length distribution. Glass fibre breakage is more important in the plasticising unit (Gupta et al., 1989) than downstream in the runner and mould cavity. It is a function of fibre concentration (Denault et al., 1989; Akay et al., 1995; Tremblay et al., 2000). Vincent (2009) measured the fibre length distribution in a plaque mould for two reinforced polypropylenes, containing 30 wt% of short glass fibres and 30 wt% of long glass fibres (Fig. 20.1). The average length of the long fibres, initially 12 mm, reduces to 0.87 mm. The short fibre length, initially 0.56 mm, reduces to 0.41 mm. Fibre breakage for the long-fibre composite is important. Figure 20.2 shows a plaque 150 ¥ 150 ¥ 3 mm, with seven ribs 25 mm apart from each other. The rib thicknesses from the entrance to the tip are 1, 3, 3, 2, 2, 3 and 3 mm. The rib heights are 9, 12, 9, 12, 9, 12 and 9 mm.
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0.5 0.45 GFC (SFT) L = 0.41, Li = 0.50 mm
0.4 0.35
Frequency
0.3 0.25 0.2
GFL (LFT) L = 0.87, Li = 12 mm
0.15 0.1 0.05 0
0
1
2
3
4 5 Length (mm)
6
7
8
9
20.1 Fibre length repartition in a 30 wt% long and short glass fibrereinforced polypropylene moulded plaque.
20.2 Geometry of the plaque with seven ribs: the size is 150 mm ¥ 150 mm ¥ 3 mm and the ribs are spaced at 25 mm.
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The material is a 40 wt% long glass fibre reinforced polypropylene (pellet length 12 mm). Table 20.1 shows the average length at different positions for two mould filling flow rates. There is not a large difference between the rib and other regions in the part. The average length reduces by about 20% along the flow direction. The sensitivity to the flow rate is negligible.
20.2.2 Fibre concentration Most authors have found small variations along the flow direction, around 1 to 2 wt% for a total amount of fibres of 30 wt% (Hegler and Menning, 1985; Kubat and Szalanczi, 1974). In the thickness directions, the fibre concentration is higher in the core than in the skin of the moulding, especially for a high fibre loading of 50 wt% (Kamal et al., 1986; Akay and Barkley, 1991). For the same part shown in Fig. 20.2, and with the same material, Fig. 20.3 shows the fibre concentration. The concentration is higher in the ribs, except in the first one, than in the plaque, but overall the deviation from the concentration of the semi-product, 40 wt%, is not very large.
20.2.3 Fibre orientation Fibre orientation depends on the type of flow. In a shear flow, a single fibre rotates, but spends most of the time aligned with the flow direction. In a Table 20.1 Average length in mm for different positions in the part shown in Fig. 20.2 Position
Q = 9 cm3 · s–1 Q = 122 cm3 · s–1
First rib Plaque near first rib Plaque between fourth and fifth ribs Plaque near fifth rib Fifth rib
1200 1200 1000 900 1000
37.4
40.8
41.8
42.4
1300 1300 900 1000 1000
44.8
43.5
39.4
39.4 38
40 37.3
40 38.8
39 39.2
39.9 39.6
38.9 40.4
39.9 39.5
41.6 40.8
20.3 Fibre weight concentration (wt%) in the part shown in Fig. 20.2.
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highly concentrated suspension, when fibres are interacting, fibres are mostly oriented in the flow direction. In an elongational flow, fibres get oriented in a stable position either parallel to the flow with a positive elongation rate, or perpendicular with a negative elongation rate. Interactions with neighbouring particles may disturb this orientation. In a real cavity filling situation, the flow is a complex combination of shear and elongation. Shear reaches a maximum in the skin, and elongation in the core. Elongation always occurs at the junction between the sprue or runners and the cavity itself, because the cross-section of the flow increases. This is why a skin–core structure is often observed in this region. Downstream, if the flow is shear dominated, fibres in the centre are reoriented in the flow direction, but the core region may appear again in case of increase of cross-section. Figure 20.4 shows polished cross-sections of long glass fibre mouldings. In the surface, fibres appear mainly oriented in the flow direction, whereas in the centre, they are perpendicular to it. They are mostly parallel to the plan of the part. Short fibre composites show the same type of orientation. When the fibres are long enough, more than around 500 mm, they can be slightly curved. Otherwise, they appear as straight rods. In order to be precise about the orientation, it is necessary to quantify it. The fibre orientation distribution function y (p, t) is defined by the probability y (p, t) dp of finding a fibre oriented between p and p + dp, where p is a unit vector aligned with a fibre (Prager, 1957). the second-order orientation tensor a2 is easier to use in comparing two orientation patterns (Hand, 1961). It is defined as the spatial average of the double tensorial product of p, and it is symmetrical and positive definite: aij =
Úp y (p, tt))pi p j dp
20.1
The trace of the tensor is equal to 1. For random orientation in space, the diagonal terms are equal to 1/3, and for random planar orientation, to 1/2. For perfect orientation in direction i, aii = 1, and the other diagonal values are zero. A diagonal tensor means that the reference frame axes are the principal axes of the tensor. In order to obtain the orientation distribution function y or the secondorder orientation tensor components, each fibre orientation in a given volume must be determined. Several techniques can be used (see, for instance, Clarke and Eberhardt, 2002 for a review), but the most widely used is to observe carefully polished cross-sections, such as those shown in Fig. 20.4. Cylindrical fibres making a certain angle with respect to the cutting plane appear as ellipses. The measurement of the orientation and length of the semi-axes of the ellipse leads to the vector p components in 3D. The image analysis technique must be accurate enough to separate touching fibres or to measure nearly circular fibre cross-sections, and corrections must be
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Flow 2 3 1
493 µm (a) 2
1
3 Flow
493 µm (b)
20.4 Pictures of a short fibre-reinforced polymer moulded plaque. The thickness is in the vertical direction: (a) flow direction perpendicular to observation plane; (b) flow direction parallel to large side of picture.
applied for long fibres for which the probability of intersecting the edge of the observed field is higher. Another important correction is necessary because the probability of observing fibres perpendicular to the observation plane is higher than when they are parallel to it (Bay and Tucker, 1992). Other techniques can be used, such as confocal laser scanning microscopy (Eberhardt and Clarke, 2001) and X-ray microtomography (Shen et al., 2004).
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This last technique allows the 3D reconstruction of the fibres in a given volume. It is very powerful, as not only fibre orientation but also length, local concentration and eventually curvature can be evaluated.
20.2.4 Skin–core structure A review of observations in various part geometries, such as plaque or centre gated discs, with different composites can be found in Papathanasiou (1997). The skin–core structure is often observed, the relative thickness of the skin and core layers depending on the processing conditions. At the flow front, fibres are mostly tangential to the front. This is why weld line regions exhibit weaker mechanical properties. Figure 20.5 shows the gapwise evolution of the orientation tensor component in the flow direction, in a 50 wt% glass fibre reinforced polyarylamide plaque, for four cavity thicknesses (Vincent et al., 2005; Vincent, 2009). The skin–core structure exists for the two largest thicknesses, but vanishes for the smallest ones. The shape of the plaque entrance, where the core is created, and the high shear rate generated by a small flow gap are responsible for this difference.
20.3
Models
20.3.1 Fibre orientation: Jeffery theory Jeffery (1922) considered a single rigid ellipsoidal particle with very small dimensions. Thus, rate of deformation can be considered as homogeneous 1
0.8 1.1 mm
0.6 axx
1.7 mm 3 mm
0.4
5 mm
0.2
0 0
0.2
0.4 0.6 Dimensionless thickness
0.8
1
20.5 Orientation tensor component in the flow direction for four plaque thicknesses from 1.1 to 5 mm (after Vincent, 2009).
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around the particle, and he supposed that this particle was immersed in a Newtonian fluid. The following time evolution equation of the unit vector p aligned with the particle axis was thus obtained: ddpp = W p + l [e p – (e : p ƒ p ) p ] dt
20.2
where e is the rate of strain tensor and W is the vorticity tensor, both defined as functions of v, the velocity field:
e (v) = 12 (—v + —vt )
20.3
W((v) = 12 (—v – —vt )
20.4
l is a function of the aspect ratio of the particle b = L/D, with L the length and D the diameter of the fibre:
l=
b2 – 1 b2 + 1
20.5
Typically, short fibres have a diameter between 10 and 20 mm, and a length between 100 and 500 mm, so that b is of the order 5 to 50, and l is larger than 0.92. In this theory, a particle rotates periodically in a simple shear flow, with a period of rotation of 1ˆ Ê Tf = 2p Á b + ˜ b¯ e Ë
20.6
where e = 2 S eij2 is the is the magnitude of the strain rate tensor. This i, j
relation has been experimentally validated (Moses et al., 2001). If the particle is infinitely long (slender body), the particle does not rotate any more but tends to orient in the flow direction. In elongation flows, equation 20.2 shows that a particle tends to a stable equilibrium position, parallel or perpendicular to the flow direction when the elongational rate is positive or negative, respectively. The kinetics of orientation are almost independent of the fibre aspect ratio when it is high. Later, Bretherton (1962) theoretically demonstrated that any rigid body of revolution has a motion in shear flow identical to an ellipsoid, meaning that the theory could be extended to fibre motion.
20.3.2 Fibre orientation: Folgar–Tucker model Jeffery’s equation is valid for dilute suspensions, when the fibre volume fraction f 1/b2. In the semi-concentrated regime, 1/b 2 f 1/b,
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hydrodynamic fibre interactions occur, whereas in the concentrated regime (f 1/b), fibre–fibre contacts also appear. Fibre-reinforced thermoplastics are often in this latest regime. Since in the concentrated regime one cannot follow each fibre, the orientation distribution function, y(p, t), defined in the previous section, is used. The equation of conservation of y is of the Fokker–Planck type and is written (by neglecting fibre Brownian motion, in the dilute case) ∂y Ê ddppˆ + ∂ y =0 ∂t ∂p ÁË dt ˜¯
20.7
For concentrated regimes, one introduces a pseudo-Brownian diffusion, giving ∂y ∂2y Ê ddppˆ + ∂ Áy ˜ = Dr 2 = 0 ∂t ∂p Ë dt ¯ ∂p ∂p
20.8
The distribution function is a complete and accurate representation of the orientation state. However, solving this equation requires significant computational resources, and for industrial applications, it is better and easier to use the second-order orientation tensor, a2, defined in equation 20.1. Passing from an orientation distribution function y(p, t) to a2 implies a loss of information. Thus, one may use higher-order tensors, such as a4, the fourth-order orientation tensor: a4 =
Úp
p ƒ p ƒ p ƒ p y ((pp ) dp
20.9
and may reconstruct y (p, t), as suggested by Advani and Tucker (1987), from both tensors. To directly solve orientation motion based on the second-order orientation tensor, Lipscomb et al. (1984) provided the following equation, after volume averaging: da2 = Wa2 – a2 W + l (ea2 + a2 e – 2e:a4 ) dt
20.10
For concentrated solutions, Folgar and Tucker (1984) chose a phenomenological approach. They added a term to Jeffery’s equation using an analogy with rotary Brownian diffusion to account globally for these complex interactions. They obtained: da2 = Wa2 – a2 W + l (ea2 + a2 e – 2e:a4 ) + 2Cie (I – 3a2 ) dt
20.11
where Ci is an empirical constant called the interaction coefficient. This model has been extensively used, and is the standard model. Two
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questions arise when treating this equation: how to compute a4 from a2? And what are the admissible values for CI? Closure approximations The fourth-order tensor a4 appears in the time evolution equation for a2, thus a closure approximation is needed to approximate a4 as a function of a2 (since to compute a4, one needs also a6). Tests carried out in simple flows (shear, elongation, or a simple combination of both) showed that the closure approximation has a large influence on the quality of the result. The simplest approximations are the quadratic closure (which gives exact results when fibres are perfectly aligned) and the linear closure (exact for random orientation). The hybrid approximation is a linear combination, depending on the orientation, of the two previous ones (Advani and Tucker, 1987). One other class of closure approximations includes fitted parameters in simple flows, like the orthotropic (Cintra and Tucker, 1995; Wetzel and Tucker, 1999) or the natural closures (Dupret and Verleye, 1998). In these approaches, the fourth-order tensor is written as a linear function of the invariants of the second-order orientation tensor. Coefficients of the different functions corresponding to each approximation are obtained by fitting to the analytical solution of the steady-state orientation distribution function for several flow situations: shear, elongation and combinations of both. Performance studies of closure approximations in shear flows show that linear closures may provide non-physical oscillations, which does not happen with quadratic or hybrid closures. The transient state is generally overestimated (with longer times), whereas the stationary one is underestimated (with shorter times); fibre alignment prediction is very unidirectional. In elongation flows, linear closure may also give unrealistic values, whereas quadratic or hybrid approximations provide better results, even if they overestimate the transient region. Orthotropic closures always provide the best results and are the most commonly used. Interaction coefficient: theoretical, numerical and experimental determination For shear flows, the interaction coefficient value is very important in making good predictions of the steady-state solution of the orientation tensor: if CI is high, the orientation becomes isotropic, whereas for a weak CI, fibres tend to align in the flow direction. On the other hand, in elongation flows, CI influences the orientation transient solution: if CI is high, fibre orientation attains the steady state more rapidly than for lower values. In a shear flow, when CI is around 10–4, a11 is close to 1, meaning that fibres are very well oriented in the flow direction. When CI increases, a11
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decreases, and for Ci = 1 the orientation becomes nearly isotropic (a11 = 1/3). The fibre aspect ratio b has a negligible influence on the steady-state attained value, even though Hinch and Leal (1973) showed that for very weak diffusion (a very small interaction coefficient) it may become important. the value of Ci can be determined through theoretical, numerical or experimental fitting. In this last case, it can be difficult to find only one Ci that fits in the whole flow range. Bay and Tucker (1992) used an empirical approach, based on experimental values for injected discs and plaques for different polymers and concentrations, to propose: Ci = 0.0814e–0.7148 f b
20.12
On the theoretical side, Ranganathan and Advani (1991) proposed a relationship between the interaction coefficient and the distance between fibres, h: Ci = K L h
20.13
where K is a constant determined experimentally and L is the fibre length. Direct numerical simulation has also been used to estimate Ci values. Yamane et al. (1994) modelled fibre–fibre interaction for a Newtonian fluid in shear flow using lubrication forces. The authors obtained very low interaction coefficients (10–7 < Ci < 10–4). Fan et al. (1998) developed a numerical approach to take into account fibre–fibre interactions, leading to an interaction coefficient that has a tensorial form, which will be discussed in the next section. Phan-Thien et al. (2002) extended the previous approach by using the trace of the interaction coefficient tensor, and they proposed the following form for Ci: Ci = 0.03(1 – e–0.224 f b)
20.14 –3
The values obtained seem of a good order of magnitude (10 < Ci < 10–2). Furthermore, one obtains the interesting result that the interaction coefficient increases with the fibre volume fraction and the fibre aspect ratio. This is a point of discussion, since high volume fractions do not often lead to isotropic orientations, which are the result of Folgar and Tucker’s model when one considers a high Ci. The models discussed in the next section will enlighten this fact.
20.3.3 Fibre orientation: recent models with anisotropic fibre interaction In the previous standard model, an isotropic rotary diffusion term was added by Folgar and Tucker. The diffusivity is proportional to the magnitude of the rate of deformation (scalar), through the constant Ci. Experimental data has shown that, for concentrated suspensions, the kinetics of orientation are
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much slower than are predicted by Folgar and Tucker’s standard model. To avoid this effect, Wang et al. (2008) introduced the reduced strain closure (RSC) model. The authors write a2 as a function of its eigenvalues Li and eigenvectors ei, and suppose that the eigenvalue kinetics are slowed by a factor of k and that the eigenvector kinetics are unchanged. The tensorial material derivative of a2 is then recalculated, resulting in the following variation of the Folgar and Tucker model: da2 = Wa2 – a2 W dt + l (ea2 + a2 e – 2e ::[[a4 + (1+ (1 + k )( )(l4 – m4 a4 )]) + 2k Cie (I – 3a2 ) 20.15 the fourth order tensors l4 and m4 are functions of the eigenvalues and eigenvectors of a2: 3
l4 = S L i (ee i e i e i e i ) i =1
20.16
3
m4 = S (ee i e i e i e i ) i =1
20.17
Parameter k is determined by fitting experimental data. For short fibre reinforced thermoplastics, it ranges from 0.05 to 0.2 (Wang et al., 2008). To avoid the use of closure approximations, Wang et al. (2008) have also derived a Fokker–Planck equation including this slow kinetics phenomenon. Nevertheless, it remains computationally expensive when compared with the tensorial form. Using a different approach, Férec et al. (2009) obtained a similar model. The authors have considered that orientation kinetics become slower with semi-concentrated suspensions because of hydrodynamic and fibre–fibre interactions. The force generated by fibre interactions was modelled using a linear hydrodynamic friction coefficient proportional to the relative velocity at the contact point and weighted by the probability for contacts to occur. Starting from this point, the authors obtained the following orientation evolution equation: da2 = Wa2 – a2 W + l (ea2 + a2 e – 2e:a4 ) dt (eb2 + b2 e – 2e:b4 ) + 2 f f M q e (I – 3a2 ) + fM
20.18
is a parameter related where f is the average number of contacts per fibre, M to drag, and q is a dimensionless interaction coefficient. b2 and b4 are the second-and fourth-order interaction tensors. b2 is given by
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b2 = 3p (a2 – a4 : a2 ) 8
20.19
and the fourth-order interaction tensor, b4, is computed from the second-order one through a quadratic closure approximation. Férec et al. (2009) have demonstrated that their model can be rewritten as a RSC model type. Globally, the RSC model behaves well for short fibre reinforced materials. For long fibre thermoplastics, it quantitatively generates higher flow aligned fibres which do not agree with the experimental values. Thus, rotary diffusion models have been lately developed (Ranganathan and Advani, 1991; Fan et al., 1998; Phan-thien et al., 2002; Koch, 1995; Phelps and Tucker, 2009). Ranganathan and Advani (1991) proposed a model in which the interaction coefficient is inversely proportional to the average interfibre spacing. Diffusion is isotropic and not really applicable to long fibre reinforced materials. Fan et al. (1998) and Phan-Thien et al. (2002) developed a rotary diffusion anisotropic model by replacing the interaction coefficient Ci by a secondorder tensor C, computed by performing direct numerical simulations of a REV of a concentrated suspension undergoing a simple shear flow (Beaume, 2009). C was then determined using the steady-state solution, not exploiting dynamic behaviour. Koch (1995) obtained also an expression for the tensor C using a mechanistic approach, by considering the influence of hydrodynamic fibre–fibre interactions on the orientation development in the semi-dilute regime. This tensor is of the form C=
nL3 [b (e : a :e ))II + b (e: a :e )] 1 6 2 6 e ln 2 b 2
20.20
where n is the number of fibres per unit volume, b1 and b2 are parameters computed by fitting with analytical orientation values in elongation flows, and a6 is the sixth-order orientation tensor. The author was then led to the following orientation equation: da2 = Wa2 – a2 W + l (ea2 + a2 e – 2e:a4 ) dt + e [2C – 2 (t (trr C )a2 – 5(Ca2 + a2 C ) + 10 a4 :C ]
20.21
Since the sixth-order tensor needs to be computed, the Koch model is more computationally costly. It does not provide better results than Folgar and Tucker’s model for long fibre reinforced thermoplastics, and the quality of results is also dependent on the ratio between b1 and b2 (if b1 is much larger than b2, diffusion becomes mostly isotropic and the model close to Folgar and Tucker’s one). Recently, Phelps and Tucker (2009) developed a phenomenological anisotropic rotary diffusion model, using also a second-order tensor to describe
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interaction, by allowing the diffusion term to depend on the orientation state and on the rate of deformation tensor, using the final expression b b C = b1 I + b2 a2 + b3a22 + 4 e + 52 e 2 e e
20.22
The five scalar parameters b1, …, b5 are fitted in order to get steady-state orientation solutions in simple shear flow and various elongation flows, ensuring stable-steady state positive eigenvalues, as well as physical solutions. The so-called ARD-RSC (anisotropic rotary diffusion – reduced strain closure) model is thus written: da2 = Wa2 – a2 W + l (ea2 + a2 e – 2e :[a4 + (1 – k )(l4 – m4 a4 )]) dt + e (2[ (2[C – (1– (1– k ) m4 :C ] – 2k ((tr tr C ) a2 – 5 (Ca2 + a2 C ) + 10[a4 +(1– +(1– k )(l4 – m4 a4 )]:C }
20.23
This latest model has been shown to be predictive for long fibre thermoplastic composites, even though there are a large number of parameters to fit.
20.3.4 Rheological models Generic form with rheological coefficients Np and Ns In the most general case, we consider that the extra stress tensor is the sum of the contribution of the fluid and of the particles (Batchelor, 1971), and that a reinforced polymer is a suspension of rigid particles in a Newtonian fluid. In this case, most theories have derived the following expression for the stress tensor (Tucker, 1991):
s = – pI + 2hI [e + N p e : a4 + N s (ea2 + a2 e )]
20.24
where hI, Np and Ns are parameters depending on the fluid viscosity, fibre aspect ratio, fibre orientation and fibre concentration. They can be obtained from rheological experiments that are difficult to perform in the concentrated case. On the other hand, theoretical expressions obtained for each of these parameters depend on the concentration domain. It is most often the dilute and semi-concentrated regime which limits greatly their wide application. Three type of approaches are used in the literature to obtain Np and Ns: the slender body theory, where a particle is considered as an infinitely thin fibre (of negligible thickness) (Batchelor, 1971; Dinh and Armstrong, 1984; Shaqfeh and Fredrickson, 1990); the derivation of known results for suspensions of ellipsoidal particles, valid for finite fibre aspect ratios (Hinch and Leal, 1973; Lipscomb et al., 1988; Hand, 1961); and the approximation based on the theory of Doi and Edwards for concentrated suspensions, where the Brownian
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motion of the suspension is not neglected and a diffusion coefficient appears in the stress tensor expression. In all the above theories, Np is larger than Ns for high aspect ratios. Slender body approximation and ellipsoidal particle theories In the approximations based on the slender body theory, hI is equal to the fluid viscosity and Ns = 0. They are valid for dilute (for f 1/b2) (Batchelor, 1970) or for semi-concentrated suspensions (Batchelor, 1971; Dinh and Armstrong, 1984; Shaqfeh and Fredrickson, 1990). For dilute suspensions, Batchelor (1970) proposed the following expression: Np =
b 2f 3ln (b )
20.25
To extend the approach to the semi-concentrated regime, authors have considered hydrodynamic and fibre–fibre interactions, as well as lubrication forces. Hence, Batchelor (1971) gave Np =
b 2f È ˘ Ê fˆ 9 Ílln (22 b ) – ln Á1 + 2b ˜ – 1.5˙ p¯ Ë ÎÍ ˚˙
20.26
whereas Dinh and Armstrong (1984) considered the influence of orientation by providing Np as Np =
b 2f Ê 2 hˆ 3ln Á ˜ Ë D¯
20.27
where h represents the characteristic distance between two neighbouring fibres, a distance that depends on the particles’ orientation. A similar expression has been obtained by Shaqfeh and Fredrickson (1990). Ranganathan and Advani (1991) proposed a modification to the Batchelor (1970) model using a corrective factor function of the fibre aspect ratio, providing better results but very sensitive to this correction. Other expressions are based on ellipsoidal particle theories (Lipscomb et al., 1988; Hinch and Leal, 1973) where Ns is different from zero but much smaller than Np, and hi is different from the fluid viscosity. For dilute suspensions, they take the form: Np =
f b2 1 + 2f 2[ln (2b ) – 1.5]
Ns =
f 6 ln (22b ) – 11 1 + 2f b2
20.28
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hi = h (1 + 2f) Extensions of this model have also been proposed, to improve the results when going towards the semi-concentrated regime. Model with interaction tensor for concentrated suspension Based on the work of Dinh and Armstrong (1984) for slender fibre suspensions, Férec et al. (2009) have derived the following expression for the total stress tensor, including a term that is a function of the interaction tensor:
s = – pI + 2hI [e + N p e :a4 + N b e :b4 ]
20.29
with Np =
f D2 XA 12 p
Nb =
2f 2 D 2 K 3p
20.30
where XA is the parallel drag coefficient to the fibres, dependent on the nature of the fibre–matrix contact, and k is a dimensionless geometric factor. For very low fibre volume fractions, Nb becomes insignificant and the total stress reduces to the Dinh and Armstrong (1984) model.
20.4
Computation of fibre orientation in injection moulding
We present computational examples of fibre orientation in injection moulding for which experimental data exist. The various moulds are geometrically complex enough to describe problems encountered in real industrial processes. In the first example, we check whether the skin–core effect is accurately computed in a rectangular plaque. Moreover, the influence of the orientation–rheology coupling and of the interaction coefficient Ci is analysed by computing the orientation near the inlet gate. The second example is a U-shape with thin walls and ribs, which is used in the automotive industry. It shows the ability to compute the appearance and location of weld lines. The third example is the plaque with seven different ribs (see Fig. 20.2) which gives information on the orientation state in small parts. First of all, the numerical methods are briefly described. The methodology is that used in Rem3D software and is based on an Eulerian approach (the whole injection cavity is meshed) and a stabilised Galerkin method, using a continuous approximation of the orientation tensor coupled to flow (generalised Stokes behaviour), the thermal equation, the flow front evolution (using a level set method) and fibre orientation equations.
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20.4.1 Numerical methods A weakly coupled approach is used, meaning that the problem is solved in two steps. The first step involves the solution of the mechanical problem, assuming that the polymer is incompressible and neglecting gravity and inertia: — · s = 0, — · v = 0
20.31
with an isotropic orientation at the first step or an orientation tensor determined at the previous time step for the other instants. The orientation is taken into account through expression 20.24 or 20.29. A classical mixed finite element method (Pichelin and Coupez, 1998) is used, in which the orientation tensor is implicitly considered. The fluid is assumed to follow an incompressible shear-thinning behaviour, represented by Carreau–Yasuda and Arrhenius laws for the viscosity h: È Ê ˆa ˘ h = h0 (T ) Í1 + Áh0 e ˜ ˙ ÍÎ Ë t c ¯ ˙˚
m –1 a
È Ê1 1 ˆ˘ , h0 (T ) = h0 (Trref ˜˙ ef ) exp ÍbT Á – Ë T T ref ¯ ˚ re Î 20.32
The parameters occurring in these laws are given in Table 20.2 for the fluid studied. As viscosity depends on temperature, the energy equation resolution is coupled and solved: Ê ∂T ˆ rC p Á + u · —T ˜ = — · (kt—T ) + h he 2 ∂ t Ë ¯
20.33
where r is the volume density, Cp the specific heat, kt the thermal conductivity, and the last term represents the viscous dissipation. These parameters are Table 20.2 Parameters used in equations 20.32 and 20.33 for PAA50, Stamax P30YM240 and P40YM243: Cp is the specific heat, k is the thermal conductivity
tc h0(Tref) a m Tref bT kT Cp r
PAA50
P30YM240
P40YM243
0.154 MPa 570 Pa.s 0.55 0.3 549 K 7764 mole.K 0.3 W/m.K 1766 J/kg.K 1522 kg/m3
0.0411 MPa 252 Pa.s 1 0.22 523 K 4450 mole.K 0.25 W/m.K 2180 J/kg.K 1000 kg/m3
0.02626 MPa 636.6 Pa.s 1 0.271 493 K 4450 mole.K 0.15 W/m.K 3100 J/kg.K 1000 kg/m3
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considered to be constant. Finally, heat conduction towards the mould walls and viscous heating are also taken into account. The second step involves the computation of fibre orientation with velocity field obtained in the previous step, through the resolution of the evolution equation 20.11. Numerical resolution of Folgar and Tucker’s equation has been performed by Kabanémi and Hétu (1999) using the fourth-order Runge–Kutta method, by Martinéz et al. (2003) with the method of characteristics, and by Pichelin and Coupez (1999) and Redjeb et al. (2005) with a space–time discontinuous Galerkin scheme. Miled et al. (2008) have proposed a standard Galerkin method associated with a RFB or SUPG stabilisation, which prevents inaccurate oscillations due to the hyperbolic character of this equation. A moving interface (such as fluid/air) is also calculated at each time step by solving a convection equation (Ville et al., 2010) associated with a signed distance function which defines the fluid domain (its value is positive in the fluid and negative in the empty region): ∂a + v · —a = 0 ∂t
20.34
Finally, all parameters and equations are extended to the whole computational domain, as the weak formulation of the finite element method ensures the continuity of stress, velocity, orientation tensor and temperature (Batkam et al., 2004).
20.4.2 Representation of orientation The problem is now to get practical information from the computed orientation tensor. In an orthonormal Cartesian basis, the diagonal term aii of the orientation tensor quantifies the fibre alignment along the main axis. As already mentioned in the first section, a11 = 1 means that all the fibres are oriented along the direction i1. Moreover, a11 = 0 indicates that all the fibres are perpendicular to the axis i1, that is, they belong to the plane. (i2, i3). The components aij with i ≠ j quantify the asymmetry of the distribution of orientation relative to the direction ii or ij. For isotropic orientation in 3D, the components are aii = 13 while aij = 0 for i ≠ j. Another way to describe the computed orientation is to plot the ellipsoid associated with the eigenvalues, i, and eigenvectors, ei, of the orientation tensor a2 (Advani and Tucker, 1987; Altan et al., 1990). The eigenvectors and eigenvalues of the orientation tensor give, respectively, the direction and length axes of the ellipsoid as shown in Fig. 20.6. In order to determine the degree of anisotropy, a von Mises effective orientation, av, can also be computed:
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3
1 l2 e 2 l3 e 3
20.6 Ellipsoid representation of orientation tensor a2.
av =
2 2 2 (aa11 – a22 )2 + (aa22 – a33 )2 + (aa11 – a33 )2 + 6 (a12 + a13 + a23 ) 2
20.35 It is null for an isotropic orientation and 1 if all fibres are oriented along a principal direction.
20.4.3 Rectangular plaque with inlet gate We consider the injection moulding of a three-dimensional plaque (Fig. 20.7(b)), and in particular we study the orientation development near the injection gate. Dimensions of the plaque are given in Fig. 20.7(a). Only half of the plaque will be considered in the simulation, since the geometry presents a symmetry plane. The polymer is a polyarylamide (Solvay Ixef 1022) reinforced with 50% weight (31.6% volume) glass fibres. The fibre aspect ratio is considered constant and equal to 10. The injection is done at a flow rate of 20 cm3/s and the initial polymer temperature is 270°C. The mould temperature is kept constant at 130°C, and the mould is filled in 2.70 s. The computational time step is 0.002 s. The orientation distribution is analysed after 1300 time iterations, almost at the end of cavity filling. In order to reduce the computational time, the long circular channel has not been taken into account. The inlet of the cavity is just after the last curve
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200 mm 3 40 mm
A Thickness = 5 mm (a)
A¢
1
2
Cut plane x2 = 0:
x2 = 0
x1 = 128
Position A x1 = 111
Position B x1 = 108 (b)
Position C Position D x1 = 1015 x1 = 94
20.7 (a) Mould schematic view and (b) zoom on the inlet with positions of sensors.
of the feeding channel. The computations are made on a half-mould with symmetric boundary conditions on the plane (i1, i2). The problem is now to impose an initial orientation at the cavity inlet as we do not take into account the cylindrical channel. Usually one considers an isotropic initial orientation. But the flow in the channel is dominated by shear deformations, which are known to orient fibres mainly in the flow direction (namely the i1 axis) at the entrance of the triangular gate. Nevertheless, the two curves in the cylindrical channel can disturb this orientation. So, computations are made for two different initial conditions (isotropic case, aii = 13 ; unidirectional case, a11 = 1) and a hybrid closure approximation. Moreover, two values of the interaction coefficient (Ci = 0.001 and 0.04) and two values of the rheological coupling coefficients (Np = 0, 100 and Ns = 0) will be tested. Figure 20.8 shows cross-sections in the (i1, i2) plane at the end of filling and the distribution of the diagonal orientation tensor components. We notice that the tensor is almost diagonal, so the (i1, i2, i3) axes are the principal axes of the orientation tensor. Also, the skin–core effect initiated at the end of the gate is preserved until the end of the plaque. Computed values of a22 are very small, except at the junction between the divergent region and the plaque. This means that fibres are parallel to the (i1, i3) plane in the areas where experimental observations are made. This is in agreement with the observations, and we will focus on the a11 component which quantifies the
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1 a11
0.9 0.8
a22
0.7 0.6
a33 0.4
Vpmax
0.3 0.2 0.1 0
2 3
1
20.8 Isovalues of a11 at the end of filling in a cross-section (i1, i2, i3 = 0) for the PAA50 with isotropic initial orientation, Np = 0 and CI = 0.001.
degree of alignment with the flow direction. The triangular gate region is divergent in the plane (i1, i3) and convergent in the plane (i1, i2) (see Fig. 20.7). The negative elongation rate along the i3 axis dominates over the positive one along the i1 axis and an orientation perpendicular to the i1 direction develops in the core. This orientation is transported at the junction between the gate and the plaque so that at the entrance of the plaque, we obtain a skin/core orientation, with a skin oriented in the flow direction, and a core perpendicular to it. These calculations show how a gate can lead to a skin/core structure in moulded parts. The computations are qualitatively in agreement with the observations. In Fig. 20.9, the evolution of a11 with x2, the plaque thickness, is shown at the four positions A, B, C and D where the experimental measurements were made (see Fig. 20.7(b)). Computations were performed with an isotropic initial orientation, showing the following: 1. A skin/core structure is rapidly formed in zone A, and it remains even in D after the junction gate-plaque. As shown in the experiments, the skin/core structure is less significant for position C than for position D. 2. For a partially coupled calculation (Np = 0), when the interaction coefficient CI increases, near the surface fibres are less oriented in the flow direction. For example, in the first position A for the lower wall (x2 = 2.5 mm), one gets a11 ~ 0.7 for CI = 0.04 and a11 ~ 0.9 for CI = 0.001.
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1.0 0.9 0.8 0.7
a11
0.6 0.5 0.4 0.3 CI CI CI CI
0.2 0.1 0.0 2.5
= 0.001; Np = 0 = 0.001; Np = 100 = 0.04; Np = 0 = 0.04; Np = 100
3.0
3.5
4.0
4.5
5.0 X1 (a)
5.5
4.5 X1 (b)
5.0
6.0
6.5
7.0
1.0 0.9 0.8 0.7
a11
0.6 0.5 0.4 0.3 CI = 0.001; Np = 0 CI = 0.001; Np = 100 CI = 0.04; Np = 0 CI = 0.04; Np = 100
0.2 0.1 0.0 2.5
3.0
3.5
4.0
5.5
6.0
6.5
20.9 Values of orientation tensor coefficient a11 for the four sensors A (a), B (b), C (c), D (d) for PAA50 and isotropic initial orientation: CI = 0.001, 0.04; Np = 0, 100.
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a11
0.6 0.5 0.4 0.3 CI = 0.001; Np = 0 CI = 0.001; Np = 100 CI = 0.04; Np = 0 CI = 0.04; Np = 100
0.2 0.1 0.0 2.5
3.0
3.5
4.0 X1 (c)
4.5
5.0
5.5
1.0 0.9 0.8 0.7
a11
0.6 0.5 0.4 0.3 0.2 0.1 0.0 –2.5
CI = 0.001; Np = 0 CI = 0.001; Np = 100 CI = 0.04; Np = 0 CI = 0.04; Np = 100 –2.0
–1.5
–1.0
–0.5
0.0 X1 (d)
0.5
1.0
1.5
2.0
2.5
20.9 Continued.
This can be explained by the fact that shear is dominant near the walls, and it tends to orient fibres in the flow direction. The interaction term tends to disorient fibres, and its effect increases with CI. The orientation in the core does not really change when CI varies.
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3. An increase of Np from 0 to 100 (that is, comparison between coupled and partially coupled calculations) leads to complex tendencies in the skin regions. In the lower skin region, that is for x2 = 2.5 mm, a11 decreases in positions A and B, but the change is negligible in positions C and D. At the opposite skin, there is no noticeable effect for positions B and D. The trends are opposite in region A (a11 increases) and C (a11 decreases), which points out the complex coupling effect on the flow velocity. 4. In the core region, taking the rheological coupling into account leads to a small increase of a11, except in position D for CI = 0.04. In position D, the lowest value of a11 is for Np = 100 and CI = 0.04. Computations realised with unidirectional initial orientation (see Fig. 20.10) show that: 1. For the weakly coupled calculations (Np = 0) and CI = 0.04, a skin/core structure is formed as for the isotropic initial orientation. Nevertheless, the core is less oriented perpendicular to the flow direction: we get a11 around 0.3–0.4 for positions A, B and C, instead of 0.2–0.3. The behaviour is completely different for CI = 0.001: orientation is in the flow direction on the whole thickness at positions A and B. a11 begins to decrease a little in the core in position C, and in D the orientation is isotropic (a11 = 0.5). 2. When Np goes from 0 to 100, in the skin and core regions, a11 decreases a little, or remains nearly constant, depending on the position and the value of CI. 3. The influence of the initial orientation is not very important in the last probe D, especially at the surfaces, except for CI = 0.001 and Np = 0 in the core: orientation is 2D isotropic in plane (i1, i2) for a unidirectional initial orientation (a11 = 0.5), whereas a transverse orientation is formed with an isotropic initial orientation (a11 = 0.25). Numerical results have been compared to the experiments (Redjeb, 2007). The measured points are in between the results obtained with both initial orientation, but the agreement is better for the unidirectional initial orientation. This is not surprising, as a circular channel orients fibres mainly in the flow direction. In the skin regions, the agreement is quite good with CI = 0.04 for the four locations A, B, C and D. The coupled calculation does not improve the quality of the agreement significantly. In the core region, agreement is better for CI = 0.001. These results show that the interaction coefficient may depend on the orientation. However, the computed skin/core orientation at the beginning of the plaque remains more important with respect to the measurements.
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a11
0.6 0.5 0.4 0.3 CI = 0.001; Np = 0 CI = 0.001; Np = 100 CI = 0.04; Np = 0 CI = 0.04; Np = 100
0.2 0.1 0.0 2.5
3.0
3.5
4.0
4.5
5.0 X1 (a)
5.5
6.0
6.5
7.0
1.0 0.9 0.8 0.7
a11
0.6 0.5 0.4 0.3 CI = 0.001; Np = 0 CI = 0.001; Np = 100 CI = 0.04; Np = 0 CI = 0.04; Np = 100
0.2 0.1 0.0 2.5
3.0
3.5
4.0
4.5 X1 (b)
5.0
5.5
6.0
6.5
20.10 Values of orientation tensor coefficient a11 in the four areas A (a), B (b), C (c), D (d) for PAA50 and unidirectional initial orientation: CI = 0.001, 0.04; Np = 0, 100.
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1.0 0.9 0.8 0.7
a11
0.6 0.5 0.4 0.3 0.2 0.1
CI = 0.001; Np = 0 CI = 0.001; Np = 100 CI = 0.04; Np = 0 CI = 0.04; Np = 100
0.0 2.5
3.0
3.5
4.0 X1 (c)
4.5
5.0
5.5
1.0 0.9 0.8 0.7
a11
0.6 0.5 0.4 0.3 0.2 0.1 0.0 –2.5
CI = 0.001; Np = 0 CI = 0.001; Np = 100 CI = 0.04; Np = 0 CI = 0.04; Np = 100 –2.0
–1.5
–1.0
–0.5
0.0 X1 (d)
0.5
1.0
1.5
2.0
2.5
20.10 Continued.
20.4.4 U-shaped part with thin walls and cross-ribs The studied part is presented in Fig. 20.11(a). The computations have used a mesh generated by a topologic mesher (Gruau and Coupez, 2005, or Coupez, 2010) governed by a natural metric in order to capture the thin walls. This
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(a)
(b)
20.11 (a) Geometry of a cross with thin walls; (b) focus on the weld line located on the wall.
metric takes into account, for each area of mould, a specified mesh size in each principal direction, ensuring an appropriate number of elements in the whole thickness of the part. The injected material is a polypropylene referenced as STAMAX P30YM240. This material is modelled by a Carreau–Yasuda law for the rheology and an Arrhenius law for thermo-dependency (the parameters are given in Table 20.2). The polymer is injected at 250°C and we assume that the mould is fully regulated at 40°C. Initially, the air inside the cavity is also considered to be at a temperature of 40°C.
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To compute the fibre orientation, one takes a fibre aspect ratio b = 10, an interaction coefficien CI = 0.001, a hybrid closure approximation and a weakly coupling (Ns, Np = 0). The orientation is analysed after a total filling time of 3.7 s. The time step is 3 ¥ 10–4s. The dynamics of filling are important because it helps to understand the dynamics of fibre orientation. The flow is fully symmetrical and it starts from the inlet to fill the bottom, the walls and the crossing central ribs simultaneously. In the beginning, the wall filling is slightly delayed compared to central ribs. This delay disappears and is reversed as soon as the cavity is half-filled. The cross-ribs have two weld lines as fronts from the flank and ribs intersect twice. The first weld line is located on the sidewall after crossing the first rib. The second weld line is located at the second row of crossing ribs, as the flow front from the sidewall is faster and has time to penetrate the rib. We focus now on the first weld line which is easily detected by the computation. Figure 20.11(b) is a photograph of the part showing the weld line. The injection gate is located at the bottom right and the presence of fibres at the surface allows seeing this line which is located in the middle of the wall. This line is slightly oriented in the direction of flow. Tracing the isosurfaces of components of the orientation tensor (see Fig. 20.12), it is possible to find this weld line. The orientation on the side wall becomes unidirectional along the i1 direction: a11 increases while a22 remains relatively low. At the intersection of the two fronts in the region where the ribs are connected to the wall, we can observe a decrease of component a11 along the entire weld line and an increase of component a22. Therefore, the weld line is well located and predicted where there is a disturbance of the orientation tensor.
20.4.5 Rib-shaped part with thin walls The studied part is presented in Fig. 20.2. It is a plaque with seven ribs and it is filled from one side. The variation of fibre concentration inside the part described in the second section is not considered in our model although it is important in terms of mechanical properties. The part has a plane of symmetry and computations are made on only half of the cavity. These computations are made for aspect ratio b = 30, CI = 0.023 and Ns, Np = 0. The fluid is a Stamax P40YM243 composed of a polypropylene matrix and 40% of glass fibres. The total filling time is around 1 s. Finally, computations are compared to experimental observations. In Figs 20.13 and 20.14, the ellipsoids associated with the orientation tensor are plotted and are more or less elongated according to fibre orientation. They are also coloured by a von Mises scalar (equation 20.35) in order to give information on the anisotropy of fibre orientation. In Fig. 20.12, the skin/
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(a)
(b)
20.12 Isovalues of a11 and a22 on the wall where the weld line is located.
core phenomenon is observed in plaque as shown in experiments. However, with the Folgar and Tucker (1984) model, an isotropic orientation is obtained in the core instead of an orientation perpendicular to the flow motion in the experiment. In the skin region, the orientation along the flow motion is observed in both experiments and computations. For the ribs, the orientation changes during filling: first during rib filling, a core/skin orientation is observed; secondly, there is a ‘disorientation’ of fibres once the rib is filled. Finally, only the base of the rib maintains a non-isotropic orientation. In Fig. 20.13, we compare the simulation with the experimental observation when the mould is fully filled (the experimental picture is slightly larger than
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vonMises OrientationPO I Frin
0.995135 0.895622 0.796108 0.696595 0.597081 0.497588 0.398054 0.298541 0.199027
0.0995139 4.80143e-007 y
x z Time: 0.230 s.–Inc: 0.4595
GLview inova 2010-08-13 AT: Rem3D simulation Frin: vonMises OrientationPO [3D element]
20.13 Ellipsoids representing orientation tensor coloured by von Mises scalar for the plaque with ribs: isotropic orientation (white), unidirectional orientation (black).
the image obtained by the simulation). We deduce that the simulation allows reproducing the increase of the thickness of the core area at the base of the ribs. Moreover, the numerical model reproduces the arch drawn orientation. As in experimental observations, a homogeneous orientation is recovered inside the rib. However, this orientation is isotropic in simulations, whereas the fibres are oriented in a direction parallel to the rib in experiments. Moreover, the arch orientation depends on the time of packing, which was not taken into account in our simulations. Finally, Rem3D software recovers reasonably faithfully the orientation encountered in the experiments. Only the areas having an isotropic orientation are not in agreement with the experiments. To overcome this discrepancy, a tensorial interaction coefficient CI has to be introduced as prescribed by anisotropic rotary diffusive models. In this way, the anisotropy of the plaque will be taken into account (the vertical direction is smaller with respect to the two others).
20.5
Conclusions
Fibre structure in reinforced moulded components is directly related to the process. Fibre length degradation takes place mostly in the plasticising unit but the fibre length distribution is not yet considered in our modelling. Fibre concentration can be considered as constant to a first approximation even though, in the plaque with ribs, we measured some important fluctuations inside the part. Therefore the evolution of fibre concentration may have
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(a)
(b)
20.14 (a) Final arch drawn orientation near the third rib; (b) comparison with the experiment.
an influence on the evolution of fibre orientation during the process. Fibre orientation varies a lot throughout the part, especially in the thickness. A core region with fibres transverse to the flow direction is almost always created
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in the gate region, because of high elongation rates. Near the surfaces, fibres are predominantly oriented in the flow direction. Quantification of fibre orientation is time consuming, with several causes of errors. Modelling of flow-induced fibre orientation is usually carried out supposing an equivalent viscous behaviour for the composite. This gives valuable information for mould design, and especially gate location. The closure approximation has a larger influence on the results than does the interaction coefficient. Full 3D computations give a precise kinematics description in the gate, at the flow front, which increases the precision of the orientation calculation. Coupling between rheology and orientation becomes important, raising the issue of the validity of constitutive equations based on dilute or semi-dilute Newtonian suspension, and of the determination of rheological parameters. The numerical computations describe reasonably well the orientation encountered in the experiments. The discrepancy could be overcome by making the model more complex, but the number of parameters increases and there is no simple way to get them. Then, both the experimental setup and numerical procedures (inverse analysis) have to be developed to identify these parameters for the polymers used in industrial processes.
20.6
References and further reading
Advani S G, Tucker C L (1987), ‘The use of tensors to describe and predict fiber orientation in short fiber composites’, J Rheol, 31, 751–784. Advani S G, Tucker C L (1990), ‘Closure approximations for three-dimensional structure tensors’, J Rheol, 34, 367–386. Akay M, Barkley D (1991), ‘Fiber orientation and mechanical behavior in reinforced thermoplastic injection moldings’, J Mater Sci, 26, 2731–2742. Akay M, O’Regan D F, Bailey R S (1995), ‘A model to predict the through-thickness distribution of heat generation in cross-ply carbon-fiber composites’, Comp Sci Technol, 55, 109–118. Altan M C, Selcuk S S, Guceri I, Pipes R B (1990), ‘Numerical prediction of threedimensional fiber orientation in hele-shaw flows’, Polym Eng Sci, 30, 848–859. Arroyo M, Avalos F (1989), ‘Polypropylene/low density polyethylene blend matrices and short glass fibers based composites. I. Mechanical degradation of fibers as a function of processing method’, Polym Comp, 10(2), 117–121. Avérous L, Quantin J C, Crespy A, Lafon D (1997), ‘Evolution of the three-dimensional orientation of glass fibers in injected isotactic polypropylene’, Polym Eng Sci, 37(2), 329–337. Batchelor G K (1970), ‘Slender body theory for particles arbitrary cross section in Stokes flow’, J Fluid Mech, 44, 419–440. Batchelor G K (1971), ‘The stress generated in a non dilute suspension of elongated particles by pure straining motion’, J Fluid Mech, 46, 813–829. Batkam S, Bruchon J, Coupez T (2004), ‘A space-time discontinuous Galerkin method for convection and diffusion in injection moulding’, Int J Forming Processes, 7, 11–33.
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Bay R S, Tucker III C L (1992), ‘Stereological measurement and error estimates for three-dimensional fiber orientation’, Polym Eng Sci, 32(4), 240–253. Beaume G (2009), ‘Modelling and numerical simulation of a complex fluid flow’, PhD thesis ENSMP, http://tel.archives-ouvertes.fr/tel-00416435/fr/ Bretherton F P (1962), ‘The motion of rigid particles in a shear flow at low Reynolds number’, J Fluid Mech, 14, 284–304. Chin W, Liu H, Lee Y (1988), ‘Effects of fiber length and orientation distribution on the elastic modulus of short fiber reinforced thermoplastics’, Polym Comp, 9(1), 27–35. Cintra J, Tucker C L (1995), ‘Orthotropic closure approximation for flow induced fiber orientation’, J Rheol, 34, 1095–1122. Clarke A, Eberhardt C (2002), Microscopy Techniques for Materials Science, Woodhead Publishing, Cambridge, UK. Coupez T (2010), ‘Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing’, J Comp Phys, doi:10.1016/j.jcp.2010.11.041. Davidson N C, Clarke A R (1999), ‘Extending the dynamic range of fibre length and fibre aspect ratios by automated image analysis’, J Microsc, 196, 266–272. Denault T, Vu-Khanh T, Foster B (1989), ‘Tensile properties of injection molded long fiber thermoplastic composites’, Polym Comp, 10, 313–321. Dinh S, Armstrong R (1984), ‘A rheological equation of state for semiconcentrated fibre suspensions’, J Rheol, 28, 207–227. Dontula N, Ramesh N S, Campbell G A, Small J D, Fricke A L (1994), ‘An experimental study of polymer–filler redistribution in injection molded parts’, J Reinf Plast Comp, 13, 98–110. Dupret F, Verleye V (1998), ‘Modeling the flow of fiber suspensions in narrow gaps’, in Advances in the Flow and Rheology of Non-Newtorian Fluids, Siginer D A, De Kee D, Chhabra R P (eds), Elsevier, Amsterdam, 1347–1398. Eberhardt C, Clarke A R (2001), ‘Fibre orientation measurements in short glass fibre composites; I, Automated, high angular resolution measurement by confocal microscopy’, Comp Sci Tech, 61, 1389–1400. Eriksson P A, Albertsson A C, Boydell P, Prautzsch G, Månson J A (1996), ‘Prediction of mechanical properties of recycled fiberglass reinforced polyamide 66’, Polym Comp, 17(6), 830–839. Fan X, Phan-Thien N, Zheng R (1998), ‘A direct simulation of fibre suspensions’, J Non-Newtonian Fluid Mech, 74, 113–135. Férec J, Ausias G, Heuzey M C, Carreau P J (2009), ‘Modeling fibre interactions in semi-concentrated fibre suspensions’, J Rheol, 53, 49–72. Folgar F, Tucker C L (1984), ‘Orientation behavior of fibers in concentrated suspensions’, J Reinf Plast Comp, 3, 98–119. Franzen B, Klason C, Kubat J, Kitano T (1989), ‘Fibre degradation during processing of short fibre reinforced thermoplastics’, Composites, 20(1), 65–76. Gruau C, Coupez T (2005), ‘3D tetrahedral, unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric’, Comp Meth Appl Mech Eng, 194, 4951–4976. Gupta V B, Mittal R K, Sharma P K, Menning G, Wolters J (1989), ‘Some studies on glass fibre-reinforced polypropylene. Part II: Mechanical properties and their dependence on fiber length, interfacial adhesion and fiber dispersion’, Polym Comp, 10(1), 8–15. Hand G (1961), ‘A theory of dilute suspension’, Arch Rat Mech Anal, 7, 81–86. Hegler R P, Menning G (1985), ‘Phase separation effects in the processing of glass bead and glass fiber filled thermoplastics by injection molding’, Polym Eng Sci, 25, 395–405. © Woodhead Publishing Limited, 2011
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Hinch E, Leal L (1973), ‘Time dependent shear flows of a suspension of particles with weak Brownian rotations’, J Fluid Mech, 57, 753–767. Jeffery G B (1922), ‘A theory of anisotropic fluids’, Proc Roy Soc London, A102, 161–179. Kabanémi K K, Hétu J F (1999), ‘Modelling and simulation of nonisothermal effects in injection moulding of rigid fibres suspensions’, J Inj Molding Technol, 3, 80–87. Kamal M R, Song L, Singh P (1986), ‘Measurement of fibre and matrix orientations in fibre reinforced composites’, Polym Comp, 7(5), 323–832. Koch D L (1995), ‘A model for orientational diffusion in fiber suspensions’, Phys Fluids, 7, 2086–2088. Kubat J, Szalanczi A (1974), ‘Polymer–glass separation in the spiral mold test’, Polym Eng Sci, 14, 873–877. Lipscomb G G, Keunigs R, Marucci G, Denn M M (1984), ‘A continuum theory for fiber suspensions’, Proc. IX Int. Congress on Rheology, 2, 497–503. Lipscomb G, Denn G, Hur M, Hur D, Boger D (1988), ‘The flow of fibre suspensions in complex geometries’, J Non-Newtonian Fluid Mech, 26, 297–325. Martinéz M A, Cueto E, Doblaré M, Chinesta F (2003), ‘Natural element meshless simulation of flows involving short fiber suspensions’, J Non-Newtonian Fluid Mech, 115, 51–78. Miled H, Silva L, Agassant J-F, Coupez T (2008), ‘Numerical simulation of fiber orientation and resulting thermo-elastic behaviour in reinforced thermoplastics’, in Mechanical Response of Composites, Camanho P (ed.), Springer, Berlin. Moses K B, Advani S G, Reinhardt A (2001), ‘Investigation of fiber motion near solid boundaries in simple shear flow’, Rheol Acta, 47, 63–73. Papathanasiou T D (1997), ‘Flow-induced alignment in injection molding of fiber-reinforced polymer composites’, in Flow-induced Alignment in Composite Materials, Papathanasiou T D and Guell D C (eds), Woodhead Publishing, Cambridge, UK, 112–165. Phan-Thien N, Fan X J, Tanner R I, Zheng R (2002), ‘Folgar–Tucker constant for a fibre suspension in a Newtonian fluid’, J Non-Newtonian Fluid Mech, 103, 251–260. Phelps J H, Tucker C L (2009), ‘An anisotropic rotary diffusion model for fiber orientation in short- and long-fiber thermoplastics’, J Non-Newtonian Fluid Mech, 156, 165–176. Pichelin E, Coupez T (1998), ‘Finite element solution of the 3D mould filling problem for viscous incompressible fluid’, Comp Meth Appl Mech Eng, 163, 359–371. Pichelin E, Coupez T (1999), ‘A Taylor discontinuous Galerkin method for the thermal solution in 3D mould filling’, Comp Meth Appl Mech Eng, 178, 153–169. Prager S (1957), ‘Stress–strain relation in a suspension of dumbbells’, J Rheol, 1, 53–62. Ranganathan S, Advani S G (1991), ‘On modeling fiber–fiber interactions of flowing suspensions in homogeneous flows’, J Rheol, 35, 1499–1522. Redjeb A (2007), ‘Numerical simulation of fibre orientation in injection of reinforced thermoplastics’, PhD thesis ENSMP, http://tel.archives-ouvertes.fr/tel-00332719/fr/ Redjeb A, Silva L, Laure P, Vincent M, Coupez T (2005), ‘Numerical simulation of fibre orientation in injection moulding process’, 21st International Polymer Processing Society Meeting, Leipzig, Germany. Shaqfeh E, Fredrickson G (1990), ‘The hydrodynamic stress in a suspension of rods’, Phys Fluids A, 2, 7–24. Shen H B, Nutt S, Hull D (2004), ‘Direct observation and measurement of fiber architecture in short fiber–polymer composite foam through micro-CT imaging’, Comp Sci Technol, 64, 2113–2120.
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Tremblay S R, Lafleur P G, Ait-Kadi A (2000), ‘Effects of injection parameters on fiber attrition and mechanical properties of polystyrene molded parts’, J. Inj. Molding Technol, 4(1), 1–7. Tucker C L (1991), ‘Flow regimes for fiber suspensions in narrow gap’, J Non-Newtonian Fluid Mech, 39, 239–268. Ville L, Silva L, Coupez T (2010), ‘Convected level set method for the numerical simulation of fluid buckling’, Int J Num Meth Fluids, doi:10.1002/fld.2259. Vincent M, Giroud T, Clarke A, Eberhardt C (2005), ‘Description and modeling of fiber orientation in injection molding of fiber reinforced thermoplastics’, Polymer, 46, 6719–6725. Vincent M (2009), ‘Flow induced fiber micro-structure in injection molding of fiber reinforced materials’, in Injection Molding, Technology and Fundamentals, Kamal M R, Isayev A I, Liu S J (Eds), Carl Hanser Verlag, Munich 253–272. Wang J, O’Gara J F, Tucker C L (2008), ‘An objective model for slow orientation kinetics in concentrated fiber suspensions: Theory and rheological evidence’, J Rheol, 52, 1179–1200. Wetzel E D, Tucker C L (1999), ‘Area tensors for modeling microstructure during laminar liquid–liquid mixing’, Int J Multiphase Flow, 25, 35–61. Yamane Y, Kaneda Y, Dio M (1994), ‘Numerical simulation of semi-dilute suspensions of rodlike particles in shear flow’, J Non-Newtonian Fluid Mech, 54, 405–421.
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21
Modelling composite reinforcement forming processes
P. B o i s s e and N. H a m i l a, Université de Lyon, France
Abstract: In this chapter various finite element approaches for the simulation of woven reinforcement forming are presented. Some are based on continuous modelling, while others, known as discrete or mesoscopic approaches, model each yarn of the fabric. The intermediary semi-discrete approach is also examined. During this approach the shell finite element interpolation formula maintains the continuity of the displacement field, but the internal virtual work is obtained as the sum of tension, in-plane shear and bending cells, of all the woven unit cells within the element. When using continuous approaches, the necessity of taking the strong specificity of the fibrous material into account causes difficulties. The main goal of the hypoelastic and hyperelastic constitutive models presented in this chapter is to describe this specific mechanical behaviour. During discrete and semidiscrete approaches the directions of the yarns are ‘naturally’ followed because the yarns themselves are modelled. The advantages and drawbacks of the different approaches are also discussed. Key words: fabric, textiles, preforming, draping, meso/macroscopic approaches, hypoelasticity, hyperelasticity.
21.1
Introduction
Textile composites offer a number of attractive properties; they possess a high capacity to conform to complicated contours, they are suitable for manufacturing components with complex shapes, and they offer a greater flexibility in processing options compared to metals and even to their nonwoven counterparts. A double curved composite preform shape can be accomplished through the forming of an initially flat fibrous reinforcement. This reinforcement can be dry (i.e. without resin) such as in the preforming stage of liquid composite moulding (LCM) processes (Advani, 1994; Parnas, 2000). These processes, especially the resin transfer moulding (RTM) process, can be used to manufacture highly loaded composite parts for aeronautical applications such as helicopter frames (Dumont et al., 2008) and motor blades (de Luycker et al., 2009). During these LCM processes the resin is injected into a preform which can be doubly curved such as those shown in Fig. 21.1 (Woven Composites Benchmark Forum, 2004) and Fig. 21.2 (European project ITOOL (ITOOL, Allaoui et al., 2011)). In the case of thermoset or 651 © Woodhead Publishing Limited, 2011
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21.1 Preform analysed in the double dome international benchmark.
21.2 Tetrahedral shape form.
thermoplastic prepregs, the resin is present within the reinforcement during the forming stage, but exists in a weak state. In thermoset prepregs this is because the resin it is not yet polymerized, and in thermoplastic prepregs it
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is because the process is performed at a high temperature. The resin is not hardened and the forming is mainly led by the reinforcement. The in-plane shear strain of the woven reinforcements can be very high during the forming of double-curved parts. The orientation and density of fibres in the deformed shape directly influences the permeability of reinforcement and becomes crucial in predicting the mechanical behaviour of the final part (Hammami et al., 1996; Loix et al., 2008). The yarns are made up of several thousands of fibres and their mechanical behaviour is very singular and specific. In order to develop the composite forming simulation codes, therefore, large deformations of the textile reinforcements are subject to a sustained effort to simulate and model them. The approach chosen to model the forming of textile composite reinforcements depends on the scale at which the analysis is made. The analysis of the deformation can be made by considering and modelling each of its yarns (or fibres) and their interactions such as contact with friction. Approaches made using this level of analysis are called discrete or mesoscopic. The disadvantage of these approaches is that the number of yarns (and even more so the number of fibres) is high and the interactions are complex. The alternative continuous approaches consider a continuous medium juxtaposed with the fabric, the mechanical behaviour of which is equivalent to those of the textile reinforcement. The disadvantage of using continuous approaches is that this mechanical behaviour is complex. It concerns large strains and strong anisotropy and evolves a great deal during the forming. Throughout this chapter continuous and discrete approaches for composite reinforcements forming simulations will be introduced. First a mesoscopic approach during which each yarn of a woven cell is modelled with shell elements with contact and friction is presented, followed by two continuous approaches based respectively on a hyperelastic and a hypoelastic model. Finally a semi-discrete approach, which is an intermediate method between the continuous and discrete approaches, is presented. The advantages and drawbacks of the different approaches are also discussed.
21.2
A mesoscopic approach
In mesoscopic approaches, the modelling measures each fibre bundle (yarn). This approach can also be described as a discrete approach since there are a discrete number of yarns in the preform. The modelling of each yarn must be simple enough to render the simulation of the whole reinforcement forming possible. The interactions between warp and weft directions are precisely taken into account by considering the contact and friction caused by relative motions between the yarns (Pickett et al., 2005; Duhovic and Bhattacharyya, 2006). This modelling could be performed at the microscopic level, i.e. by considering each fibre as a beam. Nevertheless, the massive number of fibres
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per yarn (some thousands in the case of composite reinforcements), and therefore the huge amount of contacts with friction which have to be taken into account, lead to a very costly computation. For this reason, although this approach is promising, only very small elements of the fabric have been modelled to date (Durville, 2005; Miao et al., 2008). The mesoscopic approach considers the continuous behaviour of each yarn. This continuous behaviour must take the fibrous nature of the yarn into account in order to ensure that rigidities in bending and transverse compression remain very small in comparison to the tensile stiffness. The modelling of the woven unit cell must be as accurate as possible but simple enough to allow the computation of the forming process of the entire reinforcement. Figure 21.3(a) shows the finite element model used for discrete simulations of forming processes (216 degrees of freedom (DOF)). It is compared to another FE model of the unit cell used in Badel et al. (2008) (Fig. 21.3(b))
(a)
(b)
21.3 Modelling of a unit cell of a plain weave at the mesoscopic level: (a) 216 dof FE model; (b) 47214 dof FE model.
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to analyse the local in-plane shear of a plain weave unit cell (47214 DOF). A simulation of the forming of a composite reinforcement can be based on the model of Fig. 21.3(a), but not on those of Fig. 21.3(b) because of the computational cost. In the simplified unit cell (Fig. 21.3(a)) each yarn is described by few shell elements and the contact friction and possible relative displacement of the yarns are considered. The bending stiffness of the yarns is independent of the tensile rigidity and very much reduced in comparison to the results given by plate theories. The membrane behaviour is based on a hypoelastic model. A stress rate — is related to the strain rate D by a constitutive tensor C. To avoid rigid body rotations which can affect the stress state, the derivative — , called the objective derivative, is the derivative for an observer who is fixed with respect to the material. Because this requirement is not uniquely defined there are several objective derivatives. This objective derivative is often based on a rotation Q characterizing the rotation of the material. The rate constitutive equation (or hypoelastic law) has the form: — = C:D
21.1
with
(
)
∑ Êd ˆ — = Q · Á Q T · · Q ˜ · Q T = + · – · Ë dt ¯
21.2
∑
and is the spin corresponding to Q , i.e. = Q · QT . It has been shown (Badel et al., 2008) that the rotation Q must be the rotation of the fibre in the case of fibrous materials. This is specific to fibrous materials and differs from common objective derivatives such as Green–Naghdi (Green and Naghdi, 1965) and Jaumann (Dafalias, 1983). The current fibre direction can be determined from the gradient tensor F . Assuming that the initial position of the fibre is f 10 = e10 : f1 =
F · e10
21.3
F · e10
The other basis vectors f 2 and f 3 of the orthonormal frame {f i} are obtained from the material transformation of e 02 : f2 =
F · e 02 – (F F · e 02 · f 1 ) f 1 F·
e 02
– (F F·
e 02
· f1) f1
, f 3 = f 1 ¥ f 2 , = f i ƒ e 0i
21.4
The transverse properties of the shell are identified through a picture frame test (Gatouillat, 2010). Figure 21.4 shows the results of a hemispherical
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21.4 Simulation of a hemispherical forming using the unit cell model of Fig. 21.3(a).
forming simulation based on the modelling of the unit woven shell shown in Fig. 21.3(a). This meso modelling has been used in Gatouillat (2010) for the simulation of a ‘double dome’ benchmark (Woven Composites Benchmark Forum, 2004) and to form simulations where the continuity of the textile reinforcement is no longer ensured because of strong loads on the blank holders.
21.3
Continuous approaches
Continuous approaches consider the fibrous material as a continuum in average at the macroscopic scale. The purpose is to make use of the standard finite element codes for the analysis of fibrous media (Spencer, 2000; Peng and Cao, 2005; ten Thije et al., 2007). The formulations employed within a commercial code, i.e. ABAQUS/Explicit, for stress calculations are presented comprehensively in the following. The algorithm adopted and the formulations used within a user material subroutine VUMAT are then shown.
21.3.1 Hypoelastic model The equations governing the mechanical behaviour are given in equations 21.1–21.4. They use the rotation of the fibre. A membrane assumption is used. The Green–Naghdi’s frame (GN) is the default work basis of ABAQUS/
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Explicit. Its unit vectors (e1, e 2 ) in the current configuration are updated from the initial orientation axes, (e10, e 02 ), using the proper rotation R: e1 = R · e10 , e1 = R · e 20
21.5
In the current configuration, the unit vectors in the warp and weft fibre directions are respectively: f1 =
F · f 10
F · f 02
F · f1
F · f 02
, f2 = 0
21.6
(e10, e 02 ) and ( f 10, f 02 ) are assumed to coincide initially (Fig. 21.5). Two orthonormal frames based on the two fibre directions are defined: g (g1, g 2 with g1 = f1, and h (h1, h 2 ) and with h 2, f 2 (Fig. 21.5). The strain increment d from time tn to time tn+1 is computed by the code in the GN frame. It is expressed in the two frames g and h: h d = degab ga ƒ gb = deab h a ƒ hb
21.7
where a and b are indexes taking value 1 or 2. The fibre stretching strain and the shear strain in the two frames are calculated: g g de11 = g1 · d · g1 de12 = g1 · d · g 2
21.8
g deg22 = h 2 · d · h 2 de12 = h1 · d · h 2
21.9
The axial stress component and shear stress components are then computed: g g g g ds11 = E g de11 ds12 = Gde12
21.10
e20
e20=f20
g2
e2
f2=h2
q2
e10=f10
e10 q1
f1=g1 e1
h1
21.5 Fibre axes and gN axes in initial state and after deformation.
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21.11
where Eg and Eh represent the tensile stiffness in the warp and weft fibre directions and G is the in-plane shear stiffness of the fabric. (They are not constant; G especially depends strongly on the in-plane shear.) The stresses are then integrated on the time increment from time tn to time tn+1: n +1/2
g n +1 g n g (s12 ) = (s12 ) + ds12
n +1/2
h n+ h n h (s12 ) +1 = (s12 ) + ds12
g n +1 g n g (s11 ) = (s11 ) + ds11 h n +1 h n h (s11 ) = (s11 ) + ds11
n +1/2
n +1/2
21.12 21.13
The addition of the stresses in the warp and weft frames gives the stress in the fabric at time tn+1: n +1 = (g )n +1 + ( h )n +1
21.14
More detail on this approach can be found in Khan et al. (2010). This approach is used to simulate the forming of a double dome shape corresponding to an international benchmark (Woven Composites Benchmark Forum, 2004). An experimental device has been constructed in INSA Lyon in order to perform this forming (Fig. 21.6). The woven fabric is a commingled glass/polypropylene plain weave which was tested in the material benchmark study conducted recently (Cao et al., 2008). The computed and experimental geometries after forming are compared in Fig. 21.6. The measured and numerical draw-in and shear angles agree (Figs 21.7 and 21.8) (Khan et al., 2010).
21.3.2 Hyperelastic model For this approach (Aimène et al., 2010) a potential is defined which aims to reproduce the non-linear mechanical behaviour of textile composite reinforcements. The proposed potential is a function of the right Cauchy Green deformation tensor and structural tensor invariants defined from the fibre directions. This potential is based on the assumption that tensile and shear strain energies are uncoupled, and is the sum of three terms: W = W 1 ((II1 ) + W 2 (I 2 ) + W s ((II12 )
21.15
This assumption (that tensile and shear strain energies are uncoupled) is made for the sake of simplicity. The independence of tensile behaviour relative to in-plane shear has been shown experimentally (Buet-Gautier and Boisse, 2001). The other hypotheses are probably less true, but there is very little data available on the couplings. The structural tensors l ab are defined from the two unit vectors in the warp and weft directions f10 and f 20 in the reference configuration:
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b
a
d
659
ex ey
1 2
f 3 4 5 6
g 7 8 9 10
h
l (a) Shear angle (deg.) (Avg: 75%) +4.244e+01 +3.775e+01 +3.306e+01 +2.837e+01 +2.368e+01 +1.899e+01 +1.430e+01 +9.609e+00 +4.918e+00 +2.268e–01 –4.464e+00
Y Z
X (b)
21.6 Location of points for material draw-in and shear angle measurement.
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Experimental
30
Material draw-in (mm)
25 20 15 10 5 0 A
b
c
d Ex Ey F Material draw-in points
g
h
l
21.7 comparison of material draw-in for the draped double dome.
Numerical
Experimental (OSM)
45
Shear angle (deg.)
40 35 30 25 20 15 10 5 0 1
2
3
4 5 6 7 Shear angle locations
8
9
10
21.8 comparison of shear angle of numerical and experimental outputs.
l ab = f a 0 ƒ f b 0
21.16
The two first terms W 1 and W 2 are the energies created by the tensions in the yarns. They are also functions of invariants I1 and I2 respectively themselves, depending on the right Cauchy–Green strain tensor C = F T · F and the structural tensors l aa :
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I1 = ttrr (C · L11) = l12 I 2 = ttrr (C · L 22 ) = l22
661
21.17
where la is the deformed length of a fibre with a unit initial length in the direction a. The third term W s is a function of the second mixed invariants of C . I12 = 1 tr t (C · l11 · C · L 22 ) = cos2 q I1 I 2
21.18
The second Piola–Kirchhoff stress tensor is derived using this equation for the potential: I I È ˘ È ˘ s = 2 Í∂W – 12 ∂W ˙ l11 + 2 Í∂W – 1122 ∂W ˙ l 22 ∂ I I ∂ I ∂ I I ∂ I 1 12 ˚ 1 2 12 ˚ 1 Î 1 Î 2 È I ˘ + 2 Í 12 ∂W ˙ (l l12 + l 21 ) I I ∂ I Î 1 2 12 ˚
21.19
The potential has to vanish in a stress-free configuration. Polynomial functions of the invariants are discussed in this chapter. The global form of the proposed potential energy is given by: r
s
t
1 A (I i +1 – 1) + S 1 B (I j +1 – 1) + S 1 C I k i 1 j 2 k 12 i =0 i + 1 j =0 j + 1 k =1 k
W (C (C) = S
21.20 To determine the constants Ai, Bj and Ck, three experimental tests are necessary: two tensile tests in the warp and weft directions and one in-plane pure shear test. The proposed hyperelastic model is implemented in a user routine VUMAT of Abaqus/Explicit and it is then applied to membrane elements. The simulation of a hemispherical punch forming process is performed in the case of strongly unbalanced twill. The warp rigidity is 50 N/yarn and the weft rigidity is 0.2 N/yarn. The experimental results in terms of a deformed shape are shown in Fig. 21.9(a) together with the results of the simulation in Fig. 21.9(b) and (c). The hyperelastic model is described in detail in Aimène et al. (2010).
21.4
The semi-discrete approach
This approach (Boisse et al., 1997; Hamila et al., 2009), which is relatively intermediate between the continuous and discrete approaches, obtains the textile composite reinforcement from a set of unit woven cells which bear the loads of their neighbouring cells (Fig. 21.10).
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21.9 deformed shape of an unbalanced fabric with an initial orientation of fibres 0°/90): (a) experimental shape; (b) Simulation without shear rigidity; (c) Simulation with shear rigidity.
f1
f2
21.10 loads on a unit woven cell.
∑ ∑ ∑
The tensions T1 and T2 are the resultants of the respective loads on warp and weft yarns in the directions f1 and f 2 of these yarns (Fig. 21.11(a)). The in-plane shear moment Ms is the moment resulting from the loads on the unit woven cell at its centre, in the direction normal to the fabric (Fig. 21.11(b)). The bending moments M1 and M2 are the moments resulting from the loads on the warp and weft yarns respectively (Fig. 21.11(c)).
Figures 21.10 and 21.11 have been drawn to represent a plain weave for simplicity, but the type of weaving can vary. In any virtual displacement field h such as h = 0 on the boundary with prescribed displacements, the principle of virtual work compels the virtual
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T1
T2
Ms
T2
Ms
T1
663
M2
M1
M1
M2
Tensions
In-plane shear moment
bending moments
(a)
(b)
(c)
21.11 load resultants on a unit woven cell.
inertial work to be equal to the difference between virtual internal and external works: Wext ( h) – Wint ( h) = Wacc ( h)
21.21
t s b Wint (h ) = Win intt (h ) + Win intt (h ) + Win intt (h )
21.22
with
s b where Wint t ( h), Win intt ( h) and Wint ( h) are the virtual internal works of tension, in-plane shear and bending respectively: ncell
Wint t ( h) = S
p
e11 ( h) pT1 p L1 + p e 22 ( h) pT2 p L2
21.23
p
g ( h) p M s
21.24
p
c11 ( h) p M 1 p L1 + p c 22 ( h) p M 2 p L2
21.25
p =0
ncell
Wins t ( h) = S
p =1
ncell
Winbt ( h) = S
p =1
where ncell denotes the number of woven cells, L1 and L2 the lengths of unit woven cells in the warp and weft directions, e11 ( h) and e 22 ( h) the virtual axial strain in the warp and weft directions, g ( h) the virtual in-plane shear angle, i.e the virtual angle variation between warp and weft directions, and c11 ( h) and c 22 ( h) represent the virtual curvatures of the warp and weft directions. The quantity A is denoted pA when it concerns the unit woven cell number p. e11 ( h), e 22 ( h)), g ( h), c11 ( h) and c 22 ( h) are functions of the gradient of the virtual displacement field. T1, T2, Ms, M1 and M2 are the load results on the woven cell as presented above and in Fig. 21.11. Experimental tests specific to textile composite reinforcements are used to obtain these mechanical properties. The biaxial tensile test gives the tensions
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T1 and T2 as functions of the axial strains e11 and e22 (Kawabata et al., 1973; Buet-Gautier and Boisse, 2001); the picture frame or the bias extension test gives the shear moment Ms as a function of the angle variation g between warp and weft yarns (Wang et al., 1998; Potter, 2002; Potluri et al., 2006; Lomov et al., 2006; Launay et al., 2008; Cao et al., 2008); and the bending test gives the bending moments M1 and M2 as respective functions of c11 and c22 (Kawabata, 1986; de Bilbao et al., 2010). Virtual tests, for example 3D simulations of the deformation of a unit woven cell submitted to elementary loadings such as biaxial tensions or in-plane shear (Badel et al., 2008, 2009) are an alternative to these experimental tests. The approach presented in equations 21.1–21.5 assumes that the internal load state in the material is given by the membrane and bending resultant loads. It is restricted to thin reinforcements. This is true for a large section of composite reinforcements. The five load resultants T1, T2, Ms, M1 and M2 can depend on the five kinematic quantities e11, e22, g, c11 and c22. Such knowledge is generally not available (and probably not often necessary). When considering in-plane shear, some studies have shown that the shear force can depend on the tension state (Lomov and Verpoest, 2006; Launay et al., 2008). Such data is not usually available, however, and it is assumed that the picture-frame (or the bias-extension test) gives Ms only depending on g. A three-node shell finite element M1m2m3 made up of ncell woven cells is shown in Fig. 21.12. The vectors k1 = am2 and k2 = Bm3 in the warp and weft directions respectively are defined. The internal virtual work of tension on the element (equation 21.23) defines the element nodal tensile internal forces Fintet : ncell
S p e11 (h ) pT1 p L1 + p e 22 (h ) pT2 p L2 = eT Finintttee
21.26
p =1
M5
M3 k2 k2
M2
k1
A
k1 M6
b M1
k1
Ms M1 T1
M2 T2
M4
21.12 Three-node finite element made of unit woven cells.
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The internal tensile force components are calculated from the tensions of T1 and T2: (Fintet )ij
Ê ˆ l1 l2 ˜ Á = ncell B1ij T1 + B2ij T2 2 2˜ Á k1 ÁË k ˜¯ 2
21.27
where i represents the index of the direction (i = 1 to 3) and j is the index of the node (j = 1 to 3). B1ij and B2ij are strain interpolation components. They are constant over the element because the interpolation functions are linear in the case of the three-node triangle. The internal virtual work of in-plane shear on the element (equation 21.24) defines the element nodal tensile internal forces Finset : ncell
S
p
p =1
se g (h ) p M s = eT Fin int
21.28
The internal in-plane shear force components are calculated from the inplane shear moment: (Finset )ij = ncell Bg ij M s (g )
21.29
In order to avoid supplementary degrees of freedom and consequently to improve numerical efficiency, the bending stiffness is taken into account within an approach without a rotational degree of freedom (Onate and Zarate, 2000; Sabourin and Brunet, 2006). In these approaches the curvatures of the element are computed from the positions and displacements of the nodes of the neighbouring elements (Fig. 21.12). The internal virtual work of bending on the element (equation 21.25) defines the element nodal bending internal forces Finbet : ncell
S
p =1
p
c11 ( h) p M 1 p L1 + p c 22 ( h) p M 2 p L2 = eT Fibe ntt n
21.30
The internal bending force components are calculated from the bending moments M1 and M2: Ê L L ˆ (Finbet )km = ncell Á Bb1km M 1 1 2 + Bb2km M 2 2 2 ˜ ÁË k1 k 2 ˜¯
21.31
The details of the calculations of the internal loads can be found in Hamila et al. (2009). Figure 21.13 shows the simulation of the deep drawing of a woven reinforcement with a cylindrical punch (Boisse et al., 2011). It uses the semi-discrete approach presented above. Height-independent blank holders
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(a) 0.000
Shear angle 12.5 25.0 37.5
Shear angle 12.5 25.0 37.5 50.0
50.0
(c)
(b)
(d)
21.13 Forming with a cylindrical punch: (a) geometry of the tools; (b) tensile stiffness only; (c) small bending stiffness; (d) higher bending stiffness.
are shown in Fig. 21.13(a), and Fig. 21.13(b) shows the computed deformed shape of the reinforcement when only the tensile stiffness is taken into account. There is no wrinkle and the computed shear angles are very large. They reach 70°. Figures 21.13(c) and 21.13(d) show the computed deformed shape when all the rigidities (tensile, in-plane shear and bending) have been considered. The bending stiffness is 10 times larger in Fig. 21.13(d) than in Fig. 21.13(c). There are many wrinkles in both cases, especially between the blank-holders. These wrinkles are less numerous and their size is moderately larger when the bending stiffness increases. The simulation of the hemispherical forming of a very unbalanced fabric is shown in Fig. 21.14 (Boisse et al., 2011). There is a 250° ratio between the tensile stiffness in the warp and weft directions. A 6 kg ring was used as blank-holder to avoid reinforcement wrinkling in the curved zone (Fig. 21.4(a)). The experimental shape obtained after forming is shown in Fig. 21.14(e) (Daniel et al., 2003). In the warp-direction image, large fabric sliding is observed relative to the die, whereas, contrastingly, in the weft direction (the weaker direction) no edge movement is depicted and the yarns were subjected to large stretch deformations. A 6 kg ring was used as blank-holder to avoid reinforcement wrinkling in the curved zone (Fig. 21.14(a)). The shapes after forming are shown in Figs 21.14(b), (c) and (d) for
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Punch R = 60 mm Square fabric Side = 330 mm
Blank holder 15 mm 170 mm
R = 15 mm Die
Die
121 mm (a)
(b)
(c)
(d)
(e)
21.14 Forming of an unbalanced textile reinforcement: (a) geometry of the tools; (b) tensile stiffness only; (c) tensile and in-plane shear rigidities; (d) tensile + in-plane shear + bending rigidities; (e) experimental forming.
different types of simulations. The tensile stiffness is taken into account only in Fig. 21.14(b). There is no wrinkle, but the asymmetry of the shape in the warp and weft directions is correctly obtained. In Fig. 21.14(c) tension and in-plane shear strain energies, but not bending stiffness, are taken into
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account. A great number of small wrinkles are depicted. The wrinkles are much bigger in Fig. 21.14(d) where the bending stiffness is added. The shapes of these wrinkles conform fairly well with the expectations of the experiment. Near the central point of the preform, squares drawn on the fabric prior to forming become rectangles with a length:width ratio of 1:8 (Fig. 21.14(e)). This extension ratio is correctly computed in the three cases shown in (Fig. 21.14(b), (c) and (d)). It depends on the tensile rigidities which are taken into account in all three cases. Interestingly, a fishnet algorithm which ignored the mechanical properties would lead to the same deformation in both warp and weft directions and the ratio in the central part would remain equal to 1.
21.5
Discussion and conclusion
The discrete (or mesoscopic) approach is both attractive and promising. The very specific mechanical behaviour of the textile material due to the contacts and friction between the yarns and to the change of direction is implicitly taken into account. If some sliding occurs between warp and weft yarns, they can be simulated. This is not possible when using the continuous approaches, which consider the textile material as a continuum. This is an important point because it can be necessary to prevent such a sliding in a process. Nevertheless, the main drawback of the discrete approach is the necessary compromise that must be made between the accuracy of the model of the unit woven cell and the total number of degrees of freedom. The modelling of the unit cell must be accurate enough to obtain a correct macroscopic mechanical behaviour, but the number of degrees of freedom of each cell must remain small in order to compute a forming process for which there will be thousands of woven cells. The continuous approach is the most commonly used method in composite reinforcement forming today. Its main advantage is that standard shell or membrane finite elements can be used. It is only the mechanical behaviour which has to be specified in order to take the very particular behaviour of textile materials into account. Many models exist, but none of them are clearly defined yet. The semidiscrete approach aims to avoid the use of stress tensors and directly define the loading onto a woven unit cell using the warp and weft tensions and by in-plane shear and bending moments. These quantities are simply defined on a woven unit cell and above all they are directly measured by standard tests on composite reinforcements.
21.6
Acknowledgements
The work reported here has been carried out partly thanks to the scope of the LCM3M and MECAFIBRES projects by the National French Agency for Research ANR.
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669
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Advani SG (1994) Flow and Rheology in Polymeric Composites Manufacturing. Elsevier, Amsterdam. Aimène Y, Vidal-Sallé E, Hagège B, Sidoroff F, Boisse P (2010) A hyperelastic approach for composite reinforcement large deformation analysis. Journal of Composite Materials 44(1): 5–26. Allaoui S, Boisse P, Chatel S, Hamila N, Hivet G, Soulat D, Vidal-Sallé E (2011) Experimental and numerical analyses of textile reinforcement forming of a tetrahedral shape. Composites: Part A, 42: 612–622. Badel P, Vidal-Sallé E, Maire E, Boisse P (2008) Simulation and tomography analysis of textile composite reinforcement deformation at the mesoscopic scale. Composites Science and Technology 68: 2433–2440. Badel P, Gautier S, Vidal-Sallé E, Boisse P (2009) Rate constitutive equations for computational analyses of textile composite reinforcement mechanical behaviour during forming. Composites: Part A 40: 997–1007. Boisse P, Borr M, Buet K, Cherouat A (1997) Finite element simulations of textile composite forming including the biaxial fabric behaviour. Composites: Part B Engineering Journal 28B: 453–464. Boisse P, Hamila N, Vidal-Sallé E, Dumont F (2011) Simulation of wrinkling during textile composite reinforcement forming. Influence of tensile, in-plane shear and bending stiffnesses. Composites Science and Technology, 71: 683–692. Buet-Gautier K, Boisse P (2001) Experimental analysis and modeling of biaxial mechanical behavior of woven composite reinforcements. Experimental Mechanics 41: 260–269. Cao J, Akkerman R, Boisse P, Chen J et al. (2008) Characterization of mechanical behavior of woven fabrics: Experimental methods and benchmark results. Composites: Part A 39: 1037–1053. Dafalias YF (1983) Corotational rates for kinematic hardening at large plastic deformations. Transactions of the ASME, Journal of Applied Mechanics 50: 561–565. Daniel JL, Soulat D, Dumont F, Zouari B, Boisse P, Long AC (2003) Forming simulation of very unbalanced woven composite reinforcements. International Journal of Forming Processes 6(3–4): 465–480. de Bilbao E, Soulat D, Hivet G, Gasser A (2010) Experimental study of bending behaviour of reinforcements. Experimental Mechanics 50: 333–351. de Luycker E, Morestin F, Boisse P, Marsal D (2009) Simulation of 3D interlock composite preforming. Composite Structures 88: 615–623. Duhovic M, Bhattacharyya D (2006) Simulating the deformation mechanisms of knitted fabric composites. Composites: Part A 37: 1897–1915. Dumont F, Weimer C, Soulat D, Launay J, Chatel S, Maison-Le-Poec S (2008) Composite preform simulations for helicopter parts. International Journal of Materials Forming Suppl 1, 847–850. Durville D (2005) Numerical simulation of entangled materials mechanical properties. Journal of Materials Science 40: 5941–5948. Gatouillat S (2010) Approche mésoscopique pour la mise en forme des renforts de composites tissés. PhD thesis, INSA Lyon, France. Green AE, Naghdi PM (1965) A general theory of an elastic–plastic continuum. Archives of Rational Mechanical Analysis 18: 251–281. Hamila N, Boisse P, Sabourin F, Brunet M (2009) A semi-discrete shell finite element
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for textile composite reinforcement forming simulation. International Journal of Numerical Methods in Engineering 79: 1443–1466. Hammami A, Trochu F, Gauvin R, Wirth S (1996) Directional permeability measurement of deformed reinforcement. Journal of Reinforced Plastic Composites 15:. 552–562. ITOOL ‘Integrated Tool for Simulation of Textile Composites’. European specific targeted research project, sixth framework programme, aeronautics and space, Kawabata S (1986) The Standardization and Analysis of Hand Evaluation, 2nd edn. The Textile Machinery Society of Japan, Osaka, Japan. Kawabata S, Niwa M, Kawai H (1973) The finite deformation theory of plain weave fabrics, part I: the biaxial deformation theory. Journal of the Textile Institute 64(1): 21–46. Khan MA, Mabrouki T, Vidal-Sallé E, Boisse P (2010) Numerical and experimental analyses of woven composite reinforcement forming using a hypoelastic behaviour. Application to the double dome benchmark. Journal of Materials Processing Technology 210: 378–388. Launay J, Hivet G, Duong AV, Boisse P (2008) Experimental analysis of the influence of tensions on in plane shear behaviour of woven composite reinforcements. Composites Science and Technology 68: 506–515. Loix F, Badel P, Orgéas L, Geindreau C, Boisse P (2008) Woven fabric permeability: from textile deformation to fluid flow mesoscale simulations. Composites Science and Technology 68: 1624–1630. Lomov SV, Verpoest I (2006) Model of shear of woven fabric and parametric description of shear resistance of glass woven reinforcements. Composites Science and Technology 66: 919–933. Lomov SV, Willems A, Verpoest I, Zhu Y, Barburski M, Stoilova Tz (2006) Picture frame test of woven composite reinforcements with a full-field strain registration. Textile Research Journal 76(3): 243–252. Miao Y, Zhou E, Wang Y, Cheeseman BA (2008) Mechanics of textile composites: Micro-geometry. Composites Science and Technology 68: 1671–1678. Onate E, Zarate F (2000) Rotation-free triangular plate and shell elements. International Journal for Numerical Methods in Engineering 47: 557–603. Parnas RS (2000) Liquid Composite Molding, Hanser Gardner Publications, Cincinnati, OH. Peng X, Cao J (2005) A continuum mechanics-based non-orthogonal constitutive model for woven composite fabrics. Composites: Part A 36: 859–874. Pickett AK, Creech G, de Luca P (2005) Simplified and advanced simulation methods for prediction of fabric draping. European Journal of Computational Mechanics 14: 677–691. Potluri P, Perez Ciurezu DA, Ramgulam RB (2006) Measurement of meso-scale shear deformations for modelling textile composites. Composites: Part A 37: 303–314. Potter K (2002) Bias extension measurements on cross-plied unidirectional prepreg. Composites: Part A 33: 63–73. Sabourin F, Brunet M (2006) Detailed formulation of the rotation-free triangular element ‘S3’ for general purpose shell analysis. Engineering Computations 23(5): 469–502. Spencer AJM (2000) Theory of fabric-reinforced viscous fluids. Composites: Part A 31: 1311–1321. ten Thije RHW, Akkerman R, Huétink J (2007) Large deformation simulation of anisotropic
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material using an updated Lagrangian finite element method. Computer Methods in Applied Mechanics in Engineering 196: 3141–3150. Wang J, Page JR, Paton R (1998) Experimental investigation of the draping properties of reinforcement fabrics. Composites Science and Technology 58: 229–237. Woven Composites Benchmark Forum (2004) http://www.wovencomposites.org/index. php?show=comment&catgid=4&topicid=37
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Index
ABAQUS/Explicit, 400, 422, 425, 513, 656, 661 ABAQUS Finite Element Analysis, 258 ABAQUS/Standard, 400, 513, 532 acid digestion, 35 adaptive mesh refinement, 260 AEROTISS, 134 Aggregation/Orientation (AO) model, 42 air jets, 98 AMIRA, 520 angle interlock patterns, 106–7 2.5D pattern, 108 ply-to-ply interlock pattern, 107 sample, 107 anisotropic rotary diffusion – reduced strain closure model, 629 Ansys CFX CFD software, 261–2 Antman’s theory, 465 API, 249 Aquacore, 148 aramid fibres, 9–15 Aramis software, 275 areal weight, 135–6 ASTM Standard D1894, 401, 411–12 AutoCAD, 275 auxilliary technologies, 141–4 displacers, 144 feeding devices, 143 tape feeder, 143 pick and place units for local reinforcements, 143 local fibre patch on a beam preform, 144 winding devices, 141–2 problem of slopes with wound 90° layers, 142 sample, 142 ‘back-projection method,’ 568 ballistic conductors, 33 Beer–Lambert, 567 bending behaviour composite reinforcements, 367–93 fabric bending response, 369 during shaping, 367–8 improved cantilever test, 375–82
device description, 375–6 inverse identification, 381–2 post-processing of the mean lines, 376–9 test interpretation, 379–81 macroscopic bending models and bending tests, 370–4 Cantilever test, 371 Grosberg’s parameters computed from the KES-FB2 test, 374 Kawabata Evaluation System – Fabric Bending test, 372 KES-FB2 test result, 373 bending length, 371 bending moment, 369, 371 bending test, 377 interlock carbon fabric, 383–7 average bending response under gravity, 387 curvature vs bending length, 385 mean lines for bending lengths, 384 moment vs bending length, 386 moment vs curvature, 386 Peirce’s modulus G vs bending length, 385 relative standard deviation of maximal deflection vs bending length, 384 sample, 383 non-crimp fabric, 388–91 carbon non-crimp fabric, 388 experimental vs simulated profiles, 391 inverse optimisation, 390 loading curve – experimental result and Dahl’s model fitting, 389 bias extension test, 552 experimental results, 294–8 balanced twill-weave fabric behaviour, 295–6 load vs plain weave displacement curves, high–low temperature and high–low loading speed, 296 normalised shear force from bias extension tests for balanced twill weave, 297 normalised shear force from bias extension tests for plain weave, 295
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Index normalised shear force from bias extension tests for unbalanced twill weave, 298 plain-weave fabric behaviour, 294–5 tensile force vs balanced twill weave cross-head displacement, 296 tensile force vs plain weave crosshead displacement, 294 tensile force vs unbalanced twill weave cross-head displacement, 297 unbalanced twill-weave fabric behaviour, 296–8 experimental setups, 286–94 16-yarn sample, 290 bias extension sample in oven, 287 fabric specimen illustration under biasextension test, 288 grips used at NU’s bias extension tests, 289 sample preparation, 288–90 sample size and process condition, bias extension tests, 290 shear angle contour, 292 shear angle determination, 290–2 shear angle plot, zone C vs plain weave fabric displacement, 291 shear force determination, 292–4 tested fabrics, bias-extension test, 288 unbalanced twill weave fabric, bias extension tests, 289 load displacement curve, 553 bias extension tests, 286–98 biaxial braid, 136–7 different weave types, 137 surface quality, 137 biaxial tensile behaviour, 488–91 biaxial tensile properties composite reinforcements, 306–29 analytical model, 318–24 experimental analysis, 308–18 numerical modelling, 324–8 biaxial tensile test, 518–19 biaxial tension, 226 Bingham Power Law model, 402 Boolean values, 207 boron, 19–20 ‘bottleneck effect,’ 440 boundary conditions, 474–6, 517–18 hierarchical organisation of composite, 474 induction of rigid bodies at ends of components, 474–6 rigid bodies for boundary condition definition, 475 problems, 474 boundary element method, 595 braided fabrics, 213–19 biaxial braid geometry, 215–17 two-axial braid geometry, 216 braided structure coding, 213–15 examples, 218–19 triaxial braid geometry, 218
673
braiding, 165 braided fibre structure, 166 braiding technologies characteristics and properties of braided textiles, 136–44 auxilliary technologies, 141–4 hybrid braids, 144 textile variations on circular braiders, 136–40 textile variations on 3D braiders, 141 composite reinforcements, 116–55 fundamentals, 116–20 braiding machine principle and machine elements, 118 open and closed braid, 117 sample of braiding machine, 119 weave, flat braid and tubular braid, 117 further processing, 149–51 braided profiles, 151 cutting, folding and preform mounting, 150–1 manufacturing steps for JF profile, 150 stabilisation of the preform, 149–50 stacked curing, 151 key parameters for using braiding machines, 134–6 areal weight and layer thickness, 135–6 braiding angle and coverage rate, 135 limitations and drawbacks, 154–5 Mandrel technologies, 144, 146–9 technologies for preforming, 120–34 circular braiding, 120–5 other braiding technologies, 128–33 overbraiding of mandrels, 125–8 three-dimensional braiding, 128–33 typical applications, 151–4 contoured hollow bodies, 152 crash structures, 152, 153 3D braided filler noodles, 154 3D braided stator blade axle, 153 3D braids, 152–4 multi-spar flap, 152 profiles, 151–2 Brinkman equation, 227, 442 Brownian diffusion, 624 Brownian motion, 40 Brunauer, Emmet and Teller (BET) technique, 433–4 bundle model, 59–63 C-glass, 17 cantilever test, 371, 375–82, 493 device description, 375–6 bending test, 377 mean lines from bent shapes, 378 new flexometer, 376 inverse identification, 381–2 residual vector, 382 post-processing of the mean lines, 376–9 bending moment, 378 test interpretation, 379–81
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Index
bending response from curvature and moment computing, 381 bending test with flexometer on nonelastic fabric, 380 capillary tube models, 437–8 carbon nanotube reinforcements, 32–5 composites, 32–47 performance and applications, 46–7 current costs, 35 physical properties, 34 polymer composites, 35–45 aggregating CNTs flow modelling, 42–4 Aggregation/Orientation (AO) model, 45 CNT composite systems modelling, challenges and trends, 44–5 early development, 35–6 functionalised and non-aggregating CNTs flow modelling, 36–41 number of journal article publications per year, 36 orientation model fitting, 0.05%, 0.2%, 0.33% functionalised nonaggregating CNTs, 41 processing techniques, 36–8 production steps, 37 single-walled CNTs (SWNTs) different types (zig-zag, chiral, and armchair), 33 structure and properties, 32–4 synthesis and costs, 34–5 TEM of bundle SWNT, 34 Carreau–Yasuda law, 642 Cartesian coordinate system, 217, 533 Catia V5, 508 Cauchy Green deformation tensor, 556–7, 658, 660 Cayley-Hamilton theorem, 557 CELPER, 520 centrifugation, 35 ceramic composites general features, 53–6 carbon fibres, 55–6 fiber properties, 54 non-oxide fibres, 53–5 oxide fibres, 53 ceramic reinforcements composites, 51–82 general features, 53–6 fibre/matrix interfaces, 73–4 crack deflection diagrams, 75 fibre-matrix bonding and composite strength and toughness relationships, 73 tensile stress-strain curves, 76 mechanical behaviour, fibres and interfaces composite influence, 74–81 damage tolerance and fracture toughness, 78 fatigue behaviour, 78–9 multiple length scales structure micrograph, 78
reliability, 79–81 strength data statistical distribution, 2D woven SiC/SiC composite, 81 strength density functions, 80 strength distribution, 2D woven SiC/SiC composite, 81 tensile stress-strain behaviour, 76–8 ultimate failure, 79 mechanical behaviour at high temperatures, 64–73 creep at high temperatures, 70–3 creep curves obtained on Hi-Nicalon and Hi-Nicalon S SiC-based fibres, 71 Nextel fibres comparative creep rate, 70 Nextel multifilament strands tensile strength retention, 66 SEM micrograph, 65 SiC-based SA3 fibres creep, 72 strength degradation and oxidation at high temperature, 64–6 strength retention, SiC-based Hi-Nicalon single filaments and multifilament tows, 65 stress-rupture time data, single SiCbased filaments, 67 stress-rupture time diagrams, Hi-Nicalon single fibre and multifilament tows, 67 subcritical crack growth, 66–70 subcritical crack growth constants, 68 statistical features, 57–64 fibre interactions influence by random load sharing, 62 filament-tow relations, 63–4 multifilament tows, 58–63 single fibres, 57–8 statistical distributions of strains-tofailure, 62 tensile load-strain curves, SiCbased multifilament tows, 59 tensile properties at room temperature, 58 chemical vapour deposition, 19–22 circular braiding, 120–5 conventional braiders, 120–1 advanced bobbin carrier, 122 bobbin carrier, 121 typical machine configuration, 122 radial braiders, 121, 123–5 bobbin carrier, 124 braider at the University of Dresden, 126 effect of baffle, 125 Herzog radial braider, 123 circular weaving, 99 CLogger class, 241 closure approximations, 625 compaction, 496 compaction curve, 338–44 compaction stress vs volume fraction, 339 empirical models, 342–4 micro-mechanics model, 341–2
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Index complex shapes, 112 sample, 112 Snecma composite fan blade, 113 composite reinforcement analytical model, 318–24 balanced plain weave glass fabric biaxial behaviour: experimental vs analytical, 323 global and local reference systems, 320 predictions and comparison, 322–4 theoretical background, 319–22 unbalanced plain weave glass fabric biaxial behaviour: weft tensionstrain, experimental vs analytical, 323 unit cell geometry, 319 bending properties, 367–93 bending behaviour during shaping, 367–8 bending behaviour of composite reinforcements, 368–70 improved cantilever test, 375–82 macroscopic bending models and bending tests, 370–4 test results, 383–91 biaxial tensile properties, 306–29 analytical model, 318–24 experimental analysis, 308–18 numerical modelling, 324–8 braiding technologies, 116–55 characteristics and properties of braided textiles, 136–44 fundamentals of braiding, 116–20 further processing, 149–51 future trends, 155 key parameters for braiding machine use, 134–6 limitations and drawbacks, 154–5 Mandrel technologies, 144, 146–9 technologies for preforming, 120–34 typical applications, 151–4 carbon fibres, 22–7 carbon fibres from cellulose, 22 carbon fibres from PAN, 23 carbon fibres from pitch, 26–7 carbon fibres from polyacrylonitrile (PAN), 22–6 commercially available PAN-based carbon fibres properties range, 25 first-generation PAN-based carbon fibre and recent generation comparison, 25 typical properties, PAN and pitch-based carbon fibres, 24 ceramics, 51–82 ceramic fibres: general features, 53–6 fibre/matrix interfaces: mechanical behaviour influence, 73–4 fracture strengths: statistical features, 57–64 mechanical behaviour, fibres and
675
interfaces composite influence, 74–81 mechanical behaviour at high temperatures, 64–73 chemical vapour deposition (CVD) monofilaments, 19–22 reactors schematic views, boron and silicon carbide fibres, 20 SiC fibre cross section made by CVD, used to reinforce metal matrix, 21 continuous models for mechanical behaviour analysis, 529–63 continuum mechanics-based nonorthogonal model, 533–5 fibre-reinforced hyperplastic model for woven composite fabrics, 558–62 general fibre-reinforced hyperelastic model, 556–7 non-orthogonal constitutive model, 535–7 specific application for plain weave composite fabric, 537–52 validation of non-orthogonal model, 552–6 experimental analysis, 308–18 biaxial tensile device, Buet and Gautier and Boisse and Boisse et al., 308 biaxial tensile device, Carvelli et al. and Quaglini et al., 310 biaxial tensile device, Luo and Verpoest and lomov et al., 309 biaxial tensile devices, 308–12 biaxial tensile test, cruciform specimen with free ends and tow fixed ends, 310 biaxial tensile tests of twill 2 x 2 carbon textile reinforcement, 316 biaxial tensile tests of twill 2 x 2 glass–PP textile reinforcement, 318 clamping systems, 312–13, 314 composite reinforcement specimen, biaxial tensile tests, 313 experimental results, 315–18 multiaxial multiply carbon stitched preforms, 317 strain components maps during biaxial tensile test, plain weave technical textile, 311–12 strain field measurement, 313–15 fibres, properties and microstructures, 3–30 fineness, units, flexibility and strength, 4–6 flow modeling, 588–613 analytical solution, 591–4 application examples, 603–13 governing flow equations, 589–91 numerical solution, 594–603 friction properties, 397–428 experimental data, 418–21 modeling of themostamping, 422–8 testing methodologies, 411–18
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676
Index
theory, 401–11 glass fibres, 17–19 compositions, densities and mechanical properties, 18 glass fibres are coated with a size, silane, 18 materials comparison, 6–8 bulk materials properties, compared to fibres and fibre-reinforced composites, 7 mechanical behaviour, 488–98 bending, 493–5 biaxial tensile behaviour, 488–91 compaction, 496 coupling, 497 friction identification, 497–8 in-plane shear, 491–3 mesoscopic approach for study of mechanical behaviour, 486–522 finite element modelling, 511–19 future trends, 522 geometric modelling, 503–11 simulations, use and results of finite element, 519–22 three scales of composite reinforcements, 487 yarn mechanical behaviour, 498–503 microscopic approaches for study of mechanical behaviour, 461–85 application examples, 477–82 goals, 463–5 modelling approach, 465–77 numerical modelling, 324–8 biaxial tension vs strain curves, three different woven reinforcements, 329 biaxial tension vs strain curves, three plain weave reinforcements with different crimp, 328 glass twill 3 x 1 reinforcement: biaxial tension vs strain curves, 327 glass twill 3 x 1 reinforcement: transverse strain map, biaxial loading with k = 1, 327 glass twill 3 x 1 reinforcement: unit cell FE mesh, 327 organic fibres, 8–17 aramid fibres, 9–15 broken Zylon fibre, 14 compression kink band in Dyneema fibre, 16 Dyneema high modulus polyethylene fibres, 16 high-modulus polyethylene fibres, 15–17 Kevlar fibre bent around another fibre, 13 molecules used to produce organic fibres, 10 schematic representation of liquid crystal solution, 12 thermoplastic fibres, 8–9
typical properties, most commonly commercially available organic fibres, 11 permeability properties, 431–51 future trends, 451 permeability measurement methods, 445–50 permeability tensor, 432–6 saturated permeability modelling for fibre preforms, 436–44 unsaturated permeability modelling, 444–5 small-diameter ceramic fibres, 27–9 oxide-based fibres examples, 28 oxide fibres, 27 silicon carbide fibres, 27–9 three generations SiC-based fibres examples, 29 transverse compression properties, 333–60 future trends, 359–60 inelastic models, 351–9 inelastic response of fibrous materials, 344–51 woven fabrics, 89–115 applications, 113–14 future trends, 114–15 overview, 89–90 technology description, 90–9 woven fabric definition, 100–13 X-ray tomography analysis of mechanical behaviour, 565–85 analysis of structure, 571–7 finite element simulations, 578–85 composites carbon nanotube reinforcements, 32–47 carbon nanotubes (CNTs), 32–5 performance and applications, 46–7 polymer composites, 35–45 textile reinforcement, geometry modelling, TexGen, 239–63 applications, 257–62 future trends, 262–3 implementation, 240–6 modelling theory, 246–54 rendering and model export, 254–7 WiseTex, textile reinforcement geometry modelling, 200–35 generic data structure, internal geometry description, 202–4 geometrical description, 204–25 pre-processor for mechanical properties prediction, 225–32 compression, 226 compression resin transfer moulding, 335 Computational Fluid Dynamics, 253 confocal laser scanning microscopy, 621 constant friction, 423–5 constitutive modelling, 578–82 contact element, 468–9 contact-friction interactions, 467–72 algorithmic aspect, 471–2
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Index contact detection within fibre assembly, 467–70 determination of material particles constituting a contact element, 469 generation process of contact elements, 468–70 linearized contact conditions, 470 proximity zone between 2 fibres, 469 mechanical model for contact and friction, 470–1 friction mechanical model, 471 normal contact mechanical model, 470–1 continuous approach, 656–61 hyperelastic model, 658–61 hypoelastic model, 656–8 contour woven fabrics, 110 sample, 111 Coulomb friction coefficient, 401, 403 Coulomb friction model, 374 coupling, 497 coverage rate, 165 creep curves, 72 creep test, 337 crimp height, 207 critical fibre, 59 CTexGen class, 241 CTextile class, 241 CTextileWeave class, 241 CTextileWeave2D class, 241 CTextileWeave3D class, 241 CYarn class, 241 cyclic test, 337 Dahl’s model, 374 damage-insensitive behaviour, 77 damage-sensitive stress-strain behaviour, 76 Darcy flow theory, 336 Darcy’s law, 435, 446, 590, 594 Darcy’s velocity, 590 delamination fracture, 175–83 calculated vs measured delamination fracture toughness values, 182 crack bridging and failure of z-binders under mode I load, 179 crack bridging and failure of z-binders under mode II load, 180 crack bridging toughening process, 178 effect of z-binder content, 176 factors controlling modes I & II delamination toughness of 3D composites, 177 modes I and II R-curves, 181 digital elements, 464 digital image correlation (DIC) methods, 313–15 displacement control, 412 div-conform, 599, 601 div-non conform, 601 double-wall fabric, 111–12 drag force, 438 drapability, 109–10
677
dry fibre volume mesh, 255–6 cross-section meshed with rectangular/ triangular mesh generating technique, 255 Dyneema fibre, 15 E-glass, 17 elastic modulus, 187–90 effect of increasing z-pin content on the tensile modulus of carbon/epoxy composites, 190 effect of z-binder content on the normalised tensile modulus, 188 electric arc discharge method, 34–5 electromagnetic interference (EMI) shielding, 46 electron microscopy, 38 techniques, 55 elementary beam theory, 341 elementary representative cell, 573 empirical models, 342–4 equibiaxial extension, 480 Eshelby’s equivalent inclusion principle, 229 extensometers, 313 fabric mechanical properties, 110 failure strength, 190–3 normalised tensile strength, 192 fatigue properties, 193–5 tensile fatigue life curves for stitched and unstitched carbon/ epoxy composite, 194 FE software, 324 fibre concentration, 619 fibre weight concentration, 619 fibre distortion, 223 fibre length, 617–19 30 wt% of short and long glass fibres, 618 average length in mm, 619 geometry of plaque with seven ribs, 618 fibre orientation, 551–2, 619–22 models with anisotropic fibre interaction, 626–9 schematic illustration, 551 short fibre-reinforced polymer moulded plaque, 621 fibre-reinforced thermoplastics, 616–47 computation in injection moulding, 631–45 ellipsoidal representation, 634 ellipsoids representing orientation tensor, 645 geometry of cross with thin walls, 642 isovalues of a11, 636 isovalues of a11and a22, 644 mould schematic representation, 635 numerical methods, 632–3 parameters, 632 rectangular plaque with inlet gate, 634–41 representation of orientation, 633–4 rib-shaped part with thin walls, 643–5
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678
Index
U-shaped part with thin walls and crossribs, 641–3 values of orientation tensor coefficient a11 for four areas, 640–1 values of orientation tensor coefficient a11 for four sensors, 637–8 models, 622–31 anisotropic fibre interaction, 626–9 Folgar–Tucker model, 623–6 Jeffery theory, 622–3 rheological models, 629–31 observations, 617–22 fibre concentration, 619 fibre length distribution, 617–19 fibre orientation, 619–22 skin-core structure, 622 tensor component orientation, 622 fibres composite reinforcement, properties and microstructures, 3–30 carbon fibres, 22–7 chemical vapour deposition (CVD) monofilaments, 19–22 fineness, units, flexibility and strength, 4–6 glass fibres, 17–19 materials comparison, 6–8 organic fibres, 8–17 small-diameter ceramic fibres, 27–9 filtration, 35 filtration velocity, 435 finite element method, 324, 595–6 discretisation, 597–8 finite element modelling, 511–19 hyper-elastic models, 515–16 hypo-elastic models, 512–15 local frame of fibre bundle, 514 parameter identification for hypoelastic model, 516–19 boundary conditions, 517–18 inverse method and optimisation procedure, 518–19 local validation, 519 longitudinal Young modulus, 516–17 Young modulus determination from tensile testing, 517 simulations, use and results, 519–22 ballistic impact, 522 constitutive behaviour, 520 forming defects, 520–1 permeability computations, 520 permeability tensor, 521 shear angle during bias extension test, 521 finite element simulations, 578–85 constitutive modelling, 578–82 validation of FE simulations, 582–5 equi-biaxial tension and pure in-plane shear, 584 in-plane shear of the glass plain weave fabric, 583
finite elements, 597 LCM flow simulation, 599–601 triangular non-conforming finite element, 600 finite strain beam model, 465–7 adaptation of stiffness to model macrofibres, 466–7 illustration, 466 fire retardation, 46 firmness, 109 flexometer, 375 mechanical module with a sample of noncrimp fabric, 376 floating catalyst method, 35 flow modeling analytical solution, 591–4 comparison of injections at constant inlet pressure, 595 resin flow, 592 resin flow in RTM, 594 resin flow with resin injected at constant flow rate, 593 resin flow with resin injected at constant pressure, 592 schematic of rectangular mould, 591 application examples, 603–13 complex parts, 610–13 convergence of mesh refinement for elliptic flow through isotropic material, 610 convergent analysis on radial divergent flow, 604 3D flows, 604–10 flow front positions calculated for different mesh sizes, 609 flow front progression, 607 numerical RTM flow simulation, 612 predicted vs measured flow fronts, 608 schematic diagram of through-thickness flow front, 605 schematic drawing of spherical central injection, 607 three dimensional model for simulation of experimental flow, 606 trunk fender for finite element process simulation software, 611 composite reinforcements, 588–613 flow equations, 589–91 mould filling process, 589 schematic representation of a rectangular mould with fibrous reinforcement, 591 numerical solution, 594–603 example of mould filling simulation, 603 fill factor associated with finite element in the mesh, 602 finite element discretisation, 597–8 finite elements for LCM flow simulation, 599–601 flow front advancement, 601–3
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Index mass conservation and div-conform approximation, 598–9 potential formulation, 596–7 FlowTex software, 227–9 Fokker-Planck equation, 40, 624, 627 Folgar–Tucker model, 623–6 closure approximations, 625 interaction coefficient, 625–6 four-directional fabrics, 99 friction, 497–8 composite reinforcement, 397–428 process parameters of tool-fabric friction, 399 sample of thermostamping process, 398 experimental data, 418–21 dynamic coefficient of friction, 420, 421 fabric/fabric friction, 421 predicted fluid-film thickness, 421 test conditions for Hersey investigation, 419 tool/fabric friction - plain weave, 419–21 modeling of themostamping, 422–8 testing methodologies, 411–18 displacement control, 412 displacement-control friction testing apparatus, 412 load control, 412–18 theory, 401–11 effect of tool-temperature shift term on coefficient of friction, 409 Hersey numbers for Twintex, 407 Hersey numbers vs experimental friction coefficient, 407 Power-law parameters for polypropylene, 405 Stribeck curve for a commingled glass–polypropylene four-harness satin-weave fabric, 405 theoretical Stribeck curve, 404, 408 gas permeability measurement, 450 Gebart and Berdichevsky formula, 227 geometry modelling, 503–11 beam and shell or membrane modelling for 2D fabrics, 503–5 deep drawing modelling, 505 solid shell modelling, 504 consistent mesoscopic models, 508–9 zones for consideration in mesoscopic 3D geometry, 509 description of reinforcement types, 204–25 braided fabrics, 213–19 measured and simulated geometry comparison, biaxial braid, 220 NCF quadriaxial WiseTex model, 225 non-crimp fabrics, 219–25 surface image comparison, triaxial braid with WiseTex model, 221 woven fabrics (2D and 3D), 204–13 generic data structure, internal geometry description, 202–4
679
cross-sections defining a yarn shape in a unit cell, 203 textile micro-VR worlds, 205 hexahedral mesh, 510–11 permeability calculations, 511 identification of consistent 3D model, 509–10 3D consistent geometrical model of twill, 510 pre-processor for mechanical properties prediction, 225–32 analytical models and structural analysis, 229–30 FlowTex software, 227–9 meso-FE models gallery, 233 ply-to-ply interlock reinforced composite, 231 reinforcement deformability, 225–7 reinforcement permeability calculation, 228 transformation into meso-FE, 230–2 structure of 3D mesoscopic models, 507 interpenetration in the 3D articulated segment model, 507 TexGen, textile reinforcement for composites, 239–63 applications, 257–62 future trends, 262–3 implementation, 240–6 modelling theory, 246–54 rendering and model export, 254–7 unit cell for 2D fabrics, 505–6 RUCs for plain weave, 506 WiseTex, 200–35 generic data structure, internal geometry description, 202–4 WiseTex software family, 234 glass fibres, 17–19 compositions, densities and mechanical properties, 18 glass fibres are coated with a size, silane, 18 global shear angle, 275 Graphical User Interface, 240, 243–4 TexGen windows user interface screenshot, 243 weave wizard weave pattern dialogue, 244 Green–Lagrange strains, 466, 538–9 Green–Naghdi’s frame, 656–7 Green’s theorem, 596 Grosberg’s bending model, 372–4 Hagen–Poiseuille solution, 437 Hersey number, 403, 411 Hi-Nicalon, 29 Hi-Nicalon type S, 55 high-heat treatment carbon fibres, 56 hourglass control method, 325 HXA7241 Octree Component C++, 241 hybrid braids, 144 hyperelastic model, 515–16, 658–61 deformed shape of an unbalanced fabric, 662
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680
Index
hypoelastic model, 512–15, 656–8 draw-in and shear angle measurement, 659 fibre and GN axes in initial and after deformation states, 657 material draw-in for draped double dome, 660 numerical vs experimental shear angle, 660 Icasoft, 292 image analysis software, 315 immediate elastic recovery, 345 impact damage resistance, 183–4 example of damage tolerance, 185 reduction in delamination damage to a 3D fibre composite, 183 in-plane fibres waviness, crimp and damage, 168–71 broken fibres caused by stitching, 173 crimping schema, 172 fibre waviness around a z-binder, 169 stochastic behaviour of defects and damage, 170–1 in-plane permeability measurement, 447–50 schematic, 448 in-plane shear, 491–3 woven fabric reinforced composites, 267–301 bias extension test experimental results, 294–8 bias extension test experimental setups, 286–94 fabric properties, 270 trellis-frame test experimental results, 280–6 trellis-frame test experimental setups, 270–80 inclusion theory, 229 Initial Graphics Exchange Format, 245 interaction coefficient, 625–6 intermediate-heat treatment carbon fibres, 56 ITOOL, 651 Jacquard technology, 94–5 illustration of process, 95 Jefferey equation, 38 Jeffery theory, 622–3 JF-profile, 150 Kawabata bending test, 370, 374, 493 Kawabata model, 507 KES-FB2 test, 372–3 fabric bending test, 372 Kevlar fibre, 11, 15, 37 deformation, 13 Kevlar yarns, 503 Knudsen effects, 450 Kozeny–Carman constant, 438 LabVIEW, 413 lapping diagram, 219 laser ablation method, 35
lattice Boltzmann, 227 Leicester notation, 219 Levenberg–Marquardt algorithm, 519 light microscopy, 38 liquid composite moulding, 333–4, 651 liquid crystal technology, 14 load control, 412–18 low-heat treatment carbon fibres, 56 LS-DYNA, 400 Mandrel technologies, 144–9 materials, 146 persistent mandrels, 146 removable mandrels, 146–9 Aquacore, 148 blindered sand, 148 evacuated sand-filled tubes, 148–9 example, 147 low melting alloys, 147–8 pressed salt, 148 wax with high melting point, 147–8 mass conservation, 598–9 triangular finite element, 598 matrix cracking, 78 matrix modelling, 472–4 connection elements between fibres and matrix, 473–4 meshing of the matrix, 472–3 mesh of volume occupied by matrix, 473 MeshTex software, 232 meso-scale model, 324, 342 mesophase pitch, 26 mesoscopic approach, 653–6 modelling of a unit cell, 654 simulation of a hemispherical forming using unit cell model, 656 mesoscopic-scale analysis, 572–4 average cross-section determination, 575 example of geometric configurations, 574 glass plain weave cross-sectional shape, 575 micro-mechanics model, 341–2 microcracks, 171–2 microcracking around a z-binder, 174 microscopic-scale analysis, 574–7 covariance analysis on sheared glass plain weave, 577 glass plain weave, 576 modelling theory, 246–54 cross-sections, 248–50 hybrid cross-section example, 250 lenticular cross-sections, 249 power elliptical cross-sections, 249 section selection for hybrid crosssections, 250 fabric geometries variety generated by TexGen, 247 interpolation between yarn sections, 250–1 yarn formed with varying cross-section, 251 intersections, 252–3
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Index actual textile vs model corrected using refine model option, 252 surface mesh showing intersection depths, 254 textile section with intersections, 253 volume mesh adjusted for small intersections, 254 yarn path, 246–8 yarn properties, 253 yarn repeats and domains, 251 two repeat vectors, 251 Monte Carlo approach, 261 Mori–Tanaka scheme, 229 multi-axial multi-ply warp knitted fabrics see non-crimp fabric Multifil method, 232 multilayer patterns, 107–8 4HS pattern, 109 sample, 108 Navier–Stokes equation, 227, 589 near-stoichiometric SiC-based fibres, 70 needles, 98–9 Newtonian matrix, 38 Newton–Raphson algorithm, 472 Nextel, 27 Nextel oxide fibres, 53 Nicalon, 29 Nomex, 11 non-crimp fabric, 219–25 bending test, 388–91 carbon non-crimp fabric, 388 experimental vs simulated profiles, 391 inverse optimisation, 390 loading curve – experimental result and Dahl’s model fitting, 389 example, 225 fibrous plies and fibre distortions by stitching, 223–5 NCF fibrous plies geometry, 224 knitting pattern coding, 219–21 NCF knitting pattern, 221 stitching yarn model, 222–3 NCF stitching yarn geometry, 222 non-orthogonal model continuum mechanics, 533–5 coordinate systems, 533 validation, 552–6 trellising test data, 552–6 24-yarn bias extension test data, 552 woven fabrics, 535–7 deformed plain weave structure with shear deformation, 535 non-oxide fibres, 64 objective derivative, 579, 655 OpenCascade, 245 optical full-field strain techniques, 315 organic fibres, 8–17 aramid fibres, 9–15
681
high-modulus polyethylene fibres, 15–17 thermoplastic fibres, 8–9 orthogonal 3D patterns, 106 illustration, 106 overbraiding, 125–8 curing tools and auxiliary braiding mandrel, 130 examples of overbraiding of extreme mandrel shapes, 129 mandrel manipulation on different braider configurations, 128 wall thickness transitions, 127 oxide Nextel fibres, 66 p-phenylene terephthalamides, 33–4 Patran, 510 permeability composite reinforcement, 431–51 future trends, 451 measurement methods, 445–50 gas permeability measurement, 450 in-plane permeability measurement, 447–50 through-thickness permeability measurement, 446–7 permeability tensor, 432–6 fundamentals of flow in porous media, 434–6 porous medium description, 432–4 saturated permeability modelling for fibre preforms, 436–44 continuum mechanics model, 437–41 historical perspective, 436–7 numerical models for permeability prediction, 441–4 photo curing, 38 photo-degradation protection, 46 Piola-Kirchoff stress tensor, 515, 557, 661 plain weave, 100–1 basket pattern, 101 fabric surface, 101 pattern, 100 plain weave composite fabric shear modulus, 544–51 bias experimental load-displacement curves of 16-yarn fabrics, 546 bias experimental load-displacement curves of 24-yarn fabrics, 547 bias extension sample, 545 contour of Green–Lagrange direct strains, 549 contravariant shear modulus, 550 deformed bias extension sample, 545 deformed mesh with contour of Green– Lagrange shear strain, 549 equivalent shear modulus, 550 experimental setup, 544 FE mesh for bias extension simulation, 548 specific application, 537–52 direct strains in trellising test, 539
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682
Index
shear angle in trellising test, 538 shear strains in trellising test, 540 tensile moduli, 539–43 equivalent tensile modulus, 543 results of uniaxial tensile tests, 542 uniaxial tensile tests, 541 plastic, 347 plasticity model, 355–6 plasticity theory, 347 Poisson’s effect, 466 Poisson’s ratio, 253, 325 poly-p-phenylenebenzobisoxazole, 14–15 polyacrylonitrile, 22–6 polyamide 6, 8–9 polyamide 6.6, 8–9 polyarylamide, 634 poly(ester terephthalate), 9 poly(meta-phenylene isophthalamide), 11 poly(p-phenylene-2,6-benzobisoxazole, 33–4 poly(para-phenylene terephthalamide), 11 polypyridobisimidazole, 14 porous medium, 432–4 fundamentals of flow, 434–6 REV example, non-crimp fabric composite and pore size distribution, 433 Power Law of Ostwald and de Waele, 404 preform, 431 PROEngineer, 508 projectile process, 98 Rachinger mechanism, 72 Raman spectroscopy, 35 rapier process, 96–8 fill insertion technologies, 98 reduced strain closure model, 627–8 regression technique, 68 representative unit cell, 503, 506, 518 representative volume element, 261, 432 reptation theory, 33–4 resin transfer moulding, 334, 651 Reynolds number, 435 rheological model, 42, 629–31 generic form, 629–30 model with interaction tensor for concentrated suspension, 631 slender body approximation and ellipsoidal particle theories, 630–1 rheology, 38 S-glass, 17 sapphire fibres, 53 satin, 102–4 5HS and 8HS patterns, 103 4HS patterns, 103 saturated permeability modelling, 436–44 continuum mechanics model, 437–41 capillary tube models, 437–8 reduced permeability as a function of fibre volume fraction for flow along fibre axis, 439 reduced permeability as a function
of fibre volume fraction for flow orthogonal to fibre axis, 440 resistance to flow models, 438–41 historical perspective, 436–7 reinforcement preform and principal axes, 437 permeability prediction of multiscale porous media, 441–4 schematic of a unit cell for numerical modelling, 442 saturation coefficient, 601 semi-discrete approach, 661–8 forming of an unbalanced textile reinforcement, 667 forming with a cylindrical punch, 666 loads on a unit woven cell, 662 loads resultant on a unit woven cell, 663 three-node finite element made of unit woven cell, 664 sensing, 46 shear, 226–7 shear test, 480–1 plain weave sample, 482 shuttle process, 96 flying shuttle loom, 97 hand loom, 97 Sic-based fibres, 67–8 silicon carbide, 19–20 Simplified Wrapper and Interface Generator, 240, 245 single-walled CNTs, 32–3 skin-core structure, 622 slender body theory, 630 smooth particle hydrodynamics, 595 solar cell applications, 46 Sourceforge, 245 spiral fabrics, 110 sample, 111 stacked preforming, 151 Standard for the Exchange of Product model data, 245 stitching, 160–2 illustration of stitching process, 162 stress relaxation test, 337 Stribeck number see Hersey number Stribeck’s theory, 404, 409 structural non-crimp knitting, 163 non-crimp fabric, 165 structural tensor, 658, 660 surface mesh, 255 single yarn surface mesh, 255 Sylramic fibres, 55 Teflon, 417 tension, 499 non-linear tensile curve for twisted fibre bundle, 499 Terzaghi’s law, 335 test loading cases composite sample, 481–2 deformed mesh of plain weave, 483
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Index deformed mesh of plain weave after global twisting, 484 slice of deformed mesh of plain weave, 484 dry fabric, 479–81 equibiaxial extension, 480 mesh of modelled plain weave sample, 482 plain weave sample before and after shear deformation, 483 shear test, 480–1 shear test for plain weave sample, 482 testing methodologies, 411–18 displacement control, 412 displacement-control friction testing apparatus, 412 normal load drops as fabric is pulled through pressure plates, 413 load control, 412–18 close-up of the components of the loadcontrol friction tester, 414 configuration of friction test, 418 experimental pull-out setup, 419 fabric holder modification, 416 original and modified test conditions, 416 pull-out force vs displacement curves, 414 schematic of fabric/fabric setup, 417 tow deformation and excess resin, 415 TexComp software, 229 TexGen, 324, 342 applications, 257–62 Chomarat150TB, 259 computational fluid dynamics predictions, 262 finite element analysis with continuum damage model, 260 textile composite mechanics, 260–1 textile mechanics, 258–9 textile permeability, 261–2 geometry modelling, textile reinforcement for composites, 239–63 future trends, 262–3 implementation, 240–6 core, 241–2 core module UML class diagram, 242 export, 245 Graphical User Interface, 243–4 python interface, 245 renderer, 243 TexGen module diagram, 241 TexGen usage, 245–6 twill weave from python script, 246 modelling theory, 246–54 cross-sections, 248–50 interpolation between yarn sections, 250–1 intersections, 252–3 yarn path, 246–8 yarn properties, 253
683
yarn repeats and domains, 251 rendering and model export, 254–7 dry fibre volume mesh, 255–6 plain weave tetrahedral yarn/matrix mesh, 258 plain weave tetrahedral yarn mesh, 257 plain weave triangular yarn areas, 257 single yarn volume mesh, 256 surface mesh, 255 voxel mesh, 256–7 yarns and matrix volume mesh, 256 TexGen ver. 3.3.2, 240 textile composite reinforcements modelling of forming processes, 651–68 continuous approach, 656–61 mesoscopic approach, 653–6 plane motor blade, 652 preform/RTM parts in NH90, 652 semi-discrete approach, 661–8 textile composites application examples, 477–82 application of loading cases to composite sample, 481–2 characteristics of model, 477 geometrical characteristics, 477 initial configuration of woven structures, 478–9 mechanical characteristics, 478 test loading cases on dry fabric, 479–81 interests and goals, 463–5 approach to mechanical behaviour, 464–5 complexity, 463 initial geometry, 463–4 microscopic approaches for study of mechanical behaviour, 461–85 application examples, 477–82 modelling approach, 465–77 boundary conditions, 474–6 contact-friction interactions, 467–72 finite strain beam model, 465–7 initial configuration of woven fabric, 476–7 modelling of the matrix and its interactions with textile reinforcements, 472–4 starting configuration, 476 thermal curing, 38 thermal oxidation, 35 thermomechanics, 356–9 equilibrium hysteresis loops, 358–9 thermoplastic fibres, 8–9 thermostamping, 270, 397 constant friction, 423–5 constant friction vs no friction effect on punch force, 424 punch force for different stamping rates, 424 modeling, 422–8 illustration of tools, 422 no friction, 423
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684
Index
effect on punch force for balanced plainweave fabric, 423 variable friction, 425–8 comparison of punch forces, 426 effect of variable friction on punch force, 427 example of linear fit, 425 tensile stresses in fabric yarns, 427 three-dimensional braiding, 128–33 design principle, 130 example of 3D braid, 131 first generation 3D braider, 131 machine setup for the T-profile, 132 second generation 3D braider, 133 tangled fibres on a 3D braider, 133 textile variants, 141 simulation of a 3D braided H-profile, 141 three-dimensional digital image correlation, 315 three-dimensional fibre composites, 157–97 basic fibre architecture, 158 current and potential applications, 159 delamination fracture, 175–83 impact damage resistance and tolerance, 183–4 in-plane mechanical properties, 187–95 elastic modulus, 187–90 failure strength, 190–3 fatigue properties, 193–5 joint properties, 195–6 effect of volume content of z-pins, 196 fatigue life improvement, 196 manufacture of 3D fibre composites, 160–7 braiding, 165 3D weaving, 160 stitching, 160–2 structural non-crimp knitting, 163, 165 tufting, 165–7 z-anchoring, 167 z-pinning, 163 microstructure of 3D fibre composites, 168–75 damage to z-binders, 172–4 modelling, 175 overview, 168 resin-rich regions, voids and microcracks, 171–2 swelling and compaction, 174–5 waviness, crimp and damage to in-plane fibres, 168–71 through-thickness stiffness and strength, 184, 186 through-thickness thermal properties, 186–7 through-thickness permeability measurement, 446–7 illustration, 447 through-thickness stiffness, 184, 186 through-thickness thermal conductivity, 186–7 effect of type and volume fraction of z-binder, 187
TinyXML, 241 tow testing technique, 63 transmission function, 321 transverse compression composite reinforcements, 333–60 compaction curve, 338–44 cyclic compaction of a plain weave fabric, 337 experimental procedure for material characterisation, 336–8 stress relaxation experiments, 338 inelastic models, 351–9 plasticity model, 355–6 thermomechanics, 356–9 viscoelastic model, 351–5 viscoplasticity, 356 inelastic response of fibrous materials, 344–51 cyclic loading of a continuous filament, 349 locked energy, 349–51 percentage work difference, 350 schematic of stress-volume fraction response, 348 trellis-frame test experimental result, 280–6 balanced twill-weave fabric behaviour, 285–6 normalised shear force vs global shear angle of balanced twill weave fabric, 286 normalised shear force vs global shear angle of unbalanced twill weave fabric, 287 plain-weave fabric behaviour, 281–5 shear force normalised by frame length vs shear angle, 283 shear force normalised by inner fabric area vs shear angle, 284 shear force normalised using energy method vs shear angle, 285 shear force vs shear angle, 282 shear force vs shear angle comparison for plain weave fabric, 283 shear force vs shear angle with linkage amplification, 281 unbalanced twill-weave fabric behaviour, 286 experimental set-up, 270–80 experimental displacement weave, 277 frame design and clamping mechanism, 271–3 frame size and test parameters, 273 free body diagrams, link BC and link BAF, 279 global vs theoretical strain, 278 picture frame at UML schematic diagram, 279 picture frame geometry, 280 picture-frame tests tested fabrics, 271 picture frames, 272
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Index relationship between optically measured shear angle and shear angle of frame, unbalanced twill weave, 276 sample preparation, 273–4 shear angle determination, 274–7 shear angle measurement, 275 shear force determination, 277–80 von Mises strain field, 276 yarn specimens removed from arm region, 274 trellising test, 552–6 FEM model, 554 numerical vs experimental data for 30 in2 samples, 556 results of 30 in2 samples, 554 schematic, 560 trellising shear frame with a testing sample, 553 tri-axial fabrics, 99 Triangle, 241 triaxial braids, 138 typical surface quality, 138 tubular fabrics, 110–11 tufting, 165–7 through-thickness reinforcement of a T-shaped fabric, 166 Twaron fibre, 11, 37 twill weave, 101–2 examples, 102 2X1 pattern, 102 Twintex, 213, 398, 414, 418, 422, 488 two-dimensional digital image correlation, 315 Tyranno fibres, 29 Tyranno SA3, 55 UAZ, 163 UD-braids, 134–5, 139–40 braiding principle, 139 compression properties, 140 effect of braiding, 140 Unified Modelling Language, 241 unsaturated permeability modelling, 444–5 alternative methods, 445 relative permeability, 444–5 vacuum assisted resin transfer moulding, 335 van der Waals forces, 32 viscoelastic model, 351–5 compaction stress for continuous filament mat, 354 stress during compaction and relaxation, 355 stress relaxation for dry and lubricated continuous filament mats, 353 viscoplasticity, 356 Visualisation Toolkit, 243 voids, 171–2 resin-rich regions and voids within a z-pinner composite, 173 volumetric swelling, 174–5 voxel, 567 voxel mesh, 256–7
685
plain weave voxel mesh example, 258 warp, 89 beam technology, 92–3 sample, 93 creel technology, 91–2 sample, 92 warp head, 93–5 harness frame technology, 93–4 harness heddles, 94 Jacquard technology, 94–5 illustration of process, 95 warp zones, 205 water jets, 98 Weave Wizard, 243 weaving, 160 schematic representation, 161 ±45° weaving technology, 99 weft, 89 Weibull modulus, 6, 57 ‘wet compression,’ 335 WiseTex, 324, 342 textile reinforcement geometry modelling for composites, 200–35 generic data structure, internal geometry description, 202–4 geometrical description, reinforcement types, 204–25 geometrical model, mechanical properties prediction pre-processor, 225–32 woven fabric, 204–13 composite reinforcement, 89–115 applications, 114–15 future trends, 115 2D/3D weave structure coding, 204–7 multilayered weave matrix coding, 206 definition, 100–13 carpet fabric pattern, 107 2D patterns, 100–4 3D patterns, 104–10 properties of 3 basic 2D pattern, 104 properties of 3D patterns, 108–10 specific shapes, 110–13 woven fabric characterisation, 113 examples, 213 2D woven fabrics geometrical models examples, 214 ply-to-ply interlock reinforcement crosssections, 215 triaxial braids coding example, 215 geometric model, 204–13 crimp interval for internal geometry calculation, 210 elementary crimp interval, 208 mathematical description of internal structure, 213 technology description, 90–9 basic weaving loom, 91 crossing warp head, 93–5 fabric width, 99
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Index
fill insertion, 96–9 other processes, 98–9 other weaving technologies, 99 take-up systems, 95–6 warp presentation, 90–3 weaving loom schema, 91 woven fabric composites bias extension test experimental results, 294–8 balanced twill-weave fabric behaviour, 295–6 plain-weave fabric behaviour, 294–5 unbalanced twill-weave fabric behaviour, 296–8 bias extension test experimental setups, 286–94 sample preparation, 288–90 shear angle determination, 290–2 shear force determination, 292–4 fabric properties, 270 fabric parameters, 271 woven fabrics used in study, 270 in-plain shear properties, 267–301 bias-extension test apparatus, 269 fabric properties, 270 normalised shear force vs shear angle of balanced twill weave, 300 normalised shear force vs shear angle of plain weave, 299 normalised shear force vs shear angle of unbalanced twill weave, 300 trellising-shear test apparatus, 269 specific fibre-reinforced hyperelastic model, 558–62 geometric structure and deformation modes, 558 shear strain energy density vs I8 in trellising test, 562 strain energy density vs (I4 – 1) in uniaxial tensile test, 560 trellising load curve, 561 uniaxial tensile stress–strain curve, 560 trellis-frame test experimental results, 280–6 balanced twill-weave fabric behaviour, 285–6 plain-weave fabric behaviour, 281–5 unbalanced twill-weave fabric behaviour, 286 trellis-frame test experimental setups, 270–80 frame design and clamping mechanism, 271–3 sample preparation, 273–4 shear angle determination, 274–7
shear force determination, 277–80 woven structures initial configuration, 478–9 computed configuration, 478 plain weave and twill weave, 479, 480 twill weave, 479 wx Widgets, 243 X-ray microtomography, 621 X-ray scattering, 55 X-ray tomography analyses of textile reinforcement structure, 571–7 mesoscopic-scale analysis, 572–4 microscopic-scale analysis, 574–7 analysis of mechanical behaviour of composite reinforcement, 565–85 analysis of structure, 571–7 finite element simulations, 578–85 finite element simulations, 578–85 constitutive modelling, 578–82 validation of FE simulations, 582–5 principle, 566–8 schematic of tomography equipment, 567 woven textile composite reinforcements, 568–71 device for in situ mechanical loading, 571 imaging of woven reinforcements in unloaded state, 569 in situ mechanical loading, 570–1 tomography scans, 569–70 yarn mechanical behaviour, 498–503 bending, 502 3D reconstruction of X-ray tomography imaging of carbon twill, 498 friction identification, 502–3 in plane shear, 503 tension, 499 transverse behaviour, 500–2 yarn dimensions, 211 Young modulus, 6, 35, 53, 253, 321, 326, 516–17 z-anchoring, 167 illustration, 167 z-binders, 172–4 microcracking, 174 z-pinning, 163 prepreg composite, 164 Zylon fibre, 14
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