COMPOSITE SHEET FORMING
COMPOSITE MATERIALS SERIES
Series Editor: R. Byron Pipes, Center for Composite Materials, University of Delaware, Newark, Delaware, USA Vol. Vol. Vol. Vol. Vol. Vol.
1 2 3 4 5 6
Vol. Vol. Vol. Vol.
7 8 9 10
Friction and Wear of Polymer Composites (K. Friedrich, Editor) Fibre Reinforcements for Composite Materials (A.R. Bunsell, Editor) Textile Structural Composites (T.-W. Chou, Editor) Fatigue of Composite Materials (K.L. Reifsnider, Editor) Interlaminar Response of Composite Materials (N.J. Pagano, Editor) Application of Fracture Mechanics to Composite Materials (K. Friedrich, Editor) Thermoplastic Composite Materials (L.A. Carlsson, Editor) Advances in Composite Tribology (K. Friedrich, Editor) Damage Mechanics of Composite Materials (R. Talreja, Editor) Flow and Rheology in Polymer Composites Manufacturing (S.G. Advani, Editor)
Cover illustration- Arrow diagram for a [0,9012s blister fairing, 61max- 35.3%, (for details see p. 233).
t~Zmax = - 2 5 . 9 %
Composite Materials Series, 11 COMPOSITE SHEET FORMING edited by
D. Bhattacharyya Department of Mechanical Engineering, School of Engineering, University of Auckland, Auckland, New Zealand
1997 ELSEVIER Amsterdam
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Lausanne
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New
York
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Oxford
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Shannon
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Tokyo
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands
ISBN ISBN
0-444-82641-6 (Vol. 11) 0-444-42525-X (Series)
9 1997 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright and Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U S A - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside the USA, should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands.
PREFACE Sheet forming is one of the most used processes in metal forming and hence it is not surprising to see considerable efforts being put into adapting or modifying the existing sheet metal forming techniques to suit the needs of forming composite sheets. With the increasing availability of various types of fibre-reinforced polymeric sheets, especially with thermoplastic matrices, the scope of using such materials is rapidly expanding in the automobile, building, sports and other manufacturing industries beyond the traditional areas of aerospace and aircraft applications. Their good structural and anti-corrosion properties with available design versatility give them many advantages over traditional metallic sheets. However, to realise the full potential of these materials, it is often necessary to redesign the component with subsequent manufacturing in mind. For economic competitiveness it is also desirable to develop rapid manufacturing techniques suitable for different types of composite sheets. Thus the understanding of their fundamental deformation behaviour, along with the capability of developing analytical models, is essential for efficient and defect-free forming of this relatively new generation of materials. This book contains twelve chapters and attempts to cover different aspects of sheet forming including both thermoplastic and thermosetting materials. In view of the expanded role of fibre-reinforced composite sheets in the industry, the book also describes some non-traditional applications, processes and analytical techniques involving such materials. It has often been noted by the Editor and his research colleagues that familiarity with the basic principles and ideas of sheet metal forming is somewhat lacking among the researchers of composite sheet forming. Although there are fundamental differences in nature between metallic and composite sheets, there are many concepts and techniques, originally developed for sheet metal forming, that can be successfully utilised for composites. Hence the first chapter has been dedicated to a brief introduction to the principles of sheet metal forming. The next two chapters introduce the various forms of materials, manufacturing techniques and the fundamentals of computer simulation. Chapter 4 describes the different aspects of thermoforming of continuous fibre-reinforced thermoplastics and the following chapter studies the shear and frictional behaviour of composite sheets during forming. Grid strain analysis is an established technique for sheet metals; Chapter 6 explores the possibility of applying this method in continuous fibre-reinforced polymeric sheets. For an efficient manufacturing process development, finite element modelling and an understanding of rheology play extremely important roles; the next two chapters address these topics with the fundamental concepts reviewed and the recent developments discussed. Chapter 9 introduces the theory of bending, which is an
vi
Preface
integral mechanism of most sheet forming processes, of thermoplastic composite sheets and shows a novel way of determining both longitudinal and transverse viscosities through vee-bend tests. A significant expansion of the usage of composite materials is taking place in biomedical areas and Chapter 10 discusses the thermoforming of knitted fabric-reinforced thermoplastics for load-bearing, anisotropic bio-implants. Chapter 11 studies the forming of thermoset composites, a new technique of manufacturing composite components, and highlights the phenomenological similarities with those found with thermoplastics. The last chapter introduces roll forming, a commonly used rapid manufacturing process for sheet metals, and discusses the possibility of applying it economically for continuous fibre-reinforced thermoplastic sheets. I would like to thank Dr R. B. Pipes, editor-in-chief of this Elsevier series on composite materials, for inviting me to organise and edit this particular volume. Thanks are also due to the authors of the various chapters for making their learned contributions and complying with the necessary publication schedule. Finally I wish to thank the University of Auckland, the German Science Foundation, the University of Kaiserslautern and my family for their support during my sabbatical leave which allowed me to complete the final phase of the editing work. D. Bhattacharyya
LIST OF CONTRIBUTORS
S.G. ADVANI
Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA B. Tomas ASTROM
Department of Aeronautics, Division of Lightweight Structures, Royal Institute of Technology, S-IO0 44 Stockholm, Sweden D. B H A T T A C H A R Y Y A
Composites Research Group, Department of Mechanical Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand G.R. CHRISTIE
Composites Research Group, Department of Mechanical Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand T.S. CREASY
Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA (currently at University of Southern California, Los Angeles, CA,
USA) J.L. D U N C A N
Composites Research Group, Department of Mechanical Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand R.J. DYKES
Composites Research Group, Department of Mechanical Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand K. F R I E D R I C H
Institute for Composite Materials Ltd. (IVW), University of Kaiserslautern, 67663 Kaiserslautern, Germany M.O. G H A F U R
Polydynamics Inc., 1685 Main St. West, Suite 305, Hamilton, Ontario, Canada L8S 1G5 vii
viii
List of contributors
Timothy G U T O W S K I
Laboratory for Manufacturing and Productivity, Massachusetts Institute of Technology, Cambridge, MA 02139, USA M. HOU
Centre for Advanced Materials Technology, Department of Mechanical Engineering, University of Sydney, Sydney, N S W 2006, Australia B.L. K O Z I E Y
Polydynamics Inc., 1685 Main St. West, Suite 305, Hamilton, Ontario, Canada L8S 1G5 J. KREBS
Institute for Composite Materials Ltd. (IVW), University of Kaiserslautern, 67663 Kaiserslautern, Germany Haorong LI
Laboratory for Manufacturing and Productivity, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Patrick J. M A L L O N
Mechanical and Aeronautical Engineering Department, University of Limerick, Limerick City, Ireland S.J. M A N D E R
Composites Research Group, Department of Mechanical Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand (currently at McKinsey & Company, Sydney, NSW, Australia) T.A. M A R T I N
Composites Research Group, Department of Mechanical Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand J. M A Y E R
Biocompatible Materials Science and Engineering, Department of Materials, Swiss Federal Institute of Technology, ETH Zurich, Wagistrasse 23, 8952 Schlieren, Switzerland S.P. McENTEE
Composites Research Unit, University College, Galway, Ireland G.B. McGUINNESS
Composites Research Unit, University College, Galway, Ireland F.A. M I R Z A
Faculty of Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7
Lbt of contributors
ix
Adrian M. M U R T A G H Engineering Department, Cambridge University, Trumpington Street, Cambridge CB2 1PZ, UK C.M. 6 BR,~DAIGH Composites Research Unit, University College, Galway, Ireland S.M. P A N T O N
Composites Research Group, Department of Mechanical Eng&eer&g, University of Auckland, Private Bag 92019, Auckland, New Zealand S.F. SHULER Department of Mechanical Eng&eering, University of Delaware, Newark, DE 19716, USA (currently at General Electric Co., Pittsfied, MA, USA) J. V L A C H O P O U L O S
Faculty of Eng&eering, McMaster University, Hamilton, Ontario, Canada L8S 4L7 E. W I N T E R M A N T E L
Biocompatible Materials Science and Eng&eering, Department of Materials, Swiss Federal Institute of Technology, ETH Zurich, Wagistrasse 23, 8952 Schlieren, Switzerland
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CONTENTS
A more detailed contents list is given at the beginning of each chapter.
Preface
v
List of Contributors
vii
Chapter 1 (J.L. Duncan and S.M. Panton) Introduction to sheet metal forming
1
Abstract 1 1.1. Introduction 2 1.2 Introduction to plastic flow theory 2 1.3. Forming characteristics of sheet metals 1.4. Forming limits for sheet metal 12 1.5. Industrial sheet metal forming 15 1.6. Bending and spring-back 19 1.7. Superplasticity 23 References 25
Chapter 2 (B. Tomas Astr6m) Thermoplastic composite sheet forming: materials and manufacturing techniques Abstract 27 2.1. Introduction 28 2.2. Constituents 29 2.3. Properties 48 2.4. Manufacturing techniques Acknowledgement 72 References 72
60
xi
27
Contents
xii
Chapter 3 (B.L. Koziey, M.O. Ghafur, J. Vlachopoulos and F.A. Mirza)
Computer simulation of thermoforming 75 Abstract 75 3.1. Introduction 76 3.2. Sheet production 77 3.3. Thermoforming simulation 3.4. Concluding remarks 88 References 88
78
Chapter 4 (K. Friedrich, M. Hou and J. Krebs)
Thermoforming of continuous fibre/thermoplastic composite sheets Abstract 92 4.1. Introduction 92 4.2. Experimental details and procedures 96 4.3. 2-D stamp forming 100 4.4. 3-D stamp forming 137 4.5. 3-D diaphragm forming of GF/PP laminates 4.6. Summary 159 Acknowledgements 160 References 160
91
146
Chapter 5 (A.M. Murtagh and P.J. Mallon)
Characterisation of shearing and frictional behaviour during sheet forming Abstract 163 5.1. Introduction 164 5.2. Transverse fibre flow 170 5.3. Intra-ply shear 173 5.4. Inter-ply slip 177 5.5. Friction during thermoforming References 214
197
Chapter 6 (T.A. Martin, G.R. Christie and D. Bhattacharyya)
Grid strain analysis and its application in composite sheet forming 217 Abstract 217 6.1. Introduction 218 6.2. Large strain analysis 218 6.3. Method of least squares fitting 224 6.4. Forming a composite spherical dome
226
163
Contents
xiii
6.5. Forming a composite blister fairing 230 6.6. Draping theory of textile fabrics 234 6.7. Diagnostic applications 238 6.8. Concluding remarks 241 References 244
Chapter 7 (C.M. O Brfidaigh, G.B. McGuinness and S.P. McEntee) Implicit finite element modelling of composites sheet forming processes
247
Abstract 248 7.1. Introduction 248 7.2. Modelling of composite sheets during forming 254 7.3. Numerical s o l u t i o n s - plane stress problems 258 7.4. Central indentation of a composite sheet ~ the shear-buckling problem 7.5. Experimental comparisons - - diaphragm forming 286 7.6. Conclusions of plane stress analysis 303 7.7. Numerical s o l u t i o n s - plane deformation problems 305 7.8. Conclusions of plane deformation analysis 315 Acknowledgements 318 Nomenclature 318 References 319
Chapter 8 (S.G. Advani, T.S. Creasy and S.F. Shuler) Rheology of long fiber-reinforced composites in sheet forming
323
Abstract 324 8.1. Introduction 324 8.2. Rheological properties 329 8.3. Rheological measurement techniques 348 8.4. Why the rheological properties are important and how to use them in sheet forming 356 8.5. Outlook 366 References 367
Chapter 9 (T.A. Martin, S.J. Mander, R.J. Dykes and D. Bhattacharyya) Bending of continuous fibre-reinforced thermoplastic sheets Abstract 371 9.1. Introduction 372 9.2. Development of an idealised viscous bending model 9.3. Experimental procedures 380
371
374
263
xiv
Contents
9.4. Results and discussion 382 9.5. Modified constant shear rate tests 9.6. Conclusions 399 Acknowledgements 399 References 400
392
Chapter 10 (J. Mayer and E. Wintermantel) Thermoforming processes for knitted-fabric-reinforced thermoplastics: new manufacturing techniques for load-bearing, anisotropic implants 403 Abstract 404 10.1. General aspects of anisotropic biomaterials for load-bearing implants 10.2. Knitted-carbon-fiber-reinforced composite materials 405 10.3. Net-shape forming of knitted fabrics for load-transmitting implants shown for an ulnar osteosynthesis plate 419 10.4. Deep drawing of knitted-fiber-reinforced organo-sheets 428 10.5. Discussion 432 10.6. Summary and conclusions 435 Acknowledgements 435 References 436
Chapter 11 (H. Li and T. Gutowski) The forming of thermoset composites
441
Abstract 441 11.1. Introduction to thermoset forming 442 11.2. Kinematics 446 11.3. Thermoset forming experiments and forming limit analysis 11.4. Concluding remarks 468 References 471
455
Chapter 12 (S.J. Mander, S.M. Panton, R.J. Dykes and D. Bhattacharyya) Roll forming of sheet materials
473
Abstract 474 12.1. Introduction 474 12.2. Roll forming equipment and tooling 476 12.3. Conventional form roll design 483 12.4. Computer-aided design in roll forming 489 12.5. Deformation analysis of roll forming 491
404
Contents
12.6. Roll forming of thermoplastic material 12.7. Concluding remarks 512 Acknowledgements 513 References 513 Author Index Subject Index
517 525
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498
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Composite Sheet Forming edited by D. Bhattacharyya 9 Elsevier Science B.V. All rights reserved.
Chapter 1
Introduction to Sheet Metal Forming J.L. D U N C A N and S.M. P A N T O N Department of Mechanical Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand
Contents Abstract 1 1.1. Introduction 2 1.2. Introduction to plastic flow theory 2 1.2.1. The principal element 2 1.2.2. Yielding of isotropic materials 2 1.2.3. Yielding of anisotropic materials 3 1.2.4. Deformation of principal element 4 1.2.5. Deviatoric stresses 5 1.2.6. The flow rule 5 1.2.7. Anisotropic flow rules 6 1.2.8. Plastic work 6 1.3. Forming characteristics of sheet metals 6 1.3.1. Approximate constitutive relationships 11 1.3.2. The relation between tensile data and multiaxial stress conditions 1.4. Forming limits for sheet metal 12 1.5. Industrial sheet metal forming 15 1.5.1. Stretch forming 15 1.5.2. Deep drawing 17 1.6. Bending and spring-back 19 1.6.1. Bending without tension 19 1.6.2. Bending under tension 22 1.7. Superplasticity 23 References 25
11
Abstract
This chapter is intended to provide a concise introduction to plastic flow theory and the mechanics of sheet metal forming. The influence of material properties on sheet metal "formability" is considered with specific reference to the properties that can be determined from a simple tensile test. The instabilities which limit different forming operations are described and the forming limit diagram introduced. Simple
2
J.L. Duncan and S.M. Panton
models are derived for some practical forming processes, and the forming conditions that result are described. A brief introduction to superplastic forming is given.
1.1. Introduction Sheet metal forming covers a wide variety of operations and consequently many different forming conditions. If we consider a small element of the sheet, as shown in fig. 1.1, it is observed that there are a number of common factors in the state of stress and strain in most of these processes. These can be summarised in the following way: 1. the sheet is formed by tractions transmitted through the sheet; 2. at least one principal stress is tensile; 3. with very few exceptions, the through thickness stress is zero and consequently a state of plane stress exists; 4. the process is limited by the instabilities of local necking or wrinkling (buckling). In this chapter, we first give an introduction to sheet metal forming and in section 1.2 look at some fundamentals of plastic flow theory which is the basis for studying the mechanics of forming. In section 1.3, we examine how the material properties of sheet metal can be described in a way that helps to characterise the formability of the sheet material. In section 1.4, the instabilities which limit the different sheet forming operations will be examined and in section 1.5, we develop simple models for some practical forming processes.
1.2. Introduction to plastic flow theory 1.2.1. The principal element
The state of stress at a point can be expressed in terms of the magnitude and orientation of three principal stresses as shown in fig. 1.2; in what follows, cr3, is the stress normal to the sheet, which, as mentioned, is usually zero. 1.2.2. Yielding of isotropic materials
Plastic yielding is the transition from small and recoverable elastic deformation to irreversible, permanent deformation. In a uniaxial test, the instantaneous value of the yield stress is referred to as the flow stress, of.
Fig. 1.1. A small element taken from a deforming sheet.
Introduction to sheet metal form&g
3
O3
(Yl
(Y2
Fig. 1.2. The principal stresses at a point.
Two well-known criteria for the onset of plastic yielding are (i) the Tresca and (ii) the von Mises yield criteria. The Tresca yield criterion states that yielding will commence when the magnitude of the maximum shear stress is equivalent to the maximum shear stress at yield in a uniaxial test. Relating the maximum shear stress to the principal stresses through Mohr's circle of stress, it can be shown that Tresca's yield criterion can be stated in terms of the principal stresses and the flow stress, 0"1 --0"3 = O f
(1.1)
for the case of al > a2 > a3. As originally stated, the von Mises criterion was based on the assumption that the distortional strain energy in a general state of stress reaches a critical value at yield. One could, however, state that yielding occurs at a critical value of the root-meansquare average of the the maximum shear stresses in an element. This gives the same result, which can be expressed mathematically as (0-1 -- 0"2) 2 "+- (0"1 -- 0"2) 2 + (0"1 -- 0"2) 2 = 20-2
(1.2)
It may be seen from either criterion that in metals, yielding is fundamentally related to the shear stresses in an element and that there is not a great difference between the two criteria. The stress conditions in sheet metal forming are generally those of plane stress and the two criteria can be compared graphically in fig. 1.3. It may be noted that each gives identical results for conditions of uniaxial stress and equal biaxial stress, but for other conditions the von Mises yield criterion predicts higher stresses for the onset of yielding.
1.2.3. Yielding of anisotropic mater&& In the above section, it was assumed that yielding is independent of the orientation of the principal axes. In anisotropic materials this assumption is not valid. A common form of anisotropy is for the material properties in the thickness direction to vary from those in the plane of the sheet. This type of variation of material properties with direction can be characterised by the constant, R which is the ratio of width
4
J.L. Duncan and S.M. Panton
(Yl Tresca
i,,
__.
if2
V
_ h .
Mises
Fig. 1.3. Comparison of the Tresca and von Mises yield criteria
to thickness strain in a uniaxial tensile test. In this case a suggested plane stress yield locus [1] is (1.3)
O'~ + O'~ + R(ff 1 --if2) a = R(o'f) a
where a is a constant related to the crystal structure of the material (typically about 6 for body-centred cubic materials and 8-10 for face-centred cubic materials). 1.2.4. Deformation of a principal element
A principal element will deform without shear and the strain increments which result are principal ones. With reference to fig. 1.4, the strain increments are, de1
-
da/a
(1.4)
de 2 - db/b de1 - dc/c
A metal deforming elastically undergoes a small volumetric change; during plastic strain, however, the volume change in a typical metal is very small and is often neglected. Therefore, (a + da)(b + db)(c + dc) - abc - 0 2
2
c+~
1
3
1"~
Fig. 1.4. Deformation of a principal element.
~
"~3
Introduction to sheet metal forming
5
Neglecting the product terms of small quantities one obtains da/a + db/b + de/c = 0 so that the sum of the principal strain increments is zero, i.e. (1.5)
del + de2 + de3 = 0 1.2.5. Deviatoric stresses
The principal stresses may be considered as the sum of two components, namely the hydrostatic stress (o"h) and the deviatoric stresses 0"{, 0"~ 0"~ as shown in fig. 1.5. This leads to the following expressions for the deviatoric stresses 0"~ -- 0"1 - 0"h l 0"2 - - 0"2 -- 0-h
0"~ - 0"3 - 0"h
(1.6)
where 0"h = (0-1 + 0"2 + 0"3)/3. 1.2.6. The flow rule The relation between stress and the strain increment during plastic deformation of isotropic materials is given by the flow rule, (1.7)
de]/0"{ - d e 2 / 0 " 2 ' - d e 3 / 0 " 3 ' - d~.
i.e. the principal strain increments are in the same proportion as the deviatoric stresses, .
!
del "de2"de3
/
/
(1.8a)
"0-1 " 0 2 " 0 3
The constant, d)~, indicates the magnitude of the strain increments and and its value is not determined by the stress state. For a given yielding state of stress, the material may not deform, in which case 3~ -- 0 or in a non-strain-hardening material it may deform appreciably at constant stress. In this respect, plastic deformation differs considerably from the elastic state where there is a one-to-one relationship between stress and strain. Furthermore the same strain state, del, de2, de3 may arise from a number of different stress states which differ only in their hydrostatic component. t
~3
03 IL
~h
+
-
t
Ot
a2
Fig. 1.5. Deviatoric and hydrostatic stresses.
02 ~h
~h
6
J.L. Duncan and S.M. Panton
1.2.7. Anisotropic flow rules
An alternative interpretation of the flow rules is that the vector representing the plastic strain increments is normal to the yield surface (the principle of normality). It follows that in addition to the flow rules described in (1.8a), there are flow rules which correspond to the anisotropic yield criteria described in eq. (1.3). For a situation of plane stress, the anisotropic equivalent of eq. (1.8a) is given [1] by: d e 1 " d e 2 "" (0"1) a - 1 - R ( f f 1 - 0 " 2 ) a - 1 " (0"2) a - 1 -
R(cr 2 -0"1) a-1
(1.8b)
1.2.8. Plastic work
The forces acting on the faces of the principal element shown in fig. 1.4 are ~r1.bc, crz.ac and a3.ab, hence the work done in deforming the element is given by d W = er1bc. da + ~rzac. db + cr3ab, dc
As the volume of the element is abc, from eq. (1.4) we can show that the work done per unit volume is dW/vol = eqdel + crzde2 + cr3de3
(1.9)
This may be rewritten in the form, d W/vol = a . de
(1.1 O)
where a is a generalised stress function called the equivalent, representative or effective stress and de is the equivalent or effective strain increment. It can be shown, using plastic incompressibility and the flow rules, that O" - - 1[(0" 1 -- 0"2) 2 -+" (0" 2 -- 0"3) 2 q- (0" 1 -- 0"3)2]) 0.5
(1.1 la)
and, if the element is deforming, O" : O f
(1.11b)
The equivalent strain increment is de
- - ( 2 [ ( 8 1 - 82) 2 + (82 - 83) 2 -q- (g 1 - 63)2]) 0.5
(1.12)
The equivalent stress and equivalent strain permit material properties determined from a uniaxial test to be applied in problems involving general states of stress. 1.3. Forming characteristics of sheet metals
The most basic test to characterise material behaviour is the tensile test, fig. 1.6. In this section we describe information which can be determined from this test, and explain how it may be related to other stress states which exist in practical forming operations. In the tensile test, a testpiece with a known cross-sectional area and gauge length is subjected to an increasing tensile load until the specimen fails. The extension of the
Introduction to sheet metal forming
7
F
T
[ " I]
i
Lgth
Gauge ~ 1
Currentx-sectional area= A
t [ 1
Originalx-seetior~l area = A o
T
engineering strain = 8L / L
F
truestrain= In ((L+b'I_.)/L) engineering stress= F / A o true stress = F / A
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fig. 1.6. Tensile test. gauge length, 8L, is recorded for a given load, F From this information, the engineering stress-engineering strain curve and the true stress-true strain curve may be determined. Initially we consider the engineering stress-strain curve. Different metals exhibit vastly differing stress-strain characteristics, but one may identify three or four distinct regions in the engineering stress-strain curve, fig. 1.7 illustrates the following regions. R e g i o n I. In this region of the stress-strain curve, the deformation is approximately uniform throughout the gauge length and the material remains elastic. The stressstrain relationship is linear and if the load were removed the specimen would return to its original length. In metals, the extension at the limit of elasticity is typically about one part in one thousand. Compared with other materials such as stone or concrete, metals and their alloys have a high stiffness and when they are compared with composites such as wood or with most plastics, they are very stiff. R e g i o n II. Eventually the elastic limit will be reached and the material will begin to
yield plastically. The behaviour of the material in the vicinity of the yield point is complex; however, in fig. 1.8 we observe two distinct types: (a) discontinuous and (b) H'
IH
IV
i i ! i
1 i
', |
,
i
Fig. 1.7. Engineering stress-strain curve.
8
J.L. Duncan and S.M. Panton
(a)
0,)
Fig. 1.8. (a) Discontinuous and (b) continuous yield point phenomena.
continuous yielding. Discontinuous yielding is associated with ageing and with ludering, in which bands of plastically yielded material become visually apparent within regions which are predominantly elastic. Region III. Beyond the elastic limit and any discontinuous stage, the material will deform plastically and in a homogeneous fashion. The increase in the flow stress of the material during this period is known as work-hardening; however, the crosssectional area also decreases so that the engineering stress-strain curve does not give an accurate picture of the rate of hardening. Region IV. Eventually a point will be reached on the stress-strain curve at which the engineering stress reaches a maximum. Beyond this point, small imperfections will act as initiation sites for necking. Deformation will be concentrated in these regions and eventually lead to failure of the strip. We describe this process in more detail in section 1.4. When interpreting a stress-strain curve, there are various pieces of information which can be observed routinely; these are illustrated in fig. 1.9 and are: 9 yield strength indicating the limit of elastic deformation; 9 the work-hardening characteristic of the material which can be quantified by the ratio of the ultimate tensile strength (UTS) to the yield strength, af; 9 the elastic elongation; 9 the yield point elongation, if present; 9 the maximum uniform elongation Total elongation is a widely used technological indicator of the ability of the material to stretch, but some care must be exercised because this is a function not only of material behaviour but also of specimen geometry. In the same material, shorter, wider testpieces will give greater total elongation than long, narrow ones. As we apply a tensile force to a specimen, there will be strain not only in the axial direction, but also in the width and thickness directions. For an isotropic material, these transverse strains would be equal; prior processing of the sheet may result, however, in directional material properties, and these are characterised by the ratio of width strain to thickness strain as shown in fig. 1.10. This has been denoted by, R, in eq. (1.3) above and it is also known as the r-value. It may be noted that a high rvalue will tend to retard the thinning of the material under tension.
Introduction to sheet metal forming
9
Work Hardening UTS / YS typically in the range of
1.2- 2.5 Yield strength typically 150-600 MPa
v
Maximum uniform elongation
l ~ ~ e elongation typically ~ II-"1 around0.1-0.3% II ~ yield point elongation [ I ifpresent
To tal Elongation
Fig. 1.9. A typical engineering stress-strain curve for metals and alloys. F
Thickness
Width
r = width strain / thickness strain
Typically Steels I to 2.5 Aluminium 0.6 to 1.0
F Fig. 1.10. Determination of r-value.
In fig. 1.6, we refer to both true stress-strain and engineering stress-strain curves. The engineering curve is widely used, particularly in specifying material properties. In modelling and analysing the mechanics of sheet forming processes, it is necessary to use the true stress-strain curve. The distinction is as follows. As the specimen will strain in the width and thickness directions as well as axially, the cross-sectional area of the specimen will change; the true stress (the axial load divided by the current cross-sectional) area will differ from the engineering stress (the axial load divided by the original cross-section) particularly when the extension increases to 10% or more. We may also note that for any increase in load, the true strain increment is the resulting change in length divided by the current length (as opposed to the
10
J.L. Duncan and S.M. Panton
engineering strain increment which is the resulting change in length divided by the original length). If we integrate the true strain increments we obtain true strain defined as;
e~- f de- f dL/L-ln(L/Lo)
(1.13)
If we plot true stress against true strain, fig. 1.11, we see a curve that is very different from the engineering stress-strain diagram; it shows that strain hardening continues at a diminishing rate, but it does not cease at the maximum load. One characteristic of sheet metal behaviour that is often referred to is "formability". There is no precise definition of formability as in general usage, it is taken to mean the response of a particular material to a given forming process; this response is often measured in terms of the percentage of rejects in running the sheet in the press shop. As formability depends on the material, the type of deformation process and the state of the tooling used, it is not a single-valued quantity for any batch of sheet. Variability is a major factor in the perceived formability, but apart from this, the measurable properties that can be derived from the tensile test which are considered to enhance formability are: 9 high strain hardening as indicated by the UTS/yield ratio or the slope of the true stress-strain curve; 9 resistance to thinning in terms of a high r-value. Typical values of these properties are given in table 1.1 for different grades of steel sheet commonly used in sheet metal forming. In general, discontinuous yielding and yield point elongation diminish formability and the effects of temperature and strain rate may also need to be considered.
"
w
g Fig. 1.11. True stress-strain curve. TABLE 1.1 Typical material properties for cold-rolled, low-strength steel Commercial quality
Drawing quality (DQ)
DQ special killed
Interstitial free
230 320 1.35 0.20 1.0 30-35
200 310 1.55 0.21 1.2-1.5 38-42
180 300 1.75 0.22 1.6-1.8 40-45
150 320 2.25 0.24 2.0-2.2 45-48
Yield strength (MPa) Ultimate tensile strength (MPa) UTS / YS n r Total elongation (%)
Introduction to sheet metal forming
11
1.3.1. Approximate constitutive relationships
In this section we will look at simple approximations to the uniaxial stress-strain curve. Figure 1.12a shows a rigid-perfectly plastic approximation which is particularly useful for approximate load calculations. In fig. 1.12b, we consider an elasticperfectly plastic model used in studying small strain processes such as bending and springback. In fig. 1.12c, a power hardening law has been fitted to the true stresstrue strain curve over the plastic range. In terms of the uniaxial stress and strain, this can be expressed as (1.14)
al = Ke'~
where K is a coefficient indicating the strength of the material and n is the strain hardening index. It is found that this expression will fit true stress-strain curves of annealed sheet very well except at very low values of strain near initial yielding. In fig. 1.12d, we see a model for the stress-strain-rate characteristics of strain-rate sensitive materials such as superplastic alloys. These empirical relations are presented in order to show that tensile test results, although containing much data, can be summarised by simple models; in certain situations even a model as simple as the rigid-perfectly plastic may give quite acceptable results.
1.3.2. The relation between tensile data and multiaxial stress conditions
The uniaxial stress-strain properties determined from tensile testing as shown in fig. 1.13 can be used in the analysis of the multiaxial stress conditions in general sheet forming operations. In the elastic region (I) the stress-strain characteristics are defined by the wellknown generalised Hooke's laws. The pertinent material properties are Young's modulus and Poisson's ratio. o=-Ke"
v v
o
Log o
(b)
e "-
Bi~m
(d) Log t
Fig. 1.12. Approximate constitutive relationships: (a) Rigidly perfectly plastic, (b) elastic-perfectly plastic, (c) power hardening, (d) non-Newtonian fluid.
12
.
J.L. Duncan and S.M. Panton
.
t,f
.
|~
~Work Hardening/
(II) C,e.,neralised - I Hooke's l (I)
a
-[ w
s
~ "
Fig. 1.13. Relating tensile test data to multiaxial states of stress. The conditions for initial yield (II) are determined using either of the yield criteria described in section 1.2. In the plastic region (III) the stress conditions continue to satisfy the yield criterion and the generalised stress function, a, is equal in magnitude to the instantaneous value of the uniaxial yield strength, i.e. the flow stress of the material, af. The strains in the uniaxial and generalised process are el and e as defined above. For an isotropic material, the generalised stress-strain curve and the uniaxial curve are identical. 1.4. F o r m i n g
l i m i t s for s h e e t m e t a l
The forming of sheet metal is limited by two types of instability, tensile instability (necking) and wrinkling (buckling). In any sheet material, there will always be imperfections in the form of areas which are slightly thinner or weaker than the surrounding sheet. These form the focus of potential necks and we consider first a small geometric imperfection in a tensile test as shown in fig. 1.14. The reduction in the cross-sectional area causes a greater stress in this locality and hence a greater strain; the increased strain will reduce the crosssectional area. The deformation of the whole testpiece will remain stable so long as the force required to deform the imperfection continues to increase with extension of the specimen. At some point, the increase in strength due to strain hardening will be overcome by the reduction in cross-sectional area and deformation will become unstable. For an incompressible material, it can be shown that this occurs when dal/de1 <
O"1
AreaA+~iA ~
F
~
Area A "
Fig. 1.14. Deformation in a tensile test in the presence of a geometric imperfection.
Introduction to sheet metal forming
13
which, for a strain-hardening material obeying the power hardening law, t 7 1 - - Ke~, occurs when, e > n In the tensile strip, a neck will develop at this point which will cover a region in length roughly equal to the width of the specimen. This is termed a diffuse neck. Once a neck has developed, extension of the testpiece is accommodated by deformation in the neck; this causes an increase in the local strain-rate and with ratesensitive materials the development of necking is slowed and it becomes more diffuse. For materials having the kind of behaviour modelled in fig. 1.12d with high "m" values, such as superplastic alloys, this effect is large and the rate of growth of a neck in a material is imperceptible. In such situations, the strip stretches in a nearly uniform manner giving the useful property of permitting high strains. It is stated above that diffuse necking is associated with a maximum in the load carrying capacity of a region of the testpiece, i.e. the product of stress, trl, and area, A, reaches a maximum. Some non-metals, such as certain polymers, have a stressstrain relation which is convex upwards, as shown in fig. 1.15; the same load can be sustained at the initiation of diffuse necking at A as at B. Deformation is therefore only unstable between these points and the neck will draw out until conditions reach the point B and then the neck will travel along the testpiece in a stable manner. This phenomenon is not seen in metals although, in aged materials, Ltider's bands will travel along the strip in a roughly analogous manner. In general sheet forming processes, a diffuse neck is prevented from developing by the constraint of the surrounding material; therefore the sheet will strain in a quasistable manner until a local neck or groove develops with a width of the order of the sheet thickness, as shown in fig. 1.16. If one of the principal strains in the plane of the sheet is zero or negative, then there is a direction of zero extension in the sheet. A ." Constant 10ad line
i3
Fig. 1.15. Stress-strain curve for a polymer which will deform by drawing out a neck.
Fig. 1.16. Imperfection in a continuous sheet.
14
J.L. Duncan and S.M. Panton
local neck, as postulated by Hill [2], is most likely to develop in this direction when the strains satisfy the relation,
(1.15)
81 + 8 2 -" n
If both of the principal strains are tensile, then there is no direction of zero strain. In this case, the Marciniak model [3] shows that the shape of the yield locus can stabilise the process so that necking is delayed. The Hill and Marciniak models when combined give rise to a line in the strain diagram, fig. 1.17, which indicates the onset of local necking. This is called the "forming limit curve". The existence of this curve can also be established experimentally; the standard procedure is to form strips of various width over a hemispherical punch and observe the strain at which necking occurs by use of a circular grid. The use of differing strip widths creates strain conditions which vary from plane strain through to biaxial tension. The whole diagram, fig. 1.17, is commonly known as the "forming limit diagram" and it is widely used in the diagnosis of sheet forming problems. Strains in various regions of the sheet can be measured using some gridding technique and plotted in this diagram. As an example, a typical draw die operation as shown in fig. 1.18, the sheet is clamped around the edge and drawn in by the action of the punch. When the punch bottoms in the die, the region in the centre of the sheet will be stretched in biaxial tension as shown. In the side walls, the sheet is stretched in plane strain while
Hill
Major Principal Strain A E1
Marciniak
El+ez=n
.
9
Biaxial
Tension
/
s t I, s s, f ,,,
Minor Principal Strain
Fig. 1.17. Forming limits imposed by necking.
Stretc~ng
Plane Strain
Drawing Fig. 1.18. Press forming (with draw die).
Introduction to sheet metal forming
15
in the corners, the sheet is stretched in the radial direction and compressed circumferentially. A grid circle at each of these regions will be deformed as shown in fig. 1.19 and the strains in the part will fall within the envelope shown. The importance of leaving a safety zone between this envelope and the forming limit curve is clear. It should be accepted as inevitable that small variations in process characteristics such as friction will cause some variation in strain, and if this variation in strain causes failure of the parts, then the process should be considered incapable. For this reason the forming limit diagram is.an important process control tool. The forming limit diagram is also an important tool during die try-out and can indicate appropriate modifications to the tooling. In general, these modifications will attempt to pull the critical strains away from the forming limit curve for the material.
1.5. Industrial sheet metal forming Industrial applications cover a wide spectrum of processes and, as shown in fig. 1.18, in any one process different regions will have different straining paths. It is not possible here to deal with the whole range of industrial forming operations, but it is useful to look at the idealised cases that exist at either end of this s p e c t r u m - these are two-dimensional stretch forming and axisymmetric deep drawing. The basic mechanics of each will be introduced and from this it is possible to identify the kind of material behaviour that is most advantageous for the particular operation.
1.5.1. Stretch forming In forming shallow, smoothly contoured shells such as autobody panels the sheet is stretched over a punch in a double-acting press as illustrated in the twodimensional model in fig. 1.20. The strains necessary to achieve the desired shape Plane Strain
Equal Biaxial
, s
Drawing
""
""
(Shear)
Typical envelope of strains in a formed part Fig. 1.19. Typical envelope of strains on a forming limit diagram.
16
J.L. Duncan and S.M. Panton
Fig. 1.20. Two-dimensionalmodel of stretch forming. are often quite small, but in order to ensure that the shape of the part is properly fixed without excessive spring-back and in some cases to gain strength in the part by strain hardening the sheet, the strains are often much greater than those required just by geometry. In this type of forming, the part required is contained within the trim line shown; the edges of the sheet under the blankholder are used to provide the appropriate tension to stretch the sheet over the punch and this material is discarded after trimming. The restraint at the flange is created in two ways: one is by friction between the sheet and the blankholder and the other by means of the draw bead shown. In a draw bead, the sheet is bent and unbent as it passes through the bead; the plastic work done results in additional tension on the downstream side. The difference between the two methods of restraining the sheet is that the friction effect will depend on both the coefficient of friction and the blankholder force while the draw bead will create a tension which is proportional to the yield stress and the thickness of the sheet, as may be inferred from section 1.6. In a three-dimensional stretch forming die, the important feature is that the flange is not clamped rigidly, but is allowed to move inward in a controlled fashion. In order to obtain sufficient tension to stretch the sheet over the punch, the tension in the side wall between the blankholder and the punch must be very high and to prevent tearing here the material must have good strain-hardening properties. In the two-dimensional case, the minimum strain will be at the centreline. Due to friction between the sheet and the punch, the tension will build up away from the centre. The equation governing this behaviour can be summarised with the aid of fig. 1.21. If the sheet element subtends an angle, d4~, and is being stretched by a tension, T, (force per unit length in a direction perpendicular to the plane of the diagram), then equilibrium in the radial direction gives that q = T/R
(1.16)
where, q, is the tool contact pressure and R the local radius of curvature of the tool. In the tangential direction, dT --/zq. Rd~b = / z T . dq~
(1.17)
The tension in the sheet is T -- tcr
(1.18)
Introduction to sheet metal forming
17
Fig. 1.21. Element of a sheet being stretched over a punch.
Differentiating and substituting, we obtain
d T / T =/x(dcr/~r + dt/t) dqb
(1.19)
The first term in the brackets in eq. (1.19) is the rate of strain hardening and the second term is the rate of thinning. In order to obtain good stretching over the punch without excessive thinning in the most highly stressed regions, we require a low coefficient of friction and good strain hardening. The latter requirement often means that the sheet must be in a fully annealed condition and referring to table 1.1, it is clear that IF steel will behave well in this application. (In this case, formability is traded off against strength as well as cost.)
1.5.2. Deep drawing In drawing an axisymmetric cup as shown in fig. 1.22, most of the deformation takes place in the outer region or flange which is drawn in to create the cylindrical walls of the cup. The blankholder is used to keep the flange flat and avoid wrinkling; some frictional restraint is inevitable but this is kept a small as possible.
To
BlankH o l ~
Punch ]
--~..T,+OT,
Die
To (a)
Fig. 1.22. Deep drawing a cylindrical cup.
(b)
18
J.L. Duncan and S.M. Panton
The equilibrium equation for an element in the flange shown in fig. 1.22b is
dT~o To-T~o= T~
r
0
(1.20)
The stress state in the flange is illustrated in the yield locus in fig. 1.23a. At the outside, A, a state of uniaxial compression will exist; at the inner part of the flange, B, the radial tension, T~, causes the radial stress to approach the yield stress in uniaxial tension, ~rf. The corresponding strain paths are shown in fig. 1.23b. It may be seen that the strains in the centre of the part are small, indicating that in deep drawing most of the deformation is confined to the flange. The forming limit curve is also shown in this diagram and it will be seen that necking is unlikely to limit the process. Furthermore as the level of the forming limit curve depends on the strain-hardening index, n, a low value of n can be accommodated and it is possible to deep draw fully cold-worked sheet. In the diagram in fig. 1.23b, the left-hand diagonal is a path of constant thickness deformation; in drawing the flange, its thickness will be unchanged or slightly increased. It may also be shown that a high r-value is advantageous in deep drawing as it increases the strength of the cup wall, which would deform by stretching and thinning, and decrease the strength of the flange. This brief examination of two idealised processes at the extremes of the spectrum of practical forming operations shows that different material properties such as the strain-hardening index, n, and the normal anisotropy ratio, the r-value, influence different processeses in different ways. In general, the magnitude of the tractions or tension that is developed in the sheet can be understood by examining the loading
r,
A
(a)
Fig. 1.23. Axisymmetricdeep drawing.
TO
(b)
Introduction to sheet metal forming
19
paths in a stress diagram such as fig. 1.23a and the deformation of the sheet and the limits imposed by necking anticipated from the forming limit diagram, as in fig. 1.23b. 1.6. Bending and spring-back Sheet bending is generally considered as a plane strain-plane stress process and the variables are indicated in fig. 1.24. The moment, M, and the tension, T, are values per unit length along the bend. Except for very small radius bends, the strain distribution is assumed to be linear, i.e. el - ln(1 + y/R)
(1.21)
where y is the distance from the neutral axis. Since y/R is small, el is approximately equal to y/R [4].
1.6.1. Bending without tension The neutral axis is at the mid-plane, and the strain distribution is as shown in fig. 1.25. The stress distribution depends on the stress-strain relationship for the material, and it is convenient to express this in terms of the major stress and strain, i.e. 0"1 = f(el) (82
(1.22)
- - 0" 3 - - 0 ) .
z"
~.':.~~~..-~:
t~2
....
9
t
/
,'-,'l
"
'~,~',,
-:'~::~:~:~.:~
"
_
"
,4
"l
/ I / s /
I / I
I
/
',.
0
." R
." /
I 9
/
I
Fig. 1.24. Variables in sheet bending.
, ; : ~
M
20
J.L. Duncan and S.M. Panton 01 - E'81
M
~
,~ 9
(Plastic/ power hardening)
Fig. 1.25. Stress and strain distributions in bending. For an elastic-perfectly plastic material: (i) the stress in the elastic region is given by or1 = E ' e l
(1.23)
where E ' = E/(1 - v 2) and v is Poisson's ratio. (ii) the stress in the plastic region is given by o1 = S
(1.24)
where for a perfectly plastic, von Mises material of constant flow stress, S is related to the flow stress (~rf) by S = (4/3) ~ If the material is elastic, the following well-known equation applies M I I = cqly = E ' I R
(1.25)
At the limiting elastic curvature, ~r1 = S at y = t/2 and hence the limiting elastic curvature and the corresponding moment are given by, (1/R)E -- 2 S / E ' t , ME -- St2/6
(1.26)
As the curvature increases, the material will progressively contain more material in the plastic region, fig. 1.26. From the right-hand drawing of fig. 1.26c, the fully plastic moment, can be determined as Mp = St2/4
(1.27)
A typical moment-curvature characteristic is shown in fig. 1.27. It may be noted that as Mp = 1.5ME, the change in curvature upon unloading A(1/R) = 1.5 (1/R)E. Hence A(1/R) = 3S/E't
(1.28)
Introduction to sheet metal forming
21
Increasing Moment v
PI
.Elastic
M,
M P
Limiting Elastic
Elastic- Plastic
Fully Plastic
Fig. 1.26. Stress distributions with increasing moment and curvature in an elastic-perfectly plastic material.
9
I
l
I
I
I
I
I
I
I
t
I
Mp
l
1/R
I
Curvature
E't
A (l/R)
Fig. 1.27. The moment-curvature diagram for an elastic-perfectly plastic sheet.
Unloading the plastic moment from mp to zero is associated with a change in bend angle A0 and a change in radius AR (fig. 1.28); the length of the neutral axis will remain unchanged, hence
(A(1/R)/(1/R)) + (A0/0) = 0
(1.29)
Combining (1.28) and (1.29), we can obtain the change in bend angle or spring-back as
AO = 3(S/E') (R/t)O
(1.30)
Equation (1.30) shows that the spring-back angle is proportional to the ratio of yield stress to elastic modulus. For annealed sheet, this is typically around 1 to 1,000; however, for hard temper sheet it may increase to 1 in 300 or greater. Equation (1.30) also shows that spring-back is proportional to the ratio of radius to sheet thickness. For small radius bends, such as are typical in roll forming, this may be around 2. For large radius bending, such as may be typical with autobody panels, this may be much higher, and spring-back will be large. Because of the variability in S, tl~e process may
22
J.L. Duncan and S.M. Panton
,
.... 9 --~~"-':~;i~::..'-.',.:m. ~ ~ i ~ : ~ . . , , ~.~i~ii:.'.'~..~..~ i ~ ~ "~'~:~!~....
! 9
,r3,
/
i i i
/
/ R 0
/
/
/
/
/ /
/
/
/
R+Z~R
Fig. 1.28. Spring-back of a sheet upon unloading of a moment. be difficult to control. A technique for avoiding these problems is discussed in the next section. If the unloading process from a fully plastic moment is elastic, there will be a residual stress distribution in the sheet, as shown in fig. 1.29. This is an idealised representation, but sheet metal that has been subjected to various bending, unbending and straightening processes is liable to have significantly more complex residual stress distributions. 1.6.2 9 Bending under tension
Equation (1.30) indicates that if a sheet is given a gentle curvature by the application of a moment, the spring-back would be large. To obtain accurate shape as, for example, in curving an exterior panel for an aircraft, the sheet is curved by stretching over a former, as shown in fig. 1.30. It will be seen from the stress distributions shown in fig. 1.30, that as the tension increases, the m o m e n t required to maintain a curvature less than the limiting elastic curvature will remain constant until the outer fibre stress reaches the yield stress S. When the tension has reached that required to stretch the sheet plastically, i.e.
Fig. 1.29. Residual stresses upon unloading.
Introduction to sheet metal forming
(a)
23
t
M
M
(b) (i)
S
(u)
......
(Iv) ~
S
(c) M Me
w
~9
T
St Fig. 1.30. (a) Bending under tension. (b) Stress distribution under increasing tension. (c) Moment required to maintain the curvature as the strip tension increases.
T - S t , the moment has reduced to zero and the spring-back due to bending will also be zero. In practice, the sheet will be stretched plastically by a few per cent and will then take up the radius of the former accurately.
1.7. Superplasticity The characteristic behaviour of metals and their alloys is that at room temperature they can be deformed plastically and will strengthen or work harden as they do so. The resistance to flow depends largely on the amount of deformation imposed and not on the rate of working. At higher temperatures, such as those used in hot forging, their strength will diminish and remain constant during deformation; they do not strain harden, but the flow strength is dependent on the rate of working. As the temperature increases, metals typically have a well-defined melting point and change from a solid with an appreciable, but still low, rate-sensitivity to a low-
J.L. Duncan and S.M. Panton
24
viscosity liquid. They do not display the semi-fluid highly viscous behaviour of materials such as glass or thermoplastics at the temperatures used for glass-blowing and blow-moulding. The ability to draw out a thin filament of glass from a molten pool or to blow a tube of molten polyethylene out to many times its diameter can be shown to depend on the fact that the resistance to flow is very nearly proportional to the rate of extension, i.e. these materials behave as ideal viscous liquids and imperfections do not become catastrophic. Although highly viscous behaviour is not a characteristic of metals, even at elevated temperatures, certain alloys do show moderate viscosities under unusual conditions. This phenomenon has been termed superplasticity. There are a few alloys which can be produced in a very fine grain condition, less than 10 microns, and this state is stable at temperatures up to about half of their absolute melting temperature. If these alloys are hot worked, they do not strain harden, but have a flow stress which obeys a law of the kind, O- - - B ~ m
where m, the strain-rate sensitivity index, is in the range 0.3 ~ 0.8. The alloys that have received most attention are Zn 22%A1 which is formed at 250~ and Ti 6%A1 4%V which is superplastic between 900 and 980~ If these materials are deformed in an elevated temperature tensile test, extensions of several hundred per cent are possible. The flow stress at these temperatures is low compared with typical hot-working strengths in metals but it is still higher than those observed in thermoplastics in moulding and blow-forming. Vacuum-forming can be used, but for reasonable production times, pressure forming is necessary. In designing superplastic forming processes, it is necessary to keep the strain rate within the range at which the rate-sensitivity index is highest; this may only extend over about a decade in the vicinity of 10 -3 s -1 . A useful example is illustrated in fig. 1.31 which shows the two-dimensional pressure forming of a sheet into a corner. If the sheet is clamped at A and B, the first stage of the process can be considered as uniform stretching of the sheet until it just touched the side-walls of the die; assuming incompressibility, the thickness at this instant is
A / / /
//////7-//t3
/ /
Fig. 1 . 3 1 . 2 - D p r e s s u r e f o r m i n g of a sheet into a c o m e r .
Introduction to sheet metalforming
25
tA -- (2~/2/:r)to F r o m this point onwards, the sheet is often assumed to stick to the die wall without further thinning and the current thickness, t, of the unsupported sheet tangent at the height, h, can be calculated by geometry. If the strain rate prescribed for the process is k0, the membrane stress is cr = p h / t and the required pressure is p -
tB ' /h
If it is desired to form into a small corner radius, the final value of h is small and the pressure could be high. As indicated, this usually precludes vacuum-forming. In the above analysis, the thickness is determined by geometry and diffuse necking is neglected. This is often an acceptable approximation in analysing superplastic process; however, it becomes less valid as the strain-rate sensitivity index decreases; more elaborate calculation may then be necessary.
References
[1] Hosford, W.F. & Caddell, R.M., Metal Forming: Mechanics and Metallurgy, Prentice Hall, New Jersey (1983). [2] Hill, R., The Mathematical Theory of Plasticity, Oxford University Press, London (1950), p235. [3] Marciniak, Z. & Kuczynski, K., Int. Journ. Mech. Sci., 9 (1967), p609. [4] Marciniak, Z. & Duncan, J.L., Mechanics of Sheet Metal Forming, Edward Arnold, London (1992).
This . Page Intentionally Left Blank
Composite Sheet Forming edited by D. Bhattacharyya 9 Elsevier Science B.V. All rights reserved.
Chapter 2
Thermoplastic Composite Sheet Form&g: Materials and Manufactur&g Techniques B. T o m a s , ~ S T R O M
Department of Aeronautics, Division of Lightweight Structures, Royal Institute of Technology, S-IO0 44 Stockholm, Sweden
Contents Abstract 28 2.1. Introduction 28 2.2. Constituents 29 2.2.1. Matrices 29 2.2.1.1. Polymer morphology 30 2.2.1.2. Structural aspects influencing properties 31 2.2.1.3. Thermoplastic polymer matrices 34 2.2.2. Reinforcements 38 2.2.2.1. Glass fibers 39 2.2.2.2. Carbon fibers 40 2.2.2.3. Organic fibers 41 2.2.2.4. Other fibers 42 2.2.2.5. Reinforcement-matrix interaction 42 2.2.3. Preimpregnated reinforcements 43 2.2.3.1. Solvent impregnation 44 2.2.3.2. Melt impregnation 45 2.2.3.3. Powder impregnation 46 2.2.3.4. Commingling 47 2.2.3.5. Comparison of preimpregnated reinforcement types 47 2.2.3.6. Molding compounds 48 2.3. Properties 48 2.3.1. Matrices 48 2.3.1.1. Thermal and rheological properties 49 2.3.1.2. Mechanical properties 52 2.3.2. Reinforcements 54 2.3.2.1. Thermal properties 54 2.3.2.2. Mechanical properties 54 2.3.3. Composites 55 2.3.3.1. Prediction of thermal properties 55 2.3.3.2. Prediction of mechanical properties 56 2.3.3.3. Experimentally determined thermal properties 57 2.3.3.4. Experimentally determined mechanical properties 58 27
28
B.T. flstrd'm
2.4. Manufacturing techniques 60 2.4.1. Prepreg lay-up 61 2.4.2. Prepreg consolidation 61 2.4.2.1. Vacuum-bagconsolidation 61 2.4.2.2. Matched-die consolidation 62 2.4.2.3. Double-belt-press consolidation 63 2.4.2.4. Tape laying 64 2.4.2.5. Intermittent matched-die consolidation 65 2.4.3. Sheet forming 65 2.4.3.1. Matched-die molding 66 2.4.3.2. Rubber-die molding 67 2.4.3.3. Hydroforming 68 2.4.3.4. Deep drawing 68 2.4.3.5. Diaphragm forming 69 2.4.3.6. Folding 70 2.4.3.7. Roll forming 70 2.4.3.8. Matched-die molding of GMT 71 Acknowledgement 72 References 72
Abstract
This chapter provides an introduction to thermoplastic composite sheet forming and the raw materials commonly used. First the relevant basics of polymer physics are introduced in order to provide a rudimentary background to the matrices' macroscopic properties and processing requirements, whereupon different reinforcement types and preimpregnated reinforcement forms are introduced. Second the thermal, rheological, and mechanical properties of common sheet forming raw materials and their composites are discussed and representative properties given. Finally a comprehensive yet brief overview of different composite sheet forming techniques, including blank consolidation processes, is presented. 2.1. Introduction
On the face of it, many structurally capable composites are shell-like components that one way or another ought to be manufacturable through conformation of a flat raw material blank into a complex-shaped final component. It should therefore come as no surprise that the topic of thermoplastic composite sheet forming is one of intense research, which has resulted in a number of forming techniques. Since the concept of thermoplastic composite sheet forming forming of a flat piece of material into an arbitrarily curved c o m p o n e n t - is identical to sheet metal forming, a few of the techniques investigated have been borrowed from this relatively mature manufacturing field, while others emanate from thermoset composites manufacture. Only a minority of the techniques discussed herein have genuinely been invented or developed exclusively for sheet forming of thermoplastic composites. While all forming techniques discussed in this chapter have proven technically feasible, widespread commercial success is rare. However, given the significant research and development
Thermoplastic composite sheetforming
29
efforts thermoplastic composite sheet forming is seeing, there should be little doubt that all this work gradually will lead to commercial successes in the near future. This chapter aims to give the novice to the topic of thermoplastic composite sheet forming an introduction to the raw materials used, their properties, and common forming techniques. The first two sections of the chapter introduce the constituents most commonly used and then go on to discuss the thermal and rheological properties so relevant to manufacturing. Since sheet-formed components are most likely to be used in structural applications, the mechanical properties of both constituents and their composites are also briefly treated. The final section of the chapter presents a comprehensive overview of sheet consolidation and sheet forming techniques, but since several of these techniques will be more thoroughly treated in other chapters of this volume this treatment is brief.
2.2. Constituents
The constituent materials that make up, or constitute, a composite are normally considered to include at least the matrix and the reinforcement. Oftentimes the matrix is not homogeneous but rather mixed with some performance-enhancing additive. Likewise, the reinforcement is normally surface treated or coated with some performance-enhancing substance. These substances may or may not be considered constituents on their own right. This section of the chapter covers the constituents most relevant to thermoplastic sheet forming and describes available types and family characteristics.
2.2.1. Matrices
The matrix of a composite has several functions: it is a binder that holds the reinforcement in place, it transfers external loads to the reinforcement, and it protects the reinforcement from environmental exposure. Moreover, the matrix redistributes the load to surrounding fibers when a fiber fractures and supports the fibers to prevent buckling in compression. Polymer matrices clearly dominate in composites applications, although other matrices are used to a limited degree in specialized applications; these other matrix types are not covered in this chapter. To fully appreciate differences between candidate polymer matrices in terms of properties and processing requirements, it is essential to at least have a conceptual understanding of polymer physics and chemistry. However, this treatment is primarily intended for readers with little chemistry background and the following sections therefore refrain from delving too far into the field of polymer science; the interested reader is referred to the literature specializing in this topic. The following treatment thus merely aims to provide a basic understanding of the polymerscience aspects most relevant to composites properties and their manufacturing requirements, and in particular matrices and issues pertaining to thermoplastic sheet forming.
B.T. ~lstrO'm
30
2.2.1.1. Polymer morphology A polymer is a high-molecular-weight compound that is composed of a multitude (poly) of repeated small segments (mers). Polymers are organic compounds primarily based on carbon and hydrogen atoms bound to each other by primary, or covalent, bonds. Carbon is capable of forming four covalent bonds located tetrahedrally around the atom. Carbon atoms may form covalent bonds to each other both through single and double bonds, the former being referred to as saturated and the latter unsaturated. (Carbon atoms can also bond to each other through triple bonds, but such bonds are seldom encountered in polymer science.) In contrast, hydrogen can form one covalent bond only. The least complex polymers are the hydrocarbon polymers which contain carbon and hydrogen only. The simplest hydrocarbon polymer is polyethylene (PE), the repeating unit of which is illustrated in fig. 2.1, where "n" indicates a large number (103-106) of repeating mers to form a practically useful polymer and " - - " indicates covalent bonds. It is important to note that the two-dimensional rendition of the structure in the figure in reality is three-dimensional due to the tetrahedrally arranged bonds of the carbon atoms. Another important and common hydrocarbon polymer is polypropylene (PP); see fig. 2.2. Several other hydrocarbon polymers besides PE and PP exist, although they are rarely of interest in composites applications. However, if one includes other elements than carbon and hydrogen, virtually endless combination possibilities arise. Apart from carbon and hydrogen, the most common elements are oxygen, nitrogen, sulfur, fluorine, chlorine, and silicon. As long as only carbon makes up the polymer backbone, one talks of carbon-chain polymers, whereas polymers having some non-carbon backbone atoms are referred to as heterochain polymers.
H
H
I
I
-(- C-- C-)-n I
H
I
H
Fig. 2.1. Repeating unit of PE. H
H
H
H
I
I
I
I
-(-- C - C-)-n I
H H-C-H
-(.- C-- C-)- n I
H
I
CH 3
I
H Fig. 2.2. Two ways of illustrating the repeating unit of PP.
Thermoplastic composite sheetforming
31
2.2.1.2. Structural aspects influencing properties There are strong relationships between the molecular structure, or configuration, of a polymer and its macroscopic properties in both solid and liquid form. These relationships may be hierarchically divided into intramer, intramolecular, and intermolecular structure. The next three subsections describe some molecular aspects of relevance and their influence on the polymer's macroscopic properties. Intramer structure. The properties of the polymer are to a large degree influenced by the structure of the mer, i.e. what elements are present and how they are bound to one another. Single bonds allow rotation around the bond axis (torsion), while double bonds do not. Double bonds and cyclic structures, such as the common aromatic ring (see fig. 2.3), in the backbone and large side groups leave the molecule inflexible and bulky, thus impeding molecular motion which significantly influences macroscopic properties. Polyisoprene is a good example to illustrate the strong influence of apparently subtle differences in mer configuration (see fig. 2.4); cis-polyisoprene (natural rubber) is soft whereas trans-polyisoprene (gutta percha) is a hard plastic. Note that these two mer configurations are different since the double bond does not allow rotation. The cis-prefix denotes that the backbone continues on the same side of the double bond and the trans-prefix that the backbone extends on different sides, cisPolyisoprene and trans-polyisoprene are said to be geometric isomers. Intramolecular structure. While the mer may be fully defined, there are still many combinations in which it may form a polymer. Taking PP as an example, the methyl (CH3) group may be located in different positions; see fig. 2.5. Isotactic PP has all methyl groups on the same side, syndiotactic on alternating sides, and atactic in a random manner. Even though single bonds permit rotation, these three forms of PP are different (recall the tetrahedral arrangement of the four bonds of the carbon atom). The regular structures of isotactic and syndiotactic PP, which are referred to
_ cIC'~c -C
I
II
O#
C-
I
Fig. 2.3. Two ways of illustrating the repeating unit of an aromatic group.
H I
H-C-H I
C I
-C-H I
H
H H I
C I
H-C-
I
H
I
H-G-H I
H I
H-GI
C
C
-C-H
H
I
I
I
H
Fig. 2.4. Repeating unit of c/s-polyisoprene (left) and trans-polyisoprene (right).
B.T. ~tstrO'm
32 H I
-CI
H
H I
H I
H I
H I
H I
C -C - C -C-
C-
CH3H
CH3
I
I
I
I
CH3H
I
H
CH3H
H
-C-
H
C -C-
C -C-
C-
H
CH3H
H
CH 3
I I
H I I
I
I
I
I
I
I
H
I
I
H
CH3H
H
-C-
C -C-
C -C-
H
CH3H
I
I
H
I
I
I
I
H
I I
H I
I
H I CI
CH3
Fig. 2.5. Isotactic (left), syndiotactic (center), and atactic PP (right).
as stereoisomers, translate into quite different macroscopic properties from those of atactic PP. Of great significance to the macroscopic properties of a polymer is the number of repeat units or, alternatively, the molecular weight. However, there is no way of producing polymers of a single molecular weight and in reality it may vary by three to four orders of magnitude within the same sample. While all polymer examples so far discussed have been linear with small side groups only, branching may occur. Further, no attention has been paid to the ends of the molecules, which obviously must deviate from the idealized polymer structure, but since the number of repeating units is so large possible effects of the end groups are most often ignored. The molecules may also contain impurities, i.e. some flaw inconsistent with the idealized polymer structure. To improve certain properties of a polymer it is possible to polymerize it from more than one type of monomer. Two different repeating units (here denoted A and B) thus may be polymerized into alternating (ABABABABABA), block (AAABBBBBBAA), random (ABBABAABBBA), or graft fashions, where the latter has a backbone made up entirely of one mer with branches made up entirely of the other one attached. Intermolecular structure. The interaction between molecules is a function of intramer and intramolecular structure. It was earlier pointed out that the intramolecular bonds are covalent. There are also secondary bonds, e.g. van der Waals, hydrogen, and dipole-dipole bonds, which act between molecules. The reason for referring to them as secondary bonds is that they are an order of magnitude weaker than the covalent bonds and therefore are of secondary albeit s i g n i f i c a n t - importance. At a sufficiently high temperature a polymer sample melts; the individual molecules of the melt are randomly arranged, interlaced, and undergoing constant rearrangement. The covalent bonds ensure the integrity of the molecules while the molecular movement caused by the heating dwarfs the secondary bond forces. If the polymer sample then very quickly is cooled, or quenched, into a solid the random molecular arrangement is essentially frozen in place by secondary bonds. However, if the polymer sample is instead cooled slowly, the individual molecules may, under certain conditions, attempt to align themselves into a regular crystal formation. The crystal conformation is preferred since it represents the lowest possible energy state for the molecules. Due to the precise close-packing of the molecules in the crystal state, the crystal structure has higher density than the random molecule arrangement, which is usually referred to as amorphous or glassy. Further, since the crystal conformation is preferred due to the lower energy state, energy is released upon crystallization, i.e. it is an exothermal process.
Thermoplastic composite sheetforming
33
In reality it is not possible to achieve complete crystallinity and amorphous areas therefore partly surround the crystals; polymers possessing the ability to crystallize are consequently referred to as semicrystalline. The degree to which a polymer may crystallize is to a large degree dependent on the molecule being regular in structure and flexible. This is the reason why isotactic and syndiotactic PP may form crystals whereas atactic PP may not, and why PE, due to its extremely simple, regular, and flexible structure, may achieve the highest degree of crystallinity of any polymer. Likewise, molecules that do not contain double bonds and cyclic structures in the backbone or bulky side groups are more likely to achieve a greater degree of crystallinity than if they had these features. Although many polymers of relevance for use as composite matrices are semicrystalline, most polymers are unable to achieve any appreciable degree of crystallinity under any circumstances. The resulting degree of crystallinity in a semicrystalline polymer is enhanced by higher temperature and pressure during crystallization as well as by lower molecular weight. From a manufacturing point of view, especially the former two dependencies are highly relevant. The temperature dependency of the degree of crystallinity is usually assessed in terms of cooling rate; see fig. 2.6. As the figure shows, it is possible to obtain either a more or less entirely amorphous structure through quenching of the melt or a highly crystalline structure through very slow cooling. Upon gradual heating of a semicrystalline polymer the amorphous regions melt before the crystalline regions and a melt with the random molecular arrangement is regained. The melting of the crystalline phase consumes energy in order to dislodge the molecules from their preferred low-energy state and the process is therefore endothermal. The reason why it is possible to reversibly go from solid to melt to solid etc. is that only secondary bonds act between molecules. A polymer with such a reversible behavior is called thermoplastic. In theory the melting and solidification cycles can be repeated an infinite number of times without affecting the polymer. In reality parts of some molecules will react chemically, i.e. covalent bonds will be destroyed or
50 40-
._r m m
L)
3020 10 0
~ 10 o
~ 101
~ 102
~ .... 103
C o o l i n g Rate [~
Fig. 2.6. Crystallinity of polyetheretherketone (PEEK) matrix in carbon/PEEK composite as function of cooling rate. Note the logarithmic cooling rate scale. Redrawn from reference [1].
34
B.T. flstr6"m
created, and the polymer and thus its properties will eventually degrade through oxidation, chain scission, etc. Under certain conditions covalent bonds may form between molecules. One of the more notable locations that readily become sites of such crosslinks is the unsaturated carbon-carbon double bond in the polymer backbone, which through a chemical reaction may open up, leaving a (saturated) single covalent bond within the molecule and creating new covalent bonds to other molecules. Starting from a polymer liquid consisting of molecules capable of forming crosslinks, any of a number of means may be employed to set off a chemical reaction which creates covalent crosslinks to neighboring molecules, thus creating a gigantic three-dimensional molecule. As crosslinks are formed the polymer liquid gradually loses its ability to flow since the molecules no longer can slip past one another as in the thermoplastic melt. The three-dimensional network translates into a lower energy state than the random molecular orientation of the liquid state, so the crosslinking process is (just like crystallization) exothermal. Since the three-dimensional network created is bound together by covalent bonds (in contrast to the previously mentioned intermolecular secondary bonds) it may not be melted through reheating. The type of polymer having the ability to crosslink is called thermoset. In conclusion there are two very different polymer f a m i l i e s - thermoplastics and thermosets. Thermoplastics consist of long molecules with only secondary bonds in between molecules and therefore may be melted. If irregular in structure and stiff, the molecules of a thermoplastic are randomly arranged both in the melt and in the solid; the polymer is amorphous. On the other hand, if the molecules are regular in structure and flexible, the molecules, while randomly arranged in the melt, may form crystals as the thermoplastic solidifies; the polymer is semicrystalline. However, the degree of crystallinity of a semicrystalline polymer is dependent on cooling rate; with a high cooling rate the solid polymer ends up more or less amorphous, whereas with a low rate it will be partly crystalline and partly amorphous, i.e. semicrystalline. Initially thermosets also consist of long molecules with only secondary bonds between them. However, under certain conditions, such as the presence of carboncarbon double bonds in the molecular backbone and chemically reactive substances in the bulk resin, covalent bonds may form between molecules, resulting in solidification of the resin. Since the intermolecular covalent bonds cannot be broken without simultaneously breaking the intramolecular covalent bonds, thermosets cannot be melted. As the molecular arrangement in solid thermosets (as well as in the liquid state) is random, thermosets are amorphous. While thermosets may be regarded as a special case in polymer science, it is a very important special case and thermosets clearly dominate over thermoplastics in virtually all composites applications. However, since the main topic of this volume is thermoplastic sheet forming, thermoset polymers will only briefly be mentioned in the remainder of this chapter.
2.2.1.3. Thermoplasticpolymer matrices While there is a vast array of engineering plastics available, only a few are used as composite matrices, since an engineering plastic with excellent properties does not necessarily make an excellent composite matrix. Issues that are of importance when
Thermoplastic composite sheetforming
35
selecting a polymer for use as composite matrix are reinforcement-matrix compatibility in terms of bonding, mechanical properties, thermal properties, cost, etc., though perhaps the most important aspect may be its processability, i.e. how easy is it to deal with it in manufacturing situations. Among the many issues that may be considered part of the processability are viscosity, processing temperature, processing time, and health concerns. The viscosity is important in achieving reinforcement impregnation, where each reinforcing fiber ideally should be surrounded by the matrix without voids present; impregnation is naturally facilitated by a low viscosity. Not-yet-crosslinked thermosets have shear viscosities at processing temperature of the order of 10~ Pa s, while melt viscosities of thermoplastics at processing temperature are of the order of 102 Pa s or higher (for comparison, the shear viscosity of water at room temperature is 10-3 Pa s), meaning that it is much easier to complete impregnation with thermosets than with thermoplastics. Once impregnation is completed, thermoplastics only need to be melted, shaped, and then cooled to achieve dimensional stability in a matter of seconds, although the temperatures required can be quite high by polymer processing standards. In contrast, thermosets need several minutes to several days for crosslinking to be completed, albeit normally at lower processing temperatures than with thermoplastics. The chemical structure of thermoplastics makes them chemically inert if processed correctly, meaning that no hazardous byproducts need to be considered. On the other hand, the molten thermoplastic and the heated machinery may cause severe burns. This relative lack of health worries is a distinct advantage over thermosets which, due to the active chemistry, give rise to considerable health concerns. A thermoplastic is usually fully polymerized when delivered from the supplier, meaning that all chemical reactions are completed and the user can concentrate entirely on physical processes, such as heat transfer and flow. However, there are some rare exceptions to this rule. The user may choose to take care of part of the polymerization starting off with low-molecular-weight prepolymer, thus avoiding the high-viscosity disadvantage during reinforcement impregnation. Courtesy of the low molecular weight, the polymer fluid may have a viscosity comparable to that of a thermoset resin. After the reinforcement is impregnated, the final polymerization process takes place and the molecular weight thus drastically increases. Depending on the type of polymer, the high-molecular-weight polymer may or may not decompose into lower-molecular-weight polymer molecules upon remelting. One of the main features of amorphous thermoplastics is that they are dissolvable in common industrial solvents. This means that the reinforcement can be impregnated with a low-viscosity solution, thus avoiding the problem of high melt viscosity, but it also means that the solidified polymer (and composite) is not solvent-resistant. For solvent-impregnated reinforcement, the residue solvent that was not completely driven off after impregnation is a serious concern since it impairs the quality of the composite. Amorphous thermoplastics do not shrink much when they solidify, which translates into good surface finish. Semicrystalline polymers usually have good solvent-resistance due to the crystallinity which prevents dissolution of the entire molecular structure. The crystallinity also improves high-temperature performance and long-term phenomena, such as
36
B.T. AstrO'm
creep. If the crystallinity is too low, these benefits are not seen and if it is too high the material loses toughness and becomes brittle, although it usually gains in stiffness. Hence, there is an optimum degree of crystallinity. When processing semicrystalline polymers one thus must consider, and preferably control, the cooling rate (see fig. 2.6), which rarely is an issue with amorphous matrices. Useful semicrystalline polymers have 5 to 50 volume percent crystallinity, with an optimum of 20 to 35 percent for composites applications [2]. Semicrystalline polymers shrink more than amorphous ones upon solidification; the higher the final crystallinity, the higher the density change between melt and solid. Due to the difference in shrinkage between the amorphous and the crystalline regions, the surface of semicrystalline thermoplastics is generally not as good as for amorphous ones. Since solvents normally cannot be used to dissolve semicrystalline polymers (there are some rare exceptions), reinforcement impregnation is extremely difficult. The following subsections aim to give a brief overview of some of the more common thermoplastic polymers used as composite matrices; table 2.1 shows their respective repeating units. The characteristics discussed below are merely family characteristics and large variations are the rule. Polyethylenes. PE can be both commodity (not intended for structural applications) and engineering plastic depending on grade, but is rarely used as composite matrix due to low temperature tolerance and low mechanical properties. However, PE fibers may be used as composite reinforcement. PE has the highest degree of crystallinity of any polymer due to its simple, regular, and flexible molecular structure. Polypropylenes. Just like PE, PP is both commodity and engineering plastic depending on grade. PP is the chemically least complex and cheapest polymer commonly used as composite matrix. In structural composites applications PP is usually reinforced with glass fibers. In recent years PP has become the most common thermoplastic matrix in mass-produced structural composites applications, including automobile components, such as various engine-room parts and seatback frames. Polyamides. One of the best-known thermoplastic polymer families is the polyamides (PA). While Nylon originally was the (Du Pont) trade name for PA fibers, "nylon" (not capitalized) gradually has become an accepted designation for PAs regardless of manufacturer. In contrast to PE and PP, PAs may be used at moderately increased temperatures thus greatly improving its usefulness as matrix. PAs are categorized by the presence of amine groups (--CONHm). A number of different PA grades, e.g. PA 6, PA 6,6, PA 6,10, and PA 12, are available, where the numbers indicate the number of carbon atoms in the repeating unit (see table 2.1); the properties naturally vary accordingly. The biggest drawback of PAs is that they are hygroscopic, i.e. absorb water. In composites applications PAs are normally reinforced with glass fibers and used in applications similar to glass/PP, but where higher temperature tolerance and improved mechanical properties are required. Thermoplastic polyesters. Although perhaps chiefly recognized as thermoset resins, polyesters also are available in thermoplastic forms, e.g. poly(ethylene terephthalate) (PET) and poly(butylene terephthalate) (PBT). The properties of
Thermoplastic composite sheet forming
TABLE
37
2.1
Comparison
of thermoplastic
Polymer
candidates for composites
Mer Structure H
Polyethylene, PE
H
I I~ ~CmC I HI H H
Polypropylene, PP
H
I I mC~C~ I I H
CH 3
0 Polyamide 6, PA 6
H
II C
Polyamide 12, PA 12
(CH2) 5~
~
H
II
I
H
H
0
I
I
II
~N~
Poly(phenylene sulfide), PPS
I
N
0
Polyamide 6,6, PA 6,6
Poly(ethylene terephthalate), PET
applications
(CH2)6~N~
O~
(CH2)2~ O~
C~
0 (CH2)4~
II
C~
O
O
[C[
C
-~s_
_~~_~o_~o_ o
Poly(ether ether ketone) PEEK
o Poly(ether sulfone), PES
II
o
o
o
II
/c~~.~
Poly(ether imide), PEI
CH,
_N,~~.~o_~ i~_~ o_~ :~_~ ,,
O
O O
II
Poly(amide imide), PAI
II
c
_~ ~_~o_~ II
o
38
B.T. ~lstr6"m
PET and PBT are similar to those of PAs, but lacking the hygroscopic disadvantage. In composites applications thermoplastic polyesters are reinforced with glass fibers and used in applications similar to glass/PP and glass/PA. Poly(phenylene sulfides). The most common member of the poly(arylene sulfide) family is poly(phenylene sulfide) (PPS), which has good tolerance to most chemicals and fire. PPS exhibits intermediate mechanical properties and temperature-tolerance. In composites applications PPS is reinforced with glass or carbon fibers and used in high-performance applications. Polyketones. While there are numerous aromatic polyketones, such as poly(ether ketone) (PEK), poly(ether ketone ketone) (PEKK), etc., the most common is poly(ether ether ketone) (PEEK). The polyketones possess high mechanical properties, high temperature-tolerance, good solvent resistance, and a high price. In composites applications PEEK is reinforced with glass or carbon fibers and used in critical high-performance applications. Polysulfones. Polysulfone (PSU), poly(ether sulfone) (PES), and poly(aryl sulfone) (PAS) are high-performance amorphous polymers with good tolerance to high temperatures and fire. These properties naturally come at a high cost and the melt viscosities are very high. Since polysulfones are amorphous, they are not resistant to all solvents although their resistance to many chemicals nevertheless is very good. In composites applications polysulfones are reinforced with glass or carbon fibers and used in critical high-performance applications. Thermoplastic polyimides. The polyimide family includes poly(ether imide) (PEI), polyimide (PI), and poly(amide imide) (PAI), which are all amorphous. Polyimides have the highest temperature-tolerance of the thermoplastics mentioned herein. Despite being amorphous they are very tolerant to solvents and environmental exposure and offer very good mechanical properties with the disadvantages of very high viscosities and high cost. In composites applications the members of the polyimide family are reinforced with glass or carbon fibers and used in critical highperformance applications.
2.2.2. Reinforcements The reinforcement is the constituent that is primarily intended to carry the structural loads the composite is subjected to. The reinforcement therefore to a significant degree determines stiffness and strength of the composite as well as several other properties. Composite reinforcement may be of different forms, but only fibrous reinforcement is relevant to thermoplastic sheet forming. A fiber, or filament, has a length-to-diameter ratio that approaches infinity and a diameter of the order of 10-5 m. All common fibers are manufactured in a drawing process, where the liquid raw material is drawn from an orifice. The drawing process ensures that the molecules of fibers organic in origin are aligned and parallel to the drawing direction, translating into significantly higher strength and stiffness in the axial direction than transverse to it. The most common types of fibrous reinforcement used in composites applications are glass, carbon, and aramid. However, it is not only the fiber type that
Thermoplastic composite sheet forming
39
is of significance; equally important is its configuration or form, e.g. the fibers may be discontinuous or continuous, randomly arranged or oriented, all aligned or in fabric form, etc.
2.2.2.1. Glass fibers The major ingredient of glass fibers is silica (SiO2), which is mixed with varying degrees of other oxides. The mixture is melted and extruded through minute holes in a platinum-alloy plate, or bushing. The glass filaments (fibers) vertically emerging from the bushing are drawn at high speed and are then quenched by air or water spray to achieve an amorphous structure. A protective coating is applied to the filaments before they are gathered together and wound onto forming packages. The wet glass is then dried. A group of collimated glass fibers is called a tow, strand, or yarn and an assembly of collimated tows is called a roving. When the tows are wound onto forming packages, as well as when tows are joined into rovings, they are usually rotated to provide a so-called twist, which promotes integrity and thus simplifies subsequent handling, while at the same time making impregnation more difficult since it becomes more difficult to spread the fibers prior to impregnation. The diameter of the individual filaments varies between 5 and 24 ~tm and is in composites applications commonly in the range 10-20 ~tm. The size of glass tows and rovings is given by tex number (weight in grams of 1,000 m) or yield in yd/lb (in North America). Common tex numbers are 600, 1,200, 2,400, etc., while common yields are 900, 450, 225, etc. (1 yd/lb ~ 500,000/tex). Several different glass compositions are available, the most common being E and S glass, where "E" denotes electrical and "S" high strength. E glass offers excellent electrical properties and durability and is considered a general-purpose grade that heavily dominates consumption. S glass and R glass are similar and offer improved stiffness and strength as well as high-temperature tolerance. Not surprisingly, the latter glass types are considerably more expensive than E glass. ECR glass (_c.orrosion resistant) and C glass are similar and, while having properties similar to E glass, offer improved corrosion-resistance. Since there are several different glass compositions, there is naturally a large variation in properties. Nevertheless, characteristics that tend to be shared by all glass fiber types are: 9 Good mechanical properties 9 High temperature-tolerance 9 Positive coefficient of thermal expansion (CTE) 9 Good electrical properties 9 Good corrosion-resistance 9 Moisture-sensitive 9 Abrasive 9 Inexpensive A characteristic that sets glasses apart from other reinforcements used in composites applications is that they are amorphous (i.e. glassy) and therefore are isotropic.
40
B.T. ~lstr6"m
2.2.2.2. Carbon fibers
Carbon fibers are commercially manufactured from three different precursors: rayon, polyacrylonitrile (PAN), and petroleum pitch. In the first two cases, the starting point of the carbon-fiber manufacturing process is textile fibers, whereas fibers are spun directly from the melt when pitch is the starting point. The fibers are initially drawn and oxidized at temperatures below 400~ to crosslink them to ensure that they do not melt during subsequent processing steps; drawing and oxidizing may occur concurrently. The fibers are then carbonized above 800~ in a process called pyrolysis, i.e. in the absence of oxygen, to remove noncarbon elements and create fibers virtually consisting of carbon only. Graphitization is then carried out at temperatures above 1,000~ to further eliminate impurities and enhance crystallinity. During both carbonization and graphitization further drawing may be employed to enhance orientation within the fibers. After graphitization fibers are surface treated and size is applied. The carbon atoms are covalently bonded together in so-called graphene layers which are held together by secondary bonds. The properties of carbon fibers are the result of the strong covalent carbon-carbon bonds within the graphene layers and it is therefore mainly the degree of orientation of these layers that determines the properties of the fiber. A higher temperature during graphitization promotes orientation of the graphene layers in the fiber direction, thus resulting in a higher tensile modulus. Carbon fibers are supplied in tows denoted 3K, 6K, 12K, etc., depending on the number of individual filaments they contain, where for example 3K stands for 3,000 filaments. Carbon fiber tows receive little or no twist. Filament diameters vary between 4 and 11 ~tm and is commonly around 7 ~m. Carbon fibers are often quantitatively referred to as "ultra-high modulus", "high modulus", "intermediate-modulus", "high strength", etc., where the borders between categories gradually change due to the rapid development of new fiber types. Properties significantly depend on precursor type and heat treatments used. While carbon fibers have the highest strength and stiffness of any compositereinforcement candidate, they generally only provide high strength or high modulus (in the same fiber). The span in properties of different carbon fiber types is vast, but among the properties they tend to share (at temperatures relevant to polymer composites) are: 9 Outstanding mechanical properties 9 High temperature-tolerance 9 Negative longitudinal CTE, positive transverse CTE 9 Electrically conductive 9 Excellent environmental resistance 9 Insensitive to moisture 9 Brittle 9 Expensive to very expensive The negative longitudinal CTE is due to bending of the predominantly longitudinally aligned graphene layers, for which reason the negative CTE increases in numerical value as the modulus (and graphene layer orientation) increases. Whereas the con-
Thermoplastic composite sheetforming
41
ductivity of carbon may be advantageous in some cases it is often a disadvantage. Carbon fibers thus may cause galvanic corrosion of metallic inserts and loose carbon particles suspended in the air may easily short out electrical and electronic equipment. In manufacturing operations dealing with unimpregnated carbon fibers which may be abraded, it is therefore necessary to provide costly shielding of electrical and electronic equipment.
2.2.2.3. Organicfibers While several different organic fiber types have been used as composites reinforcement, the category is dominated by aramid fibers. Kevlar is often assumed synonymous with aramid, but is in fact just the (Du Pont) trade name of the most common of a few commercially available aramid fiber types. Aramids, short for aromatic polyamides, are members of the PA family; fig. 2.7 illustrates the repeating unit of Kevlar (cf. PAs, table 2.1). Aramid fibers are manufactured in a process called solution spinning. The polymer powder is dissolved in sulfuric acid and is extruded through small holes, or spinnerets, into a narrow air gap. The fibers are quenched in a water bath to solidify the fibers and wash off most of the acid. The fibers are further washed, dried under tension, and then wound onto spools. Since aramid fibers are not brittle a protective size is not necessary. When the polymer solution is extruded through the spinneret, the molecules align with the direction of shear and the subsequent quenching ensures that the orientation remains in the final fiber. The degree of orientation may be further enhanced by heat treatment under tension resulting in improvements in longitudinal modulus. Due to its high degree of crystallinity and rigid molecular structure the temperaturetolerance of aromatic polyamide is very good for an organic material. The fiber diameter is typically 12 lam and tows, which are not twisted, consist of anything from a couple of dozen to several thousand fibers per tow. Tows are normally designated either by the number of fibers or the denier count (weight in grams of 9,000 m). Characteristics of aramid fibers when used as composites reinforcement include: 9 Very good mechanical properties, especially toughness and damage tolerance 9 Moderate temperature-tolerance 9 Negative longitudinal CTE, positive transverse CTE 9 Good electrical properties 9 Fair corrosion-resistance 9 Very moisture-sensitive 9 Tough 9 Expensive
-C
0 I1_0~_
0 H II I _.0~_ C-N
Fig. 2.7. Repeating unit of Kevlar.
H I N-
42
B.T. ~lstrO'm
While possessing several attractive properties, aramid also gives rise to a few notable difficulties. From a design point of view it is important to realize that the strength in compression is only a fraction of that in tension and that the fiber-matrix compatibility generally is poor. The outstanding toughness of aramid creates a problem in that fibers are very difficult to cut and machining of aramid-reinforced composites therefore requires special tools and techniques. Polyethylene fibers. High-modulus PE fibers may be manufactured through a process called gel spinning, which is a variant of the solution spinning process used for aramids. Since the tensile properties of both PE and aramid fibers are dictated by the properties of the covalent bonds of the molecular backbones, their mechanical properties are similar, but due to the lower density of PE fibers their specific strength and modulus are higher and comparable to carbon-fiber properties. The main drawbacks of PE fibers are poor matrix compatibility and poor temperature-tolerance, which likely makes PE fibers impossible to use with thermoplastic matrices.
2.2.2.4. Other fibers Several specialty fibers are used in different applications, for example offering extra-high temperature-tolerance, radar transparency, etc. Fibrous reinforcements used in polymer-matrix composites applications include boron and ceramic fibers, metal wires, and natural fibers, such as jute and wood. None of these fiber types is common in sheet forming applications.
2.2.2.5. Reinforcement-matrix interaction Conceptually a fiber-reinforced composite consists of transversely isotropic fibers and isotropic matrix with a perfect bond in between. Reality, however, is significantly more complicated. For a composite to be able to support external loads, fibers and matrix must cooperate.'It is often assumed that the fiber-matrix bond should be perfect, i.e. have the same properties as the matrix, and as strong a bond as possible is indeed often desirable to improve, for example, interlaminar shear strength, delamination resistance, fatigue properties, corrosion-resistance, etc. However, in some load cases a weak bond actually may be preferable; damage tolerance of a composite with a brittle matrix is usually enhanced by a weak fiber-matrix bond [3]. Whatever the load case, the fiber-matrix interface is of crucial importance to the properties of a composite. Manufacturers of brittle fibers, such as glass and carbon, apply a size, or finish, to the fibers to protect them from damage during subsequent handling, such as spinning, weaving, etc. Since a single reinforcement roving may contain tens of thousands of fibers and thus may be difficult to handle, the size also may be intended to promote tow integrity during such handling. However, since a fiber coating may interfere with the creation of a strong interface, and promotion of tow integrity is detrimental to reinforcement impregnation, the size may be seen as a necessary evil. Consequently the size in some instances is removed, e.g. burned off, when subsequent handling has been completed. To enable application of the size to every fiber, it is in glass manufacture applied before the fibers have been gathered into a tow,
Thermoplastic compositesheetforming
43
whereas a carbon-fiber tow must be spread out as widely as possible before application. A good bond between fiber and matrix may be formed through several mechanisms that all require wetting of the fibers by the resin. The preferred bonding mechanism is chemical (covalent) bonding, whereas mechanical interlocking may work well if the fiber surface is irregular, which is the case with some carbon and organic fiber types. Assuming that the fiber is porous, interdiffusion of polymer molecules into the fibers offers another possibility, whereas electrostatic attraction (secondary bonds) probably does not offer a strong enough bond to alone be responsible for satisfactory bonding [4]. However, despite over thirty years of intense research, the true mechanisms behind most successful fiber-matrix bonds are not well understood. Moreover, successful recipes are jealously guarded trade secrets. For glass fibers a coupling agent is applied to the fiber surface to enhance fibermatrix compatibility. Well-functioning coupling agents are available for thermosets such as unsaturated polyesters and epoxies, whereas coupling agents for high-temperature thermosets and all thermoplastics are not as efficient or even nonexistent. However, much work is underway to develop coupling agents for thermoplastic matrices. For carbon fibers the situation is rather different in terms of achieving fiber-matrix compatibility. Instead of applying a coupling agent the fiber is treated so as to promote its reactivity and compatibility with the matrix. Most commonly the fiber surface is oxidized to create surface oxygen groups that can form covalent bonds with the matrix. It appears that mechanical interlocking also is important with carbon fibers, since bonding generally is less good for high modulus fibers which have a higher degree of orientation of the graphene layers and thus smoother surfaces. For organic fibers the situation is further different in that no size is needed to protect the tough fibers and that no successful coupling agent has been developed. A common misconception is that the properties of the matrix are uniform throughout the composite, but various studies have proved the existence of a matrix property gradient from the fiber surface into the matrix bulk, a so-called interphase region (not to be confused with the interface). Without such an interphase there would be a huge discontinuity in modulus at the fiber-matrix interface. It has proven to be advantageous to some properties to have an intermediate modulus or ductile interphase [4]. With some semicrystalline matrices the fibers act as nuclei for crystallization, meaning that the interphase may become highly crystalline and hence stiffer than the bulk matrix.
2.2.3. Preimpregnated reinforcements One of the most complex, difficult, and not least important aspects of composites manufacturing is the impregnation of the reinforcement, particularly if the matrix is highly viscous. Consider a carbon-fiber-reinforced composite with a fiber volume fraction of 0.6, an average fiber diameter of 7 ~tm, and a volume of 1 ml (a cube with 10-ram sides). This volume contains 16 km of fiber, 0.4 ml of matrix, and 0.34 m 2 of interface. If the matrix is a high-viscosity molten thermoplastic it is
44
B. T. ,4str6m
not difficult to imagine that it is extremely difficult to evenly spread 0.4 ml of matrix onto a fiber surface area equivalent to five and a half pages of normal writing paper. Invariably, the impregnation will be imperfect and result in some dry fibers and entrapped gas as well as nonuniform fiber distribution. This is the background to the widespread use of preimpregnated reinforcement (prepregs) in manufacturing of high-cost and high-performance composites. Commercial prepregs are manufactured with dedicated machinery under well-controlled conditions and the result is low void content and reasonably uniform fiber distribution, and the prepregs often contain matrices that are not for sale except in prepreg form. The significant convenience of prepregs naturally comes with a significant disadvantage in terms of high cost. Prepregs are available in several forms depending on reinforcement and matrix as well as intended use. With continuous and aligned reinforcement the major impregnation forms are solvent impregnation, melt impregnation, powder impregnation, and commingling. Preimpregnated discontinuous and randomly arranged reinforcement is common in compression and injection molding processes. Such molding compounds, which may be melt or powder impregnated, are not referred to as prepregs although they conceptually belong to the same category; they are further discussed in section 2.2.3.6.
2.2.3.1. Solvent impregnation
Recalling earlier sections of this chapter, amorphous thermoplastics are not resistant to all solvents and thus may be dissolved and used in solvent impregnation. The polymer is dissolved to significantly lower its viscosity and thus facilitate reinforcement wetting and impregnation. The reinforcement is led into a solvent bath where the combined efforts of surface tension and the fact that the reinforcement is guided over rollers or bars ensures impregnation. Emerging from the bath, the impregnated reinforcement goes through nip rollers that carefully meter the reinforcement-tosolution ratio, whereupon the impregnated reinforcement goes into a drying oven, where the solvent is driven off and recovered (see fig. 2.8). Thermoplastic prepregs are not sticky and no backing paper is used; rolled-up prepregs are stored at room temperature. Correctly performed solvent impregnation produces intimately impregnated reinforcement (see fig. 2.9a). Both rovings and fabrics may be impregnated in
Fig. 2.8. Schematic of solvent impregnation.
Thermoplastic composite sheetforming
45
Fig. 2.9. Cross-sections of different prepreg types. (a) Solvent- or melt-impregnated. (b) Powderimpregnated with polymer sleeve. (c) Commingled. Reinforcing fibers are black and matrix is gray. Matrix powder particles and fibers typicallyhave significantlylarger diameter than reinforcing fibers.
this fashion. Regardless of resin type, residue solvent in the matrix presents a problem since it may be seriously detrimental to the properties of the composite. 2.2.3.2. Melt impregnation Ideally, melt impregnation is the preferred impregnation process, since it completes the ultimately desired intimate fiber wetting without introduction of a solvent (see fig. 2.9a). Unfortunately it is also the most difficult impregnation technique due to the high viscosities involved; with thermoplastics the resin has the consistency of chewing gum. In this case surface tension is of little help, and the first impulse may be to somehow increase the transverse pressure to force the matrix into the reinforcement. However, an increased pressure also compacts the fiber bed further and thus makes it even less permeable, so this tactic has the opposite effect. Some melt-impregnation techniques for thermoplastics involve the application of molten matrix onto the reinforcement right before the nip between the spread-out reinforcement and a rotating roller. With this approach the pressure forces the matrix through the reinforcement towards the lower pressure on the other side of the reinforcement, i.e. along the direction of the negative pressure gradient. Following several rollers like these the process is finished off with nip rollers to ensure that the reinforcement-matrix ratio is correct. The newly fabricated prepregs are cooled by calendering rolls and the matrix solidified. Melt-impregnation techniques of this type, which are carefully guarded by patents, are probably only used to impregnate parallel-fiber tows or yarns, since attempts at impregnating fabrics likely would prove extremely difficult due to the restrictions of tow cross-overs. An alternative melt-impregnation technique capable of impregnating fabrics employs a double-belt press (DBP) (see fig. 2.10). A DBP essentially consists of
46
B.T. ~IstrO'm
Fig. 2.10. Schematic of double-belt-press prepregging of continuous fabric with thermoplastic matrix. two steel belts that feed the material through the machine while applying lateral pressure to the material. In the first section of the machine the steel belts are heated, while in the latter section the belts are cooled. A reinforcement fabric sandwiched between two polymer films is fed into the machine. The heat within the press melts the polymer films and a pressure gradient applied by the belts causes the molten polymer to impregnate the fabric, which finally is cooled before exiting the press. Since the residence time within the double-belt press is long, even highly viscous thermoplastics are capable of percolating the reinforcement structure. An interesting concept is offered by the thermoplastic long discontinuous-fiberreinforced (LDF) material. In this melt-impregnated prepreg product, the fibers are aligned as in an ordinary melt-impregnated prepreg, but the fibers are discontinuous, allowing forming not possible with continuous-fiberreinforced prepregs.
2.2.3.3. Powder impregnation By grinding the solid matrix into a fine powder, the reinforcement can be impregnated using a slurry or a fluidized powder bed. A slurry employs a liquid, often water, to disperse the powder and impregnate the reinforcement with the lowviscosity aqueous solution, whereupon the water is driven off. The slurry may contain some kind of agent to promote adhesion between matrix powder and fibers, or the powder-impregnated reinforcement may be pulled through an oven to slightly melt the powder onto to the fibers. A fluidized powder bed contains a powder cloud that is kept fluidized by circulating air. Correctly performed, electrostatic attraction and/or friction ensures that the reinforcement passing through the powder cloud is properly filled with resin particles. The use of an oven to fuse the matrix onto the fibers is common. Another option to prevent the powder from being shaken out of the reinforcement during subsequent handling involves enclosure of the powder-filled yarn in an extruded polymer sleeve of the matrix material (see fig. 2.9b). The main advantages of powder-impregnated reinforcements is that they are flexible (unless subsequently excessively melted) and that they are cheaper than solventand melt-impregnated prepregs. The main drawbacks are that the reinforcement is
Thermoplastic composite sheet forming
47
not completely melt-impregnated, which thus has to be undertaken during composite component manufacturing, and that the powder rarely is evenly dispersed within the reinforcement. The final melt impregnation is often difficult to satisfactorily achieve and the results are liable to be less good than if melt-impregnated prepregs had been used. Another drawback specific to slurry-impregnated materials is that they may suffer from residue adhesion agent translating into problems similar to those encountered with solvent-impregnated prepregs. Powder-impregnated reinforcement, which tends to be unidirectional, is rarely referred to as prepreg, but rather powder-impregnated tow or yarn. However, not only yarns but also fabrics may be powder-impregnated; resin powder is first distributed onto one side of the horizontal fabric and the powder thermally fused in place, whereupon the other side of the fabric receives the same treatment. Such a powder-impregnation process may be followed by a DBP to produce a fully melt-impregnated fabric.
2.2.3.4. Commingling Commingled reinforcement consists of mechanically commingled (combined) reinforcing fibers and fibers spun from a thermoplastic resin (see fig. 2.9c). The advantages and limitations of commingled prepregs are the same as for powderimpregnated yarns (with the difference that no worry of residues from a slurry impregnation is warranted). Commingled reinforcement is unidirectional and is usually referred to as commingled tow or yarn.
2.2.3.5. Comparison of preimpregnated reinforcement types A qualitative comparison of the products of the aforementioned impregnation methods would consider flexibility, quality of impregnation, and cost. Flexibility is highly desirable to allow conformation to curved shapes and to allow for use in textile processes. High quality of impregnation is of course always desirable and likewise cost should be kept as low as possible. In general solvent- and melt-impregnated prepregs possess limited flexibility. Not surprisingly, high quality of impregnation goes hand in hand with high cost in solvent- and melt-impregnated prepregs. The main advantages of powder-impregnated and commingled reinforcements is that they are flexible enough to be used in textile manufacturing processes such as weaving, braiding, etc. and that they are, on a relative scale, inexpensive. Their main drawback is that the reinforcement is not melt-impregnated, which creates processing difficulties at a later stage. (It deserves to be pointed out that also narrow tapes of solvent- and melt-impregnated prepregs may be used in very specialized weaving processes.) There are relatively few thermoplastic prepregs on the market; available combinations include glass-fiber-reinforced PP, PAs, thermoplastic polyesters, as well as carbon-fiber-reinforced high-performance thermoplastics. Fiber volume fractions of unidirectionally or fabric-reinforced thermoplastic prepregs normally range from 0.35 to 0.6.
48
B.T. flstr6"m
2.2.3.6. Molding compounds Many techniques to manufacture composites with more or less randomly oriented and often discontinuous reinforcement also use some form of preimpregnated reinforcement as raw material. However, when preimpregnated reinforcement does not contain oriented and essentially continuous fibers, it is usually not referred to as prepreg but rather as molding compound. With molding compounds it may be more appropriate to talk about reinforced resin rather than impregnated reinforcement, since the fiber content tends to be considerably lower than in prepregs. Glass-mat-reinforced thermoplastic (GMT) is available in sheet form reinforced with randomly oriented fibers, which may be discontinuous or continuous. The most common technique to manufacture GMT is to use a DBP (see fig. 2.10), where two random-fiber mats and three polymer films or layers of extruded molten polymer are sandwiched before entering the press. Since this type of more or less completely meltimpregnated GMT ends up being a few millimeters thick and thus stiff, it is usually stored fiat instead of being rolled up. In another technique to manufacture GMT, which is similar to paper-making, chopped fibers, resin powder, and additives are dispersed in a slurry, which is deposited onto a moving belt where the water is driven off. This type of GMT thus consists of a porous fiber structure containing matrix powder, which, if desired, may be more or less completely melt-impregnated in a DBP. GMT is also available with part of the reinforcement continuous and oriented; alternatively, randomly reinforced GMT may be combined with continuous-fiberreinforced prepregs right before molding. Fiber volume fractions are normally 0.1-0.3 and fiber lengths in the range 10-30 mm unless the reinforcement is continuous. The commercial incarnation of GMT is massively dominated by glass-reinforced PP. 2.3. Properties The emphasis of this section 2.3 is on properties relevant to manufacturing and basic mechanical properties and intentionally refrains from delving into the quagmire of mechanical properties such as impact, fracture, fatigue, creep, etc. The intention of this section is merely to give the reader a feeling for typical properties of constituent materials on their own and in composite form. The properties quoted herein are from several different sources and thus may be slightly contradictory, which illustrates the significant differences existing between material formulations, processing conditions, test methods, etc. Unless otherwise noted, all properties are for room-temperature (RT) conditions.
2.3.1. Matrices For a matrix to be able to perform its tasks of supporting and protecting the primarily load-bearing reinforcement a number of properties are of relevance. The most pertinent properties usually are moduli and strengths in tension, compression, and shear, while also ultimate strain and fracture toughness may be important. The
Thermoplastic composite sheetforming
49
properties of the matrix usually determines the environmental tolerance of a composite; tolerance to elevated temperature and aggressive environments, such as UV radiation, oxygen, solvents, water, etc., thus are of paramount importance. As with most property comparisons of such broad material families indeed, as with all quantitative information of this section 3 the information should be seen as indicative only.
2.3.1.1. Thermal and rheological properties At sufficiently low temperatures an amorphous polymer is a glassy solid and any macroscopic deformation is likely due to stretching of secondary bonds and angle deformation of covalent bonds and only involves segments consisting of a few atoms. As temperature increases a region of rapid loosening of the secondary bonds is encountered and significantly larger segments of the molecules become free to move through rotation of covalent bonds. The temperature where these changes take place is referred to as the glass-transition temperature, Tg (see fig. 2.11), and is accompanied by a decrease in stiffness of several orders of magnitude. As the temperature increases further, amorphous thermoplastics often have a so-called rubber plateau where the stiffness does not decrease significantly and deformations are due to molecules sliding past one another. Further heating leads to complete melting. Thermosets, on the other hand, which are always amorphous, tend to have an extended rubber plateau, but since covalent bonds hold the polymer network together the polymer never melts. (However, if the temperature increases enough, the polymer will start to degrade, i.e. covalent bonds will be broken or formed.) Semicrystalline thermoplastics also show a drop in stiffness at Tg since the amorphous regions lose so much stiffness (see fig. 2.11). However, the crystalline regions remain unaffected and act as physical crosslinks and the polymer keeps much of its macroscopic stiffness. Not until the crystalline melting point, Tin, is reached does the polymer completely melt from a macroscopic viewpoint. Since the transition temperatures (rig and Tin) are functions of the molecular mobility in the bulk polymer it is easy to appreciate that a more rigid molecular
lO,
"--" 102 ~ 101 ,
i
i
alline i ~~morphous
100 10-1
-~ rr 10.2 10-3 , 50
~ o w t T~100
_-~i 9 Cr~
L molecular\ weight \ 150
High i \ molecular \ weight i I --. ; T~ 250 200
Temperature [~
Fig. 2.11. Relaxation modulus of polystyrene (PS) as function of temperature. Note the logarithmic stiffness scale. Approximate positions of Tg and Tm added. Redrawn from reference [5].
B. T. .~str6"m
50
structure and bulky side groups require a higher temperature to permit the same degree of mobility. Consequently, rigid thermoplastics with bulky side groups and strong secondary (intermolecular) bonds have high transition temperatures. Likewise, the very strong (intermolecular) covalent bonds of thermosets cause them to have higher rig s than many thermoplastics. For amorphous thermoplastics the maximum continuous use temperature consequently is slightly below Tg, whereas for thermoplastics with appreciable degree of crystallinity temperatures in excess of Tg and even close to Tm may be permissible for limited periods of time. Table 2.2 illustrates indicative transitions temperatures for some unreinforced, or neat, polymers as well as common or recommended processing temperatures, Tproc. In the unstressed state the molecules of a high-molecular-weight polymer liquid are heavily intertwined. For the liquid to flow requires that molecules move in relation to one another and the resistance to flow in a high-molecular-weight polymer liquid therefore is significant. However, if the liquid is sheared, the molecules are gradually aligned, fewer entanglements remain, and the resistance to flow (i.e. the viscosity) is reduced. On the other hand, if the molecular weight is moderate, i.e. the molecules are not excessively long, the resistance to flow is lower since there are not as many entanglements. In this case shear is unlikely to significantly lessen the degree of entanglement and thus will not affect the viscosity much. Besides temperature, the molecular weight of a polymer is therefore the single most important factor in determining its viscosity. The viscosity and the so-called shear-thinning, or pseudoplastic, tendency (i.e. that the viscosity is reduced by higher shear rates) increase rapidly with molecular weight. Since a higher temperature manifests itself in increased molecular mobility it strongly facilitates flow. A thermoplastic melt, which consists of high-molecular-weight molecules, is usually shear-thinning (see fig. 2.12, which shows the viscosity of PEEK). As further illustrated by the figure there is also a significant temperature-dependency. Table 2.2 gives typical zero-shearrate shear viscosities, r/0, for a few polymers only, since viscosity data usually are quite difficult to come by in the literature. However, the behaviors exhibited in fig. TABLE 2.2 Transition temperatures, processing temperatures, and viscosities of selected neat polymers [6-13]. The shear viscosities quoted are the zero-shear-rate values at the given temperatures
PP PA 6 PA 12 PA 6,6 PET PPS PEEK PEI PES PAI
Tg (~
Tm (~
Tproc (~
rio (Pa s)
-20--5 50-70 45 55-80 80 85 145 215 225 245-275
165-175 225 180 265 245-265 285 345
> 185 225-290 180-270 270-325 260-310 300-355 360-400 350-425 340-380 < 400
101-102 (230~
380 (38ooc) 103 (360~ > 105 (340~
Thermoplastic composite sheetforming
51
A 700 o
600 o.
500 7
.,~
400
o~
300
~
200 e.-
r
100 0 10 0
~ 101
~ 10 2
~.....~ 10 3
Shear Rate [l/s]
Fig. 2.12. Shear viscosity of molten PEEK as function of shear rate and temperature. Note the logarithmic shear-rate scale. Data from reference [10].
2.12 can be assumed indicative for thermoplastics while the magnitudes and degrees of temperature and shear-rate dependencies naturally vary from one polymer to another. In terms of most composites manufacturing operations a low viscosity is desirable, meaning that it may be tempting to increase the processing temperature based on the temperature dependency displayed in fig. 2.12. Although this temperature-dependency indeed is exploited for this purpose, it cannot be done indiscriminately. As previously mentioned, solid polymers eventually degrade through chain scission and crosslinking. This is even more so for molten thermoplastics since the molecular mobility is so much greater than in solid form, as well as the fact that the higher temperature stimulates chemical reactions. Figure 2.13 illustrates the allowable exposure time before onset of degradation for molten PEEK as function of temperature. The tolerance to degradation of PEEK is greater in the absence of oxygen, which is a
120 ~ ' 100 .n E ,80 0
"u E
60-
GI9 40 o 9 20
.E_ I-
0 360
In Air I
380
.....
I
400
,,,
I
420
~-- Temperature [~
Fig. 2.13. Time to surface degradation for PEEK as function of temperature in presence and absence of oxygen. Data from reference [14].
B.T. flstr6m
52
trait shared with most thermoplastics which readily oxidize if given the opportunity. The general trends exhibited in fig. 2.13 can be assumed indicative for thermoplastics. Heat transfer in polymers is due to thermal agitation across intramolecular and intermolecular bonds; the stronger the bond, the higher the conductivity. The coefficient of thermal conductivity (CTC) therefore increases with molecular weight, molecular alignment, and crystallinity. The CTC may increase or decrease with temperature depending on polymer. The specific heat, or heat capacity, arises from the freedom of movement of the molecules and thus decreases with crystallinity and increases with temperature, i.e. molecular mobility, and accordingly increases rapidly as Tg is passed. It was previously mentioned that since the crystalline morphology represents close-packing of the molecules the density of the crystal is higher than in the amorphous phase (see fig. 2.14). The figure further shows that the density dependency on temperature changes considerably as Tg is passed. This behavior may also be assessed in terms of CTE. The CTE of polymers tends to be a linearly increasing function of temperature both below and above Tg, but with a stronger temperaturedependency above Tg. Closely related to the CTE is the total volumetric shrinkage from melt (processing temperature) to solid (service temperature), which if not taken into account may lead to poor dimensional stability and sink marks. Table 2.3 gives thermal conductivity, k, specific heat, Cp, and CTE, c~, of some neat polymers. The table illustrates that the thermal properties do not vary drastically between polymers; this is essentially true also for thermosets. Since the data of table 2.3 apply at room temperature it is important to recall from the discussion above what happens as the temperature increases.
2.3.1.2. Mechanical properties Since crystals are stiffer than amorphous regions, stiffness increases with degree of crystallinity (see fig. 2.11). However, higher stiffness may also be obtained using
E3
I _
i
Liquid
r~
Fig. 2.14. Polymer density as function of temperature.
Temperature
Thermoplastic composite sheet forming
53
TABLE 2.3 Thermal properties of selected neat polymers [6,15,16]
PP PA 6 PA 12 PA 6,6 PPS PEEK PEI PES
k
G
(W/m ~
(kJ/kg ~
10-6 ~ -~
0.11-0.17 0.24 0.21-0.31 0.24 0.29 0.25 0.07 0.26
1.8-2.4 1.67 1.26 1.67 1.09 1.34
80-100 80-83 61-100 80 49 40-47 47-56 55
1.0
polymers featuring increased molecular rigidity, bulky side groups, and to a lesser degree increased molecular weight, although it should be remembered that such configurational traits impede crystal growth. Thus, if both high stiffness and high temperature-tolerance is desirable there will be a tradeoff between high transition temperatures and the degree of crystallinity, i.e. stiffness. Significantly, the thermoplastics with the highest mechanical properties and temperature-tolerance owe these traits to stiff molecular structures, which also deprive them of the ability to form crystals to any significant degree, i.e. they are amorphous. Thermosets typically are stiffer than thermoplastics due to the three-dimensional molecular structure bound together by covalent bonds. The strength of a material may be measured in so many different ways that sweeping statements about strength are difficult to make. However, the strength of polymers is highly dependent on and increases with the strength of intramolecular and intermolecular bonds, with the degree of crystallinity and, to a lesser degree, with molecular weight. Consequently, courtesy of the three-dimensional molecular structure, thermosets tend to have higher strength than thermoplastics. In general the relatively weak intermolecular forces of thermoplastics translate into a ductile material with high strain to failure, toughness, and damage-tolerance, since the molecules to a certain degree can slip relative to each other without rupturing covalent bonds. Since crystalline regions act as physical crosslinks, ductility decreases with increasing degree of crystallinity. Correspondingly, thermosets tend to be brittle and have low strain to failure, toughness, and damage-tolerance since the covalent bonds cannot yield much. Table 2.4 gives indicative mechanical properties of neat polymers. The table significantly illustrates that the differences in mechanical properties between polymers with the exception of the failure strain are not as significant as one might expect. Several properties other than thermal and mechanical may be of importance, depending on the intended application, e.g. electrical properties, optical properties, and tolerance to environmental exposure, but these are considered beyond the scope of this chapter.
54
B.T. ~lstrO'm
TABLE 2.4 Mechanical properties of selected neat polymers [16]. p denotes density, E elastic (Young's) modulus, a strength, and e strain to failure
PP PA PPS PEEK PEI PES PAI
p (kg/m3)
E GPa
cr (MPa)
e (%)
900 1,100 1,360 1,260-1,320 1,270 1,370 1,400
1.1-1.6 2.0 3.3 3.2 3.0 3.2 3.7-4.8
31-42 70-84 84 93 105 84 93-147
100-600 150-300 4.0 50 60 40-80 12-17
2.3.2. R e i n f o r c e m e n t s
While the composite matrix is normally responsible for properties such as temperature and environmental tolerance, the reinforcement primarily determines the composite's mechanical properties. The reinforcement typically has strength and modulus one to two orders of magnitude greater than polymer matrices. 2.3.2.1.
Thermal properties
Table 2.5 gives representative thermal properties of some reinforcement types. The table illustrates a couple of interesting differences in reinforcement properties. First, glass fibers are isotropic for reasons previously discussed. However, perhaps the most relevant property in this context is the negative longitudinal C T E of aramid and carbon fibers; while another is the high longitudinal CTCs of carbon, which for some carbon types even exceed the CTCs of the best metal conductors. The table further gives approximate m a x i m u m use temperatures Tmax. 2.3.2.2. M e c h a n i c a l p r o p e r t i e s
Table 2.6 gives representative mechanical properties of c o m m o n reinforcements, illustrating the vast range of reinforcement properties available. While table 2.4 illustrates that there are no excessively large variations in mechanical properties of TABLE 2.5 Thermal properties of selected reinforcements [16-20]. Indices l and t denote properties in longitudinal and transverse fiber directions, respectively kl
kt
(W/m ~ E glass S-2 glass Kevlar 49 (aramid) Carbon (PAN) Carbon (pitch)
Cp
(kJ/kg ~
0.87
0.87
0.041 7-70 100-520
0.048
0.825 0.737 1.42 0.7-0.9 0.7-0.9
otI
Tmax (~
ott
(10-6 ~ -~) 5.0 2.9 -2.3 -0.5--0.7 -0.9--1.6
5.0 2.9 41 7-10 7.8
350 300 250 600 500
Thermoplastic composite sheet forming
55
TABLE 2.6 Longitudinal tensile properties of selected reinforcement types [16,18,21,22]. The abbreviations for carbon
fiber types stand for high strength/strain, intermediate modulus, high modulus, and ultra-high modulus, respectively
E glass S-2 glass Kevlar 49 (aramid) Kevlar 149 (aramid) Carbon (HS/S) Carbon (IM) Carbon (HM) Carbon (UHM)
P (kg/m 3)
El (GPa)
~l (GPa)
6l (%)
2,520-2,620 2,490 1,440 1,470 1,700-1,900 1,700-1,830 1,750-2,000 1,870-2,000
73 86 131 186 160-250 276-317 338-436 440-827
3.4 4.5 3.6-4.1 3.4 1.4-4.93 2.34-7.07 1.9-5.52 1.86-3.45
4.88 5.7 2.8 2.0 0.8-1.9 0.8-2.2 0.5-1.4 0.4-0.5
matrices (with the exception of failure strain), there are certainly most significant differences in the mechanical properties of different reinforcement types. It is also noteworthy that the stiffer the fiber, the lower the strain to failure; it is thus not possible to have both high strain to failure and high modulus. The qualitative categorization of the multitude of carbon fibers in table 2.6 is obviously somewhat arbitrary and certainly partially contradictory, but it is nevertheless quite common in the composites industry.
2.3.3. Composites All properties discussed above pertain to the constituents on their own. When the constituents are combined into a composite they all lend some degree of their own properties to the composite; this is after all the underlying concept of a composite. Even if one does not take the overall part geometry into account, there are numerous variables in composites design, such as constituent types, fiber contents, fiber orientations, etc. It has therefore proved to be quite convenient to be able to approximately predict composite properties. The following sections will briefly look at basic relationships to predict some thermal and mechanical properties of the simplest possible composite form: the lamina. A lamina consists of a flat (or curved) assembly of unidirectional fibers or a fabric impregnated with a matrix (see fig. 2.15). Although the properties of unidirectionally reinforced composites of any crosssection may be predicted using these relationships, they will likely prove inadequate when trying to predict properties of composites with more complex geometries, as well as laminated composites composed of laminae stacked at different angles. While laminate theory is considered beyond the scope of this chapter the interested reader should encounter little problem locating one of the many textbooks covering this topic.
2.3.3.1. Prediction of thermal properties To enable determination of lamina properties from those of the constituents, several assumptions may be made; among the more common assumptions are that
B.T. /lstrSm
56
Fig. 2.15. Composite lamina.
matrix and fibers are isotropic. While glass fibers really are isotropic, carbon and aramid fibers clearly are not and it may also be questionable whether the matrix really is isotropic. Through micromechanics-based considerations one may derive expressions to determine the thermal properties of laminae. The longitudinal CTEs may be expressed as [23]:
Vfolf Ef @ VmOlmEm =
(2.1)
v j E j + VmEm
where V is volume fraction and indicesfand m refer to fiber and matrix, respectively. The transverse CTEs may similarly be written [23]:
o~, = Vfotf (1 + vf) + Vmotm(1 + Vm) - (Vfvf + Vmvm)O#
(2.2)
where v is Poisson's ratio and c~t is given by eq. (2.1). Equivalent expressions for the CTCs are [24,25]:
kt = Vfkf + Vmkm kfkm
(2.3) (2.4)
= Okm + Vmk
while the heat capacity may be expressed as [25]:
Up=
Vf pffpf -k- VmPmCpm Vfpf + Vmpm
(2.5)
2.3.3.2. Prediction of mechanical properties Micromechanics-based predictions of lamina mechanical properties are significantly more common than the thermal relationships quoted above. Employing the same assumptions as in the previous section, one may estimate the modulus of elasticity (Young's modulus) in the longitudinal direction as: Et = VuET + VmEm
(2.6)
Thermoplastic composite sheetforming
57
Equation (2.6) is sometimes called the parallel model and constitutes an upper bound for the modulus of a composite. The modulus in the transverse direction may be estimated as: Et -
ETEm
(2.7)
V:Em + VmE: Equation (2.7) is sometimes referred to as the serial model and may be regarded as a lower bound for the modulus of a composite. The parallel model may also be used to estimate the major (in-plane) Poisson's ratio and the serial model correspondingly may be employed to estimate the (inplane) shear modulus, although Poisson's ratios and shear moduli of fibers rarely are known. Comparisons with experimental data have shown that the parallel model yields reasonably accurate results, whereas the serial model predicts moduli considerably deviating from experimental data. It is also possible to predict the lamina strengths based on micromechanics considerations although such expressions require extensive assumptions, including that all fibers are perfectly parallel, have exactly the same properties (i.e. fail simultaneously), and are perfectly bonded to an isotropic matrix. In longitudinal tension one may express the strength as: =
+ Zm m
(2.8)
If the fibers are much stiffer than the matrix (which holds true for all common fiber-reinforced polymers) and the fiber volume fraction is significant (which is the case for all structural composites) the latter term in eq. (2.8) may be omitted. Since this strength expression is based on such highly ideal assumptions it is to be regarded as an upper bound for the strength; indeed it predicts strengths significantly higher than experimentally determined ones. A variety of equivalent expressions for transverse, shear, and flexural strengths have been derived and may be found in the pertinent literature, including textbooks on mechanics of composite materials (e.g. references [26] or [27]). Such textbooks also discuss the assumptions and accompanying weaknesses of the micromechanics models above, including other and more refined models, and invariably also describe how to determine the effective properties of laminates. Prediction of laminae and laminate properties using the equations above as well as other and more refined models no doubt is convenient in the early design stages. It cannot be overemphasized, however, that such predicted properties rarely are sufficient to finalize a design and experimentally determined properties are usually a necessity.
2.3.3.3. Experimentally determined thermal properties Since predictions rarely can replace experimentally determined properties entirely, this section and the next therefore attempt to give some examples of experimentally determined properties. Figure 2.16 shows the temperature-dependency of the thermal properties of unidirectionally reinforced carbon/PEEK, which probably is the most well-characterized thermoplastic-based composite to be found in the open
B.T. AstrSm
58
0.8-
8
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E
i
0.6
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i
0.5
i
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i
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i
0.3
.
1620
.
{
6
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i
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i i
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.............................[..................................~
1580
.................................. ,,.................................... 9 i................................ ~................................. i ................ i i i I ' i i t *
1.4
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0 a~ ". "~
..................................i....................................~...................................T...........
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i
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60
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--
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i i i , t i , ..................................i..................................i ...................................i ..............,..................~...............................i..............., ................... i i i ~ i ! J ! i i i i ......................... ~.................................i ................................ l ................i ................................i............J..................
i
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i
i
,
i~
j
f
1.0 0.8 0.6
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0.4
.-.
0
...............J.............................~.............t ................... 0 . 2
100
Temperature
200
Tg
300
T,,, 400
0
[~
Fig. 2.16. Thermal properties of unidirectionally reinforced carbon/PEEK as functions of temperature at Vf = 0.61. Values for CTE in excess of 150~ are estimated. Tg and Tm indicated in graphs. Data from reference [21.
literature. Thermal properties of other material systems are difficult to find in the literature and one therefore often has to resort to using eqs. (2.1) through (2.5).
2.3.3.4. Experimentally determined mechanical properties In analogy with the previous section, this section also attempts to provide some representative property data for thermoplastic composites. Table 2.7 gives properties of composites manufactured from E glass/PP, E glass/PA 12, and carbon/PEEK prepreg. The glass/PP and carbon/PEEK composites were molded from unidirec-
59
Thermoplastic composite sheet form&g
TABLE 2.7 Mechanical properties of composites molded from E glass-reinforced PP, E glass-reinforced PA 12, and high-strength carbon-reinforced PEEK prepregs [2,28,29]. r is shear strength, G shear modulus, and I L S S interlaminar shear strength, while index l denotes longitudinal, t transverse and tensile, c compressive,f flexural, and 2. out of plane Matrix: Reinforcement type:
PP E glass
PA 12 E glass
PA 12 E glass
Reinforcement form:
UD
UD weave
Balancedweave
0.35 1,480 620
0.55 1,900 710 90
0.52 1,850 350 350
570
800 160
500 500
D
p
flit ott
rrtc rrtc
rrtf rrtf rt• rlt
"t't_l_ Ett Ell
Etf Etf Glt Gt! Gt_I_ ~31t Vl_l_ Vtl
(kg/m3) (MPa) (MPa) (MPa) (MPa) (MPa) (MPa) (MPa) (MPa) (MPa) (GPa) (GPa) (GPa) (GPa) (GPa) (GPa) (GPa)
27.5 22
44 7.2 38 8
26 26 22 22
(%)
ett ILSS
(%)
(MPa)
0.61 2,130 80 1,100 220
120 175 80 137 9.4
5.1 5.1 3.2 0.33 0.32 0.04 0.40 0.40
21t_l_ V_l_t
e,
PEEK High-strength carbon UD
2.1
1.5 2.2 47
1.6 1.6 42
tionally reinforced m e l t - i m p r e g n a t e d prepregs, while the glass/PA 12 composites were m o l d e d f r o m p o w d e r - i m p r e g n a t e d fabric prepregs of two different types. The prepreg d e n o t e d U D weave was a c r o w f o o t weave with 90 percent of the fibers in the warp direction and 10 percent in the weft direction, whereas the prepreg d e n o t e d balanced weave was a satin weave (with equal a m o u n t s of fibers in the two orthogonal directions). Table 2.8 illustrates how the increased handleability of c o m m i n g l e d materials is also likely to lead to composite properties inferior to those achievable with melti m p r e g n a t e d prepregs. The d r a w b a c k of c o m m i n g l e d feedstock is that it m u s t be m e l t - i m p r e g n a t e d during processing, which is a non-trivial p r o c e d u r e with such a high-viscosity p o l y m e r as P E E K . Similar tendencies should be expected with powd e r - i m p r e g n a t e d reinforcement.
B.T. AstrO'm
60
TABLE 2.8 Properties of composites molded from melt-impregnated and commingled carbon/PEEK feedstock [2] Product form
O'lt (MPa)
Elt (GPa)
Olc (MPa)
Elc (GPa)
o'tt (MPa)
Ett (GPa)
Melt-impregnated Commingled
> 1,172 > 782
132 127
1,175 865
113 115
91 64
8.9 8.9
Table 2.9 provides properties for components compression molded from glass/PP GMT. For all product forms listed, with the exception of the last one which has half the reinforcement longitudinally oriented, the properties should be reasonably isotropic in the plane.
2.4. Manufacturing techniques The term sheet forming may generously be thought of as including both sheet formation, or rather consolidation of the raw material into a so-called blank, and forming of this blank, or sheet, into a curved component. To date virtually all types of sheet forming have employed prepregs and with this raw material form one may think of the route from prepreg to final component as consisting of three basic steps: prepreg lay-up, prepreg consolidation, and sheet forming. In some cases two or even all three of these steps may be combined into one operation, but it is more common that all three steps are separate. The remainder of this chapter introduces the most common routes to achieve these steps, but the order of the techniques described in no way implies any preference in terms of technical or economical feasibility. It deserves to be pointed out that most of the techniques discussed below are known by more than one name and the designations used herein are not necessarily the most common ones. TABLE 2.9 Properties of composites molded from glass/PP GMT [30]. For the last GMT type (Wf --- 0.43) half of the reinforcement is longitudinally oriented. W denotes weight fraction. GMT type
Wf
Fiber length (mm)
p
O'lt Elt (kg/m 3) (MPa) (GPa)
Elt (%)
O'lf Elf (MPa) (GPa)
Powder-impregnated
0.25 0.30 0.35 0.40
12 12 12 12
1,070 1,120 1,160 1,220
75 86 96 105
4.0 4.5 5.2 5.9
3.3 3.0 2.8 2.7
130 140 150 165
4.8 5.5 5.9 6.4
Melt-impregnated
0.20 0.30 0.40 0.43
continuous continuous continuous continuous
1,020 1,130 1,190 1,210
55 85 105 250
3.5 4.5 6.0 10.5
1.8 1.8 1.7 1.7
90 110 140 160
3.5 4.5 5.5 8.5
Thermoplastic composite sheetforming
61
2.4.1. Prepreg lay-up The most straightforward way of laying up the prepreg plies to form a composite no doubt is to cut the prepreg by hand and likewise to lay it up by hand, ply by ply. Due to the research and development status of most sheet forming operations, this fully manual procedure is indeed the most common, although semiautomated or fully automated machinery used in the aerospace industry when dealing with thermoset prepregs certainly could be used with thermoplastic prepregs as well. With such machinery wide prepreg sheets would be automatically cut to size and then robotically stacked in the desired fashion. However, for such procedures to make financial sense, relatively large productions volumes would be necessary. In contrast, with continuous and quasi-continuous consolidation processes the prepreg lay-up becomes an integral part of the consolidation process. In such continuous processes the prepreg plies are continuously uncoiled from rolls and gradually consolidated, as further discussed in the following sections.
2.4.2. Prepreg consolidation Once the prepreg plies have been laid up the stack may be consolidated in several different fashions. The basic requirement on any consolidation technique for thermoplastic prepregs is to apply sufficient pressure to maintain the molten reinforcementmatrix mass in the desired shape for the time required to allow itJto become dimensionally stable. With thermoplastic prepregs this time is of the order of tens of seconds to a couple of minutes, since the part only needs to be cooled for the matrix to solidify. It has been shown that full consolidation of two thermoplastic layers may be achieved in a fraction of a second provided the layers are at sufficiently high temperature and instantaneously are brought into intimate contact. With thermoplastic resins the strength between two layers brought into intimate contact at a temperature in excess of Tg is governed by diffusion of polymer molecules across the interface between the two layers. Given sufficient time this diffusion process will lead to an interfacial strength equal to that of the virgin resin. The time to reach the virgin strength is strongly dependent on temperature, since higher temperatures promote diffusion. This healing process is referred to as autohesion, i.e. self-adhesion between two layers of the same thermoplastic resin. While autohesion has been achieved in very short time frames in laboratory composites consolidation experiments, it is rare that similarly short time frames are achieved in real processing operations. In most manufacturing situations it is not the time required to achieve virgin material strength over an interface that limits the processing rate; the limits are usually set by the time required to heat and cool the material.
2.4.2.1. Vacuum-bag consolidation Following established procedures for thermoset composites manufacture, it is naturally possible to vacuum-bag consolidate a prepreg stack, possibly using an autoclave. Figure 2.17 illustrates how the prepregs have been laid up onto a mold and then covered by a vacuum bag. Vacuum is then drawn under the bag to compact
62
B.T. flstr6m
Fig. 2.17. Schematic of vacuum-bag consolidation.
the prepreg stack with (the external) atmospheric pressure. The vacuum is maintained throughout a complete heating and cooling cycle and results in a fully consolidated laminate. Heating is likely achieved through placing the entire vacuumbag assembly, including mold, in an oven and - - following complete melting of the prepreg matrix - - subsequent removal of the assembly from the oven and back into open air to cool off. If an autoclave is used, higher compaction pressures may be achieved through pressurization of the internal atmosphere of the autoclave; heating would likely be achieved through heating of the same internal atmosphere. Whether an autoclave is used or not, the mold naturally may incorporate provisions for independent heating and cooling to increase process efficiency. Although vacuumbag consolidation is certainly technically feasible, it likely does not make economical sense in anything but laboratory-scale experiments due to the large degree of timeconsuming manual labor and - - if an autoclave is used - - the expensive equipment involved. However, hand layup of thermoset-based prepregs onto a contoured mold followed by vacuum bagging and autoclave consolidation is an established technique to manufacture shell-like composites that also has proven feasible with thermoplastic prepregs. In this case prepreg lay-up and conformation to the mold are combined into one step that is followed by vacuum-bag consolidation (with or without autoclave) without further forming. 2.4.2.2. Matched-die consolidation
A much more common and economically more sensible consolidation technique is to employ one or more hydraulic presses and consolidate the prepreg stack between parallel platens, possibly using a picture-frame mold to prevent resin bleeding from the edges of the laminate. One incarnation of this consolidation technique employs an oven to heat the prepreg to a temperature in excess of the softening point of the matrix, whereupon the prepreg stack is quickly transferred to a press equipped with a mold having a temperature below the softening point of the matrix. The press then very rapidly closes to apply sufficient consolidation pressure during cooling (see fig. 2.18). This consolidation technique is best suited for fully melt-impregnated prepregs and cycle times of less than a minute may be achieved assuming that the matrix is completely melted. The reason for the short cycle times is that with fully meltimpregnated prepregs consolidation is only a matter of squeezing the prepreg plies together to eliminate inter-ply gaps and then to allow for autohesion to occur before the matrix resolidifies.
Thermoplastic composite sheetforming
63
Fig. 2.18. Schematic of matched-die consolidation. For material forms that are not already fully melt-impregnated, e.g. powderimpregnated and commingled prepregs as well as unimpregnated reinforcement layers interleaved with resin films (called film stacking), the processing requirements are quite different. With such materials matrix flow relative to the fibers must be permitted to achieve full fiber wetting and a completely consolidated laminate. Due to the very high matrix viscosities involved this required time frame is significantly longer than that needed for consolidation of melt-impregnated prepregs. To provide such extended time for flow, either two presses and two sets of molds - - one heated and one cooled or a mold which allows both heating and cooling are required. With two presses the prepreg stack is placed in the heated mold, which then is closed to apply slight pressure until the desired equilibrium temperature (in excess of the softening point of the matrix) is reached within the material. When the specified molding temperature has been reached, molding pressure is applied for a given length of time. The heated mold is then opened and the molten material is rapidly transferred to the cooled mold where it is consolidated under pressure. With one mold having both heating and cooling capabilities the prepreg stack is placed in the unheated mold in the press which is then closed. As in the previously described technique the material is gradually heated under slight pressure until the desired temperature is reached when full molding pressure is applied; the mold is then cooled to consolidate the material. (These two techniques may be combined employing one mold and two presses. In this case the material remains in the mold which is moved from the heated to the cooled press.) Neither of these two techniques appear to have much potential for mass production of flat laminates due to the expensive machinery (two presses) and long cycle times; the former technique may possibly achieve cycle times of the order of ten minutes, whereas the latter has cycle times of the order of an hour.
2.4.2.3. Double-belt-press consolidation A DBP offers the only realistic means of continuously consolidating prepregs into wide laminates (see fig. 2.19). The prepregs enter the press and are heated under pressure until the matrix is molten, whereupon the laminate is cooled under pressure so as to exit the DBP fully consolidated. In this case prepreg lay-up and consolidation are essentially performed simultaneously since the incoming prepregs are
64
B.T. flstr6m
Fig. 2.19. Schematic of double-belt-press consolidation.
uncoiled from rolls of material through the forward motion of the bands of the DBP. On-line consolidation is a term that is sometimes used to describe such simultaneous lay-up and consolidation. This consolidation method is the only technique that appears to be economically feasible for large-scale blank manufacture. Linear consolidation speeds are of the order of a few millimeters per second. Any prepreg form may be continuously consolidated into a laminate with a DBP, but processing rates are likely to be lower with not yet melt-impregnated material forms.
2.4.2.4. Tape laying A significantly less common but nevertheless technically feasible technique to online consolidate flat laminates is tape laying. In this case prepreg in tape form is unrolled from a spool and continuously laid up onto the mold (see fig. 2.20). Both the mating surfaces are locally heated and joined under pressure so as to achieve continuous consolidation, or welding. If both the previously laid ply and the incoming prepreg tape have completely melted surfaces when they are joined, no separate consolidation step may be needed and the laminate thus is ready for demolding as soon as lay-up is completed. However, one may instead aim only for partial consolidation to ensure that the prepregs stay in place in relation to one another and then achieve full consolidation in a separate processing step, such as through any of
Fig. 2.20. Schematic of tape laying.
Thermoplastic compositesheetforming
65
the aforementioned consolidation techniques or concurrently with one of the forming techniques described later in this chapter. If only partial consolidation is the aim, the lay-up can proceed much more rapidly with the same amount of heating, thus increasing the processing rate. Tape laying may also be used to produce curved components, thus enabling one-step layup--consolidation-forming. However, due to the localized and highly nonuniform heating history of such a component most attempts at such one-step manufacture have resulted in considerable residual stresses causing component warpage. It is therefore common to combine tape lay-up with subsequent consolidation whether the component in question is a blank or a final component. 2.4.2.5. Intermittent matched-die consolidation The final consolidation technique described herein is intermittent matched-die consolidation as illustrated in fig. 2.21. In this technique the incoming prepregs are heated in an oven so as to melt the matrix (the oven may possibly be replaced by a press with a heated mold to mimic the two-press consolidation procedure described above). The molten material is then indexed to enter a press with a cooled open-sided mold, where it is consolidated. The mold then opens, the material is indexed, and the consolidation repeated. Since this consolidation technique is quasi-continuous it may hold a promise of being economically feasible for large-scale blank manufacture. 2.4.3. Sheet forming Once a fully consolidated blank manufactured through any of the aforementioned lay-up and consolidation techniques is available, there is a range of possible means to form it into an arbitrarily shaped component. Virtually all prepregs contain continuous and oriented fibers, which from a manufacturing point of view are inextensible and embedded in a lubricating liquid resin. The issue of conforming and subsequently reconsolidating a previously consolidated blank (or possibly an unconsolidated prepreg stack) is a non-trivial exercise. Figure 2.22 illustrates possible deformation modes of the plies that make up the blank (or the unconsolidated stack) and the minimum requirements in terms of flow mechanisms to achieve this mode of deformation. Since the blank predominantly is made to conform to the mold and the inextensible fibers allow very little flow, the blank is normally almost the
Fig. 2.21. Schematic of intermittent matched-die consolidation.
66
B.T. AstrO'm
Fig. 2.22. Hierarchy of deformation modes for stacked reinforcement plies and corresponding required flow mechanisms. Redrawn from reference [2].
same size as the final component (often larger) and the thickness change between blank and final component is small. In shaping of doubly curved components there is a pronounced risk of wrinkling of the plies during conformation. The wrinkling tendency is significantly reduced if the plies to be formed have shape and size resulting in as little excess material around the edges (that needs to be trimmed off in a secondary operation following molding) as possible. Another important means of reducing the risk of wrinkling is to keep the plies under slight tension during molding. In most of the following technique descriptions such tensioning devices are neither explicitly mentioned nor shown in the figures, but would nevertheless likely improve results in all techniques when forming doubly curved components. Correctly optimized most of the sheet forming techniques discussed below ought to be able to both form the final component and consolidate a previously unconsolidated prepreg stack in the same step. In practice this has proven difficult to achieve in many (but not all) techniques and it appears as if most work on sheet forming utilizes preconsolidated blanks as raw material. Use of preconsolidated blanks is clearly economically disadvantageous, but generally simplifies and speeds up processing and results in improved consolidation in the formed component. For reasons of overall process economy, one-step forming and consolidation ought to be an area most worthy of research.
2.4.3.1. Matched-die molding Matched-die compression molding, also known as stamping due to its close similarity with sheet metal stamping, employs matching metal dies, or molds, mounted in
Thermoplastic composite sheetforming
67
a press (see fig. 2.23). The general processing steps are similar to those in matched-die consolidation described above. An oven is used to heat the blank to a temperature in excess of the softening point of the matrix, whereupon it is quickly transferred to a mold having a temperature below the softening point of the resin. The press then very rapidly closes to force the blank to conform to the mold and then maintains sufficient pressure during component cooling to ensure that the material is reconsolidated in its new shape. Due to its short cycle times and similarity with compression molding of both sheet metal and thermoset composites, matched-die compression molding is seeing quite some interest, particularly from the automotive industry. Significant drawbacks of this technique are very expensive molds and that forming is so rapid that it is likely that only moderately curved components are manufacturable when the feedstock contains continuous and aligned fibers.
2.4.3.2. Rubber-die molding Rubber-die molding is closely related to matched-die molding; fig. 2.24 illustrates the concept. Rubber-die molding reduces the risk of wrinkles in the part through more evenly applied pressure. Due to the poor heat transfer characteristics of rubber the matrix solidifies less rapidly upon die contact than with metal dies and more deeply drawn components are thus manufacturable. To facilitate molding of more complex geometries the rubber block may be contoured to more or less match the shape of the solid lower mold half. The technique employs significantly cheaper molds than matched-die molding.
Fig. 2.23. Schematic of matched-die molding.
Fig. 2.24. Schematic of rubber-die molding.
68
B.T. ~lstrO'm
2.4.3.3. Hydroforming Another quite similar molding technique is hydroforming. In this process a liquid is contained by a flexible membrane that is capable of conforming to the shape of the other mold half (see fig. 2.25). Following mold closure the liquid is pressurized to force the blank to conform to the other mold half. For the same reasons as with rubber-die molding the cooling is relatively slow, thus allowing for longer forming times and consequently more complex contouring. Also hydroforming reduces the risk of wrinkles in the part through the hydrostatically applied pressure, which may be significantly higher than in rubber-die molding. Hydroforming employs cheaper molds than both the aforementioned techniques.
2.4.3.4. Deep drawing In deep drawing the material to be formed is mounted in a frame that keeps the material under slight tension until forming is completed (see fig. 2.26). As the molten material leaves the preheater it is placed over a female "mold", essentially consisting of a hole the shape of the projected area of the final product. A male mold then rapidly punches the pliable material through the hole of the female mold. This crude forming technique likely produces components with poor external surface finish and allows limited control of material movement during forming. Nevertheless, it is a low-cost method with lower tooling costs than the aforementioned techniques and has proven its commercial feasibility in, for example, manufacturing of hinge covers for civilian aircraft cargo areas.
Fig. 2.25. Schematicof hydroforming.
Fig. 2.26. Schematicof deep drawing.
Thermoplastic composite sheetforming
69
2.4.3.5. Diaphragmforming The only technique that has been developed exclusively for thermoplastic composites manufacturing is diaphragm forming, which may be seen as a refined version of conventional autoclave consolidation. The process has been investigated by numerous different organizations and the results are encouraging. Since diaphragm forming has been investigated by many, the technical solutions vary significantly; two of the more common versions are described in the following. The blank to be formed is placed between two flexible diaphragms. The diaphragms, but not the blank which remains free-floating, are clamped around the entire perimeter using a clamping frame and the air is evacuated between the diaphragms. Thus far in the description most versions of diaphragm forming remain the same; the main differences are in heating and forming techniques. In one diaphragm forming version the diaphragm-blank sandwich is placed in an oven and heated to a temperature in excess of the softening point of the matrix. The diaphragm-blank sandwich is then rapidly placed onto a one-sided female mold and vacuum is drawn in the space between the lower diaphragm and the mold and pressure is applied above the upper diaphragm to force the blank to conform to the mold (see fig. 2.27). Either vacuum and pressure or just one of them may be used in forming. Since the mold is normally unheated the component solidifies as it gradually comes in contact with the mold. In another incarnation of diaphragm forming the diaphragm-blank sandwich is placed on top of the mold and then this entire assembly is placed in an autoclave, which often is purpose-built. The internal atmosphere of the autoclave is then heated to melt the matrix whereupon the combined forces of vacuum below the lower diaphragm and pressure above the upper diaphragm make the blank conform to the mold. When forming is completed the heated gas surrounding the mold and the diaphragms is replaced with cool (ambient) air to solidify the component. More rapid processing may be achieved if the mold incorporates heating and cooling options. The first diaphragm material was superplastic aluminum, but most later work has employed polymer films, particularly PI, and sheet rubber. Rubber diaphragms have an advantage in that they may be reused a few times, whereas aluminum and polymer films may not be reused since they become permanently deformed. It is important that the processing temperature is such that the diaphragms allow large plastic deformations, which for polymer diaphragms translates into temperatures between
Fig. 2.27. Schematic of diaphragm forming.
70
B.T. ~lstrO'm
Tg and Tm. PEEK-based prepregs are thus very well compatible with PI diaphragms, whereas at the processing temperatures common with PP they do not allow sufficient deformation. With rubber diaphragms this is not a concern since they are pliable at all realistic forming temperatures, but instead the processing temperature must be below the upper use temperature of the diaphragms. Since low pressures are involved, simple tooling may be used. Materials include sheet metal which allows rapid temperature changes and, for experimental work, plaster and wood molds, which have their predominant advantage in the low cost. The main characteristic of diaphragm formed components is that they may be deeply drawn and quite complex, especially when forming in an autoclave. This is partly due to the diaphragms keeping the material both under tension and under slight lateral compression and partly due to the matrix being pliable for a long time (basically until it comes into contact with the mold in the first technique and for any desired time in the latter technique). Due to its versatility the scope for commercial applications of diaphragm formed components should be great, but the commercial acceptance of the technique remains to be seen. Components that appear to be nearing commercial reality are glass/PP blister fairings and leading edges for jet engines to replace hand-laid-up and autoclave-consolidated carbon/epoxy components. 2.4.3.6. Folding Folding is a sheet forming technique different from the previously discussed ones in that it only involves localized heating and forming. Blanks may be folded through heating along a line and subsequent bending along this line (see fig. 2.28). Despite the fact that continuous fibers are likely to buckle or fracture, folding even of honeycomb sandwich panels, where the core ends up being crumpled has proven commercially viable in manufacturing of for example parts for aircraft interiors. 2.4.3.7. Roll forming Roll forming, which just like matched-die molding has been borrowed from sheet metal forming, is the only potentially continuous sheet forming technique. In roll forming several consecutive pairs of contoured rollers, normally four or more, gradually deform the molten blank to the desired shape (see fig. 2.29). The rollers, which are driven, are normally unheated and they consequently gradually cool the component. Roll forming is potentially capable of manufacturing any constant cross-section geometry and the products may be curved if desired. Nevertheless, so far thermoplastic roll forming has been used to manufacture hat and Z shapes only. Compared to most of the previously mentioned manufacturing techniques, little
Fig. 2.28. Schematicof folding.
Thermoplastic composite sheet forming
71
Fig. 2.29. Schematic of roll forming.
work on roll forming of thermoplastic composites appears to be ongoing despite the feasibility of the technique having been proven.
2.4.3.8. Matched-die molding of GMT In composites applications the term sheet forming is usually used to refer to conformation of a flat blank reinforced with aligned and continuous reinforcing fibers to a mold. In contrast, in matched-die molding using GMT as raw material the materialflows (rather than conforms) to fill the mold, since there are normally no aligned and continuous fibers to effectively prevent flow and also due to the fact that the fiber content is lower than in prepregs. While matched-die molding of GMT therefore may be slightly beyond the scope of this chapter, it is the only technique to manufacture shell-shaped structural thermoplastic composites that is currently in widespread commercial use and it has therefore been included for reference. The technique employs presses and molds similar to the previously described technique of matched-die molding. The main difference from fig. 2.23 is that in this case the heated raw material does not already cover most of the mold; instead the so-called charge covers only about half the mold area and the closing mold forces the charge to flow to fill the mold (see fig. 2.30). Since the GMT flows to fill the mold a component molded from GMT may have varying thickness and, for example, bosses and ribs, which are features difficult to obtain with matched-die molding of continuous-fiber-reinforced blanks. Since the material flow is notable the reinforcement is oriented in the direction of the flow and properties therefore may vary significantly within a component. In applications where increased stiffness is required GMT with varying degrees of oriented and possibly continuous reinforcement or interleaved unidirectionally reinforced prepregs may be used, thus partially bridging
Fig. 2.30. Schematic of matched-die molding of GMT.
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B.T. ~lstrO'm
the gap between matched-die molding of GMT and matched-die molding of continuous-fiber-reinforced blanks. The biggest users of GMT are automobile manufacturers, particularly European ones. GMT-based composites are notorious for their poor surface finish and are consequently often hidden from direct view in under-hood applications and in for example seat frames.
Acknowledgement The author gratefully acknowledges that the bulk of this chapter was written when he was a Visiting Senior Lecturer at the Department of Mechanical Engineering at the University of Auckland, New Zealand.
References [1] Thermoplastic Composites Materials Handbook, ICI Composites, Inc., USA, 1991. [2] Cogswell, F.N., Thermoplastic Aromatic Polymer Composites, Butterworth-Heinemann, Oxford, UK, 1992. [3] Jang, B.Z., Advanced Polymer Composites: Principles and Applications, ASM International, Materials Park, OH, USA, 1994. [4] Caldwell, D.L., Interfacial Analysis, in Handbook of Composite Reinforcements, Ed. S. M. Lee, VCH Publishers, New York, NY, USA, 1993. [5] Tobolsky, A.V., Properties and Structure of Polymers, John Wiley & Sons, New York, NY, USA, 1960. [6] Shalaby, S.W. & Moy, P., Thermal and Related Properties of Engineering Thermoplastics, in Engineered Materials Handbook, Volume 2, Engineering Plastics, ASM International, Metals Park, OH, USA, 445-459, 1988. [7] Galanty, P.G., and Richardson, J.J., Polyethylene Terephthalates (PET), in Engineered Materials Handbook, Volume 2, Engineering Plastics, ASM International, Metals Park, OH, USA, 172-176, 1988. [8] Brady, D.G., Polyphenylene Sulfides (PPS), in Engineered Materials Handbook, Volume 2, Engineering Plastics, ASM International, Metals Park, OH, USA, 186-191, 1988. [9] Data Sheet 2: Making Consolidated Sheet from Aromatic Polymer Composite, APC-2, ICI Fiberite Corporation, 1987. [10] Taske II, L.E., Personal communication, BASF, Charlotte, NC, USA, 1990. [11] Fines, R.E. & Bartolomucci, J.P., Polyether-imides (PEI), in Engineered Materials Handbook, Volume 2, Engineering Plastics, ASM International, Metals Park, OH, USA, 156-158, 1988. [12] Watterson, E.C., Polyether Sulfones (PES, PESV), in Engineered Materials Handbook, Volume 2, Engineering Plastics, ASM International, Metals Park, OH, USA, 159-162, 1988. [13] Fitzpatrick, J.E., Polyamide-imides (PAl), in Engineered Materials Handbook, Volume 2, Engineering Plastics, ASM International, Metals Park, OH, USA, 128-137, 1988. [14] The Thermal and Oxidative Thermal Degradation of APC-2, ICI Composites, Inc. [15] Modern Plastics Encyclopedia '95, McGraw-Hill, New York, NY, USA, 1995. [16] Hancox, N.L. & Mayer, R.M., Design Data for Reinforced Plastics- A Guideline for Engineers and Designers, Chapman & Hall, London, UK, 1994. [17] Anon., Fibers, in Engineered Materials Handbook, Volume 1, Composites, ASM International, Metals Park, OH, USA, 360-362, 1987. [18] Bunsell, A.R., Fiber Reinforcement, in Handbook of Composite Reinforcements, Ed. S. M. Lee, VCH Publishers, New York, NY, USA, 199-217, 1993. [19] Morgan, R.J. & Allred, R.E., Aramid Fiber Composites, in Handbook of Composite Reinforcements, Ed. S. M. Lee, VCH Publishers, New York, NY, USA, 5-24, 1993.
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[20] Diefendorf, R.J., Carbon/Graphite Fibers, in Engineered Materials Handbook, Volume 1, Composites, ASM International, Metals Park, OH, USA, 1987. [21] Pigliacampi, J.J., Organic Fibers, in Engineered Materials Handbook, Volume 1, Composites, ASM International, Metals Park, OH, USA, 54-57, 1987. [22] Miller, D.M., Glass Fibers, in Engineered Materials Handbook, Volume 1, Composites, ASM International, Metals Park, OH, USA, 45-48, 1987. [23] Shapery, R.A., Thermal Expansion Coefficients of Composite Materials Based on Energy Principles, Journal of Composite Materials, 2 (1968) pp. 380-404. [24] Rosen, B.W., Thermomechanical Properties of Fibrous Composites, Proceeding of the Royal Society of London, A319 (1970) pp. 79-94. [25] Taylor, R., Thermophysical Properties, in International Encyclopedia of Composites, Ed. S. M. Lee, VCH Publishers, New York, NY, USA, 5, 1-7, 1991. [26] Eckold, G., Design and Manufacture of Composite Structures, Woodhead Publishing, Cambridge, UK, 1994. [27] Daniel, I.M. & Ishai, O., Engineering Mechanics of Composite Materials, Oxford University Press, New York, NY, USA, 1994. [28] Plytron GN 638 T, Unidirectional Glass-Fibre/Polypropylene Composite, Borealis, Stathelle, Norway, 1995. [29] Product Information Vestopreg, HOls AG, Marl, Germany, 1994. [30] Berglund, L.A. & Ericson, M.L., Glass Mat Reinforced Polypropylene, in Polypropylene: Structure, Blends and Composites, Volume 3, Composites, Ed. J. Karger-Kocsis, Chapman & Hall, London, UK, 202-227, 1995.
This . Page Intentionally Left Blank
Composite Sheet Forming edited by D. Bhattacharyya 9 Elsevier Science B.V. All rights reserved.
Chapter 3
Computer Simulation of Thermoforming B.L. K O Z I E Y and M.O. G H A F U R Polydynamics Inc., 1685 Main St. West, Suite 305, Hamilton, Ontario, Canada L8S 1G5
J. V L A C H O P O U L O S and F.A. M I R Z A Faculty of Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7
Contents Abstract 75 3.1. Introduction 76 3.2. Sheet production 77 3.3. Thermoforming simulation 78 3.3.1. Finite element formulations 80 3.3.1.1. Membrane formulations 80 3.3.1.2. Thick sheet formulations 82 3.3.2. Material behaviour 83 3.3.2.1. Non-linear elastic models 83 3.3.2.2. Visco-elasticmodels 85 3.3.3. Thermoforming simulation examples 86 3.4. Concluding remarks 88 References 88 Abstract This chapter provides a description of the application of the finite element method to the simulation of the thermoforming process. The objective of thermoforming simulation is the provision of a rational means of mold design and to also allow for the design of "optimal" final parts using the minimum amount of material. This can be achieved by comparing the simulated behavior using various materials and mold configurations with varying process conditions. This eliminates the need to perform inefficient and expensive "trial-and-error" procedures. A "state-of-the-art" review of the finite element method in the simulation of thermoforming is given. The review covers the details of the membrane and thick sheet finite element formulations as well as non-linear elastic (Ogden, Mooney-Rivlin) and visco-elastic (K-BKZ) material constitutive relationships. Some examples, giving comparisons between simulation results and experimental values are also included. Good agreement was obtained between the predicted and measured thickness distributions. The results 75
B.L. Koziey et al.
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also indicate that the choice of material model (non-linear elastic versus visco-elastic) must be done carefully if reliable predictions of the thickness distribution are to be achieved. For straight thermoforming into shallow molds a non-linear elastic model is suitable. However, when deep drawn forming, complex mold geometry or plugassistance is involved, a visco-elastic model is required to obtain accurate predictions. 3.1. Introduction
The term "thermoforming" describes a number of related polymer processing techniques, in which thermoplastic sheets are softened by heat and subsequently formed by the application of vacuum, pressure, or a moving plug. The sheet may be stretched over a male mold (positive forming) or into a female mold (negative forming). On contact with the mold heat is lost and the material regains stiffness as it cools. The vacuum and plug-assisted forming processes are depicted in figs. 3.1 and 3.2, respectively. For a detailed description of thermoforming operations, equipment design as well as heating and cooling issues refer to Throne [1]. Geometries of thermoformed products are usually simple (boxes, food trays, various containers, refrigerator liners, but lately also computer casings). Thermoforming competes with blow molding and injection molding. The main advantages of this process are (i) the relatively low cost of thermoforming machines and the very low cost for the molds, and (ii) it is easy to form large area thin section parts. The disadvantages are (i) limited product shapes, (ii) difficulties in obtaining the required thickness distribution, (iii) it is difficult to control molecular orientation, and (iv) limitations in service temperature which may induce strain recovery or shrinkage. In thermoforming the final part thickness is typically controlled by employing differential heating of the plastic sheet and plug assistance. The critical parameters associated with these techniques for controlling the final part thickness have generally been determined by trial and error. Unfortunately this has meant that the A
plastic sheet
B
!
.... Clamp
.
Vacuum holes
// . .l
.
.
.
.
.
.
V
Vacuum
Fig. 3.1. Vacuum forming: (A) preheated clamped sheet over female mold prior to forming; (B) vacuum applied.
Computer simulation of thermoforming A
..................
Clamp
N~
Air
B
77 Air
C
"/" "
Plastic Sheet
T Vent
I Vent
Fig. 3.2. Plug-assist thermoforming: (A) preheated clamped sheet; (B) sheet stretched with plug advance; (C) pressure applied to complete forming.
development of new mold designs and evaluation of thermal process parameters has been inefficient and expensive. Furthermore, a "trial-and-error" process does not allow a quick comparison to be made between competing mold designs and different materials. The objective of computer simulation of thermoforming is the provision of a rational means of mold design and to also allow for the design of "optimal" final parts using the minimum amount of material. This can be achieved by comparing the simulated behavior using various materials and mold configurations with varying process conditions. In this chapter a review of the use of the finite element method in the simulation of the thermoforming process is presented. Because thermoplastic sheet is essential to the thermoforming process a brief description of the primary sheet production methods is given. This is followed by a description of the problems facing the thermoforming industry and what can be hoped to be gained through mathematical modeling of the thermoforming process. A "state-of-the-art" review of the finite element method in the simulation of thermoforming is then given. The review covers the details of the various finite element formulations and material constitutive relationships used. Some examples, giving comparisons between simulation results and experimental values are also included.
3.2. Sheet production Thermoplastic sheet is essential to the thermoforming process. Thermoforming sheet is loosely categorized as thin-gauge (sheet thickness less than 0.25 mm) and thick-gauge (medium-weight sheet: 0.25-1.5 mm; heavy-gauge sheet: greater than 1.5 mm). Thin-gauge sheet is usually produced by the process of calendering while thickgauge sheet is usually produced by the process of extrusion [2] using a coathangershaped die or by calendering [2]. Thermoplastic sheets are manufactured in a variety of widths and lengths and usually come in a variety of predetermined thicknesses. In
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general all thermoplastic resins that can be processed into a sheet can be thermoformed. Most of the fully developed thermoforming markets have been with amorphous [3] resins such as PVC, HIPS, PS, ABS, HDPE, PET, PETG, PP, PVDC, EVOH, PMMA. Amorphous plastics are suitable for thermoforming above the glass transition temperature Tg (usually 30~ to 60~ above Tg). These materials can be processed quite successfully over a wide temperature range. For semi-crystalline and reinforced amorphous materials such as PET and HIPS the forming window is very narrow and in general a higher forming pressure is required. For semi-crystalline materials the temperature should be just above the melting point Tm. The most common and versatile sheet manufacturing technique for thermoforming sheet production is extrusion. In extrusion, polymer pellets are drawn from a hopper into the gap between a heated barrel and a rotating screw. The mass of compacted solid pellets is transported forward and melted under the action of friction and heat. The polymer melt is then pumped through a die, usually of coathanger geometry, to produce a flat sheet which is then cooled and sized to proper gauge. The geometry is designed to allow the melt to flow downstream and laterally simultaneously so that a sheet of uniform thickness is produced. The sheet may be pulled mechanically by a roll stack as it exits the die. This not only reduces the gauge of the sheet, it also produces molecular orientation of the sheet in the machine direction. In most cases this imbalance in the orientation causes problems in the thermoforming process. This is especially true when the parts that are thermoformed are extremely deep or complicated. When the orientation is balanced biaxially on a specially configured extrusion line the strength and impact resistance of the sheet can be significantly increased. The other major process is calendering in which the molten polymer is converted into a sheet by a pair of rotating rolls. The melt forms a characteristic melt bank and spreads as it is squeezed between the rolls. Common calender roll arrangements include the L or Z configurations. A fixed gap is set between the initial rollers which is then continuously reduced between subsequent rollers until the desired gauge is achieved. Unlike sheets produced by the extrusion process those produced by the calendering method have no significant molecular orientation. A considerable amount of information is available in the literature on modeling and simulation of plasticating screw extrusion, flow through flat dies and calendering. The reader is referred elsewhere for more details [4-7].
3.3. Thermoforming simulation The current problems facing the thermoforming industry lie mainly with large wall thickness variations throughout the part with corners typically ending up as the thinner regions. Other problems include physical instabilities during inflation mainly rupture of the sheet as well as shrinkage and warpage of the final part. All these problems have an enormous effect on the performance and cost of a part. Non-uniform wall thickness variations and sheet rupture can be thought of arising for two main reasons, namely, mold geometry and process conditions. Other factors
Computer simulation of thermoforming
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affecting the wall thickness include the initial dimensions of the sheet, temperature distribution on the surface of the sheet, stretch-strain material properties of the resin, inflation dynamics, and the cooling and solidification phenomena that occur in the mold cavity. Mathematical modeling can provide valuable insights into mold design and process improvement by aiding in the selection of new mold designs, by simulating the behavior of plausible mold configurations, and providing rational means for comparing the formability of different polymeric materials in a given thermoforming process. Such capabilities eliminate the need to perform expensive trial-and-error procedures, in order to optimize the resin distribution and minimize the peripheral waste. Advances in computer technology and numerical methods have made it possible to simulate complex forming problems. The advantages of using the finite element method over any other numerical method are twofold: first, the formulation is not restricted to any particular geometry and, second, the highly non-linear behavior can be directly accommodated into the method. Many difficult aspects of finite element modeling must be addressed in the analysis. These include, but are not limited to, large strains, large deformations, non-linear material behavior, incompressibility, contact between polymer and mold wall, physical and numerical instabilities during inflation, and time-dependent material and thermal effects. While some of these issues, such as large strains and large deformations, must be dealt with rigorously in the formulation, others such as contact between the polymer and mold wall can be included in a simplified manner without adversely affecting the accuracy of the model. The determination of what aspects of the thermoforming process should be included, excluded, or simplified in the finite element formulation is largely dependent on the specifics of the problem to be analysed. For example, for shallow pressure or vacuum forming, in which the forming of the sheet often occurs very quickly, the viscous behavior of the material may not be significant. However, in plug-assisted and deep drawn thermoforming, the forming operation usually takes longer, so the inclusion of time-dependent material effects (visco-elastic) may have a significant influence on the final results. Because of the limited amount of data available these decisions have to be based on engineering judgment and practical experience. A survey of the early work on the application of the finite element method to the simulation of the thermoforming process is given by Zamani et al. [8]. Their review covers the period between 1966 and 1988. In general, these early applications were found to be successful for the problems to which they were applied. However, the usefulness of the models developed as design tools, and their applicability to general thermoforming problems was not well established because of the limited range of problems investigated. These works, however, were very important since they established the basis for much of the future research efforts. Since that time the finite element models have increased in complexity, and more rigorous and comprehensive comparisons between simulation results and actual measurements have been made. As a result the currently available finite element models have become useful design tools allowing for optimization of resin distribution and minimization of peripheral waste. Furthermore, they also provide a means
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of comparing the formability of various polymers under different temperature and pressure conditions. A "state-of-the-art" review of the finite element method in the simulation of thermoforming is given in the following sections. The review covers the details of the various finite element formulations and material constitutive relationships used. Some examples, giving comparisons between simulation results and experimental values are also included.
3.3.1. Finite element formulations In the finite element method the body to be analyzed is divided into a number of small subdivisions, or finite elements. The variables to be approximated are interpolated within each element using shape functions. Provided that the governing differential equations or a variational principle exists, finite element equations can be constructed from which an approximate solution for the process to be modelled can be obtained. Many texts are available on the finite element method and its application. For more information refer, for example, to Oden [9], Bathe [10], and Zienkiewicz and Taylor [11,12]. In the simulation of the thermoforming process there are two distinct types of finite element formulations: those which employ the membrane approximation, and those without the membrane assumption, i.e. thick sheet formulations. Each formulation has certain advantages and disadvantages. These will be described below, along with the details of each approach.
3.3.1.1. Membrane formulations In membrane formulations the bending resistance of the hot polymer is neglected and the sheet thickness is assumed to be very much smaller than its other dimensions. This assumption is quite reasonable for the bulk of the structure except possibly near clamping devices where the sheet may experience substantial bending. The majority of the finite element models formulated for the simulation of the thermoforming process employ the membrane approximation. See for example the work of Allard et al. [13], Warby and Whiteman [14], Nied et al. [15], deLorenzi and Nied [16], Taylor et al. [17], and Kouba et al. [18,19]. In general, the motion of the membrane is referred to a fixed Cartesian coordinate system. The position vectors of an arbitrary point P on the membrane surface in the current and reference configurations are denoted by ~ and ~', respectively. They are related through the expression ~ = X + 5, where 5 is the displacement vector. Since the resulting finite element equations will be non-linear they must be solved in an incremental manner. This is done by referring all variables to a previously calculated known equilibrium configuration and then linearizing the equations so that an approximate solution for the new configuration can be obtained. The solution can then be improved by iteration. Typically either the initial configuration at time 0, or the latest equilibrium configuration at time t are used as possible reference configurations. If all static and kinematic variables are referred to the initial configuration the formulation is referred to as a Total Lagrangian (TL) formulation. If all variables are referred to the last equilibrium configuration the formulation is referred to
Computer simulation of thermoforming
81
as an Updated Lagrangian (UL) formulation. The choice of which formulation to employ usually depends on the type of constitutive relation used. Since the constitutive relations typically used in thermoforming simulations are of the hyperelastic type, for example Mooney-Rivlin [20], Ogden [21], and the visco-elastic K-BKZ model the TL formulation is frequently employed. In this case the appropriate energy conjugate stress and strain measures are the second Piola-Kirchhoff stress tensor S, and the Green-Lagrange strain tensor E. The Green-Lagrange strain tensor is defined as 1
E-~(C-I)
(3.1)
where C is the right Cauchy-Green deformation tensor and I is the identity tensor. The constitutive relationship is usually expressed in terms of the principal stretches ~'i of the right stretch tensor U, or the invariants of the deformation tensor C. The invariants of the deformation tensor are easily calculated. To determine the principal stretches the eigenvalue problem for the Cauchy-Green deformation tensor, which is related to the right stretch tensor by U = C, must be solved. This yields two of the principal stretches, )~1 and ,~2, the third principal stretch ,k3 normal to the membrane surface, is determined using the incompressibility constraint )~1~,2~. 3 = 1. The incompressibility constraint in terms of the invariants of the deformation tensor is written as 13 = detC = 1. In membrane formulations the incompressibility condition is always satisfied. Typically the finite element equations are derived using the virtual work principle which requires that the rate of internal virtual work is equal to the external virtual work for an arbitrary velocity field satisfying the kinematic boundary conditions [9]. Assuming that the inertia and body force terms can be ignored the virtual work expression for the deformed membrane using a TL description is given by / S "6Eh0 d A 0 A0
dA
(3.2)
A
where A and A0 are the areas of the deformed and undeformed membranes, respectively. The applied pressure is denoted by p while t7 defines the vector normal to the surface of the deformed membrane, h0 is the initial membrane thickness and 6 is the membrane velocity vector. The finite element equations of equilibrium required for the incremental analysis are obtained through variation and linearization of the virtual work expression, eq. (3.2), followed by the introduction of the finite element discretizations. The elements used for discretization of the membrane typically [16,18] employ linear variations of the displacements within the element. In addition, the elements are usually triangular which allows curved edges to be more easily modeled. As mentioned previously the resulting finite element equations are solved using an incremental procedure. Starting from a known equilibrium configuration, usually the initial undeformed state, a small load increment is applied and the displacement increments are found. The solution is improved by applying an iterative method,
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usually the standard Newton-Raphson method. Since the geometry of the membrane can often change drastically even for small load increments the load vector and stiffness matrix are recalculated during the iteration procedure using the updated geometry while keeping the load constant. After several iterations the incremental deformations become less than some prescribed value at which point the solution is considered to have converged. In the presence of a mold, collision calculations are continuously made to determine whether a node has contacted or penetrated the mold surface. When this occurs within a prescribed tolerance the node is permanently fixed to the mold surface at the contact point. This most closely approximates the actual forming conditions in which very little slip between the polymer and mold is observed. The load is incremented and the iterative procedure repeated until the desired pressure has been reached or all nodes have contacted the mold surface. The two most extensive membrane models of the thermoforming process are those developed by deLorenzi and Nied [16] and Kouba et al. [18,19]. Both employ similar finite element formulations; however, the model developed by Kouba et al. includes both non-linear elastic and visco-elastic material models, while that developed by deLorenzi and Nied includes only non-linear elastic material behaviour. In addition, the model of Kouba et al. includes pre-stretching of the sheet in a plug-assisted thermoforming process. This cannot be simulated using the deLorenzi and Nied formulation. A more detailed description of the various material models employed in thermoforming simulation is given in section 3.3.2. 3.3.1.2. Thick sheet formulations In a thick sheet formulation the bending resistance of the sheet is not assumed to be negligible as in a membrane formulation. Consequently, in addition to inplane shearing and extension of the sheet, transverse shearing and bending effects are also included in the model. This means that the strains and stresses will not be constant through the sheet thickness but will vary. While this provides a more accurate model of the sheet deformation the formulation of a thick sheet finite element model is much more complex than a membrane formulation. The added complexity is due to the fact that, (i) the incompressibility condition in a thick sheet formulation cannot usually be satisfied exactly but must be imposed through some type of numerical constraint (e.g. penalty method or Lagrange multipliers), and (ii) a thick sheet finite element model is inherently more complex than a membrane model. Perhaps the earliest thick sheet finite element model was that developed by Oden [9] which was used to simulate the free inflation of a sheet of finite thickness assuming non-linear elastic material behavior. The model was however not extended to include a mold so the thermoforming process could not be simulated. It was not until very recently with the work of Song [22,23] and Igl and Osswald [24] that thick sheet finite element models were applied to thermoforming process simulation. Igl and Osswald used a one-dimensional shell finite element model to study the thermoformability of wood-fiber-filled polyolefin composites. Song developed an axisymmetric thick sheet finite element model which can simulate straight and plug-assisted thermoforming assuming a non-linear elastic material response. While these
Computer simulation of thermoforming
83
simulations were found to be successful the added complexity of the thick sheet formulation is only important in some problems. In typical thermoforming problems the membrane approximation was found to predict results very close to those given by the thick sheet model even for relatively thick sheets [18]. The main advantage of the thick sheet models is that localized effects, such as bending and shearing near clamping devices and in corners, and stress concentrations near a plug boundary can be predicted. In typical thermoforming operations these effects will not dominate and thus the added complexity and computational effort required by thick sheet models is difficult to justify.
3.3.2. Material behaviour
For a material model to be applied successfully in a numerical simulation it must be capable of describing the material response under actual processing conditions. Ideally it should give a good representation of the stress-strain behavior of the plastic for both small and large strains, and over a wide range of temperatures and strain rates. During inflation into the mold cavity the sheet is mainly subjected to biaxial stretching. Consequently, the elongational characteristics of the polymer play an important role in the thermoforming process. From industrial experience the strain hardening characteristics of the plastic have been found to significantly affect the final thickness distribution of the finished product. Throne [1] states that the elastic modulus for materials in thermoforming is typically between 0.07 MPa and 1.0 MPa. Strain rates observed in the process normally range from 0.1 s-~ to 10 s-1. It is generally recognized that thermoplastics in a semi-molten state possess a strong viscous component which allows them to flow when sufficient stress is applied. They also possess a significant elastic component that resists flow and imparts integrity and self-supporting properties to the sheet. As a result in finite element simulations of thermoforming the polymer behavior is typically modeled using either a non-linear elastic or a visco-elastic constitutive model.
3.3.2.1. Non-linear elastic models
There is strong experimental evidence [25-27] that when polymers are extended at relatively high strain rates, at temperatures above the glass transition temperature their behavior can be adequately modeled using constitutive equations originally developed for rubber-like materials. It has also been shown experimentally [28] that the mechanical behavior of polymer materials above the glass transition temperature is essentially incompressible. The assumption of non-linear elastic material behavior greatly simplifies the formulation of the finite element equations, but neglects the viscous component of the polymer response. In non-linear elasticity the components of the constitutive tensor are given in terms of certain material constants, and are also functions of the material strains. For more information on non-linear elasticity refer, for example, to Green and Zerna [29], Green and Adkins [30], and Eringen [31].
B.L. Koziey et al.
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For an isotropic material the stress-strain relationship is frequently defined using a strain energy function, W, given in terms of the Green-Lagrange strains. Such a material is referred to as a hyperelastic material and the stresses are given by OW S = ~ aE
(3.3)
where S is the second Piola-Kirchhoff stress tensor. Once an appropriate form of the function W has been defined the stresses can be evaluated. The determination of W must be done carefully to ensure that the frame indifference requirement is satisfied. Two different definitions of W which have been used almost exclusively in thermoforming simulations are the Mooney-Rivlin model and the Ogden model. Mooney-Rivlin model. In the Mooney-Rivlin [20] model it is assumed that the strain energy density function, W, for an incompressible material can be expressed as a polynomial function of the first two invariants, I1 and 12, of the Cauchy-Green deformation tensor, C. The generalized form of the strain energy function is given by [201 M
W - Z
N
Z
Aij(I1 - 3)i(I2 - 3) j
(3.4)
i=0 j=0 where A O. are experimentally determined constants. When the body is undeformed the strain energy W = 0 and constant A00 = 0. If only the first two terms, A10 and A01, are retained then the standard Mooney-Rivlin expression is obtained which is written as W -- A10(I1 - 3) + A01 (I2 - 3)
(3.5)
When A01 = 0 the model is referred to as the neo-Hookean model. The components of the stress tensor S can be calculated by substituting eq. (3.5) into eq. (3.3) and performing the required differentiation. Note that the stress normal to the membrane surface, $33, is assumed to be negligible. Ogden model. In the Ogden [21] model the strain energy density, W, is assumed to be a function of the principal stretches, instead of the invariants of the deformation tensor, and is given by m
W-
~/z__~n(,k~n + Z2" + Z3n _ 3)
(3.6)
n=l 0r
where Z l, ~'2 and )~3 are the principal stretches. Constants lZn and an are determined by fitting the model to experimental stress-strain data and can be either negative or non-integer values. However, the constants must yield a positive strain energy density function. Note that, because incompressibility is assumed, the third principal stretch, ~3 is expressed in terms of ~'1 and ~'2 using the constraint ~.1~.2~.3 = 1. The summation over n in eq. (3.6) extends over as many terms as are necessary to characterize the behavior of the material. Usually n is equal to 1, 2, or 3. As with the Mooney-Rivlin model the components of the stress tensor are determined by substituting eq. (3.6) into eq. (3.3) and performing the necessary differentiations.
Computer simulation o f thermoforming
85
Again the stress normal to the membrane surface is assumed to be negligible. Because the Ogden model is given in terms of the principal stretches instead of the invariants, 11 and I2, the physical meaning of the resulting stress-strain relations are much easier to determine.
3.3.2.2. Viseo-elastic models While non-linear elastic models effectively describe the material response under certain conditions, they must be applied with caution in actual thermoforming situations since they neglect viscous effects. For example, for relatively slow thermoforming viscoelastic effects have been found to be extremely important [32]. Hylton gives several examples of how the thermoformability of a given resin may be assessed on the basis of several visco-elastic properties. Perhaps the first finite element model to take visco-elastic effects into account is that given by Kouba et al. [19]. In their formulation the stress-strain data at different strain rates is fitted to a K-BKZ [33] constitutive equation modified by the inclusion of a Wagner damping function. The constitutive equation used is of the integral type and is given by t
s-
f
m ( t - t')h(I1,/2)B(t, t') dt'
(3.7)
where t' is the previous time, t is the current time, B(t, t') is the Finger strain tensor (left Cauchy-Green deformation tensor i.e. C -1) and m ( t - t') and h(I1,12) are the memory and damping functions, respectively. The damping function is of the form proposed by Wagner and Demarmels [34], and is written as 1
h(I1, 12) -
1 + av/(I 1 -
(3.8) 3 ) ( / 2 - 3)
where a is a material parameter and 11 and I2 are invariants of the Finger tensor B. The memory function which is related to the relaxation spectrum is given by (t--tt)
m ( t - t ' ) - Z ai e
(3.9)
ri
where ri are the relaxation times and a i are material constants and i is the number of relaxation processes considered. Assuming that pressure is applied and the deformation of the sheet starts at time t - 0 the stress in the two principal directions (i -- 1, 2) tangent to the surface of the sheet is given by t
S(t) - f m(t - t')h(I1, I2)[LZ(t, t')
-
LZ(t, t')] dt'
0
(3.10)
0
4- h(t)[LZ(t) - LZ(t)] f --00
m ( t - t') dt'
86
B.L. Koziey et al.
where Li(t, t') is the stretch ratio in the i th principal direction at time t related to time t', and L3(t, t') is the stretch ratio perpendicular to the sheet surface at time t related to t'. Temperature effects have also been included in the model because the model constants are a function of the temperature. Another approach to the inclusion of viscous effects in thermoforming is that presented by Vantal et al. [35]. They employ an approach similar to that used in metal forming. For deformation of the solid polymer under the glass transition temperature a visco-elasto-plastic constitutive relation of the form
-- kp(T)(g, e)
(3.11)
is used. Quantities 8, g and ~ are the effective stress, effective strain and effective strain rate, respectively, The temperature-dependence of the material response is included through the function kp(T). Beyond the glass transition temperature the rubbery polymer is modeled using a phenomenological visco-elastic constitutive relation. The material parameters were determined by fitting the constitutive model to uniaxial tensile data obtained for a wide range of temperatures and strain rates. In contrast to the other constitutive models described above, an updated Lagrangian finite element formulation is required since the constitutive relation is valid for small strains only.
3.3.3. Thermoforming simulation examples A finite element software package capable of modeling fully 3-D thermoforming, with or without plug assistance, has been developed and tested at McMaster University [18,36-38]. Extensive comparisons between predicted and experimental thickness distributions for both straight and plug-assisted thermoforming have been made which has established the validity of the finite element model. Experimental data on uniaxial or biaxial testing of polymers is scarce, especially at strain rates and temperatures that are within the ranges found in the thermoforming process. Nevertheless successful simulations have been carried out using both simple and complex mold geometries. For simple shallow mold geometries simulations using the non-linear elastic Ogden model are very accurate. A simulation of the vacuum forming of a simple dome-like automotive fuel tank component was performed using both the Ogden model and the visco-elastic K-BKZ model. The thickness distributions along a cut through the dome predicted by both models are plotted along with the actual thickness distribution in fig. 3.3. There is good agreement between the experimental results and both the Ogden and K-BKZ predictions. While the Ogden model is best suited for the simulation of straight thermoforming into shallow simple molds, the K-BKZ model is required when the simulation includes deep-drawn forming, complex mold geometry or plug assistance. The plug-assisted thermoforming simulation of a sheet into a complex mold was performed. The mold has a volume of about 22 liters and is used to form electronic device casings. A finite element mesh consisting of 1,536 elements was used to discretize the sheet which had an initial thickness of 6.35 mm (0.25 in.). The simulation was performed using both the Ogden and K-BKZ models. The K-BKZ model
Computer simulation of thermoforming
87
Fig. 3.3. Comparison of predicted and experimental final thickness distribution for automotive fuel tank component.
was fitted to material stress-strain data obtained for strain rates ranging from 0.01 s-1 to 10 s-1. The thickness distribution predicted using both material models is plotted in fig. 3.4, along with the experimental values for the cutting plane shown in the figure.
Fig. 3.4. Comparison of predicted and experimental final thickness distribution for film scanner casing.
88
B.L. Koziey et al.
The distribution predicted by the Ogden model is seen to be in poor agreement with the experimental data. However, the thickness distribution predicted by the KBKZ model is in very good agreement with the experimental distribution. This example clearly illustrates that the choice of material model (non-linear elastic versus visco-elastic) must be done very carefully if reliable predictions of the thickness distribution are to be obtained. 3.4. Concluding remarks A "state-of-the-art" review of the application of the finite element method to thermoforming simulation has been given. In general, the currently available models provide a rational means of mold design and can also aid in the optimization of the final resin distribution and the minimization of peripheral waste. While the existing models are capable of providing good predictions of the final part thickness, care must be taken in the selection of the material model used. Non-linear elastic models such as the Ogden model are attractive in their simplicity in comparison to viscoelastic models such as the K-BKZ model, but typically only provide reliable prediction for straight thermoforming (no mechanical assisted sheet stretching) into shallow simple molds. Future work should be targeted towards the development of less complex viscoelastic material models, inclusion of time-dependent thermal effects, and slippage of the sheet along the plug surface. In addition, more experimental data on uniaxial and biaxial stretching of polymers needs to be established at strain rates and temperatures typically experienced in the thermoforming process. References [1] [2] [3] [4]
Throne, J.L., Thermoforming, Hanser Publishers, New York, 1987. Tadmor, Z. & Gogos C.G., Principles of Polymer Processing, John Wiley & Sons, New York, 1979. Birley, A.W., Haworth B. & Batchelor J., Physics of Plastics, Hanser Publishers, Munich, 1991. Vlachopoulos, J. & Mitsoulis E., Fluid Flow and Heat Transfer in Calendering, in Transport Phenomena in Polymeric Systems, Ellis Horwood, Chichester, 1989. [5] Vlachopoulos, J., Calendering, in Concise Encyclopedia of Polymer Processing and Applications, Pergamon Press, Oxford, 1992. [6] Vlachopoulos, J., Silvi N. & Vlcek J., POLYCAD| A Finite Element Package for Molten Polymer Flow, in Applications of Computer Modeling for Extrusion and Other Continuous Polymer Processes, Hanser Publishers, Munich, 1992. [7] Vlcek, J., Perdikoulas J. & Vlachopoulos J., Extrusion Die Flow Simulation and Design with FLATCAD, COEXCAD and SPIRALCAD, in Applications of Computer Modeling for Extrusion and Other Continuous Polymer Processes, Hanser Publishers, Munich, 1992. [8] Zamani, N.G., Watt D.F. & Esteghamatian, M., Status of the Finite Element Method in the Thermoforming Process. Int. J. Num. Meth. Eng., 28 (1989) pp. 2681. [9] Oden, J.T., Finite Elements of Non-linear Continua, McGraw-Hill, New York, 1972. [10] Bathe, K.J., Finite Element Procedures in Engineering Analysis, Prentice Hall, Englewood Cliffs, NJ, 1982. [11] Zienkiewicz, O.C. & Taylor R.L., The Finite Element Method, Volume 1: Basic Formulation and Linear Problems, McGraw-Hill Book Company, London, 1988.
Computer simulation of thermoforming
89
[12] Zienkiewicz, O.C. & Taylor R.L., The Finite Element Method, Volume 2: Solid and Fluid Mechanics Dynamics and Non-linearity, McGraw-Hill Book Company, London, 1991. [13] Allard, R., Charrier J.M., Ghosh A., Marangou M., Ryan M. E., Shrivastava S. & Wu R., An Engineering Study of the Thermoforming Process: Experimental and Theoretical Consideration. J. Polym. Eng., 6 (1986) pp. 363. [14] Warby, M.K. & Whiteman J.R., Finite Element Model of Viscoelastic Membrane Deformation. Comput. Meth. Appl. Mech. Eng., 68 (1988) pp. 33. [15] Nied, H.F., Taylor C.A. & deLorenzi H.G., Three-Dimensional Finite Element Simulation of Thermoforming. Polym. Eng. Sci., 30 (1990) pp. 1314. [16] deLorenzi, H.G. & Nied H.F., Finite Element Simulation of Thermoforming and Blowmolding. Modeling of Polymer Processing: Recent Developments, Isayer, A.I. (ed.) Munich, 1991, pp. 117. [17] Taylor, C.A., deLorenzi H.G. & Kazomer D.O., Experimental and Numerical Investigations of Vacuum-Forming Processes. Polym. Eng. Sci., 32 (1992) pp. 1163. [18] Kouba, K., Bartos O. & Vlachopoulos J., Computer Simulation of Thermoforming in Complex Shapes. Polym. Eng. Sci., 32 (1992) pp. 699. [19] Kouba, K., Ghafur M.O., Vlachopoulos J. & Haessly W. P., Some New Results in Modelling of Thermoforming. SPE Tech. Papers, 52 (1994) pp. 850. [20] Mooney, M., A Theory of Large Elastic Deformation. J. Appl. Phys., 11 (1940), pp. 582. [21] Ogden, R.W., Large Deformation Isotropic Elasticity of Theory and Experiments for Incompressible Rubberlike Solids. Proc. R. Soc. Lond., A326 (1972) pp. 565. [22] Song, W.N., Mirza F.A. & Vlachopoulos J., Finite Element Analysis of Inflation of VacuumForming Processes. J. Rheol., 35 (1991) pp. 93. [23] Song, W.N., Mirza F.A. & Vlachopoulos J., Finite Element Simulation of Plug-Assist Forming. Int. Polym. Proc., 3 (1992) pp. 248. [24] Igl, S.A. & Osswald T.A., A Study of the Thermoformability of Wood Fibre Filled Polyolefin Composites. SPE Tech. Papers, 50 (1992) pp. 122. [25] Treloar, L.R.G., The Mechanics of Rubber Elasticity. Proc. R. Soc. Lond., A351 (1976) pp. 301. [26] Schmidt, L.R. & Carley J.F., Biaxial Stretching of Heat-Softened Plastic Sheets Using an Inflation Technique. Int. J. Eng. Sci., 13 (1975a) pp. 563. [27] Schmidt, L.R. & Carley J.F., Biaxial Stretching of Heat-Softened Plastics Sheets: Experiments and Results. Polym. Eng. Sci., 15 (1975b) pp. 51. [28] Ward, I.M., Mechanical Properties of Solid Polymers, John Wiley & Sons, New York, 1983. [29] Green, A.D. & Zema W., Theoretical Elasticity, Oxford University Press, New York, 1954. [30] Green, A.D. & Adkins J.E., Large Elastic Deformations, Oxford University Press, New York, 1960. [31] Eringen, A.C., Nonlinear Theory of Continuous Media, McGraw-Hill, New York, 1962. [32] Hylton, D., Laboratory Techniques for Predicting Material Thermoformability: A Review. SPE Tech. Papers, 49 (1991) pp. 580. [33] Tanner, R.I., Engineering Rheology, Oxford University Press, New York, 1985. [34] Wagner, M.H. & Demarmels A., A Constitutive Analysis of Extensional Flows of Polyisobutylene. J. Rheol., 34 (1990) pp. 943. [35] Vantal, M.H., Bellet M., Monasse B., Jammet J.C. & Andro R., PPS Tech. Papers, l0 (1994) pp. 317. [36] Kouba, K., Ghafur M.O. & Vlachopoulos J., Computer Modelling of Blow Molding. SPE Tech. Papers, 51 (1993)pp. 1861. [37] Kouba, K. & Vlachopoulos J., The Role of Rheology in Thermoforming Process: Experimental and Theoretical Consideration. SPE Tech. Papers, 50 (1992) pp. 114. [38] Vlachopoulos, J., Kouba K. & Ghafur M.O., The Role of Rheology in Thermoforming and Blow Molding, Fourth European Rheology Conference, Seville, Spain, 1994.
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Composite Sheet Forming edited by D. Bhattacharyya 9 Elsevier Science B.V. All rights reserved.
Chapter 4
Thermoforming of Continuous Fibre~Thermoplastic Composite Sheets K. F R I E D R I C H 1, M. H O U 2 and J. K R E B S 1 1Institute for Composite Materials Ltd. (IVW), University of Kaiserslautern, 67663 Kaiserslautern, Germany," 2Centrefor Advanced Materials Technology, Department of Mechanical Engineering, University of Sydney, Sydney, NSW 2006, Australia
Contents Abstract 92 4.1. Introduction 92 4.2. Experimental details and procedures 96 4.2.1. Materials employed for stamp and diaphragm forming 96 4.2.2. Preparation of pre-consolidated laminates 97 4.3. 2-D stamp forming 100 4.3.1. General remarks 100 4.3.2. Experimental details 102 4.3.3. Description of stamping process 104 4.3.4. Characterisation methods for thermoformed components 105 4.3.4.1. Microscopy 105 4.3.4.2. Characterisation of in-plane fibre movement 106 4.3.4.3. Thermal analysis 106 4.3.5. Results and discussion 107 4.3.5.1. Determination of preheating time 107 4.3.5.2. The forming temperature 110 4.3.5.3. Correlation between stamping velocity and stamping pressure 4.3.5.4. Effect of forming condition on part geometry 119 4.3.5.5. Thermoanalysis 130 4.3.5.6. Fibre movement studies 133 4.3.6. Optimised processing window for 2-D stamp forming 135 4.4. 3-D stamp forming 137 4.4.1. General remarks 137 4.4.2. Set-up of 3-D stamp forming device 138 4.4.3. Experimental procedure 139 4.4.4. Results and discussion 140 4.4.4.1. Stamp forming mechanisms 140 4.4.4.2. Stamp forming of GF fabric/PEI laminates 141 4.4.4.3. Stamp forming of UD GF/PP laminates 143 4.4.5. Recommendations for 3D stamp forming 145 4.5. 3-D diaphragm forming of GF/PP laminates 146 91
115
92
K. Friedrich et al.
4.5.1. General remarks 146 4.5.2. Experimental procedure 147 4.5.3. Assessmentand characterisation of thermoformed parts 148 4.5.3.1. Large strain analysis technique 148 4.5.3.2. Occurrence of defects and thickness variations in diaphragm formed parts 150 4.5.4. Variation of forming parameters and rating of part quality 152 4.5.4.1. The forming temperature 153 4.5.4.2. The forming ratio 154 4.5.4.3. The forming pressure 156 4.5.4.4. The forming velocity 156 4.5.4.5. The laminate thickness and lay-up 157 4.5.5. Conclusions and recommendations 157 4.6. Summary 159 Acknowledgements 160 References 160 Abstract Continuous fibre-reinforced thermoplastic polymers are a relatively young group of engineering materials compared with their thermosetting counterparts. However, due to their outstanding mechanical and thermal properties these materials are becoming increasingly attractive not only for aerospace and automotive applications. In order to produce defect-free components it is essential to gain a fundamental understanding of the effects of processing conditions on the structure and morphology of the resultant composite component. Only if the material's behaviour under the particular processing conditions is fully understood can the processing parameters be set in such a manner, that the desired microstructure and mechanical performance of the resulting component can be achieved. When forming continuous fibre-reinforced materials instabilities, such as wrinkles and buckles, may occur. In the example of the 2-D, 3-D and the diaphragm thermoforming technique it is demonstrated how the processing conditions can be optimised by individually investigating the parameters mainly governing the properties of the finished component ultimately enabling the production of defect-free parts. 4.1. Introduction Continuous fibre-reinforced thermoplastic polymers are still a relatively young group among the engineering materials compared with their thermosetting counterparts. However, due to their advantageous properties such as good thermal stability, high toughness/damage tolerance, infinite shelf-life and ease of processing, these materials are becoming increasingly attractive to various industrial sectors. In particular applications not only in the aerospace and automotive field but also in branches such as the medical, environmental and recreational industry represent potential markets. In order to exploit the advantageous features of continuous fibre-reinforced thermoplastic composites for a large array of industrial applications and products,
Thermoforming of cont&uousfibre~thermoplastic composite sheets
93
several manufacturing methods have been developed over the last decade [1]. Utilising related processes known from thermosetting composites and adapting sheet forming techniques typical of those used with metallic materials, fabrication techniques, such as compression moulding, tape winding, thermoforming, joining and pultrusion have become available (fig. 4.1). However, the introduction of advanced thermoplastic composites has also led to novel requirements being imposed on processing techniques for manufacture of high-quality intermediates. Unlike thermosets, which are usually compounded at the moulding site, fibre-reinforced thermoplastics have to be supplied in a variety of different ready-to-use intermediates in order to match the requirements of the various production techniques [2]. Embedding the reinforcements into high-viscosity thermoplastic matrices is still a demanding task. Many approaches were necessary in order to overcome the difficulties of impregnation enabling the production of intermediate material forms such as film stacked prepregs (pre-impregnated materials), commingled fibres,
Fig. 4.1. Processingwith thermoplasticcomposites.
94
K. Friedrich et al.
pultruded bands and powder impregnated bundles which are commonly used in contemporary thermoplastic manufacturing technology (fig. 4.2) [3]. In order to produce good quality parts it is necessary to initially gain a fundamental understanding of the effects of processing conditions on the structure and morphology of the resultant composite component [4--6] (fig. 4.3). Only if the material's behaviour under the particular processing conditions is fully understood, can the processing parameter be set in such a manner that the desired microstructure and mechanical performance of the laminate can be achieved. In this chapter two thermoforming techniques, stamp forming and diaphragm forming, are investigated with regard to their potential for producing defect-free
Commingled Fibers
Film Stacking
I lll
~-~ Reinforcing Fibers
Fiber Woven Fabric
Polymer Fibers ~ ' - u I u i u i u~ Polymer Films
Powder Impregnated Bundles
Pultruded Band
Fiber Bundle
~
Fiber
Polymer Powder
Polymer
olymer Sheath
Fig. 4.2. Intermediate material forms for thermoplastic composites.
Intermediate Material
Form
H
Processing Method
a.) Commingled Yarn
a.) Compression Molding
b.) Powder
b.) Tape
Impregnated
Fibers
c.) Melt Impr, Tapes
Winding
c,) Stamp Forming
H
Processing Parameters
m,.,cro,u.,r,uo,H
Mechanical
of Bulk Composite
Performance of Laminate
a.) Interlaminar a.) Temperature a.) Morphology Fracture of Crystalline b.) Pressure Polymer Matrix Toughness b.) Materials c,) Time b.) Alignment Stiffness of Fibers and Strength d.) Cooling c.) Formation of Voids [
,
.....
I
No
I
I
Fig. 4.3. Methodology for fundamental studies on processing properties of thermoplastic composites.
Thermoforming of conth~uousfibre/thermoplastic composite sheets
95
components from continuous fibre-reinforced thermoplastic prepreg materials. The particular interest in stamp forming stems from its obvious suitability for mass producing single and double curvatured parts at considerable low cycle times. Diaphragm forming on the other hand is very unlikely to match the productivity of stamp forming; however, when it comes to producing very complex-shaped structures, this technique appears to be more suitable due to its capability to accommodate inter-ply shear induced laminate thickening. Here, superplastic aluminium diaphragms like Supral TM* were originally employed for vacuum-bagging hightemperature fibre/matrix systems such as CF/PEEK (APC-2 TMt, AS4) in autoclaves. Due to the pronounced strain-rate sensitivity, consequently resulting in high cycle times, the fairly restricted stretchability and the high temperatures required for achieving maximum elongation, lately most emphasis has been placed on the use of polymeric diaphragms such as Upilex TM* [1,7-9]. Since polyimid diaphragms are not strain-rate sensitive, forming can be completed in a matter of minutes in contrast to approximately 20 minutes that are required for superplastic aluminium diaphragms. It has to be noted, however, that due to the thermal properties of polyimid diaphragms they are only suited for processing composites containing high melting temperature matrices such as PEEK (poly ether ether ketone) and PEKK (poly ether ketone ketone). As a consequence, when looking into industry branches where attributes such as high volume production, cycle times in the order of seconds, low material costs and automation are of decisive significance, these materials, despite their superior properties, will have difficulty in asserting themselves against the traditional engineering materials which are still dominating the market. According to recent trends in the composites industry, structural materials such as glass-fibre-reinforced polypropylene and polyamide do have the best future prospects for substituting metallic materials in markets such as the environmental sector where there are tremendous opportunities for glass-fibre-reinforced thermoplastics sewage pipes, scrubbers and tanks [10]. Also in the automotive industry where polypropylene is already well established as non-reinforced material the number of glass-fibre-reinforced components is continuously growing. Today, parts such as noise shields, bumper carriers, compartment capsulations and seat shells are made from glass-mat-reinforced thermoplastics (GMT) in order to name only a few [11]. An expansion to continuous glass fibre-reinforced polypropylene structures, therefore, appears to be a more than sensible venture. One of the most recent additions in the group of structural composite materials are Plytron TM~ and Tepex TMw Fundamental studies on the formability and the resulting part properties of this material, as well as mechanical properties and environmental resistance have already been conducted by a number of research groups. However, there is still far more work necessary in order to make thermoforming of such *SupralT M is an Alcan Registered Trademark. tAPC_2TMis a ICI Ltd. Registered Trademark. SUpilexT M is a UBE Ind. Registered Trademark. ~PlytronT M is an ICI Ltd. Registered Trademark. w T M is a DuPont Registered Trademark.
96
K. Friedrich et al.
materials become a repeatable and cost-effective fabrication technique capable of producing accurately predictable finished part properties at acceptable cycle times. Employing a number of differently reinforced matrices, unidirectional and fabric, and different types of reinforcing fibres, carbon and glass, the intention of this chapter is to outline the durability and limitations of both stamp and diaphragm forming. It addresses aspects such as flow processes occurring during thermoforming, consolidation quality and achievable structural properties of the resulting components. Moreover, the occurrence of instabilities in the finished components such as wrinkling and buckling in both processes is investigated and illustrated with the aim of creating a complete picture of their potentials and limitations.
4.2. Experimental details and procedures 4.2.1. Materials employed for stamp and diaphragm forming This section intends to give an introduction to the structural and physical properties of the materials employed for the investigations into stamp and diaphragm forming. Furthermore, the procedures followed for pre-consolidating laminates for later usage in the forming experiments is described and illustrated since it is felt that this step has a significant influence on the material's behaviour under thermoforming conditions and the resulting component properties. The following section also provides an introduction to all prepreg materials employed for the subsequent series of experiments, outlining their physical and thermal properties. The most important properties are summarised in table 4.1. (a) Continuous glass fibre (GF) reinforced thermoplastic composite called Plytron | with a polypropylene (PP) matrix, by ICI Ltd., UK. The nominal glass fibre content of this material is 60 wt. % and 35 vol. % glass/polypropylene. The material was supplied as a tape with a width of 240 mm and a nominal thickness of 0.47 mm. TABLE 4.1 List of materials employed for thermoforming experiments Advanced thermoplastic composites (abbreviations)
Melt temp. (~ Glass trans, temp. (~ Fibre volume fraction (%) Material form Manufacturer 2-D forming 3-D froming
GF/PP
CF/PP
GF/PP
CF/PA12 CF/PEEK GF/PEI
145-170 -20 35 Prepreg ICI Ltd. UK
163
163
176
334
20 Prepreg ICI Ltd. UK
33 Prepreg BASF Germany
40 60 Plate Enichem Italy
60 Plate Enichem Italy
9
210 50 Prepreg Ten Cate Holland
Thermoforming of continuousfibre~thermoplastic composite sheets
97
(b) Pultruded continuous carbon fibre (CF) polypropylene (PP)-tape, with a nominal thickness of 0.50 mm and a nominal fibre content of 33 vol. % carbon/ polypropylene, manufactured by ICI Ltd., UK. (c) Pultruded continuous glass fibre (GF) polypropylene (PP)-tape, with a nominal thickness of 0.50 mm and a nominal fibre content of 20 vol. % glass/polypropylene, manufactured by BASF AG, Germany. (d) Continuous carbon-fibre-reinforced pre-consolidated laminates with a polyamide (PA-12) and a polyetheretherketone (PEEK) matrix manufactured by Enichem, Italy. The nominal fibre content of these materials is 40 vol. % and 60 vol. % carbon/polyamide 12 and carbon/polyetheretherketone respectively. The uni-directional (UD) laminates with a nominal thickness of 3.2 mm were made from powder-impregnated fibre bundles. (e) Glass fibre fabric (GF) reinforced amorphous polyetherimide (PEI) manufactured by Ten Cate, The Netherlands. Here, the fibre content is 50 vol. % glass/ polyetherimide with a nominal tape thickness of 0.80 mm. The weave style of the glass fabric was 8 H satin and is illustrated in fig. 4.4. 4.2.2. Preparation of pre-consolidated laminates
An integral part of all forming processes is the consolidation of flat laminates which is either accomplished during forming (in-situ processing) or carried out prior to the actual thermoforming process. Since the properties of the moulding are essentially dependent upon the quality of the lamination, it is important to thoroughly define and control the conditions under which well consolidated material may be obtained. In principle, lamination involves two processes: (a) obtaining autoadhesion between the plies and (b) removing that which lies between them (air or water vapour) [12]. This can be achieved by using either pressure in order to dissolve the air into the melt or vacuum to extract it. For the 2-D and 3-D stamp forming experiments pre-consolidated sheets (100 x 160 mm and 128 x 128 mm) were prepared by pressurising a loose stack of plies with a predetermined stacking sequence in between two heated platens. The unidirectional and quasi-isotropic sheets with a [0~ [0~176176 and [0~176 s [13] stacking sequence were made from CF/PP and GF/PP (BASF) prepregs according to the
Fig. 4.4. Woven structure of 8-H satin glass fabric in GF/PEI composite.
98
K. Friedrich et al.
following procedure. First, a steel mould filled with several layers of prepreg material is placed in a heatable press. The stacked pile of plies is then preheated in the absence of pressure until the matrix has reached its molten state. Consolidation is achieved by applying external pressure for about 5-10 min. Finally, the mould is cooled below the melt temperature of the matrix whilst maintaining the applied pressure in order to prevent the occurrence of voids within the laminate. The consolidation temperature and pressure have to be carefully monitored throughout this process since too high a temperature and/or pressure can lead to undesired fibre misalignment caused by excessive matrix flow and fibre migration. However, on the other hand, if the temperature and pressure are set too low the resulting laminates are likely to contain high void content and poor inter-ply bonding. The resulting sheets employed within this study exhibited a thickness of 2.8-3.0 mm. A simple but effective method for characterising the movement of the reinforcements during forming is to embed tracer wires made from material that is not transparent to X-rays into the surfaces layer of a pre-consolidated laminate (e.g. copper wires). The motion of the wires during stamp forming can then be detected by X-ray analysis techniques or usage of powerful light sources for composites with transparent or opaque matrices. The technique utilised for fabricating pre-consolidated laminates with embedded copper tracer wires is illustrated in fig. 4.5. First, the copper wires are mounted onto a metal frame with a uniform lateral distance between them. Then the copper wires are pressed onto the surface of a prepreg ply using a hot press set to the melting temperature of the individual matrix in the ply. The plies with integrated copper wires thus obtained are then used for preparing sheets for subsequent fibre motion measurements. For the stamp forming experiments described here, blanks were cut to size, as shown in fig. 4.6, where L denotes the length, B the width and do the original thickness of the laminate. In order to measure the temperature profile of the heated pre-consolidated sheets during stamp forming, NiCr-Ni thermocouples were embedded into the centre layer of several laminates. An alternative method that may be employed for the production of pre-consolidated laminates is depicted in fig. 4.7. For implementing the diaphragm forming
Fig. 4.5. Schematic diagram of manufacturing pre-consolidated laminate with embraced Cu trace wires.
Thermoforming of continuousfibre/thermoplastic composite sheets
99
Fig. 4.6. Definition of sample cut from GF fabric/PEI laminate.
Metal Frame with Gasket u,apnra~m Top Plate
~(Alumini
;
j
/
~g~'~
m)
, ~ ~ O v e n Bag #2 , Stacked ~Prepreg Plies ~ M e t a l Strip ~ O v e n Bag #1 Vacuum Table (Aluminium) Outlet to Vacuum Pump
Fig. 4.7. Preconsolidation device for diaphragm forming experiments.
experiments it was decided to use vacuum rather than pressure for pre-consolidating rectangular flat sheets with different lay-ups and thicknesses. In order to consolidate a multi-ply sheet a completely uncompacted stack of plies is placed on a vacuum table. Oven bags (Melinex Type S [14]) covering the plies prevent the material from sticking to the aluminium platens and metal strips surrounding the plies during consolidation. Applying a vacuum pressure of about - 2 0 kPa the stack of plies is heated up in an oven to the matrix's melt temperature where it dwells for about 15 minutes. In the case of Plytron, the cooling rate which basically determines the
1O0
K. Friedrich et al.
extend of crystal growth in the semi-crystalline polypropylene matrix should be greater than 1~ since otherwise too high a level of crystallinity is achieved, resulting in a low fracture toughness [12].
4.3. 2-D stamp forming 4.3.1. General remarks
Thermoplastic polymers can undergo a reversible phase change from solid to liquid, thereby enabling the development of shaping and joining methods analogous to those for conventional metallic materials. When transferring a flat, continuousfibre-reinforced laminate into a 3-D shaped component a number of different deformation mechanism governing the forming behaviour of this material have to be taken into account. Unlike the monolithic metallic sheet, continuous fibre-reinforced thermoplastic (CFRT) materials are virtually inextensible in the direction of the reinforcement. For these material systems, the dominant mode of deformation, parallel to the fibres, during sheet forming is therefore shearing within the individual plies and between them, namely intra- and inter-ply shear. Additional mechanisms that occur when transferring a flat multi-ply sheet into a double curvature formed component are resin percolation through and along the layers of the reinforcing fibres, resin flow transverse to the fibres and rotation of adjacent plies relative to one another [15-19]. A schematic diagram of the deformation modes in the increasing order of shape complexity and correlated flow mechanisms shown in fig. 4.8 can be described as follows. (a) Resin percolation is fundamental to all flow processes and occurs to some extent in all types of fabrications. It is the flow of polymer matrix through and along the layers of the reinforcing fibres. This process allows layers of prepreg to be bonded together to form one sheet. Fritzer and J~iger studied the flow behaviour of matrix materials in order to optimise the processing parameters such as pressure and temperature leading to a homogeneous distribution of the matrix in the composite [20-22]. (b) Transverse flow is the process by which a prepreg spreads out to accommodate local pressure variations during forming processes. It is responsible for apparent stretching that can occur in a unidirectional laminate in perpendicular direction of the reinforcements [23]. A small amount of transverse flow is observed during any forming process at elevated forming temperatures caused by local pressure gradients that arise from small variations in the laminate thickness and mould clearances [13,24]. Transverse flow can also result from shear stresses developing between the thermoplastic material and the forming tool surface. (c) Inter-ply slip is a relative shear movement of two adjacent laminate layers. In this case the polymer matrix acts as a lubricant between neighbouring plies. When a laminate is formed onto a curved surface it has to deform in such a manner that inter-ply slip can account for the gradual change in shape. If this process is inhibited fibre wrinkling and buckling are normally the result [25-27].
Thermoforming of continuous fibre/thermoplastic composite sheets
101
Fig. 4.8. Deformation modes and flow mechanisms of continuous fibre-reinforced thermoplastic materials. [1].
(d) Intra-ply shear is needed when a shearing strain occurs in the plane of the laminate, thus allowing part conformity to complex curvature geometry. Theoretically, there is no limit to the amount of shear deformation in unidirectional laminates. Fabrics, by contrast, are usually interlocked at fibre crossover points, limiting the shear strain to the locking angle of the unit cell (trellis angle) [281. (e) Inter-laminar rotation is required for forming multi-ply laminates into double curvatured shapes. This shear action occurs in a thin "resin-rich layer" that forms at each laminate surface during consolidation due to resin migration. Most complex curvature parts require a change of initial fibre orientation between adjacent plies. For the production of 2-D and 3-D components the occurrence of inter- and intraply shearing processes is of particular interest. Only if the reinforcements can alter their orientation by shearing and rotating relative to one another during forming instabilities such as wrinkles and buckles can be prevented from occurring. Therefore, particular attention has to be devoted to these flow mechanisms in any thermoforming procedure for continuous fibre-reinforced thermoplastic composites.
102
K. Friedrich e t al.
4.3.2. Experimental details
In the series of 2-D stamp forming experiments described in the following subsections, the quality of the finished components made from fiat pre-consolidated laminates is evaluated by investigating the effect of the processing parameters such as pressure, temperature, cycle time, velocity, mould geometry and laminate architecture on the microscopic and macroscopic properties of the formed parts. For this series of experiments blanks made from CF/PP, GF/PP (BASF), CF/PA12, CF/PEEK and GF fabric/PEI were employed. The 2-D stamp forming experiments were conducted employing the matched metal die forming process as it is commonly known from sheet metal forming. For simplicity, the forming set-up was mounted into an universal testing machine that allowed closing velocities ranging from 1 to 1,000 mm/min. Forming was carried out under non-isothermal conditions, which means that the pre-consolidated laminates had to be pre-heated to processing temperature in a laboratory hand press prior to forming. Once the desired temperature was reached the laminates were then transferred into the press and formed immediately. In order to record the load conditions occurring during forming, an x - y recorder was connected to the press's pressure transducer. The alignment of the male and female moulds was realised by two pillars ensuring that both mould parts were kept aligned (fig. 4.9). The male mould was supported by two coil springs surrounding the pillars. The springs not only prevented possible damage to the pressure transducer that can occur at high
Fig. 4.9. Schematic diagram of angle mould for 2-D stamp forming.
Thermoforming of continuousfibre/thermoplastic composite sheets
103
loads but also allowed the moulds to open up automatically after each processing cycle, easing de-moulding and handling of the formed samples. In the forming series discussed in this section three pairs of right-angle vee-shaped moulds with three different bend-radii (2.5, 5.0 and 10.0 mm) were employed with a uniform clearance between male and female mould of 2.5 mm (fig. 4.10). The universal testing machine was operated as a load-controlled mechanical press under compression mode. The maximum applicable stamping pressure was controlled by pre-setting switch-off pressures. Figure 4.11 depicts the correlation of stamping pressure and stamping stroke versus time. Here, the solid line represents the load and the tinted line the stroke of the stamper. In order to form a part, the stamper was driven at a constant speed. Once it physically touches the pre-heated laminate forcing it into the cavity of the female mould, the load increased until the pre-set switchoff load was reached which immediately terminated the forming stroke. The forming
Fig. 4.11. Definition of processing parameters.
104
K. Friedrich et al.
durations thus obtained ranged from 4 to 20 s dependent on the stamping velocity (fig. 4.12). Due to inaccuracies in the press's control unit, reaching the switch-off load did not immediately lead to a halt of the stamper, but resulted in a swing-over effect which increased with increasing stamping speed. After the peak value was reached the compression pressure began to drop to an equilibrium pressure level which can be related to processes such as matrix flow, fibre migration and shrinkage of the laminate that occur upon cooling. An additional finding of this series of forming experiments was that the stacking sequence of the laminate also significantly affects the final stamping pressure. The effects of this phenomenon will be discussed in more detail in section 4.3.5.3.1. 4.3.3. Description o f stamping process
Generally, the deformation behaviour of thermoplastic composites are directly linked to their processing temperature. Amorphous polymers can be deformed when in their rubber-plastic state (above glass transition temperature Tg), whereas semi-crystalline polymers can only be deformed near or above their melting temperature. Therefore, pre-consolidated laminates produced as described in section 4.2.2 generally have to be subjected to some pre-heating procedure before deformation can take place. There are three different principles that may be utilised for preheating thermoplastic laminates: (i) conduction heating between two heated platens, (ii) infrared heating and (iii) forced convection in air or inert gas circulation ovens. In this part of the study contact heating plate were employed for heating pre-consolidated sheets because this method was found to relatively efficient on a laboratory scale [29]. The major drawback of this method, however, was found to be that the laminate may de-consolidate during heat-up, and the tendency of the molten matrix to stick to the contact surface of the heated platens. Therefore, in order to ease the 25 o 20 2: o,-~ ~9 ,__.~ 15
H
= O
E I-
\
g
10
.,-.,
5 0
200
400
600
Stamping velocity v [ mm/min ]
/
H ts~
Fig. 4.12. Closing time in relationship to stamping velocity.
h
v
800
Thermoforming of cont&uousfibre~thermoplastic compositesheets
105
handling of the heated sheets it is advisable to sandwich them between two polymeric diaphragms, i.e. polyimide films such as Kapton 200 HN. An advantageous side effect of this measure is that it normally enhances the quality of the surface finish of the finished components and it also reduces the cooling rate of the heated sheets during forming. This way the total duration at which the laminate can be deformed is significantly extended. In order to form a part, the sheets were pre-heated to processing temperature, then transferred into the stamping device and formed. Transferring should not exceed a few seconds so that the laminate retains its formability. During stamping the laminate is cooled below its glass transition temperature or melting temperature through the direct contact with the cold metal moulds. This means that de-moulding can normally take place directly after forming is accomplished. The geometry of a fiat laminate and a stamped bend is depicted in fig. 4.13, where L denotes the length, B the width, 0 the final angle of the stamped bend, d the final laminate thickness, and Ad the reduction in thickness relative to the original thickness do. Two different types of blanks (a) B = 15, L = 80 and (b) B = 60, L = 80 were used for these experiments, where the latter had copper wires embedded for subsequent X-ray analysis procedures.
4.3.4. Character&athgn methods for thermoformed components 4.3.4.1. Microscopy Optical microscopy is a suitable and inexpensive method for examining the fibre arrangement and impregnation quality of polished cross-sections of stamped samples. For this purpose, specimens cut from stamped vee-bend components L o
_-o
l
"
!
(a)
Length Direction
(b) ~
90
~ -
Fig. 4.13. Sample geometry and its definition.
/
106
K. Friedrich et al.
were embedded in epoxy resin and polished with 180, 500, 800 and 1200 sand paper, followed by polishing with 1 ~tm AL203 paste. Photographs were taken from three sections in one specimen, i.e. at both ends and the bend region. 4.3.4.2. Characterisation o f in-plane fibre movement
A qualitative study investigating the in-plane fibre movement in stamp-formed components was implemented by using X-ray analysis techniques on specimens with embedded copper tracer wires. X-rays (produced by RBVII/TYK9B, Rich, Seiteut & Co.) travel through the formed sample sensitising a film (AGFAGEVAERT D4) placed behind the sample (fig. 4.14). Due to the higher absorption of X-rays of the copper wires, these wires appear as dark lines on the film. In order to evaluate the distortion of copper wires, the lateral distance between adjacent wires has to be measured. The histogram of copper wires at different distances compared with the corresponding original alignment (standardised distance between adjacent copper wires in this experiment: 2.5 mm) indirectly indicates the magnitude of fibre migration. 4.3.4.3. Thermal analysis
The crystallinity of thermoplastic polymers mainly depends on their thermal history experienced during forming. In order to study the effect of processing conditions on the morphological properties of the matrix materials, DSC (Differential Scanning Calorimetry) is a suitable method for determining the degree of crystallinity of polymeric matrices before and after forming. For this reason, test samples (weight ~ 10 g) were cut from the stamp formed parts and heated at a rate of 10~ Principally, the crystallinity of a polymer is determined by integration of its DSC curve. The measured enthalpy AH is directly proportional to the degree of crystallinity of the matrix but also proportional to the matrix fraction in the sample. Therefore, after each DSC measurement the tested sample needs to be subjected to a TG-analysis (Thermal Gravimetry) in order to determine the pure matrix weight (WT~). With AHc as the degree of enthalpy for 100% crystallinity, the
X-ray
Film
Outer Surface
Inner Surface
Fig. 4.14. Schematicdiagram of X-ray equipment.
Thermoforming of continuousfibre~thermoplastic composite sheets
107
following equation can be applied for the calculation of the crystallinity of a polymer: C=
AH WDS C
Anc WTo
• 100 [%]
(4.3.1)
where C = crystallinity of a sample [%], W D S C - - sample weight for DSC-analysis [g], AH = enthalpy measured by DSC-analysis [J/g], WXG = matrix weight of the same measured by TG-analysis [g], and AHc =theoretical enthalpy for 100% crystallinity [J/g]. In order to acquire data for a theoretical calculation of the final angle reduction of vee-bent parts (section 4.3.5.4.3), the thermal expansion coefficients of pre-consolidated laminates along and perpendicular to the fibre direction have to be determined by TMA (Thermal Mechanical Analysis). The samples (5 • 6 • 6 mm 3) were cut from pre-consolidated plates with a diamond saw. It has to be noted that during the preparation of TMA samples the surfaces between detector and sample table has to be kept parallel. The heating rate in this study was set to 2~ with a load of 0.1 N applied to the detector which was equivalent to a pressure of 5.6 kPa. The measuring temperature range for the individual fibre/matrix system was determined by the melting temperature of the corresponding polymeric matrix.
4.3.5. Results and discussion 4.3.5.1. Determination of preheating time The contact heating method employed is a one-dimensional unsteady-state conduction procedure. Thermal conduction is an energy transportation through atomic and molecular interactions resulting from unequal temperature distributions. A simple heat transfer analysis can be used to obtain an estimate of the pre-heating time. Equation (4.3.2) describes the general three-dimensional thermal conduction problem, regarding the temperature as a function of position in space and time [30]: 8t = P-~e \~2x + ~
+ ~2z }
(4.3.2)
where T = plate temperature [~ t =time [s], X = thermal conductivity [Cal/ cm s ~ Cp = specific heat [Cal/g ~ and p = density [g/cm3]. Since the length and width of the pre-consolidated plates were much bigger than the thickness, it was assumed that the heat can only be conducted in the laminate's thickness direction. The heating process can therefore be reduced to an one-dimensional heating problem. The one-dimensional unsteady-state heat conduction model is illustrated schematically in fig. 4.15. Assuming that (a) the thickness of the two polymeric films is almost zero and therefore negligible and (b) the surfaces of the plates reached the pre-set heating temperature (Zheating) as soon as heating is initiated and retain this temperature throughout the process. The plate which had to be heated was of a thickness of 2x0 and an initial temperature of To. Then the differential equation for unsteady-state heat conduction in one direction is:
108
K. Friedrich et al.
Fig. 4.15. One-dimensional unsteady-state heat conduction model. ST_ 6t
k~ ( 8 2 T ~
-pc.
\-r
(4.3.3)
t > o, o < ~ < ~o
with the following initial and boundary conditions: dT ~=0, dt T -- Theating
x-0,
t>0
(4.3.4)
X - X0,
t> 0
T -- T o
O<.x<.xo
t -- 0
(4.3.5) (4.3.6)
Boundary condition (4.3.4) is symmetric to the middle plane of the plate (x = 0). Therefore, the temperature distribution is also symmetric to this plane and there is no energy flow through this plane. The analytical solution of eq. (4.3.3) can be written as: i-~176
Y= Z
r
[2i -- i)-~rr
cos
( 2 i - 1) 2 " x r x n
(4.3.7)
i=1 where
9
Y _~
rHeiz- T T H e i z - T O'
z
'
)~t ~ Cpp X 2'
~
n
~
x X0
In order to estimate the heating time two physical properties of the composite are needed, i.e. thermal conductivity ~. and specific heat Cp. The following equations are usually used for calculating these properties [31]: Cpc -- L ( ~ . f p f C f + ).mPmCm) Pc ~c-
Xm4U
[1 -- ~/~]/~'m "+-
1
(4.3.8)
(4.3.9)
Thermoforming of continuous fibre/thermoplastic composite sheets
109
Here, the indices m, f and r denote the matrix, fibre and composite respectively. The thermal properties of the materials used for the evaluation of the heating time are listed in table 4.2 [32-37]. Using eqs. (4.3.7), (4.3.8), (4.3.9) together with the data given in table 4.2, the temperature profile of the middle plane (x = 0) in relation to the heating time can be calculated. The time, at which the temperature of the middle plane reaches Theating is the estimated preheating time. Figure 4.16 illustrates the temperature profile of the middle plane of a CF/PP laminate as a function of heating time. For GF/PP material the heating plate was heated up to 180~ and the laminate had a thickness of 3.0 mm. In order for the middle plane to reach this temperature approximately 34 s were needed. The difference between the estimated and actually measured value was about 10 s. The reason for this discrepancy can be explained with the two polymeric Kapton films which were not accounted for in the estimation. Naturally, with increasing laminate thickness, extended heating times are needed in order to heating the middle ply of the laminate to the desired temperature (fig. 4.17). Preheating times estimated for different laminates are summarised in table 4.3. TABLE 4.2 Physical property of materials used Physical property Abbreviation
Thermal conductivity (cal/cm s ~
Heating capacity (cal/g ~
Density (g/cm 3)
CF GF PP PAl2 PEEK PEI
2.03 2.08 5.26 5.49 5.97 5.25
0.22 0.20 0.41 0.28 0.32 0.37
1.80 2.55 0.90 1.02 1.32 1.27
:3 "~
9
x x x x x x
250
L
~ ag
10- 2 10-3 10-4 10-4 10-4 10-4
T 'Heating
~J
...............
i
,
plate
-,,
q
|
i
i
{ ..............
i:i: ii:i:iiiiii.......... ;i loo i
O
I--
50
-
~ 4 -
~ .
[
~
0
10
.
.
Estimated data . data
20 Heating time [ s ]
Fig. 4.16. Heating temperature profile of CF/PP laminate.
30
j 40
110
K. Friedrich et al.
i /mHeating plate
350 --~ O ~
280 210
.~ "--" 140 E
Ill ..................i]............................................ i....................
70
~"
~ f .............................
.............................................
i
0 0
20
i......................
o o
,~
Zx
40
60
'mm' 00 "
a . a s
5.00
80
...........
1O0
Heating time [ s ] Fig. 4.17. Estimated heating time in relationship to laminate thickness.
TABLE 4.3 Estimated heating time Abbreviation
Theating
tcentral layer
(~
(s)
Fibre volume fraction (%)
Laminate thickness (mm)
CF/PP GF/PP
20 33
3.0 2.9
180
34 36
CF/PA12
40 60
3.2
220
24 16
CF/PEEK
60
3.2
380
18
GF fabric/PEI
50
2.6 3.9 5.0
280
28 62 103
4.3.5.2. The forming temperature One of the major problems in thermoforming processes is the determination of the temperature range within which a laminate can be successfully formed. A fairly simple but effective way of monitoring the temperature history in a laminate during forming under non-isothermal conditions is to embed a NiCr-Ni thermocouple into the laminate prior to forming (fig. 4.18). Naturally, this method is only feasible on a laboratory scale or in the preparation and adjusting phase for manufacturing a large batch of parts. In the following investigation a series of experiments are described in which the stamping velocity was varied from 500 mm/min down to a value at which the laminate could no longer be deformed. The temperature profile of CF/PP laminates as a function of the processing time was monitored and is illustrated in fig. 4.19. The
Thermoforming of continuous fibre/thermoplastic composite sheets
111
Hot laminate
/
----
NiCr-Ni
Thermocouple
Fig. 4.18. Temperature measurement during stamp forming.
200
i J
175 -,9
9
I ~
i .:
................................. ]- ........................................ ! ....................................................................................
'
I
!
i ......................................................................................
j
150
t
~
125 o
~.
75 5o
25
'
0
"J 0
' 10
, 20 Stamping
. . . . . 30
40
i 50
, 60
time [ s ]
Fig. 4.19. Temperature profile of CF/PP laminate during stamp forming.
upper linear bound along the black dots indicates the cooling rate of a laminate in an ambient environment which basically represents the condition where the heated laminate is not yet in contact with the cold surface of the tool. The temperature at which the cooling of the laminate starts dropping more rapidly (due to the tool/ laminate contact during closing) is defined as the stamping temperature. Its value raises with increasing stamping velocity. In two-dimensional forming such as vee-bending, inter-ply shear is one of the most important deformation mechanisms which allows the individual plies to slip relative to one another and thus prevents instabilities from occurring in the laminate [38]. In principle, inter-ply shear can be regarded as the result of a pressure/velocity gradient
112
K. Friedrich et al.
between adjacent plies of the laminate. However, if the stamping velocity is too low, the actual temperature of the laminate drops below a temperature level where interply slip can no longer occur (fig. 4.20). In this case, the shear stress acting on the plies does not exceed the shear yield stress of the matrix material, which ultimately results in fibre buckling at the inner face of the bend. In addition, fibre breakage can occur in the outer face. An example of the buckling phenomenon is represented in fig. 4.21, for which the forming temperature of the laminate was as low as 156~ When stamping [0~ and [0~176176 laminates fibre buckling can be observed at the inner face of the bend area. Stamping [0~176 laminates the velocity is still high enough to produce good quality samples. This result is related to the significant transverse intra-ply shear in the 90 ~ layers, for which a much lower yield shear stress 180
=
170
r Q.
E "" ~
~
=
160
0
150
E co
140
0
100
200
300
400
500
600
700
800
Stamping velocity [ mm/min ] Fig. 4.20. Effect of stamping velocity on stamping temperature of CF/PP laminate.
Fig. 4.21. Fibre buckling at inner bend section (CF/PP) laminate.
Thermoforming of continuousfibre~thermoplastic composite sheets
113
has to be exceeded. At stamping velocities in excess of 200 mm/min the stamping temperature of the laminates stays above 160~ This is sufficient for ensuring defectfree bend samples independent of the laminate's stacking sequences. Micrographs of polished cross-sections of the bend regions give clear evidence for the good quality of the stamp formed parts under these conditions (fig. 4.22). In this situation inter-ply slip between the individual plies of the laminate are clearly visible even at the far end of the bend samples (fig. 4.23). The surface finish of these parts can also be rated as
Fig. 4.22. Micrograph of the curved area of [0]6 CF/PP laminate.
Fig. 4.23. Inter-ply slip at the bend end of CF/PP laminate.
114
K. Friedrich et al.
satisfactory since no fibres have migrated to the surface during the forming operation. Increasing the stamping velocity, on the other hand, decreases the mould closing time, thus keeping the stamping temperature relatively high. When, in addition, the stamp pressure is set too high, large portions of matrix are squeezed into the outer range of the bend area (matrix migration) whereas the fibres accumulate in the inner bend region (fig. 4.24). The reason for this is that although the critical shear stress for inter-ply slip is exceeded locally, i.e. in the curved area, there is not enough time for this flow mechanism to take place at the two fiat ends of the vee-bend. Figure 4.25 schematically shows the possible fibre distributions after stamp forming that may be obtained with different forming temperatures. Another example of the effect of stamping temperature is given for the amorphous PEI matrix, reinforced with glass-fibre fabric. The glass transition temperature of PEI is 210~ If the temperature exceeds that level the GF fabric/PEI laminate
Fig. 4.24. Matrix migration at bend angle section of CF/PP laminate.
Fig. 4.25. Possible fibre distribution after stamp forming.
Thermoforming of continuous fibre/thermoplastic composite sheets
115
should be formable in the presence of an external load. A series of experiments shows, however, that the temperature for GF fabric/PEI laminates has to be clearly higher. The laminates had to be preheated to 280~ in order to achieve defect-free components. Figure 4.26 illustrates the temperature drop-off during stamp forming. The preheating temperature of 280~ results in a stamping temperature of the laminate of only 260~ when the stamping velocity amounts to 200 mm/min. However, this temperature is not high enough to enable inter-ply slip. As a consequence, fibrebuckling occurs at the inner bend section of the GF fabric/PEI laminates (fig. 4.27). Only at stamping temperatures of 265~ and above, which corresponds with a minimum stamping velocity of 300 mm/min, buckle-free vee-bend samples were produced.
4.3.5.3. Correlation between stamping velocity and stamping pressure 4.3.5.3.1. Influence of laminate stacking sequence Figure 4.28 shows the correlation between stamping velocity and stamping pressure in relation to the stacking sequence of CF/PP laminates. In this example, the switch-off pressure was set to 0.14 MPa. It is obvious that the resulting pressure increases with the stamping velocity. In addition, laminates with a higher number of 90 ~ layers lead to a reduced swing-over of the stamping pressure. This is caused by transverse flow in the 90 ~ layers in sample length direction [39]. The result is a remarkable reduction in the bend thickness (Ad/do) (fig. 4.29). For example, the thickness reduction at a stamping pressure of 2.5 MPa is 35% for [0~176 18% for [0~176176 but only 5% for a [0~ laminate. On the contrary, fibre transverse flow along the fibre direction does not occur in unidirectional laminates. Only a small amount of molten matrix is squeezed out in fibre direction. !
300 ~ I
~
~
t
s
i i
~
I /
I I
$ l:::
,--.,
0~
Stamping
velocity [ mm/min ]
A
206
0
600
.........
150
. . . . . . . . . . . . . . . . .
(D
"~ ~
100 50
i
~,
~--k'Ri
~
~:i
.... I
.................................................................................................................
0 0
20
40
60
80
1O0
Stamping time [ s ]
Fig. 4.26. Temperature profile of GF fabric/PEI laminate during stamp forming.
116
K. Friedrich et al.
Fig. 4.27. Fibre buckling at inner bend section of GF fabric/PEI laminate.
Fig. 4.28. Correlation between stamping velocity and stamping pressure in relation to laminate structure.
4.3.5.3.2. Influence o f fibre volume fraction The correlation of stamping velocity and pressure in unidirectional laminates as a function of fibre content and matrix material is shown in fig. 4.30. At the same switch-off pressure of 2.21MPa and stamping velocity of 300 mm/min, the resulting pressures are 4.5 MPa for CF/PP, 5.5 MPa for CF/PA12 and 6.5m MPa for C F / P E K laminates respectively. Due to the high fibre content in CF/PA12 and C F / P E E K laminates (60 vol. %), the rigidity of these two laminates is higher than that of the CF/PP laminates (only 20 vol. % CF) leading to higher pressure required for stamp forming process. The difference in
o f continuous f i b r e ~ t h e r m o p l a s t i c
Thermoforming
50
'
"
40 ._o r
30
-o r
I Z
1
aoo
|
_.,zl.....................................I
20
--
[]
10
t
~0f -
I '11
5o0
................................................. ~ ' ZX
[]
I*
400
..................................... 71 []
"0 oO
117
sheets
! Stamping velocity[ Laminate structure [ [mm/min] I 0]6 IEo,go,ol, lto,~O=ls " 100 .... E3
7-
-
t-
composite
I 01
o
I
zx
I
[]
9 I I
~
I
""
I
I
' .........
..Q c"
0 ore, 0
.................................................
i--,
............
"
-
"ID
rr
0
I
0
,
I
2
.
,
,
4
Stamping pressure [ MPa ]
i
6
,
8
Fig. 4.29. Thickness reduction of stamped bend side (stamping time: 10-15 s).
Fig. 4.30. Correlation between stamping velocity and stamping pressure in relation to matrix and fibre content.
stamping pressure between the CF/PA12 and CF/PEEK laminates can be explained by the different melting viscosities of the polymer matrix [34]. As already mentioned, the molten matrix can only migrate along the fibre direction during the forming of unidirectional laminates. This can lead to minor thickness reductions in the flat
l 18
K. Friedrich et al.
regions (fig. 4.31). At the same stamping pressure of 5 MPa, there is a reduction in leg thickness of 9% for CF/PP, 3% for CF/PA12 and 2% for CF/PEEK laminates. The sum of these results allows the conclusion that a greater reduction in thickness is linked to a higher matrix content and a lower viscosity of the molten polymer matrix. 4.3.5.3.3. Influence of laminate thickness Figure 4.32 shows the resulting stamping pressure as a function of GF fabric/PEI laminate thickness. The resulting
t Stamping velocity mm/min 200 PP CF2OO/o [] PAl2 .......
""
PAi2 PEEK
15
or) Or) t"
._o t,-
10
CF 40% CF60%
OF60% l ' J
[] []
f
9
300 6
400 9
500 I 9 I
~ *:
-X[]
~ I <> /
o
a
9
......
~ I" i & ! .Y'o ~ i i PA ~21'
,
f I
PA 12
". .O .. CO "rO
5
.,4
e.~
~-""el
_
~
K~..g~-t3"~l
to
,
o
-o
0
0
I 2
,
,
........
J
,
4 6 Stamping pressure [ MPa ]
i 8
Fig. 4.31. Influence of stamping pressure and stamping velocity on bend side thickness.
Fig. 4.32. Stamping pressure of GF/PEI laminate in relation to laminate thickness.
10
Thermoform&g of continuousfibre~thermoplastic composite sheets
119
stamping pressure decreases with increasing laminate thickness which can be explained as follows: (a) As the laminate thickness increases, the total number of plies that can slip over one another increases, too; however, increasing inter-ply slip reduces the stiffness of a laminate. (b) Thicker laminates have a greater thermal capacity than thinner ones. For this reason the stamping temperature of thicker laminates is higher than for thinner ones due to the slower cooling behaviour. Higher stamping temperatures also result in reduced stiffness of the laminate. Both factors can, in addition, reduce "swing-over" effects.
4.3.5.4. Effect of forming conditions on part geometry 4.3.5.4.1. Influence of stamping time on bend angle Additional to inter-ply slip between adjacent plies the fibres also have to bent in order to conform with the veeshape of the mould during forming. When re-heating a vee-bent part it tends to return to its original fiat state. This phenomenon is called memory or spring-back effect [29,40]. The magnitude of spring-back is mainly dependent on the stamping time (fig. 4.33). With a stamper radius of 2.5 mm and a stamping velocity of 400 mm/min, the following observations can be made for CF/PP laminates: 1. With increasing stamping time the discrepancy between mould angle (90 ~ and formed angle decreases. Here, a constant angle is achieved when the forming time exceeds 15 s, independent of the laminate's stacking sequence. 2. In a unidirectional laminate the spring-forward effect is not as strong as in laminates with 90 ~ layers. 3. In all parts, the formed angle is always smaller than the angle of the mould. 90
o
89
tl:l
o.
I
Laminate
I
[ 0 ] 6
structure '~
[0,90,0] s o I [ 0,902] s [] i ..............................................................................z~.................................................. ~ .....~..................................-.~--~................................... 2.5 mm
0 n CD t'-
Stamper radius
88
87
tH. ~
86
F0
,
0
, 10
i 20
, 30
stamping time [ s ] Fig. 4.33. Effect of stamping time on final angle of CF/PP laminate (stamping velocity: 400 mm/min;
stamping pressure: 3.26 MPa).
120
K. Friedrich et al.
In order to achieve a good conformity of the laminate's angle with the mould angle the fibres have to be arrested firmly in the matrix upon cooling. At short stamping times, with the matrix still being in its viscous state, elastic deformation of fibres cannot be prevented effectively resulting in a distorted cross-section of the vee-bent strips (fig. 4.34). This effect increased with decreasing forming times resulting in smaller effective bending angles of the parts. On the other hand, with increasing stamping time the cooling time of the stamped laminate is also extended, allowing the matrix to arrest the bent fibres resulting in a uniform angle. As mentioned earlier, 90 ~ layers in a laminate can lead to transverse flow of matrix and fibres. This effect can lead to an inhomogeneous distribution of 90 ~ layers in the fiat ends of the vee-bent parts, which is possibly the reason for not achieving a 90 ~ bend angle with these stacking sequences. Figure 4.35 illustrates the effect of stamping time and pressure on the final part angle of a [0~ laminate. For the various stamping pressures both bigger and more uniform angles were achieved with increasing stamping time. Uniform angles achieved with lower pressures are larger than that of higher stamping pressures. When stamping with a pressure of 3.26 MPa it takes about 15 s to form a uniform angle, whereas at a pressure of 5.53 MPa only 5 seconds are needed. Once a uniform angle is obtained the cross-sections of the strips do not exhibit any visible distortion as observed before (fig. 4.36). As mentioned in section 4.3.5.2, high stamping pressures lead to severe matrix flow between the fibre-rich layers in the bend resulting in excessive movement of the fibres near the bend region and consequently smaller part angles are obtained. The effect of the stamping time and mould geometry on the part angle formed at a pressure of 3.26 MPa and a velocity of 400 mm/min is illustrated in fig. 4.37. Again, the formed part angle increases with increasing stamping time reaching a constant value at stamping times in excess of 15 s. This result clearly depicts that the mould geometry, as used in this experiment, has no influence on the final angle of the veebent part. Similar dependencies of formed part angle on the stamping time were observed when forming GF/PP, CF/PA12 and C F / P E E K laminates.
.
.
.
.
d
Fig. 4.34. Schematic deformation of cross-section of bend side under short stamping time.
Thermoforming of continuous fibre/thermoplastic composite sheets
I
90
i
stamping velocity
9
El ll) o'J
89
stamping
1._.~
t,-
~
pressure
i
88
E2.
i
I
,!
i
[0]6
Laminate structure
121
400 mm/min 3.26 MPa
9
5.53 MPa
O
i t
i
"V"
t--
LL 87 t
I
86
....
0
I
10
Fig. 4.35. Final part angle of
[0]6
20
30
Stamping time [ s ]
CF/PP laminate in relation to stamping pressure and stamping time.
Pressure [Ml'a]
Press time
[s]
I
3.26
5.53
0.4
20.0
Fig. 4.36. Effect of stamping time on cross-section of bend side.
In contrast to the results obtained with unidirectional fibre reinforcements, the final part angle of GF fabric/PEI laminates as a function of the stamping time, with a stamping velocity of 700 mm/min, a pressure of 8.6 MPa and a stamper radius of 2.5 mm, is shown in fig. 4.38. Here, all angles are greater than the actual angle of the mould. Therefore, the following conclusions can be drawn: 1. The final angle obtained with fabric reinforcements is always greater than the actual angle of the mould. With increasing stamping times (more than 30 s) the resulting angles settle on a value close to 90 ~
122
K. Friedrich et al.
90
,
J
.,
Stamper radius [ mm]
9
2.5 5.0 10.0
89
Q
a
,..
88
0 r-! /k
! ~
. . . . . . . . . . . . . . . . . . . . . . . . .
T
~9
87
oEm
! 86
|
....
0
|
i
10
,,,
i
20
30
Stamping time [ s ] Fig. 4.37. Final part angle of
,~
98
[0]6
CF/PP laminate in relation to stamping time and stamper radius.
i
i
i
!
i
~TJ
a
96
i..~=
o) t-
! i i
....
i
!
I
94
i [.
i
[Warp,~ and [Weft]s
92 e.m
LL
9O 88
!..,. l i
10
"
.... - ! ,
,
i
.
2'-
[45~ ]s
20 Stamping time [ s ]
30
i
40
Fig. 4.38. Final part angle of GF fabric/PEI laminate in relation to stamping time (stamping velocity: 700 mm/min; stamping pressure: 8.6 MPa; stamper radius: 5.0 mm.
2. There is no differences observed between "weft" and "warp" stacked laminates under similar processing conditions. 3. The effect of stamping time on the formed angle decreases with increasing stamping radius. Looking at the structure of a woven fabric, the warp and weft yarns are interlaced tightly at their cross-over points; however, there is no physical bond between them. This interlocked structure does not allow transverse flow of warp or weft fibre
Thermoforming of continuousfibre~thermoplasticcompositesheets
123
bundles making it behave more like a sheet metal during forming. At very short stamping times the fabric reinforced laminates form greater angles but do not show a distortion of the cross-section of the bend legs. Due to the interlaced structure of a fabric the shear strength in warp and weft direction is low compared with the tensile and flexural strength in fibre direction. The shear strength is mainly resulting from the resistance to changes in the interlacing angle caused by friction and elastic restriction to the rotation of the interlacing angle between the warp and weft fibres [41]. In +45~ the fibres (warp and weft yarns) are orientated at an angle of 45 ~ to the stamper (fig. 4.39). The applied bending force not only leads to inter-ply shear between adjacent layers but also shear deformation between warp and weft and rotation of warp and weft in each individual layer. This way the spring-back of 45~ is hampered, making them less sensitive to stamping time and stamper radius than in laminates with "warp" and "weft" stacking sequence. In the stamp forming process described here, the deformed glass weave was kept in place through the change of phase of PEI matrix from molten and/or rubbery to glassy state. However, at very short stamping times (1-2 s), the temperature of the formed parts was still above the glass transition temperature of the PEI matrix during demoulding. Therefore, the matrix could not effectively prevent the tendency of the glass weave to spring back to its original fiat shape, leading to severe springback. On the contrary, at long stamping times (in excess of 30 s) the deformed glass weave was maintained by solidification of the polymer matrix under pressure. 4.3.5.4.2. Dependency of bending angle on other parameters Apart from the preheating temperature, the relationship between stamping velocity and pressure, the stamping time is another important forming parameter to be determined. Therefore, a series of stamp forming experiments employing different laminates' stacking sequences is described in the following paragraph. The angle measurement and optical examination of polished sections of the formed parts are utilised in order to assess the quality of the forming results.
Fig. 4.39. Deformationmechanismsof fabric-reinforcedlaminate.
124
K. Friedrich et al.
Figures 4.40 and 4.41 illustrate the effect of stamping velocity and pressure on the final part angle of stamp-formed CF/PP and G F / P P laminates at the example of three different laminate stacking sequences formed with a stamping time of 15-20 s. The optical inspection of a selection of parts shows that there are no voids to be observed in the bend region. However, none of the vee-bent parts measures an angle of exactly 90. On average, the reduction in angle is 2.0-4.0 ~ for CF/PP and 1.3-4.0 ~ for G F / P P laminates. In addition, for each laminate stacking sequence the angle reduction is also a function of stamping velocity and pressure, e.g. higher stamping velocities and pressures result in greater angle reductions. It has to be noted, that laminates with a higher number of 90 ~ plies exhibit more angle reduction and a bigger thickness reduction than unidirectional laminates. A microscopic inspection 90
89 a
9
o'J t--
Q.
stamping velocity Laminate str~cture [mm/min ] [ 0 ]61[ 0,90,0]s~[ 0.90: ]s 10o i / m 200 o 9 ................... .. 300 4, o [] 400 [] & " 500 o z~ ' ,, ~--..-~ -. ~-...........I ................
_ _ _
.....
_ _
88
87
t--
. ~
u_
86 0
2
4
6
8
Stamping pressure [ MPa ]
Fig. 4.40. Final part angle of CF/PP laminate in relation to stamping pressure and velocity. 90 a
L----.
' I I
I
89
. . . . . . . . . . . . . . . . . . . .
I i .......
l
0"} t.-
[0 6
. . . . . . . . . . . . . . . . .
[0,90,0] s
,--.=
Stamper radius[mm] h
I !
5-~
~
"
"
-i ,o0 j o . .
....
~
.... = l_1
-
88
[ 0,902]s ~ ~
c
"
[]
~
eA ~ ~ " " " ~ - . . ~ . L -
i
87
............................
U.
t .............................................
i
t
i
I
! 86
0
2
4
6
8
10
Stamping pressure [ MPa ]
Fig. 4.41. Final part angle of GF/PP laminate in relation to stamping pressure and velocity.
Thermoforming of continuousfibre~thermoplastic composite sheets
125
of some samples shows that, due to the transverse flow of fibre and matrix, the 90 ~ layers are not homogeneously distributed in the sample as in the flat laminate. The greater angle reduction in laminates with a high number of 90 ~ layers is believed to be associated with excessive transverse flow in these layers. The transverse flow mechanism in continuous-fibre-reinforced thermoplastics has been thoroughly studied by Barnes and Cogswell [39]. Additionally, it has been observed that the stamper radius has only little or negligible effect on the final part angle (fig. 4.41). The difference in angle between CF/PP and GF/PP laminate is thought to be associated with the properties of the different reinforcing fibres rather than the stamp radii. The effect of stamping velocity and pressure on the part angle of unidirectional laminates is summarised in fig. 4.42. Again, all angles are smaller than the actual angle of the mould (90~ The reduction of part angle increases with increasing stamping velocity and pressure for all materials. The angle reduction also increases with increasing matrix volume fraction. Applying the same stamping pressure, the matrix squeeze flow in CF/PP laminates (20 vol. %) is more extensive than in CF/ PAl2 (60 vol. %) and CF/PEEK (69 vol. %) laminates. Another reason for the differences in angles between CF/PA12 and CF/PEEK is thought to be associated with the different melting viscosities of the two polymers. Figure 4.43 illustrates the final part angle of GF fabric/PEI laminates as a function of stamper radius and laminate thickness. Here, the stamping pressure was set to 8.6 MPa with a velocity of 700 mm/min and a stamping time of 30--40 s. The laminates employed were either [warp]s or [weft]s. It is obvious that the formed part angle is not only dependent on the laminate thickness but also on the stamper radius; i.e. the spring-back effect decreases with increasing laminate thickness and stamper radius. The direct tool contact with the cold surface of the mould leads to a temperature gradient in the laminates consequently resulting in higher stamping temperatures in
9
e
stamping velocity PP PAl2 PAl2 PEEK mm/min CF 20% CF 40% CF 60% CF 60% 200 [] [] [] 9 300 o e 4, []
90 Dl::l:Zlg .....
a
....P A l 2 (
89
I
I
400
m
500
~
6
0
)
~
9
~
~-
isl
[]
<>
.........1.....................................................
" a A
9
I............................................. ,
r
"~ t~ 88 r-.
u_ 87
86 0
i
2
4 6 Stamping pressure [ MPa ]
I
8
]
10
Fig. 4.42. Final part angle of [0]s laminate in relation to stamping pressure and velocity.
126
K. Friedrich et al.
Fig. 4.43. Final part angle of GF fabric/PEI laminate in relation to stamper radius and laminate thickness (switch-off pressure: 2.08 MPa; stamping velocity: 700 mm/min). thicker laminates than in thinner ones. Keeping all stamping parameters constant, thicker laminates exhibit only little spring-back which explains the minor difference in mould to part angle. Another important factor influencing the spring-back is the local degree of deformation or the bend radius. When forming a large radius only little local deformation is needed to force the laminate into the cavity of the female mould meaning that only minor spring-back is observed. Figure 4.44 shows the polished cross-sections of G F fabric/PEI stamp formed samples with different stamper radii and laminate thicknesses.
Fig. 4.44. Section photomicrographs of bend samples made with different stamper radius (a) and laminate thickness (b).
Thermoforming of continuousfibre/thermoplastic composite sheets
127
4.3.5.4.3. Calculation of bending angle The results in section 4.3.5.4.2 showed that all the final part angles made from unidirectional laminates are smaller than the 90 ~ angle of the mould. This phenomenon is commonly called spring-forward effect [40] and is mainly the result of the anisotropy of thermal properties of composite materials. In particular, the difference between in-plane and out-of-plane thermal expansion coefficients is responsible for the discrepancy between the desired and actual shape of the stamped parts. Figure 4.45 shows two arc sections made from isotropic and anisotropic materials. Since the out-of-plane thermal expansion coefficient for an isotropic material is identical with the in-plane thermal expansion coefficient (Otr = d0), the thermal contraction of the arc section only results in minor
(~r
/
\\
/
// \\
//
// \ \ section / / \ \ \ \ angle / / / / \\ //
\ \
\ \
Xt
enclosed angle
(a)
a r = ~9
01 = 0 2
~r
/
/ /// / / / / / /
,
02 O1
(b)
a r > a 0 ,O1 >O2
Fig. 4.45. Thermally induced distortion in an arc section.
\
K. Friedrich et al.
128
changes of the arc's size. As a result, the section angle before and after thermal contraction remains constant (~1 = ~2) (fig. 4.45 (a)). In contrast, for anisotropic materials (in this case unidirectional fibre-reinforced composites) the out-of-plane thermal expansion coefficient is much higher than in in-plane direction (Otr>Oto). Therefore, during contraction of an anisotropic material the dimensions of the arc almost remain constant in one direction, whereas the other changes significantly. This results in a different arc size and hence a different configuration (fig. 4.45 b) which ultimately leads to the observed angle reduction. The thermal expansion coefficients of the unidirectional laminates employed for the experiments described here were determined by TMA. The results show that there is a remarkable difference between dr and c~0 (table 4.4). According to the mathematical studies by ONeill, the reduction in formed angle based on the anisotropy of the thermal properties can be calculated [42]. For simplicity, the angle section can be treated as being a sectional slice of a circular cylindrical tube. In this case it is then convenient to introduce polar co-ordinates (r, 0, z) as depicted in fig. 4.46. The following solutions can be then formulated:
(4.3.10)
u = u(r, 0), v = v(r, 0), w = w(r, O)
where u, v and w represent the r, 0 and z components of displacement in the circular section. The components of infinitesimal strain are then reduced to: Err
,u
~F ,
8zz = ~ Z '
,(,;)
EO0 - - - F
8r0---2
u-'l-
--~-~--~r--
(4.3.11)
F_.rz ~--- 80z = 0
TABLE 4.4 Thermal expansion coefficients of studied composite materials and estimated reduction in final part angle
Abbreviation
Vf (%)
CF/PP
GF/PP
Laminate structure
0/r (10-5K -1)
cto (10-6K -1)
AT (~
A ~ (degree)
20
[0]6 [0,90,0]s [0,902]s
17.9 20.0 20.6
2.6 4.8 7.3
135-140
2.1-2.2 2.3-2.5 2.4-2.6
33
[0]6 [0,90,0]s [0,902]s
14.2 14.0 14.5
12.5 7.2 9.8
135-140
1.5-1.7 1.6-1.7 1.6-1.7
14.0
-5.1
40 CF/PA12
[0]s 60
CF/PEEK
60
[01s
1.9-2.1 150-160
13.3
-5.6
2.66
-2.6
(< Tg) 7.8
(< Tg) -2.7
1.8-2.0
300-310
1 7-1.8
T h e r m o f o r m & g o f continuous f i b r e / t h e r m o p l a s t i c composite sheets
129
Y
tXr X
Fig. 4.46. Co-ordinate system for a circular part.
The stress-strain equations for an orthotropic, linear elastic solid can be reduced to" Err --1--Olr(Ts -- T ) = SlltTrr --~ s120"00 + s130"zz
Soo + Oto(T~ - T )
=
s 1 2 t T r r Jr- $220"00 -1- s 2 3 0 " z z
Szz + Otz( Ts - T ) = s13tYrr -Jr- $23t700 + s33O'zz
(4.3.12)
13rO ~ $660"r0, O'rz --" O'Oz ~ 0
where sij (i, j = 1, 2, 3 . . . . . 6) are stiffness coefficients of the material and T s - T denotes the change in temperature (T~ = stamping temperature and T = actual sample temperature when demoulding). The thermal expansion coefficients C~r,ct0 and t~z are assumed to be constant. Since the final configuration of the stamped part is subjected to zero boundary traction, only the deformation fields for nil stress are taken into account. This requires Err---Ol
r
(Ts
--
T),
Soo--Oto
Szz -- -Otz(Ts - T ) ,
F_,rO - -
(Ts-
T)
0
(4.3.13)
Equation (4.3.13) with (4.3.11) represents four equations for the three undetermined variables u, v and w. The system now appears to be over-determined, however, providing that Ts - T is uniform throughout the cross-section of the laminate, there is only one sensible solution: u-
- a r r ( T s - T)
V - - (Ol r -- Oto)rO ( T s -
w-
T)
(4.3.14)
-Ot zZ ( T s - T )
The angular displacement of the plane 0 = constant is therefore" AO -- v / r -- (Olr
- - OlO) 0
(Zs
--
T),
(4.3.15)
130
K. Friedrich et al.
From eq. (4.3.15) the angular displacement Aft caused by a temperature drop of Ts - T for the circular section (fig. 4.46) with the sectional angle can be calculated as follows: Aft = (Ctr - Oto) fl ( T~ - T)
(4.3.16)
Through simple geometrical correlation the reduction of final part angle during stamp forming is described as: A (I) = (0/r -- 12/0) L~A T
(4.3.17)
where: A 9 - part angle reduction, Otr = out-of-plane thermal expansion coefficient, or0- in-plane thermal expansion coefficient, 0 = mould angle, and A T = reduction in temperature during stamp forming. The calculated part angle derived by utilising eq. (4.3.17) and the thermal expansion coefficient data measured with TMA are listed in table 4.4. Due to the high value of the glass transition temperature (Tg = 143~ two values for the thermal expansion coefficient of CF/PEEK laminates have to be used, one below and one above Tg. Therefore eq. (4.3.17) has to be used twice for calculating A cI, for CF/ PEEK laminates. Experimental results in figs. 4.40, 4.41 and 4.42 show that the actual angle reduction corresponded well with the theoretical values calculated with eq. (4.3.17). However, an additional dependency of A~ on the stamping pressure and laminate structure has been determined. This means, additional to the differences in thermal expansion coefficients there are more factors influencing the final part angle, one of which appears to be the wrinkling of fibre bundles in the bend region (which will be further discussed in section 4.3.5.6). When forming vee-bends, the compressed fibres in the bend region can lead to wrinkling of fibre bundles, providing the compressive stresses exceed a certain limit. Other influencing factors are the non-symmetrical distribution of plies after forming, the thickness variations in the layers and the different temperature gradient in the samples in the cooling cycle. 4.3.5.5. Thermoanalysis
When cooling a molten thermoplastic matrix, crystallisation is initiated once the temperature has dropped below a certain level. However, even under ideal conditions it is impossible to achieve 100% crystallinity since some molecules are inevitable trapped in a random state when solidification occurs. The crystallisation is mainly a function of the cooling rate and the inherent structure of the polymer chains present. Therefore, determining the influence of the processing parameters on the degree of crystallinity is of great importance since they directly effected the mechanical and morphological properties of the stamp formed components. The characteristic DSC traces for CF/PP vee-bend parts (fig. 4.47) shows that the stamping velocity does not effect the morphological properties of PP matrix during stamping. Each curve shows a kink at 163~ which corresponds with the melt temperature of the PP matrix. Providing the heat of fusion, AH, of 100% crystalline PP is 190 J/g [43], a degree of crystallinity of 41.2% for the original pre-consolidated CF/PP laminate plate and 37.2-38.9% for the sample can be calculated. These values
Thermoforming of continuousfibre~thermoplastic composite sheets
131
Original material 200 mm/min 300 mm/min
r-
E"
400 mm/min
-e v
"1o
500 mm/min
Melting
0
"o ,r
,
50
,
i
,,
l,
,
1O0 150 Temperature [ ~ ]
,
j
200
Fig. 4.47. DSC traces of stamped CF/PP laminate in relation to stamping velocity.
are obtained by integrating the melting curve for the CF/PP vee-bend parts, formed with stamping velocities of 200-500 mm/min. Similar results can be achieved with GF/PP. It is therefore concluded that both stamping velocity and reinforced materials (CF and GF) do not effect the morphology of the PP matrix. The list of values obtained for the crystallinity of the PP matrix is listed in table 4.5. The effect of stamping velocity on the melting behaviour of CF/PA12 laminate is given in fig. 4.48. A comparison of materials with different fibre content (40 and 60 vol. %) shows that the volume fraction of the PAl2 matrix has no observable effect on the melting behaviour of the CF/PA12 laminates. Each curve exhibits a kink at 176~ which corresponds with the melt temperature of the PAl2 matrix. However, it is interesting to note that each melting curve of the stamped samples also shows a small peak at about 160~ (indicated by arrow) located just before the melting point. TABLE 4.5 Crystallinity of stamped composite in relation to stamping velocity Abbreviation
VU (%)
Matrix crystallinigy (%) Original material
Stamping veloity (mm/min) 200
300
400
500
CF/PP
20
41.2
38.9
38.3
37.5
37.2
GF/PP
33
42.1
40.7
42.4
42.3
41.2
CF/PA12
40 60
25.1 26.2
21.9 21.9
21.5 21.9
20.3 21.8
21.3 21.7
CF/PEEK
60
27.5
23.3
22.5
21.4
21.2
132
K. Friedrich et al.
~
#,-.--.--. Original [9 material
t~ o..,
t,., :3
200 mm/min ~~"
L_ v
300 mm/min
"0
033
~/ ~,
400 mm/min
i
i
I
i
i
50
1O0
150
200
250
Temperature [ ~ ] Fig. 4.48. DSC traces of stamped CF (Vf = 40%)/PA12 laminate in relation to stamping velocity.
This behaviour is amplified with increasing stamping velocity. The reason for this behaviour can be explained by the partial re-crystallisation of the PAl2. Before stamp forming, pre-consolidated CF/PA12 laminates are heated up to a temperature of 220~ at which all crystals are melting. Dependent on the cooling rate the molten PAl2 matrix builds new crystals during the subsequent forming process. If the cooling rate is set at too high a value, some of the molecules do not have enough time for crystallising but are solidified at random state. Only a small fraction of matrix crystallises (about 1.5-4.0%) in the DSC test when the temperature is raised from room temperature to approximately 160~ This process within the matrix leads to small peaks in the DSC trace curves. As the temperature continuously rises, the newly built crystals are getting melted together with other crystals formed during stamping. For this reason, in the calculation of the melt enthalpy (AH), i.e. by integrating the melting curve, the small amount of exothermic energy (depicted by the small peak) has to be subtracted from the integrated melt energy. The heat of fusion, AH, for a 100% crystallised PAl2 is 210 J/g [44]. The calculated crystallinity is shown in table 4.5. Compared with the pre-consolidated laminates the influence of stamping velocities, ranging from 200 to 500 mm/min, on the crystalline is about 3% to 4.5%. Figure 4.49 shows the melting curves of stamped CF/PEEK laminates at different stamping velocities. Each curve exhibits a kink at 345~ which corresponds with the melt temperature of PEEK. Setting the stamping velocity to 200 mm/min, the melting curve is similar to the one of the pre-consolidated laminate. With increasing stamping velocity a small exothermic peak appears at 165~ This implies that the amorphous portion of the PEEK in the stamped sample increases with increasing stamping velocity. For CF/PA12 the situation is similar. The difference between figs. 4.48 and 4.49 is that crystallisation and melting temperature of PEEK are farther apart. The heat of fusion, AH, of 100% crystalline PEEK is 130 J/g [36] and the
Thermoforming of continuous fibre/thermoplastic composite sheets
133
Original material 200 mm/min
t.-
-E
300 mm/min 400 mm/min
v
"0
0 "0
Melting ;"
loo
'
2oo .....
" ~ 3oo'
'
'4oo
Temperature [~ ] Fig. 4.49. DSC traces of stamped CF (Vf -- 60%)/PEEK laminate in relation to stamping velocity. calculated crystallinity of the stamped samples is 23.3% to 21.2% and 27.5% for the pre-consolidated laminates (table 4.5). A comparison of the different values is given in table 4.5. It becomes obvious that the stamping velocity affects the degree of cystallinity of advanced composites with semi-crystalline thermoplastic matrices. The crystallinity of the matrices decreases with increasing stamping velocities due to the higher cooling rates. This effect is amplified with increasing processing temperatures. However, by heating the mould the cooling rate and thus the resulting degree of crystallinity can be controlled during forming.
4.3.5.6. Fibre movement studies
The microscopic inspection of polished vee-bend sections shows that fibre movement in the bend area in form of fibre weave has taken place. The extent of fibre weave is more severe in the outer bend side than in the inner one. In order to study this phenomenon pre-consolidated CF/PP laminates with embedded copper tracer wires were stamp formed and then studied by X-ray analysis. Figure 4.50 gives an example of X-ray photographs taken from flat and stamped [0]6 CF/PP samples. The image on the left shows the original distribution of copper tracer wires in the pre-consolidated CF/PP laminate and the ones on the right the situation after stamp forming. The horizontal centre line in each of the four photographs represents the centre of the bend in the stamped part. It is obvious that on these horizontal lines the distance between the tracer wires has changed remarkably. This implies that the carbon fibre movement during forming can be qualitatively studied by simply looking at the movement of the tracer wires. Since the wires are thicker by one order of magnitude than the carbon fibres, this study can only provide a qualitative description of the behaviour of the carbon fibres during forming.
134
K. Friedrich et al.
Fig. 4.50. Movement of Cu tracer wires at the inner and outer surface of the bend section of a CF/PP laminate.
The movement of the tracer wires compared with their original situation is illustrated in fig. 4.51. The CF/PP laminate consists of six layers. The tracer wires were placed between the first and second, and fifth and sixth layer. The lower diagram represents the wire distribution in the flat laminate. Since the wires move to some extent during the preparation of pre-consolidated laminates only about 30% of them maintained their predetermined distance of 2.5 mm. The movement of other wires in this experiment varied within 2.9 and 3.1 mm. Comparing the three diagrams in fig. 4.51 it becomes obvious that stamp forming leads to further fibre movement, especially in the outer surface. When the hot laminate is pressed into the female mould,
Fig. 4.51. Histograms for distance distribution of Cu tracer wires in a
[0]6
CF/PP bend section.
Thermoforming of continuous fibre~thermoplastic composite sheets
135
Fig. 4.52. Histograms for distance distribution of Cu tracer wires at outer surface in relation to stamping pressure.
transverse flow of matrix and fibres in perpendicular direction to fibre orientation occurs and results in fibre weaving. In addition, some matrix material has migrated towards the outer bend radius, also leading to fibre weaving. Figure 4.52 illustrates the effect of stamping pressure on the weaving of tracer wires in the outer surface at different stamping pressures. As already outlined in section 4.3.5.2, high stamping pressures can result in extensive matrix migration in the outer bend region. As a result of matrix flow the movement of the Cu tracer wires increases with the stamping pressure. With a stamping pressure of 1.73 MPa the maximum and minimum distances between the tracer wires is 1.5-3.5 mm. In contrast, with a stamping pressure of 3.8 MPa the maximum and minimum distances are 1.1-3.6 mm with 50% of the them being less than 2 mm or more than 3 mm apart from one another. Based on these results it can be concluded that high stamping pressures result in an inhomogeneous distribution of the reinforcing fibres in the finished component. The influence of the stamper radius on the movement of the tracer wires in the outer surface is illustrated in fig. 4.53. Comparing the three diagrams it is fairly difficult to decide which stamper radius has more effect on the migration of the wires because the distances between them in the diagrams are all within the same range. As a consequence it can be assumed that different stamper radii have nil or negligible effect on the fibre movement.
4.3.6. Optimisedprocessing windowfor 2-D stamp form&g In fig. 4.54 the results of the stamp forming experiments are summarised. The middle rhombus represents the acceptable processing window for stamp forming
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Fig. 4.53. Histograms for distance distribution of Cu tracer wires at outer surface in relation to stamper radius.
",4,
x X//////////~,
ffl
," Fiber .~I Aoce,:,taUe =
" buckling ~ •rocessina i and/or ~ '-wi-nc/ow~ ', breakage_~
I~.;
~,~t~n~e
x/k""
I ~ thinning and //// ~ ~ - - J ~ / resin migration Y~. ~ Y~
Stamping velocity Fig. 4.54. Schematic diagram of processing window for 2-D stamp forming.
unidirectional laminates. Here fibre buckling and breakage as well as resin migration do not occur. A temperature range suitable for successful stamp forming was found to be within a few degrees above melting temperature of the individual matrix material. Stamping pressure and velocity, which are mainly dependent on the volume fraction of the matrix and its melt viscosity, should be in the range of 1.5-8 MPa for CF/PP, 2-9 MPa for GF/PP, 3-8.7 MPa for CF/PA12, 4-9.5 MPa for CF/PEEK,
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with v = 200-500 mm/min. At low stamping velocities (< 200 mm/min) and pressures the fibres tend to buckle or break in the bend region. In contrast, at too high a stamping velocity (> 500 mm/min resulting in a higher stamping pressure) or at too high a pressure, excessive thinning and matrix migration perpendicular to the fibre direction is likely to occur (e.g. the matrix was squeezed out to the outer radius region). The reduction of the final angle of the stamped vee-bend parts is mainly caused by the anisotropic thermal properties of continuous-fibre-reinforced thermoplastic composites, described by the equation Aep = ( d r - Oto) OAT. However, some other factors such as unsymmetrical distribution of fibres in the laminate after forming, thickness variations in the individual layer, wrinkling of fibres in the bend region and different cooling rates in the inner and outer sections of the bend sample can also lead to a reduction of the formed angle. Fibre movement during stamp forming can be qualitatively studied by X-ray, employing samples with embedded copper tracer wires. The effect of processing conditions on the morphological properties can be studied by DSC tests. For the GF fabric reinforced amorphous PEI laminate, the forming temperature should be at least 60~ above the glass transition temperature of the PEI matrix. The suitable stamping pressure and velocity were found to be 5-9 MPa and 300-700 mm/min. The optimised stamping parameters for the studied materials are listed in table 4.6. TABLE 4.6 Optimum processing conditions for 2-D stamp forming Composite Abbreviation
Processing conditions
Vf (%) Melting temperature Stamping (~
CF/PP
time (s)
Stamping velocity (mm/min)
20
Stamping pressure (MPa) 1.5-8
180 GF/PP
33
2-9
CF/PA 12
40 60
220
CF/PEEK
60
380
GF fabric/PEI
50
280
> 15
200-500
3-8.5 4-9.5
> 30
300-700
5-9
4.4. 3-D stamp forming 4.4.1. General remarks
In order to apply the experience acquired within the 2-D stamp forming experiments to more complex shapes, in this section 3-D forming experiments of hemispherical dome-shaped components is described and discussed. The materials employed were pre-consolidated GF/PP and GF fabric/PEI laminates. The forming
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set-up is similar to the one used for the 2-D stamp forming, except that here a hydraulic driven press was used instead of a mechanical one. 4.4.2. Set-up of 3-D stamp forming device
A photograph of the forming device is shown in fig. 4.55. Here, (IV) is the 3-D mould which was directly linked with the hydraulic press (I) via a pressure transducer (III). This press was capable of generating a maximum load of 80 kN in compression stroke. In addition, it allowed opening and closing velocities of 70-250 mm/s and a compression stroke rate of 2-15 mm/s. The maximum stroke was 115 mm including 30 mm compression stroke. The stamping pressure was controlled by adjusting the air pressure. The main advantage of the hydraulic press is that "swingover" effects cannot occur as were observed with the mechanical press. Figure 4.56 shows a sketch of the 3-D stamp forming device. The male half of the mould consists of a stamper capable of moving vertically through a mid-plate guide system. The female half is a hemispherical dome-shaped cavity. The mould is designed in a manner that upon closing, the fiat laminate is clamped before the stamper forces it downwards into the cavity of the female die. The clamping force of the clamping frame was adjustable. However, determining the actual clamping force during forming is fairly difficult since the clamped area of the laminate is continuously declining during stamping. The reason for that phenomenon is to be found in the inextensibility of the reinforcing fibres which do not allow the laminate to stretch, as is commonly observed in sheet metal forming. The clamping forces
Fig. 4.55. Experimental set-up for 3-D stamp forming.
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Fig. 4.56. Working principle of hemisphere mould. R 27 I
R 27
~,~[
I
,
J R :30
I
(A)
30
,
(B)
Fig. 4.57. Geometry of three-dimensional moulds.
referred to in the following section are therefore only describing the initial clamping force. The geometrical details of the two different pairs of matched dies are given in fig. 4.57.
4.4.3. Experimental procedure For this series of stamping experiments GF/PP laminates with two different stacking sequences ([0]6 and [02,902]s) and two different GF fabric/PEI laminates ([Warp]12 or [Weft]12) were employed. The pre-consolidated circular flat laminates were of a thicknesses of 3.1 to 3.2 mm. In the preheating process the laminates were placed between two films, similar to the 2-D stamp forming experiments for easier handling. It has to be noted that in 3-D stamp forming these films are not removed prior to forming, which means that they may have an adverse effect on the quality of the stamped part. Therefore films should be used with super-elastic behaviour at the individual processing temperature of PP and PEI. From literature references [45,46] and results of a series of experiments with different polymer films, PA-66 and UpilexR-25 polyimide films have been chosen for CF/PP and GF fabric/PEI laminates in this series of 3-D stamp forming experiments described in this chapter.
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The hydraulic press employed had two working stocks, one for closing allowing velocities of 70-250 mm/s and the other one for compression with velocities of 2-15 mm/s. Based on the experimental results of 2-D stamp forming, the following processing parameters were used for the 3-D stamp forming: Preheating temperature: Closing velocity: Compression velocity:
180~ for GF/PP and 280~ or GF fabric/PEI 10 m/min 500 mm/min
4.4.4. Results and discussion 4.4.4.1. Stamp forming mechanisms
In order to fully understand the forming mechanisms involved in the stamping process it is advisable to study the individual stages of a 3-D stamp forming operation first. Figure 4.58 illustrates schematically a step-by-step deformation of a flat circular laminate as it is drawn into the cavity of a hemispheric die. Segment A represents a slice of a circular laminate before forming with section 1, 2, 3, 4 and 5 located in a single plane. Segment B can be regarded as part of a hemisphere in which 1 and 2 are part of the spherical dome section whereas 3, 4 and 5 are still flat. Segments C and D show the progressive deformation processes as the flat laminate is continuously drawn into the hemispheric cavity. This figure clearly depicts the reduction in area of the individual sections when travelling into the cavity of the mould [47,48], inducing hoop forces which resolve into compressive stresses perpendicular to the radius of the circular laminate (fig. 4.59). In general, when the stamping forces generated during forming exceed the critical buckling stress, fibre wrinkling and distortion around the periphery of the cavity and in the flange area are observed. This phenomenon is likely to increase when no clamping forces are acting in the flange area. These forces create desirable frictional forces, Fr, acting in opposite direction to the material flow when being drawn into the cavity. An additional forming mechanism occurring during 3-D forming
Fig. 4.58. Step-by-stepdeformation of a flat laminate into a hemisphere part.
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F h Hold-down load F f Frictional force Fb
Bending
/ ~ (-/, ,-
force
Fc Compressive
force
t
Tensile force
Fig. 4.59. Force analysis in 3-D stamp forming. is rotational intra-ply shear, also known as the Trellis effect [49,50]. During forming, the frictional forces are generated between the reinforcing fibres leading to stretching and shearing within the plane of the laminate. This process is called intra-ply shear. This mechanism continues until either the fibre direction coincides with the load direction or the cross-over angle between the fibre bundles reaches its minimum value. Each section also undergoes a bending deformation, e.g. a bending force Fb, when travelling from the flat flange into the convex cavity of the mould. In order to successfully transfer a laminate into a 3-D component the stamping force has to exceed all forces opposing this action as shown schematically in fig. 4.59.
4.4.4.2. Stamp forming of GF fabric/PEI laminates 4.4.4.2.1. Influence of clamping force As mentioned previously, without applying a clamping pressure the stamping force can easily exceed the critical buckling stress of the laminate which may then result in severe buckling in the flange area. Figure 4.60 shows two G F fabric/PEI samples. Apart from the clamping pressure all forming parameters were kept constant. The sample on the left is formed with a clamping pressure of only 0.1 MPa. It is obvious that this sample exhibits more buckles than the one on right. This means that the frictional force created by the hold-down pressure is not sufficient to induce intra-ply shear throughout the laminate during forming. When increasing the clamping pressure to 0.2 MPa, however, there is only little buckling visible in the flange area and the rim. When setting the clamping pressure too high, the laminate may tear along the rim of the dome section. It can therefore be concluded that clamping force is a limiting factor to stamp forming processes that employ clamping frames. 4.4.4.2.2. Influence of laminate dimensions Pre-consolidated G F fabric/PEI laminates with different initial diameter were stamp formed in order to investigate the influence of laminate dimension on the shape of a formed part; see fig. 4.61. Here, the
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Fig. 4.60. GF fabric/PEI hemisphere parts stamped by different hold-down pressure of 0.1 MPa (left) and 0.2 MPa (right).
Fig. 4.61. Effect of flat laminate dimension on part shape. smaller sample had a blank diameter of 78 mm whereas the bigger one had a diameter of only 120 mm. Suppose the forming ratio was defined as blank area/ cavity area A 1 / A , the smaller sample had a forming ratio of 2.54 and larger part one of 6.0. It can be seen that the larger sample exhibits some buckling in the flange area. This implies that the excess material in the far side of the flange has a significant influence on the occurrence of buckles. Similar results are observed when forming unidirectional reinforced composites employing diaphragms [46]. Most buckles in the flange area occur in the bisecting direction to the fibre direction which is related to compressive forces generated in this particular direction during forming. 4.4.4.2.3. Wall thickness distribution In order to investigate the wall thickness distribution of the formed part the cross-section thickness of formed hemispherical components was measured in three directions, i.e. in warp, weft and +45 ~ to the fibre direction. The normalised thickness, defined as the ratio of sample thickness to the thickness of original flat laminate, in relation with the measuring angle 0, is illustrated in fig. 4.62. The most remarkable thinning was measured at the apex of the hemisphere. This can be explained by the locally higher pressure, since the male and female die contact first at this point. It was also observed that the molten matrix and
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Fig. 4.62. Thickness distribution of stamped spherical part (GF fabric/PEI)
even fibre bundles flew away from this area. This effect creates a pressure gradient along the spherical surface ranging from the apex to the edge of the hemisphere leading to transverse matrix and fibre flow towards the lower pressure regions. The changes in wall thickness at the apex was determined to be 2% and 12% applying a stamping pressure of 12 and 20 MPa.
4.4.4.3. Stamp forming of UD GF/PP laminates 4.4.4.3.1. Effect of laminate stacking sequence Based on the different fibre alignment, the unidirectional GF/PP laminate behaves in a different manner under 3-D stamp forming conditions. Figure 4.63 shows a stamp-formed hemispherical sample and the corresponding blank ([0]8-GF/PP laminate). The horizontal line represents the direction of the glass fibre in the sample. It is obvious that the diameter of the circular laminate in the fibre direction is reduced after forming, whereas the diameter vertical to the fibre direction remains constant. This is related to the inextensibility of the reinforcing fibres and, therefore, during forming the outer regions of laminate have to travel towards the centre in order to fit the cavity of the mould. On the other hand, the fibres are free to flow in a direction perpendicular to their orientation under pressure, i.e. the unidirectional laminate is stretched perpendicularly to fibre direction, as can be seen from the marked lines in fig. 4.63. As result, the diameter vertical to the fibre direction remains constant. Similar to forming fabric-reinforced composites, there are some buckles in the bisecting direction to the fibre direction in the flange area of the samples made from continuous GF/PP. It has to be noted, however, that there is always some buckling in the formed GF/PP samples, regardless of the stamping pressure and clamping pressure applied. The reason is thought to be related to the high deformation rate in stamp forming. The results of diaphragm forming with unidirectional
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Fig. 4.63. Deformation behaviour of [0]8GF/PP laminate. laminate showed that when the deformation rates exceed a value of 7 mm/min fibre buckling occurred [45,48]. Due to the semi-transparent nature of GF/PP laminates, the glass fibre or fibre bundles can be observed clearly by putting a strong light source behind the formed sample. Figure 4.64 shows a photograph in which fibre buckling is indicated by dark lines, see flange area ("A" indicates fibre buckling). When the stamping pressure is applied the central region of the flat laminate is forced downwards into the cavity. Since no material stretching can occur in the direction of the reinforcements, they are forced to move inwards from the outer edges of the laminate. This inward displacement of the fibres is primarily resisted by intra-ply shear forces, which can be resolved into symmetric forces at +45 ~ to the fibres, resulting in buckling phenomena in this direction. In the section near the area of reduced diameter fibre weaving can also be observed in a small ribbon area from 2 to 4 mm (fig. 4.64, B). The reason for this phenomenon is thought to be related to the inter-laminar rotation, caused by a moment exerted on the fibre and prepreg layers. In addition, inter-laminar slip between different layers can also be observed in this area (fig. 4.64, C). 4.4.4.3.2. T h i c k n e s s distribution The distribution of the wall thickness in GF/PP spherical dome-shaped parts is shown in fig. 4.65. It is independent on the measuring direction and symmetrical to the central line through the apex of the part. All samples tend to thin around the apex of the hemisphere, whereas the wall thickness in the flange area remains constant. With increasing stamping pressure the wall thickness decreases. For example, the thickness reduction around the apex of the dome section works out to be 25%, 45% and 55% with a stamping pressure of 8 MPa, 12 MPa and 16 MPa respectively. In comparison to GF fabric/PEI samples (fig. 4.62) the wall thickness reduction of GF/PP laminates is more distinctive. This is related to the different fibre arrangement in GF/PP laminate and the higher volume fraction of the matrix.
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Fig. 4.64. Fibre buckling and fibre wave in a hemispherical GF/PP part.
Fig. 4.65. Thickness distribution of stamped spherical part (GF/PP).
4.4.5. Recommendations for 3D stamp forming Based on the experimental results the following conclusions can be stated for 3-D stamp forming: 1. A suitable clamping pressure is essential in order to achieve satisfactory stamp forming results. The clamping pressure is dependent on both reinforcing material and forming ratio A1/A (blank diameter/cavity diameter).
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View
/
z
wave
~
Y
x (Fiber direction)
fiber buckling Fig. 4.66. Illustration of fibre wave and buckling in 3-D forming.
2. The thickening and thinning in unidirectional reinforced laminates is much higher than in fabric-reinforced laminates. This is related to the strong transverse flow of fibre and matrix in the unidirectional material. 3. Fibre buckling in the flange area always occurs in the bisecting direction of the reinforcements regardless what type of reinforcement is used. This phenomenon is related to the occurrence of compressive forces during forming. Another defect, which is only observed in hemispherical dome-shaped parts made from unidirectional GF/PP laminates, is fibre weaving. The reason is thought to be related to the inter-ply rotation between adjacent layers of the laminate. A sketch of these instabilities is given in fig. 4.66.
4.5. 3-D diaphragm forming of GF/PP laminates 4.5.1. G e n e r a l r e m a r k s
After highlighting the particularities of 2-D and 3-D stamp forming, this chapter will focus on the advantages and drawbacks inherent in the diaphragm forming technique. The probably most fundamental difference between diaphragm and stamp or matched die forming is that normally only a single die is needed for transforming a flat laminate into a doubly curvatured component. In most cases the vacuum-bagged laminates are formed by drawing them into the cavity of a female die, either in the presence of external pressure, by evacuating the air within the die, or a combination of both. Draping the laminate over the contour of a male die in the presence of external pressure is also a possible option for producing 3-D components. The probably most obvious drawback to the diaphragm technique in comparison with stamp forming is that the mostly disposable diaphragms have to be removed from the parts after forming, which not only is time-consuming but also accounts for extra costs and waste. On the other hand, however, the vacuum-bagging
Thermoforming of continuousfibre~thermoplastic composite sheets
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technique enables near in-situ processing, meaning that pre-consolidation of flat laminates, as needed for stamp forming, becomes superfluous. In terms of consolidation quality this fact represents a considerable advantage over stamp forming with prepregs. Studies by Breuer and co-workers showed that, if the forming pressure applied during processing is below the pre-consolidation pressure of the prepreg, voids can reappear in the finished component [51]. Due to the nature of stamp and matched die forming process, high pressure gradients occurring during forming are inevitable and increase with the complexity of the die. Initially, considerable effort has been put into autoclave diaphragm forming of high-performance composites such as APC-2 | employing superplastic aluminium diaphragms like Supral | and Upilex | In economical terms, however, diaphragm forming in autoclaves is still far from asserting itself against the more cost-effective and rapid stamp or matched die forming technique. In view of the current trends in the international composites market and considering the recent developments in processing techniques of CFRTs, it becomes obvious that more attention has to be devoted to structural rather than high-performance composites. This need has been generally recognised and recent publications cover the areas from processing of high-quality, ready-to-use prepreg materials to various aspects of component manufacturing with these prepregs. This development holds a twofold advantage for the diaphragm forming technique. First and foremost, forming can be conducted without needing autoclaves which is related to the lower processing temperatures required for processing materials such as polypropylene or polyamide matrix composites. Secondly, the lower processing temperatures allow the employment of reusable elastomeric diaphragms, making this manufacturing technique more competitive from the economic and environmental point of view. The following sections of this chapter are addressing various aspects associated with the diaphragm forming technique. It highlights the effects of reusable diaphragms on the forming behaviour of CFRTs and the resulting component properties. The occurrence of instabilities such as out-of-plane buckling and in-plane wrinkling in correlation with laminate particularities, characteristic flow mechanisms and diaphragm contact is investigated and discussed for the example of hemispherical dome components. This shape, in spite of its apparent simplicity, involves all deformation modes which are prevalent in much more complex shapes.
4.5.2. Experimentalprocedure In order to provide a fundamental understanding of the response of CFRT to diaphragm forming in contrast to stamp forming conditions, a series of preliminary experiments utilising a diaphragm forming set-up as depicted in fig. 4.67 is described in the following paragraphs. For these experiments pre-consolidated Plytron (GF/ PP) laminates with various lay-ups ([0~ [+7.5~ [+22.5~ [+30~ and [+45~ were drawn into a female die in the presence of external pressure. Forming was performed under non-isothermal conditions meaning that the laminates had to preheated to processing temperature in an external oven prior to forming. For this purpose the laminate was vacuum-bagged between two silicone rubber diaphragms.
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K. Friedrich et al. Vent Holes Wooden Mould Vacuum Bag with Blank Sheet Base Plate with Compressed Air Supply
Fig. 4.67. Sketch of forming device with vacuum bag.
The main aim of the subsequent series of experiments is to investigate the influence of processing parameters on the properties of the resulting components. By varying the parameters essentially governing the forming behaviour of the composite such as forming temperature, load, velocity, ratio A1/A (blank area/cavity area) as well as the laminate's stacking sequence and blank shape it is intended to determine a guideline for diaphragm forming at the example of GF/PP (Plytron).
4.5.3. Assessment and characterisation of thermoformed parts 4.5.3.1. Large strain analys& technique A very powerful tool for getting a macroscopic description of the material's deformation behaviour without any knowledge of a constitutive relationship is the large strain analysis technique. The theory behind this technique was originally invented for investigating the strain distribution in sheet metal forming processes but has since been successfully used to study composite sheets as well [52]. In order to measure the strain distribution in a given test piece, circular/square grids were printed on the surface of each sheet prior to forming by using a silk screen process. The deformed nodal co-ordinates were used in the computation of surface strains using a software package developed at Auckland University [53]. The theory behind the strain analysis technique was originally invented for investigating the strain distribution in sheet metal forming processes. The following gives a short introduction to the mathematical background of this technique formulated by Martin and Christie [54]. Suppose a sheet of material has its surface with an array of grid points. The coordinates of each of these points in space may be measured before and after deformation, and the surface can be divided up into a number of elements, as shown in fig. 4.68a. The x, y, z co-ordinates are then a function of xl, x2 on the parametric surface. After deformation the new co-ordinates of each point may be measured with a digitiser. If no holes form in the material during deformation, then the original Xl, x2 co-ordinates for each point remain unchanged. Using this information, it is possible to find the bi-cubic surface patch which best fits the deformed data. Both the undeformed and the deformed co-ordinates of each data point can be expressed as cubic
149
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k
~ ~ X
-
....
~r
(a)
v
X
(b)
Fig. 4.68. An array of data divided up into elements, (a) before deformation and (b) after deformation.
functions on the (xl, X2) embedded co-ordinate system. Figure 4.68b shows the four elements and their nodal points after deformation. When considering large deformations it is inappropriate to use the infinitesimal strain tensors as an indicator of finite deformation. This is especially true when a constitutive relationship is to be developed for cases of large strain. If the material is treated as a continuum, the strain tensor can be developed from a purely geometric viewpoint. Consider two points on a surface in 3-D space, P and Q, with surface co-ordinates u~ and u~ + du ~ respectively. The co-ordinates of P and Q can also be written in terms of their Cartesian co-ordinates in x i and x i + d x i respectively. The squared distance between points P and Q is then 3
0~/ds 2 - ~ ( d x i ) 2
Ouoe
(4.5.1)
i=1
and expanding d x i in terms of its partial derivatives, Ox i d s i -- - ~
du
(4.5.2)
By substituting eq. (4.5.1) into eq. (4.5.2) the squared length, d s 2, m a y then be obtained [54]" OX i OX i
ds2 = 0--~
---3du'~du~ ax
(4.5.3)
If the original length of this line segment, before deformation, is represented by dS, then, OX i OX i
dS2 -" 0---~- -----~ Ox du'~ due
(4.5.4)
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K. Friedrich et al.
The change in length squared of line segment PQ after deformation is ds 2 - dS 2 = (%~ - A ~ ) du~du ~ = 2EabdU~du/~
(4.5.5)
where Eab is the Green strain tensors in two dimensions. In fact, it is more convenient to write the expression for Eab in terms of xl, x2 co-ordinates. Thus from (5.5),
1
I{OX i'_ OXi
Eo~ - -~(ao~ - Ao~) - -~ \ 0 ~ - ~
OXi oxi~ O~ O~fl (4.5.6)
O~ O~~,] Ou~ Ou~
where O~/Ou ~ represents a scaling factor between the arc length on the surface, du a, and the normalised length increment dx a. Now, because the undeformed and deformed co-ordinates, X i and x i, can be expressed in terms of Xl, x2 through a so-called least squares fitting process [54], Eab may be calculated at any point on the surface. The two principal surface strains then correspond to the eigenvalues and eigenvectors of the strain tensor, Eab [56]. In terms of the principal stretches ll, and 12, the principal strains are /~11()2 __ 1)
(4.5.7)
and E22()~2 - 1)
(4.5.8) ^
If the material is assumed to be incompressible, the third principal strain E33, may be determined using the incompressibility constraint from eqs. (4.5.9) and (4.5.10). ~.l).e~.3 -- 1
(4.5.9)
/~33().~- 1)
(4.5.10)
4.5.3.2. Occurrence o f defects and thickness variations in diaphragm f o r m e d parts The results of the preliminary forming series showed that the instabilities such as in-plane wrinkles and out-of-plane buckles only occur in distinctive areas of the diaphragm formed part which can be directly related to the lay-up of the laminate (fig. 4.69). This result corresponds well with the observations made during the 3-D
Fig. 4.69. Sketch of in-plane wrinkles and out-of-plane buckles.
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stamp forming experiments. Looking at the deformation mechanisms involved in this forming process, the following conclusions can be stated. When applying the forming pressure, the central region of the laminate is forced into the cavity of the die which principally requires an increase in the laminate's surface area (also see fig. 4.58). Since the material cannot stretch in the direction of reinforcement, the virtually inextensible fibres have to move towards the centre from the outer edges of the sheet. As a result tensile stresses are generated in the fibre direction. Simultaneously, the decrease in area when moving towards the apex of the dome section leads to compressive hoop stresses in the bisecting direction of the reinforcements which are resolved in out-of-plane buckles appearing in the direction away from the fibres. Another remarkable phenomenon which is observed when forming laminates with similar blank shape but different lay-ups is depicted in fig. 4.70. Dependent on the arrangement of the reinforcements, the outline of the formed spherical dome-shaped components differs noticeably. The unidirectional reinforced laminate, [0~ exhibits extensive transverse squeeze flow in the perpendicular direction to the fibre axis, resulting in severe thickening and thinning, whereas the [+45~ laminate, on the other hand, features a fairly constant wall thickness throughout the whole part. This result suggests that the resistance to matrix flow along the fibres is much higher than away from them. Furthermore, it becomes obvious that there is a close link between blank shape, laminate architecture, die geometry and the resulting outline of the thermoformed component. This finding is utilised later for optimising the blank shape in terms of minimum scrap. Figure 4.71 shows typical results of the large strain analysis technique for a [+45~ GF/PP dome. The direction and size of the arrows indicate the nature and magnitudes of strain respectively. The lines indicate the reinforcing fibres. In this case, due to the symmetrical shape of the component, showing a quarter is sufficient to describe its strain distribution. This graph confirms that the majority
..m,,,
I
Undeformed
[0~
I
[+7.5~ ~.~
'
~
[+22.5~
,
"
[+30~
Fig. 4.70. Influence of lay-up on the outline of the finished component.
[•176
~%
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x
Fig. 4.71. Arrow diagram for a [45]sGF/PP dome. of large compressive strain occurs in the directions away from the fibres. The big arrows in the flange area indicate in-plane wrinkling. In some cases this mechanism even led to cracks along the transverse fibre direction in the flange and the bend-over region.
4.5.4. Variation of forming parameters and rating of part quality Following the preliminary studies on the forming behaviour of the GF/PP laminates and the observed instabilities, such as in-plane wrinkling and out-of-plane buckling, now the influence of the main forming parameters on the finished quality of the components is addressed. By varying one parameter at a time within a certain range, as previously illustrated for the stamp forming technique, it is intended to define a processing window for the example of GF/PP. Since it is fairly difficult to assess the quality of a finished component by means of definite values, a rating scale of 1 to 10 is introduced here, that allows a qualitative comparison of the diaphragm formed parts. The assessment is based upon the three different criteria which are regarded to be decisive for the finished properties of the components:
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the contour of the formed part with respect to the actual contour of the mould; the surface finish of the mould facing surface (Surface A); and the surface finish of the surface turned away from the mould (Surface B). On this scale 10 denotes very little wrinkling and no buckling and an excellent die conformity of the part, whereas 1 denotes severe in-plane wrinkling and out-of-plane buckling and an unsatisfactory die conformity of the part.
4.5.4.1. The forming temperature The first forming parameter to be altered is the forming temperature. In this context it has to be mentioned that polypropylene crystallises with considerable supercooling, typically 30-40~ depending on cooling rate (fig. 4.72). The maximum in cp on cooling, therefore, occurs at a much lower temperature than on heating and, in between, the Cp will be different as the polymer is molten on cooling, but still solid on heating. Therefore, forming experiments on both the heating and the cooling cycle are discussed here. On the cooling cycle the blank was preheated to 180~ which is about 15~ above the nominal melting temperature of PP and then formed at temperatures ranging between 170~ and 110~ As shown in fig. 4.73 the GF/PP material proved to be deformable to temperatures as low as 120~ without showing any out-of-plane buckles and only negligible in-plane wrinkling. In fact, looking at the wall thickness variation, it was found that with decreasing forming temperatures the components exhibited less thickening and thinning. This result suggests that at low forming temperatures on the cooling cycle the laminates form by inter-ply slip and interply rotation rather than transverse flow. 400
Heating Cycle 9. - . . . .
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_
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TEMPERATURE [~ Fig. 4.72. Enthalpy versus temperature for Plytron.
I
200
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Fig. 4.73. Forming results on cooling cycle.
When forming on the heating cycle the laminates were preheated to a predetermined temperature and formed immediately. As can be seen in fig. 4.74 the results here were quite different from the ones on the cooling cycle. Already at forming temperatures as high as 160~ the laminates proved to be virtually undeformable, which corresponds well with the thermal behaviour of the polypropylene matrix as illustrated before. From these results it can be concluded that the forming temperature is a decisive parameter essentially influencing the quality of the forming results. Therefore, it should be well controlled and monitored throughout the thermoforming process. In the case of GF/PP, forming on the cooling cycle significantly enlarges the temperature window within which the material can be deformed. As a result transferring the preheated laminate into the press is not as critical and forming velocities do not need to be as high, which allows extra time for flow processes to take place. 4.5.4.2. The forming ratio The forming ratio A1/A2, which may be defined as the ratio of blank area to the
projected area of the die cavity, plays an important role in composite sheet forming. With regard to mass production, minimising the scrap per part significantly affects not only the economic efficiency of the manufacturing process but also the profitability of the product. In stamp forming processes that employ clamping frames the forming ratio can only be reduced to a certain extent in order to ensure that the laminate is kept under tension throughout the forming stroke. This limitation does not apply to diaphragm forming. Here, the tensional forces are induced through
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Fig. 4.74. Forming results on heating cycle.
physical contact between the stretchable diaphragms and the laminate. As a result the forming ratio can be reduced significantly. Figure 4.75 shows for the example of a [+45~ laminate how the blank shape of a hemispherical dome part was reduced to a forming ratio of 1.6. As can be seen in the flange area of the part, the material was only drawn into the die in the direction of the reinforcements, changing the circular UNDEFORMED
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.
.
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outline of the blank to a square one. Therefore, in a subsequent step the redundant corners of the circular blank were removed prior to forming in order to further reduce the forming ratio to a value of 1.48. The sketch of the formed part shows that its outline is now circular, which coincides with desired final finish. Another important aspect of reducing the forming ratio is related to the flow processes occurring in the laminate during forming. As described before, the inextensible fibres have to flow in order to achieve die conformity. The flow mechanisms mainly involved in this process are inter-ply slip and inter-ply rotation. The latter leads to shear thickening and shear thinning which is accompanied by a local change of fibre angle between neighbouring plies. Shear thickening generates compressive stresses between the fibres which can result in either local laminate thickening or outof-plane buckling. By removing material in the bisecting direction of the reinforcements, as shown in fig. 4.75, the effective area in which compressive stresses can be generated is reduced and thus the occurrence of out-of-plane buckling.
4.5.4.3. The forming pressure
The forming pressure was varied between 70 and 560 kPa. Here, the best forming results were obtained at 300 kPa: the components were completely buckle-free in both surfaces and exhibited only minor wrinkling in the bend-over area in the mould-facing surface. Also the wall thickness variation of 4-12% at this forming pressure is thought to be within a tolerable range. It is obvious that GF/PP can be formed at considerably low pressures compared with the stamp forming technique, which is probably advantageous with regard to mass production. In fact, if too high a pressure and temperature is applied for consolidating and forming, an unacceptably high level of fibre waviness, even in cross-plied laminates, is likely to occur.
4.5.4.4. The forming velocity
Due to the manual nature of the thermoforming set-up it was difficult to measure the actual drawing velocity in a diaphragm forming process. Therefore, the main aim here was to investigate whether forming can be accomplished before the laminate temperature drops below crystallisation temperature. The actual temperature of the laminate was monitored during the whole process including the pre-heating period in the oven. Since the forming process itself was finished within 2-4 seconds and a wooden mould was used, it was found that the laminate's temperature only dropped by 10-15~ on average during that particular phase. Of course, these values are essentially dependent on the thickness of the laminate, the size of the die cavity and the material and the heat capacity of die employed. Generally, the forming velocity should be adjusted in such a manner that the flow mechanisms necessary to achieve die conformity can actually take place. However, when forming under non-isothermal conditions the total duration within which forming is possible is mainly restricted by the thermal behaviour of the thermoplastic matrix. In the case of forming GF/PP on the cooling cycle, the forming velocity can be kept fairly low since the material retains its formability over wide temperature band.
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4.5.4.5. The lamh~ate thickness and lay-up
Regarding the alternation of the lay-up the main interest here was to investigate how an increase in the total number of plies of the laminate influences the finish sheets it was found quality of the parts. By forming [+45~ [(+45~ and [(+45~ that, when forming a circular dome shape, the lay-up has only a subordinate influence on the component's properties in comparison with the other parameters investigated. Even with a sixteen layer sheet it was still possible to form parts of acceptable quality. It may be advisable to reduce the forming speed when forming thick laminates in order to allow more time for inter-ply slip and inter-ply rotation. 4.5.5. Conclusions and recommendations
The results illustrated and discussed in this chapter outline the potentials inherent in the diaphragm forming process. Despite its apparent drawbacks regarding the suitability for mass production it has proved its versatility and capability of producing complex-shaped 3-D components. In contrast to stamp forming, the double diaphragm forming process does not hamper laminate thickening in areas where excessive trellis flow occurs. As a result even complex shapes like conical cups rectangular containers can be formed in absence of instabilities such as out-of-plane buckling (fig. 4.76). Another fundamental difference between the two forming techniques is that diaphragm forming shapes a laminate by drawing it into a female die in
Fig. 4.76. Selection of thermoformed components.
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contrast to matched die forming that drapes it onto a stamper. This way, even in the absence of a clamping plate, sufficient clamping forces are generated since the outer regions of the sandwiched laminate are pressed onto the flange of the die before the central part is drawn into the cavity. Furthermore, additional tensile stresses are superimposed by the diaphragms due to their ability to stretch, which can reduce or even out compressive stresses in the laminate during forming and thus prevent instabilities from occurring. It can therefore be concluded that diaphragm forming should be preferred over stamp forming when complex shaped 3-D components are formed from continuous fibre-reinforced thermoplastic sheets. However, when fabricating 2-D or simple 3-D shapes employing the more rapid stamp forming technique is probably the better choice. In order to demonstrate the commercial feasibility of diaphragm forming, a blister fairing, which is part of the thrust reversing unit of an aircraft, was manufactured. A similar part has already been produced using CF/PEEK [57,58]. However, since the material costs and the processing of this material is very expensive and complicated the parts did not assert themselves against the titanium and CF/epoxy fairings currently in use. At current stage, this component is made by draping carbon fibre cloths over a male mould bonded with epoxy resin in a 12-hour processing and curing cycle. The component, shown in fig. 4.77, was thermoformed using GF/PP and GF fabric/PA with a semi-automated set-up within a matter of seconds, by
Fig. 4.77. Blister fairing made out of GF/PP laminates.
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applying the knowledge gained through the theoretical and practical studies described in the previous sections. Mechanical tests and an economic study confirmed that both the Plytron and the Tepex components were clearly superior to the carbon fibre/epoxy variant with regards to mechanical properties and cost effectiveness [59].
4.6. Summary The main objective of this chapter was to present a comprehensive study investigating the particularities of stamp and diaphragm forming for processing continuous fibre-reinforced composite sheets. By addressing fundamental issues such as the material's response to various forming conditions it was intended to outline the difficulties involved with forming highly anisotropic thermoplastic composite sheets. Investigating the governing forming parameters such as temperature, pressure, velocity and blank shape the advantages and drawbacks of both stamp forming and diaphragm forming are illustrated and discussed in this chapter. Stamp forming proved to be a very rapid forming technique for producing 2-D components. The main flow mechanism here was identified to be inter-ply slip. Regardless of the laminate's stacking sequence, die conformity was mainly achieved through relative movement of neighbouring plies. In case of forming 3-D components, however, the situation becomes more complex. Dependent on the die geometry inter-ply rotation and transverse matrix flow are necessary in order to form a component in the absence of instabilities such as out-of-plane buckling and in-plane wrinkling. Inter-ply rotation or trellis flow consequently leads to shear thickening and thinning in the direction away from the reinforcements. Due to the nature of the matched die process, these thickness variations cannot be accommodated and the resulting instabilities seem to be inevitable. The employment of clamping frames can help to alleviate this problem, however, in terms of material and cost-effectiveness they are not very desirable. This apparent deficiency of the stamp forming technique was found to be the main advantage of diaphragm forming. When forming a vacuum-bagged sheet into a doubly curved die, the stretchable diaphragms superimpose highly desirable tensional stresses on the laminate and lower or even nullify the compressive hoop stresses that are normally observed in 3-D forming. In addition, diaphragm forming, in contrast to stamp forming, can accommodate thickness variations caused by inter-ply rotation and trellis flow which makes it more appealing for forming complex 3-D shapes. It has to be taken into account, however, that only the die-facing skin of the component fully conforms with the contour of the die. As a final conclusion it can be stated that both forming techniques feature the capability for the production of continuous fibre-reinforced components with selected shapes. However, more work is necessary in order to help these processes to assert themselves against conventional manufacturing techniques as they are currently in use. A basic guidance to achieve this goal is given in this chapter and ir a number of further references that have been published only recently [60-64].
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Acknowledgements The authors gratefully acknowledge the support of the Deutsche Forschungsgemeinschaft (DFG FR 675/7-2). Further thanks are addressed to the various companies who supplied the various testing materials. In addition, Dipl.-Ing. J. Krebs appreciates the research fellowship from the Deutscher Akademischer Austauschdienst (DAAD) allowing him to stay with the Department of Mechanical Engineering, University of Auckland, New Zealand. Dr.-Ing. M. Hou wants to acknowledge the grant of a post-doctoral fellowship by the Co-operative Research Centre-Aerospace Structures (CRC-AS), Australia. Finally, Prof. Dr.-Ing. K. Friedrich would like to mention the financial assistance of the Fonds der Chemischen Industrie, Frankfurt, Germany, for his personal research activities in 1997.
References [1] Cogswell, F.N., Thermoplastic Aromatic Polymers; Butterworth-Heinemann Ltd., Oxford, (1992), 124-139. [2] M~mson, J.-A.E., Processing of Thermoplastic-Based Advanced Composites; Kausch, H.-H., Legras, R. (ed): Advanced Thermoplastic Composties: Characterization and Processing, Hanser Verlag, Munich, Vienna, New York, Barcelona, (1993), 273-301. [3] Savadori, A., Cutolo, D., Impregnation Flow and Deformation Processing of Advanced Thermoplastic Composites, Makromol. Chem., Macromol. Symposium 68, (1993), 109-131. [4] Muzzy, J., Norpoth, L., Varughese, B., Characterization of Thermoplastic Composites for Processing; Sampe Journal, Vol.25, No. 1, (1989), 23-29. [5] Robroek, L.M.J., The Influence of a Rubberforming Cycle on Mechanical Properties of Continuous Fibre Reinforced Thermoplastics; Plastics, Rubber and Composite Applications 19, (1993), 137-141. [6] Carlsson, L.A. (ed): Thermoplastic Composite Materials; Elsevier Science Publishers, Amsterdam, (1991). [7] Mallon, P.J.; O'Brfidaigh C.M; Pipes, R.B., Polymeric Diaphragm Forming of Continuous Fibre Reinforced Thermoplastic Matrix Composites; Composites, Vol. 20, No. 1, (1989) 48-56. [8] Barnes, A.J. and Cattanach, J.B., Advances in Thermoplastic Composite Fabrication Technology; Proceedings of the Materials Engineering Conference, Leeds, (1985). [9] Ostrom, R.B., Koch, S.B. and Wirz-Safranek, D.L., Thermoplastic Composite Fighter Forward Fuselage; SAMPE Quarterly, (1989) 39-45. [10] N.N., Thermoplastics Lead Europe's Composites Recovery; Reinforced Plastics, Vol. 38, No. 6, (1994) 18-23. [11] N.N., Reinforced PP Making Headway in Cars; Reinforced Plastics, Vol. 36, No. 3, (1992) 22-24. [12] Cuff, G: Fibre Reinforced Industrial Thermoplastic Composites, ICI Ltd., private communication, (1993). [13] Hou, M. and Friedrich, K., Stamp Forming of Continuous Carbon Fiber/Polypropylene Composites, Composites Manufacturing, Vol. 2, No.l, March 1991, pp. 3-9. [14] Melinex Polyester Films, Product Information, Technical Data Sheet, MX TD 251 Issue No. 2, ICI, (1987). [15] Scherer, R., Zahlan, N. and Friedrich, K., Modelling the Interply-Slip Process during Thermoforming of Thermoplastic Composites using Finite Element Analysis, in: Composite Materials-Design and Analysis (ed. W.P. de Wilde and W.R. Blain), Computational Mechanics Publications, Southampton (1990), pp. 39-52.
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[16] Wittich, H. and Friedrich, K., Kontinuierliche Konsolidierung von thermoplast-impr~ignierten Langfaserb~indern zu Hochleistungsverbundlaminaten, 22. Internationalen AVK-Tagung, Mainz, Germany, 22-24 May (1989) 7.1-7.7. [17] O'BrAdaigh, C.M. and Pipes, R.B., Issues in Diaphragm Forming of Continuous Fiber Reinforced Thermoplastic Composites, Polymer Composites, Vol. 12, No. 4, Aug. (1991), pp. 246-256. [18] Cogswell, F.N. and Leach, D.C., Processing Science of Continuous Fiber Reinforced Thermoplastic Composites, SAMPE Journal, May/June (1988), pp. 11-14. [19] Pipes, R.B. and Smiley, A.J., Analysis of Diaphragm Forming of Continuous Fiber Reinforced Thermoplastics, J. of Thermoplastic Composites Materials, Vol. 1, Oct. (1988), pp. 298-321. [20] Fitzer, E. and J~iger, H., Die Verst~irkung der Thermoplaste Polycarbonat und Polysulfon mit Carbonfasern: Herstellung und Eigenschaften der UD-Verbundk6rper, AVK Tagung, Freudenstadt, Okt. (1985) 38.1-38.15. [21] Ye, L., Klinkmiiller, V. and Friedrich, K., Impregnation and Consolidation in Composites Made of GF/PP Powder Impregnated Bundles, Journal of Thermoplastic Composite Materials, Vol. 5, Jan. (1992), pp. 32-48. [22] Van West, B.P., Pipes, R.B. and Advani, S.G., The Consolidation of Commingled Thermoplastic Fabrics, Polymer Composites, Vol. 12, No. 6 (1991), pp. 417-427. [23] Barnes, J.A. and Cogswell, F.N., Transverse Flow Processes in Continuous Fibre Reinforced Thermoplastic Composites, Composites, Vol. 20, No. 1, January 1989, pp. 38-42. [24] Hou, M. and Friedrich, K., Thermoforming of High-Performance Composites with Thermoplastic Matrix, Engineering Plastics, Vol. 5, No. 2, 1992, pp. 86-100. [25] Tam, A.S. and Gutowski, T.G., Ply-slip during the Forming of Thermoplastic Composite Parts, Journal of Composite Materials, Vol. 23, June 1989, pp. 587-605. [26] Hou, M. and Friedrich, K., Zum Thermoformen von Hochleistungsverbund-werkstoffen mit thermoplastischer Matrix, Proc. AVK Conf., Berlin Sept. 12-14. 1991, Edit. by Erich, K., published by Arbeitsgemeinschaft Verst~irkte Kunststoffe e.V. (AVK), Germany. [27] Hou, M., Friedrich, K. and Scherer, R., Optimization of Stamp Forming of Thermoplastic Composite Bends, Composite Structures, 27(1994), pp. 157-167. [28] Suemasu, H., Friedrich, K. and Hou, M., On Deformation of Woven Fabric Reinforced Thermoplastic Composites during Stamp Forming, Composite Manufacturing, Vol. 5, No.l, 1994, pp. 31-39. [29] H6ger, A., Warmformen von Kunststoffen, Kunststoff-Verarbeitung Folge 18, Carl Hanser Verlag, Miinchen, 1971. [30] Ozisik, M.N., Heat Conduction, A Wiley-Interscience Publication, 1980. [31] Weeton, J.W., Peters, D.M. and Thomas, K.L., Engineers' Guide to Composite Materials, American Society for Metals, Carnes Publication Services, Inc., 1987. [32] Kroschwitz, J.I., Concise Encyclopedia of Polymer Science and Engineering, A Wiley-Interscience Publication, 1990. [33] Carlsson, L.A.(ed.): Thermoplastic Composite Materials, Composite Materials Series, Vol. 7, Elsevier Science Publishers B.V., Amsterdam, 1991. [34] Anon., Hostalen PP Werkstoffblatt, Hoechst AG, Stand: 1989. [35] Anon., Lexikon Werkstofftechnik, VDI-Verlag GmbH, 1991. [36] Anon., "Victrex" PEEK material instruction, ICI, PLC, 1984. [37] Anon., ULTEM PEI materials instruction, GE Plastics, 1991. [38] Scherer, R. and Friedrich, K., Experimental Background for Finite Element Analysis of the Interplyslip Process during Thermoforming of Thermoplastic Composites, Developments in the Science and Technology of Composite Materials, Fourth European Conference on Composite Materials, Stuttgart, F.R.G., Sept. 25-28, 1990, pp. 1001-1006. [39] Barnes, J.A. and Cogswell, F.N., Transverse Flow Processes in Continuous Fiber-Reinforced Thermoplastic Composites, Composites, Vol. 20, No. 1, (1989), pp. 38-42. [40] Zahlan, N. and O'Neill, J.M., Design and Fabrication of Composite Components: The Springforward Phenomenon, Composites, 20, No. 1 (1989), pp. 77-81. [41] Chou, T.W. and Ko, F.K., Textile Structural Composites, Composite Materials Series, Volume 3, Elsevier Science Publishers B.V., 1989, pp. 88.
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[42] O'Neill, J.M., Rogers, T.G. and Spencer, A.J.M., Thermally Induced Distortions in the Moulding of Laminated Channel Sections, Meth. Engng. Ind., Vol. 2, No.l, (1988), pp. 65-72. [43] Anon., Service instructions, TA 400 thermal analysis. Metteler measurement and evaluation system. Instrumente AG, Greifensee, Switzerland, (1988). [44] Anon., Hills VESTAMID, Werkstoffblatt, Hills AG, Stand: 1988. [45] Monaghan, M.R., Mallon, P.J., O'Brfidaigh, C.M. and Pipes, R.B., The Effect of Diaphragm Stiffness on the Quality of Diaphragm Formed Thermoplastic Composite Components, SAMPE Quarterly, Vol. 21, No. 4, July (1990), pp. 48-55. [46] Mallon, P.J., O'Brfidaigh, C.M. and Pipes, R.B., Polymeric Diaphragm Forming of Complexcurvature Thermoplastic Composite Parts, Composites, Vol. 20, No. 1, (1989), pp. 48-56. [47] Li, H.L., Koch, P., Prevorsek, D.C. and Oswald, H.J., Cold Forming of Plastics Part I. Draw Forming of Thermoplastic Sheets, Polymer Engineering and Science, March, Vol. 11, No. 2, (1971), pp. 99-108. [48] Broutman, L.J., Kalpakjian, S. and Chawla, J., Deep Drawability of Biaxially Rolled Thermoplastic Sheets, Polymer Engineering and Science, March, Vol. 12, No. 2, (1972), pp. 150-156. [49] Robertson, R.E., Hsiue, E.S., Sickafus, E.N. and Yeh, G.S.Y., Fibre Rearrangements During the Molding of Continuous Fibre Composites: I. Flat Cloth to a Hemisphere, Polymer Composites, Vol. 2, 1981, pp. 126-131. [50] Potter, K.D., The Influence of Accurate Stretch Data for Reinforcements on the Production of Complex Structural Moulding, Composites, Vol. 9, 1979, pp. 161-173. [51] Breuer, U., Neitzel, M., High Speed Forming of Thermoplastic Composite Sheets, Polymers & Polymer Composites, Vol. 4, No. 2, 1996, pp. 117-123. [52] Martin, T.A., Bhattacharyya, D., Pipes, R.B., Computer-Aided Grid Strain Analysis in Fibre Reinforced Thermoplastic Sheet Forming, Computer Aided Design in Composite Material Technology III, Computational Mechanics Publications, London, New York, ISBN 1-85166-781-4, pp. 143-162. [53] Martin, T.A., Bhattacharyya, D., Pipes, R.B., Deformation Characteristics and Formability of FibreReinforced Thermoplastic Sheets, Composite Manufacturing, Vol. 3, No. 3, (1992), pp. 165-172. [54] Martin, T.A., Forming Fibre Reinforced Thermoplastic Composites, Dissertation, University of Auckland, New Zealand, 1993. [55] McConnell, A.J., Applications of Tensor Analysis, Dover Publications, Inc., New York, (1957), pp. 163-167. [56] Reddy, J.N., Rasmussen, M.L., Advanced Engineering Analysis, John Wiley & Sons, New York, (1982), pp. 135-140. [57] Owens, G.A., Lind, D.J., Applications of Carbon Fibre Reinforced Thermoplastic (PEEK) Composite to Aero Engine Component Manufacture, Proc. Fibre Reinforced Composites '84, (1984), 12.1-12.11. [58] Barnes, A.J., Cattanach, J.B., Advances in Thermoplastic Composite Fabrication Technology, Proceedings of the Materials Engineering Exhibition and Conference, (1985). [59] Krebs, J., Bhattacharyya, D., Friedrich, K., Production and Evaluation of Secondary Composite Aircraft Components--A Comprehensive Case Study, Composites Part A, (1997) in press. [60] Hou, M., Ye, L., Mai, Y.-W., Advances in Processing of Continuous Fibre Reinforced Composites with Thermoplastic Matrix, Plastics, Rubber and Composites Processing and Applications, 23, 5, (1995), 279-293. [61] Hou, M., Stamp Forming of Fabric-Reinforced Thermoplastic Composites, Polymer Composites, 17, 4, (1996). [62] Breuer, U., Neitzel, M., Ketzer, V., Reinicke, R., Deep Drawing of Fabric-Reinforced Thermoplastics: Wrinkle Formation and Their Reduction, Polymer Composites, 17, 4, (1996) 643-647. [63] Hou, M., Friedrich, K., On Stamp Forming of Curved and Flexible Geometry Components from Continuous Glass Fiber/Polypropylene Composites, Composites Part A, (1996) submitted. [64] Hou, M., Friedrich, K., Adjustable Forming of Thermoplastic Composites for Orthopaedic Applications, Journal of Materials Science, Materials in Medicine, (1997) in press.
Composite Sheet FormO~g edited by D. Bhattacharyya 9 Elsevier Science B.V. All rights reserved.
Chapter 5
Characterisation of Shearing and Frictional Behaviour during Sheet Forming Adrian M. M U R T A G H Engineering Department, Cambridge University, Trumpington Street, Cambridge CB2 1PZ, UK
Patrick J. M A L L O N Mechanical and Aeronautical Engineering Department, University of Limerick, Limerick City, Ireland
Contents Abstract 163 5.1. Introduction 164 5.2. Transverse fibre flow 170 5.3. Intra-ply shear 173 5.4. Inter-ply slip 177 5.5. Friction during thermoforming 197 References 214
Abstract An introduction is given to the processes involved in the forming of thermoplastic composite and various shearing deformations are explained, including intra-ply shearing and inter-ply slip in unidirectional laminates, and the trellis mechanism in fabric materials. A consolidation rig was used to evaluated the influence of processing parameters such as pressure, temperature and time on the quality of consolidated laminates. Consolidation quality was also found to be heavily dependent on lay-up and whether or not the flow processes were restricted. Unidirectional restricted laminates were difficult to consolidate with best results achieved in low pressure regions. For (0/90) and quasi lay-ups laminate quality improved as the consolidation pressure increased with acceptable parts achieved at pressures of 500 kPa and over. Transverse flow measurements were used to obtain values of transverse flow viscosity for APC-2 material. A consolidation/shearing rig apparatus was used to carry out inter-ply slip experiments and to determine the effects of temperature, pressures and fibre orientation on the inter-layer shear stress, as a function of sliding velocity. The shearing rig is based on a computer-controlled motor-driven leadscrew. Normal pressure, increased 163
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beyond a low nominal value ( < 100 kPa), substantially increased shear stress values. Temperature effects showed the influence of the viscous matrix. Laminates in which fibres from adjoining plies lay parallel to one another showed much higher shear stress values compared with cross-ply laminates. Fibre straightening had to be taken into account in fabric specimens before slip could be initiated. The presence of additional PEEK resin layers did not decrease shear stress as expected in APC-2 laminates. Yield stress measurements were also carried out on specified materials and showed a significant level of stress had to be exceeded before true slip occurred, e.g. 1 kPa for a cross-ply laminate, 2.5 kPa for a unidirectional laminate of APC-2. Using the results obtained from the inter-ply slip tests, a model based on modified form of the Herschel-Buckley model for a power law fluid was developed for APC-2 and for 5-H satin Cetex fabric. The values for the multiplier k and exponent n were first related independently to the parameters of temperature, normal pressure and fibre angle, then a factoring technique was used to create a master model for both materials. Yield stress was incorporated in both cases: for the fabric, a modified form of the Oldroyd model was used to account for tow stretching before slip occurs. A combination solid phase/melt model for APC-2 was also generated. Friction testing was carried out on APC-2 and 5-H satin Cetex to establish the respective material's sliding behaviour as it deforms during the typical press forming process. The coefficient of friction was measured using two different test set-ups (twin platen arrangement and friction sled) according to varying parameters of sliding velocity, temperature, normal load, relative fibre angle and presence of release agent. The observed behaviour was found to be hydrodynamic in nature, with a strong adhesive element. A rubber pad/glass fabric composite interface was also investigated under non-isothermal conditions. A fiction law was developed to predict the friction coefficient between tool and part during forming (dependent on the mentioned parameters), again based on a power law fluid model, due to the establishment of a viscous resin layer at the interface. 5.1. Introduction
During the sheet forming of complex 3-D shapes from a flat laminate of fibrereinforced thermoplastic, various forming mechanisms must occur to facilitate part manufacture. For continuous fibre reinforced composites such as carbon-fibrereinforced PEEK (APC-2), there are four such mechanisms: resin percolation of the matrix amongst the fibres, transverse fibre flow, intra-ply shear and inter-ply slip of individual plies across one another. Friction between composite and forming tools may be included as a fifth mechanism. These mechanisms occur irrespective of the actual forming process or the shape of the part. However, different mechanisms may be more or less pronounced, and have a greater or lesser effect, depending on circumstances. For example the presence of flexible diaphragms in diaphragm forming allow more squeeze flow to occur at bends compared with matched-die press forming, and higher forming rates in press forming may cause high intra-ply shearing rates which may lead to buckling in the laminate. Although fundamental research has been conducted to investigate each mechanism thoroughly and in a quantitative
Shear&g and frictional behaviour dur&g sheet forming
165
fashion, the current understanding is not complete. Such an understanding is necessary to allow prediction of possible processing problems and to guarantee a successful manufacturing operation. This chapter outlines the current understanding of the forming mechanisms which are central to the successful and economic manufacture of continuous fibre-reinforced thermoplastics using a sheet forming technique. The first mechanism, resin percolation, is not in effect a shearing deformation of the reinforcing fibres but rather a movement of fluid through the fibres and will not be dealt with in detail in this chapter. The resin flow through the fibre bed for unidirectional materials is shown in fig. 5.1a. In fig. 5.1b, the flow that occurs in fabrics to fill the interstices between layers is shown. Without sufficient resin percolation, complete consolidation of the part will not occur, resulting in defects such as voids. The examination of this mechanism, which is mainly dependent on the viscosity of the matrix, has been dealt with by other researchers [1-4]. Transverse fibre flow is an important mechanism that occurs in unidirectional plies, whereby parallel fibres migrate in a sideways fashion due to a normal pressure differential being applied across the surface of the laminate. The measurement of this mechanism illustrated in fig. 5.2a is generally referred to as the transverse viscosity.
Resin flow through unldlrecflor~l fibre bed
Resin flow through fabric to fill Interstices
(a)
(b)
Fig. 5.1. Resin percolation. -3.
~'~ve~
Resin flow
fibre flow
Normal pressure
'~n'eUing' effect
(a) Fig. 5.2. Transverse fibre flow.
(b)
166
A.M. Murtagh and P.J. Mallon
In fig. 5.2b, the barrelling effect that occurs for unidirectional laminates may be observed. This effect is due to the fact that beyond a limiting deformation, fibre movement tends to "lock-up" or stall. Higher pressure towards the centre causes the central region of fibres to deform further. Resin flow may also occur at the ends of the laminate, as the squeezing pressure causes the resin to migrate along the path of least resistance, i.e. in the direction of the fibres. Intra-ply shear is a mechanism that occurs within the plane of each individual ply. For unidirectional fibre-reinforced composites, the shear mechanism entails parallel movement along the length of adjoining fibres. This may occur either through the thickness in the longitudinal (1-3 plane) or along those planes parallel to the surface plane (1-2 plane). The measurement of this mechanism is shown in fig. 5.3a and 5.3b and is normally referred to as longitudinal viscosity. In unidirectional plies where the longitudinal viscosities in the 1-2 plane and the 1-3 plane are equal, the composite is said to be transversely isotropic. Also, the properties of a laminate might infer that a single ply's response is similar to the bulk unidirectional laminate response. This is not true since shear of a laminate in the 1-3 plane results in inter-ply shear between the plies which is a weaker mechanism than intra-ply shear. For fabrics, transverse flow is restricted by the weave. Depending on the thickness of the tows/type of weave, transverse flow is either minimal or practically nonexistent when compared to unidirectional materials. On a smaller scale, if each tow can be approximated to an ellipse/oblong before consolidation, then after consolidation, this ellipse becomes more elongated and its aspect ratio becomes bigger, even though the same number of fibres exists in the tow (fig. 5.4). This elongation is essentially a transverse flow effect, but on a more localised scale than unidirectional materials. This flow is important in filling in the gaps between plies and must occur for good consolidation to occur, even for simple shapes.
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.
.
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Shearing and frictional behaviour during sheet forming
167
Fig. 5.4. Transverse flow in fabric tows.
For fabrics, with reinforcement in two or more directions, intra-ply shearing may take the form of the so-called in-plane "trellis effect". This mechanism is seen in all complex curvature shapes made from fabrics and if it does not occur for some reason or is inhibited in some way, then out-of-plane buckling will occur. The trellis mechanism is illustrated in fig. 5.5. When the force acting on the fabric acts at an off-set angle (shearing angle) to the direction of interlacing orthogonal fibre tows as shown, the effect is to cause the angle between the two sets of tows to decrease - - this angle is known as the "trellis" or crossover angle. This angle has a limit beyond which the tows cannot rotate any further and they "lock" in place. This "locking" angle is mainly a function of the weave style and thickness of the tow. Beyond this angle, further deformation may cause out-of-plane buckling in the part. The fourth mechanism that occurs when forming is that of inter-ply slip, whereby the individual ply layers slip across one another when forming a curved shape (fig. 5.6).
Fig. 5.5. Trellis effect in fabrics.
168
A.M. Murtagh and P.J. Mallon
Resin k:~yer
Ply slipno buckling
n e d , no slip buckling of Inner plies
Fig. 5.6. Inter-ply slip between the layers.
The reason this must happen is again due to the inextensibility of the reinforcing fibres. For example, when forming a 90 ~ bend, the plies closest to the bend undergo a compressive stress as the laminate deforms, and this can result in out-of-plane buckling. The outermost plies are in tension by not being able to stretch to accommodate the increased arc length on the outer surface. In practical forming operations, each ply behaves as a separate entity with shearing taking place between the plies. Intra-ply shearing of the fibres need not be included in any analysis. The slippage effect between plies is assumed to occur in a resin-rich layer existing between the plies. In this way, the flow can be described as a simple Couette f l o w - viscous flow of a fluid between two parallel plates. This neglects the fact that the thickness of the resin layer can vary considerably from being barely evident (especially for unidirectional lay-ups) to being a few fibre diameters thick in some regions. However, a reasonable assumption is that the average thickness is approximately one fibre diameter (6 lain) which allows an inter-ply slip viscosity to be determined if the shearing velocity is known. Inter-ply rotation occurs when forming double curvature components when the angle between fibre directions in adjacent plies must change during forming to accommodate the part geometry and may be regarded as a complex inter-ply slip deformation. For fabrics, inter-ply slip between plies can also occur. However, the fibres are not so inextensible as with unidirectional materials because of the "crimped" nature of the fabric prepreg and this allows a certain straightening of the fibres when forming over a bend and inter-ply slip only occurs once the fibre tows straighten and become inextensible. Conversely, the crimped nature of the tows also means that buckling is more likely to occur in those plies that experience a compressive stress. One further phenomenon that could be considered a type of flow mechanism is the frictional interaction that must occur between the forming tools and the surface of
Shearing and frictional behaviour during sheet forming
169
the thermoplastic composite. For diaphragm forming, the composite does not make any direct contact with the mould, rather, the sliding mechanism is a combination of the interaction between the composite and diaphragm material on one side, and the diaphragm and mould material on the other side. However, during press forming of thermoplastic composite parts, there is a frictional interaction between the composite and mould surface. For matched-die moulds, it may be censidered an interaction between a fluid-like, molten polymer sliding against a hard, yet smooth, metal surface. When rubber pad forming is the process involved, the friction is between two semi-compliant mediums. The dominating factor here is temperature, further complicated by the fact that most press forming processes are anisothermal in nature and temperature has been shown to have a profound effect on the frictional characteristics of polymer systems. The current understanding of this phenomenon is incomplete and further investigation is necessary to measure the effects of temperature together with other possible variables such as normal pressure, forming speed and surface texture. Figure 5.7 gives an example of a press-formed top-hat section and shows how the frictional forces build up during the forming cycle. In Step 1, the preheated laminate is transferred from the heating station to between the punch and the mould cavity. A blank holder may be used at this point to keep the laminate in position and maintain a slight tensile force on the sheet as it is being formed. As the punch begins to
Blankholder ~ Punch Laminate
/ J
|
Fig. 5.7. Friction during press forming.
170
A.M. Murtagh and P.J. Mallon
descend (Step 2), it encounters the laminate and the frictional force begins to develop between metal and composite. As the part is more fully formed (Step 3), the frictional forces reach a maximum, dependent on the normal pressure being exerted between the punch and cavity on the composite sheet. Once the part has fully formed (Step 4), all moving components come to rest and friction has no further part to play, although other flow mechanisms such as resin percolation and transverse flow can occur. This section has served to introduce each particular mechanism as it occurs for sheet forming of continuous fibre composites. The following sections will deal with each particular mechanism in detail. 5.2. Transverse fibre flow
Most research to investigate the transverse fibre flow phenomenon has been carried out using parallel-plate "squeeze flow" apparatus. Using the assumption that the resin adheres to the platen surface, a theoretical analysis relates the transverse intra-ply shear strain to the shear stress and the time scale involved [5]. This analysis also includes the concept of a yield stress value and the transverse flow viscosity may consequently be defined by FH 2 ~7 -
2VW2 L
(5.1)
where F is the applied normal force (N) at time t (s), H is the specimen thickness (m), L is the specimen length along the fibres (m), W is the specimen width across the fibres (m) and V is the transverse flow velocity (m/s). Work performed by Barnes and Cogswell [6] indicate a value of approximately 4,000 Pa s for this transverse flow viscosity. As part of the same study, the limiting transverse flow, i.e. how much the sample spreads upon the application of a force, was measured [6] and defined using a numerical technique as a function of pressure and thickness:
w0
= 1 + 0.4p1/3H
(5.2)
where Wp is the final width, W0 is the initial width, P is the pressure (MPa) and H is the thickness (m) of the sample. A similar experimental approach using a squeeze flow apparatus was taken by Mulholland [7] and a schematic of the test rig is shown in fig. 5.8. In this apparatus, two heated platens are compressed between the jaws of a servo-hydraulic 100-kN press which subjects the lay-up to transverse flow in the lateral direction. A linear variable differential transformer (LVDT) was used to record the squeeze flow as a function of time and from this the transverse flow velocity was calculated. Two sidemounted LVDTs were used to record the thickness variation of the sample as the pressure was applied. As an example, the apparent viscosity of a (0)16 sample of APC-2 as a function of time was calculated using eq. (5.1). In this example, the applied pressure was 410 kPa. Results of this are shown in fig. 5.9, together with the recorded squeeze flow velocity. Initially, the viscosity is infinity (before flow is
Shearing and frictional behaviour dur&g sheet forming
171
Fig. 5.8. Squeeze flow measuring apparatus. 400000 350000
9
'"'1
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-
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'
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-'l
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20
30
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l
50
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-0.01
'
60
Time (rain) Fig. 5.9. Transverse flow and apparent viscosity variation with time for (0)16 APC-2.
initiated) at point A. The initial flow viscosity around point B is approximately 60,000 Pa s, which slowly increases up to point C to a value of 300,000 Pa s as transverse flow occurs. This point at C represents the transverse flow "locking" effect when slight twisting in the fibres interferes with any further flow. When the consolidation pressure is increased at point C to 1 MPa, the higher pressure causes flow
172
A.M. Murtagh and P.J. Mallon
to re-start and the viscosity decreases once more to 60,000 Pa s at point D. After this, the sample was cooled and transverse flow ceased. The value of 60,000 Pa s observed for the initial flow viscosity is an order of magnitude higher than that recorded by Barnes and Cogswell [6]. However, lower consolidation pressures were used in their work. The effect of lay-up on consolidation quality was also investigated and fig. 5.10 shows three sample C-scans. Sample A ((90)8 lay-up) was constrained in the longitudinal direction but allowed to flow transversely. The B sample, (0,90,0,90)s lay-up, was unrestricted on all sides. Sample C ((0)8 lay-up) was arranged to prevent transverse flow parallel to the fibres, but allowed resin squeeze-out at the ends of the fibres. The C-scan for Sample A again shows the band of poor consolidation quality across the centre as seen previously. Sample B shows good consolidation quality and it was observed that there was minimal transverse flow in this specimen. Sample C shows that the resin squeezed out at the edges of the sample increases the void content substantially, indicated by the white regions along the top and bottom of the sample. Further evaluation of this work involved ultrasonic scanning of samples both with and without transverse flow having occurred. The effect of consolidation pressure can be seen in fig. 5.11 for eight (0)8 APC-2 samples. Scans (a,b,c,d) are for samples that had unrestricted transverse flow. Samples (e,f,g,h) were placed in a picture frame during consolidation and so squeeze flow was minimised. White areas on the scans indicate the presence of voids, and grey areas indicate better consolidation quality. The unrestricted samples show an increase in porosity as the consolidation pressure was increased. Most interesting is the white band that appears across the centre of each sample which grows thicker as the pressure is increased. A possible explanation for this would be the migration of resin from the centre of the laminate towards the edges where the squeeze flow was taking place, resulting in the creation of voids in the central region where the pressure was released. The other four samples (e,f,g,h)
Fig. 5.10. Ultrasonic scans for 8-ply APC-2 laminates.
Shear&g and frictional behaviour during sheet forming
173
Fig. 5.11. Ultrasonic scans for 8-ply APC-2 laminates consolidated at 0.41 MPa.
show the expected pattern, i.e. better consolidation quality as the pressure is increased.
5.3. Intra-ply shear A number of experimental and theoretical studies have been carried out to characterise axial and transverse intra-ply shear in unidirectional continuous fibrereinforced composite materials, specifically APC-2. For small deformations, experiments have been carried out using oscillatory flow techniques to quantify axial and transverse shear viscosities. Conventional rheometers operating in a torsional mode impose both axial and transverse shearing modes in a composite sample as shown in fig. 5.12. Ideal fibre reinforced fluid theory, first developed by Spencer [8], and adapted by Rogers [9], allowed both axial and transverse components to be separated out from experimental results on a number of samples. Work carried out by Groves [10] on APC-2 estimated the value for the axial intra-ply shear viscosity of 7,400 Pa s and for the transverse intra-ply shear viscosity of 6,100 Pa s, a ratio between the two viscosities of approximately 1.2. Cogswell [1] recorded slightly lower values, 6,000 Pa s in the axial and 3,500 Pa s in the transverse mode. At a higher angular velocity of 100 rad/s, Scobbo [11] measured an axial viscosity of 6,000 Pa s and a transverse viscosity of 3,500 Pa s. The average ratio of axial to transverse viscosity is approximately 1.5.
174
A.M. Murtagh and P.J. Mallon
Oscillating
" Axial shear
shear Fig. 5.12. Intra-ply shearing modes in oscillatory torsion.
All the studies performed indicate a highly non-linear response, with a yield stress of about 1,000 Pa. Comparison of the composite viscosity with the measured viscosity for the neat resin indicate a large increase in viscosity for the composite [11]. This would seem in part to be due to fibre twist and interference within the laminate. The temperature-dependence of melt viscosity for the composite material is the same as that for the resin. A 10~ increase in temperature reduces the viscosity by approximately 17% [5]. In fabrics, intra-ply shear in the 1-2 plane, which gives rise to the trellis effect as shown in fig. 5.13, can be investigated in purely kinematic terms to predict the trellis angle for a particular fabric strain. The effects of the resin are negligible on this
Instron
|
1. lnstron frame
5. Extension rod
2. Crosshead
6. Sample clamp
3. Load cell
7. Fabric sample
4. Top cooler
8. Environmental chamber
connection
i
_
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,, \ J O
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o(t)l Ic;;• / X /
"
\ /
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- - - - - - - 1 . ~ ,,. / \ / .,, fl ["4 \ / \ /
Fig. 5.13. Trellis-angle measurement on deforming fabric specimen.
\q
I I0
Shearing and frictional behaviour during sheet forming
157
effect. For a bi-directionally reinforced fabric orientated at 45 ~ to the shearing direction, the trellis angle can be predicted from the simple kinematic expression: cos 0 -- cos 00 exp(,kt)
(5.3)
where 00 is the initial fibre angle (45 ~ in this case), at time t = 0. This shows that movement is symmetric about the X-axis and that the fibre angle decreases as a function of applied strain rate ~ and is independent of the material constitutive law, i.e. is purely kinematic. The true axial strain (not to be confused with the engineering strain) in the sheet is equal to e l l - ln(~0t)) -- )~t
(5.4)
and we can see that the angle change is simply dependent on the axial strain. Experiments were carried out to verify this model. This involved shearing an appropriate fabric specimen (glass-fibre-reinforced polyamide, 7-H satin weave, Vestopreg G101 | at processing temperature inside an environmental chamber [12]. The fabric was gripped using special jaws which were in turn connected to an Instron straining frame, as shown in fig. 5.13. A cross marked in ink at the centre of the specimen was observed as the specimen was strained vertically. The change in angle was recorded and fig. 5.14 is a plot of true strain against predicted trellis angle (from eq. (5.3)) and the observed trellis angle. The experiment shows relatively good agreement with the theory. Differences may be attributed to uneven shearing at the ends of the specimen, which was caused by the presence of the grips which 50 Model 40
D
Specimen
30 Trellis angle (e)
20
10
'Locking'
,=,,,=
angle
0.0
0.1
0.2 True strain
0.3 I; ! 1
Fig. 5.14. Comparison of model and test values for trellis angle.
0.4
176
A.M. Murtagh and P.J. Mallon
constrained the sample from reducing in width. The shaded region indicates the spread of values for which the trellis locking angle was observed, i.e. the strain at which out-of-plane buckling was first observed visually on the specimen. Thus for G 101 fabric, the locking angle lies between 25 ~ and 30 ~ This value is in accordance with that observed by other researchers [13,14]. The ideal fibre-reinforced fluid constitutive model for one family of fibres has been extended to include two families of fibres by Johnson [15]. The original model [8] defines stress in terms of strain rate and two viscosity terms/71 and/Tt, the axial and transverse viscosities. For a fabric, a third viscosity term must be included: 1
O'11 -- ~, [4/71 -- (3/71 -- 2/72) sin 2 20 -- ~/73 sin2 40]/sin 4 0
(5.5)
all is the stress in the 1-1 direction, Z is the shear rate and 0 is the trellis angle. The three viscosity terms are denoted by/71,/72 and/73. They may not be related to simple shearing modes in any particular plane in the fabric, unlike/Tz and/Tt which can be related to shear in the 1-3 and 2-3 planes for unidirectional materials. Instead, they may be consider as "mixed-mode" parameters. Using the same experimental set-up that was used to verify the kinematic model for trellis angle prediction, it was possible to obtain an estimate for these three viscosities for the G101 specimen. Three samples were sheared at a constant shear rate at three different temperatures. Tensile stress in the 1-1 direction was calculated as a function of tensile force and the instantaneous cross-sectional area of the sample across the centre. A plot of true strain against tensile stress is shown in fig. 5.15, for 225~ 245~ and 265~ 5.00e+6
al
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"9 3.00e+6
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Exp 265~
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2
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m
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0.00
-
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0.10 True strain
0.15 ~
Fig. 5.15. G101 fabric stretch test modelled using eq. (5.5).
11
0.20
0.25
Shearing and frictional behaviour during sheet forming
177
Equation (5.5) was used to model the stress response as shown. Note that the initial elastic-viscous stretching of the specimen (around region A) is not modelled as the model assumes steady flow. The value of the three viscosity parameters r/l, 02 and 03 are given in table 5.1. Values for r/1 decrease as temperature increases, while conversely values for r/2 increase with temperature. This would not be expected for "real" viscosity components but may be possible for these "mixed-mode" parameters. The ratio between 01 and 02 decreases from 13.33 to 1.71 as temperature is raised from 225~ to 265~ At lower temperatures, an adequate curve fit was achieved by setting ~2 = 0, which simplifies the model considerably. The optimum value for 773 was found to be approximately zero for all conditions.
5.4. Inter-ply slip Inter-ply slip in unidirectional materials occurs due to the fact that the reinforcing fibres in a composite laminate may be considered to be inextensible. When forming single or double curvature shapes, where the stack of plies making up the laminate may be considered as a number of inextensible, yet flexible, plates, then a relative displacement must occur between the layers to accommodate the different path lengths of each ply around the bend. Micrographs taken at the edge of formed parts with initially ground and square edges serve to illustrate this phenomenon [12]. The total slip deformation, d, can be shown to be a function of the bend angle, 0, number of plies, N, and the thickness of each ply, t (see fig. 5.16), and is given by Ot
d - ( N - 1)~-
(5.6)
T A B L E 5.1 P a r a m e t e r s for fabric m o d e l T e m p e r a t u r e (~
O~ (Pa s)
7"]2 (Pa s)
773 (Pa s)
265 ~ 245 ~ 225 ~
1.2 e 7 3.0 e 7 4.0 e 7
7.0 e 6 3.5 e 6 3.0 e 6
0 0 0
t
0
Fig. 5.16. Ply slip a r o u n d b e n d angle 0.
178
A.M. Murtagh and P.J. Mallon
Figure 5.17 is taken from a (0,90,0,90,)s APC-2 laminate formed over a 90~ The slip occurs in the resin layer between each ply and intra-ply shear within each ply is negligible. The step-like slip effect can be seen along the edge of the laminate (for example at point A, the interface of a 0 ~ and 90 ~ ply). The only point at which no slip can be seen is at the central axis of the laminate, at the interface between two 90 ~ plies (point B). Other cross-ply laminates show similar degrees of slip fig. 5.18 shows the deformed edges of a quasi-isotropic (0/+ 45/90/-45)s 90~ part. Resin layers can be seen as dark lines between the plies. Some intra-ply shearing is evident on the top 0 ~ and adjoining +45 ~ ply, possibly due to friction against the tool during forming. The slip behaviour of unidirectional (i.e. all plies oriented at 0 ~ laminates contrasts greatly with that for cross-ply parts. In most cases, for pre-consolidated unidirectional parts, the dominant flow behaviour was intra-laminar, with no obvious resin layers existing and with the entire thickness of the laminate behaving in many respects as a single ply. This is shown in fig. 5.19 for a (0)8 90~ part where the
Fig. 5.17. Inter-ply slip for (0,90,0,90)slaminate.
Fig. 5.18. Inter-ply slip for (0/+45/90/-45)s laminate.
Shearing and frictional behaviour during sheet forming
179
Fig. 5.19. (0)8 l a m i n a t e - intra-ply shearing.
flow is even across the thickness, and there is no distinction between plies. One exception to this rule was observed with laminates pre-consolidated at low pressure (100 kPa) before forming. Figure 5.20 shows a distinct step-like behaviour (no apparent resin layers) with each ply seeming to act independently. This may be caused by the low consolidation pressure not causing sufficient fibre/fibre interaction between plies to, in effect, "fuse" the plies together. In fabrics, the inter-ply slip behaviour can differ markedly from that seen in unidirectional materials. The existence of a crimp in the fabric due to the woven nature of the tows means that the assumption of inextensibility does not hold. When a tensile force is placed across the plies during forming, the crimp allows fibres to straighten to a certain degree. The amount of possible stretch varies depending on the weave style [17]. This is shown in fig. 5.21 - - the fibre straightening factor (FSF) is the possible percentage change in length of a ply segment along its length. It is a maximum for the most highly crimped style, i.e. plain weave. To measure this elongation in a ply under tension, specifically for Cetex fabric, rectangular samples were tested in an environmental chamber and their load response to an applied displacement was measured [16]. Figure 5.22 is a plot of
Fig. 5.20. (0)8 laminate - - step-like behaviour.
180
A.M. Murtagh and P.J. Mallon
Fig. 5.21. Possible tow straightening in fabrics.
0. =E
Cetex fabrics:
-
B
Temperature 320~ Speed 0.5 mm/min I a
5-H weave 9
.
Plain weave .Model 5,.H
m. r
I-
m
0
. - ~ -
|
1
....
2 Strain
|
3
(%)
Fig. 5.22. Stretch testing of Cetex fabric specimens.
strain in the longitudinal direction against tensile stress, showing the response for both a plain weave and a 5-H satin weave sample. The behaviour shows that there is an initial strain in the samples before the stress begins to increase exponentially, i.e. a "strain-hardening" phenomenon. The plain weave allows for more tow stretching compared to the 5-H satin weave sample. The stress/strain behaviour of the 5-H satin weave sample was modelled using a curve-fitting technique as follows: r~ - 0.03061 e 2"1003(e)
(5.7)
where a = tensile stress and e = strain in specimen. This model is useful in predicting internal tensile stresses in a deforming fabric ply and can be used in conjunction with an understanding of the inter-ply slip behaviour of the material to form part a numerical model for sheet forming of Cetex fabric. As stated previously, inter-ply slip occurs in the thin resin interlayer between plies. In order to obtain a thorough and quantitative understanding of the nature of the shear stresses to be encountered in this layer during forming, a number of analyses
Shear&g and frictional behaviour during sheet form&g
181
by different researchers have been carried out. Cogswell [1] was the first to identify the mechanism for unidirectional tapes, specifically APC-2, and carried out some initial slip characterisation studies. Scherer [18] obtained experimental results from an inter-ply slip apparatus taking into account the effects of shearing velocity, temperature, pressure and lay-up for a carbon-fibre-reinforced polypropylene unidirectional composite. A model obtained using these results was used as input to a finite element analysis on the inter-ply slip process during thermoforming [19]. Xiang Wu [20] also performed pull-out tests on 16-ply uniaxial APC-2 laminates and modelled the forming of 90~ parts as approximating 3-point bending. He calculated the apparent viscosity in the resin-rich interlayer and noted an order of magnitude difference between measured values for a (0)16 laminate and predicted values for the neat resin. Groves [21] observed this viscosity increase in APC-2 using a dynamic spectrometer and one explanation proposed was possible fibre friction and interference between layers. Further investigations by Jones [22] showed that the observed non-Newtonian behaviour of the composite and the assumption of the individual plies behaving as buoyant plates was not due to inertia effects and instead was due to intra-ply shearing. Kaprielian and O'Neill [23] performed pull-out experiments on a steel plate from between stacked plies of APC-2 and were able to correlate results with a model based on the ideal fibre-reinforced fluid constitutive equation [8]. Soll [24] performed tests to optimise the forming conditions when forming simple rightangled bends using a unidirectional material and noted that a tensile force applied across the laminate during forming increased part quality by reducing fibre wrinkling. Tam and Gutowski [25] developed a linear visco-elastic model of the inter-ply slip process, modelling the fibre-rich plies as a linear elastic layer and the resin interlayer as a viscous layer. This model allows prediction of laminate stresses and displacements, and strains in the slip region. Results showed good correlation with isothermal forming experiments. Scobbo [11], using DMA analysis, observed the effects of fibre interaction between adjoining plies with fibre effects beginning to dominate visco-elastic behaviour at higher temperatures. More recently, Morris and Sun [26] have performed inter-ply slip experiments on APC-2 laminates at lower temperatures, approaching and below the melt temperature, Tm, and have modelled this solid-phase forming behaviour. The remaining part of this section will describe and summarise part of another inter-ply slip study performed by the authors [12,16,27]. This work was carried out with the aim of simulating the conditions of slip during forming in a controlled manner and examining the effects of different forming parameters (slip velocity, temperature, normal pressure and fibre orientation) on shear stresses in the interlayer. Experiments were carried out using a consolidation unit [7] together with a custom-built shearing apparatus [28]. The consolidation unit consists of two heatable platens mounted on a 100 kN hydraulic press. The shearing apparatus is made up of a motor driven leadscrew which performs the pull-out motion on the horizontally mounted sample lay-up. A 500 N load cell recorded the shearing load. A schematic of the apparatus is shown in fig. 5.23. Figure 5.24 is an overall view of the shearing apparatus and the consolidation mounted together on the hydraulic press.
182
A.M. Murtagh and P.J. Mallon
Consolidation unit
Exterior ply clamp
DC M otor
Load cell
Shearing apparatus
Fig. 5.23. Consolidation unit/shearing apparatus for inter-ply slip testing.
Fig. 5.24 Test set-up for inter-ply slip experiments.
The sample lay-up comprised of a central pull-out ply, two exterior stationary plies and intermediate plies with various fibre orientations. For the APC-2 sample shown in fig. 5.25, the lay-up is (0, 90, 0)s the central and two exterior plies are oriented at 0 ~ to the pullout direction, and the two inner plies are at 90 ~ A narrow shim of material behind the pull-out ply keeps the gap between top and bottom platen constant to avoid "pinching" of the specimen. All APC-2 and Cetex fabric specimens were first consolidated at a pressure of 1 MPa for 5 minutes prior to testing. A specified normal load (usually 100 kPa) was then applied via the hydraulic press for the duration of the pull-out test. The shearing
Shearing and frictional behaviour during sheet forming
183
Shim
Fig. 5.25. Specimen geometry for inter-ply slip testing.
force and ply displacement were recorded for a particular shearing velocity. The shear stress was calculated simply by dividing the shear force by the sheared area using Fs
r -
(5.8)
w ( L - d)
where r is the shear stress, Fs is the shearing force, L and w are given by the specimen geometry and d is the displacement of the pull-out ply at a particular time t. Figure 5.26 show the results for a particular test, where the shearing velocity has been increased in increments. Each increase in velocity causes a corresponding increase in the associated "steady" shear load, as shown. This allows the shear
0.4
250 u
(0,90, 0)S APC-2 CP 1 MPa, NP 0.1 MPa
200 A
Z v
SPT 375~
0.3 A
150
E
"o
...o
0.2
3
l_
vE
8
2 .-._,,ml
m
L.
0.1
50
Load
Velocity 0
100
200 Time (sec.)
Fig. 5.26. Velocity increments during inter-ply slip test.
300
] 400
0.0
=
,,r
184
A.M. Murtagh and P.J. Mallon
velocity/shear stress relationship to be determined from an average of single sample tests for a particular set of conditions. This technique was used throughout this test programme to obtain all the shear velocity/shear stress plots shown. The effect of processing temperature has a large effect on the shear deformation behaviour of composites. This is due to the resin viscosity, which is strongly dependent on temperature. The reinforcing fibres are unaffected. For APC-2, the melt temperature is 343~ so theoretically a laminate can be formed at any temperature above this. It is recommended that processing should occur in a window between 360~ and 400~ Isothermal conditions, and a constant resin viscosity is readily achievable with diaphragm forming, where the entire laminate is held in a heated chamber. For press forming, temperature may be uniform initially, after being transferred from the heating unit, but decreases rapidly, due to convection cooling in the air. Conduction cooling occurs once contact is made with the tool, which may be around 250~ Indeed, forming may actually occur at temperatures less than 343~ for APC-2 under certain conditions, but in this case, inter-ply shearing may result in a high level of shear stress [26]. Figure 5.27 shows the effect of varying temperature on a (0, 90, 0)s sample. As expected, raising the temperature decreases the shear stresses in the laminate. However, at higher temperatures, problems such as surface oxidation and excessive in-plane fibre wrinkling or "washing" (due to the low viscosity of the resin) may occur. For an anisothermal process such as press forming, a wide variation in temperature is unavoidable. The effect of releasing normal pressure completely during a typical slip test after consolidation is shown in the initial part of fig. 5.28. After reaching a "yield" level of 12
a
,
,,
,,,,
,,=
(0,90,0)S APC-2 CP 1 MPa NP 100 kPa 10
A
J tJ
360~ 3700C
L
l
A v
Ir
380~ 390~ 400~
A
ml|
0.0
|
i
!
0.1 Shear velocity (mmls)
Fig. 5.27. Effect of temperature on inter-ply slip of APC-2.
I
0.2
Shear&g and frictional behaviour during sheet forming
185
2.5
200 a
(0,90,90,0)S APC-2 SPT 385~ CP 1.5 MPa 2.0
150
A A
zv
Shear load
_8 loo | _m
Pressure
1.5
x
i_
1.0 m
m
50
0
0.5
0
200
400
600
800
o Z
0.0
Time (sec.)
Fig. 5.28. Variation of shearing force with normal pressure.
100 N, shear load relaxes to a reduced level of 25 N. However, once pressure begins to increase, to about 100 kPa, shearing forces recover to a higher level than the original maximum (120 N). At a constant pull-out velocity, as pressure is increased further, the shearing force continues to increase, but the magnitude of further increases is much smaller than the initial "jump" when pressure was first reapplied. The initial jump increased stress levels by 500% - - at a pressure of 2 MPa (20 times the original step increase), shear stress has only increased by a further 50%. One possible reason for this is that under no normal force, the plies are free to disengage from intimate contact and can slide semi-freely against one another. However, as soon as a nominal normal pressure is re-applied, the plies come together once again, intimate contact is re-established and viscous slip in the layer between plies resumes. Further increase in pressure does not greatly affect this viscous slip. Theoretically, for an incompressible fluid, viscous forces should be independent of normal pressure, in the absence of a pressure gradient along the direction of flow. In practice, pressure increases do increase the shear stresses, possibly due to increased fibre contact, but not significantly. This effect need not be crucial in terms of diaphragm forming, where there is always a normal pressure present in the form of a vacuum applied between the diaphragms. As the pressure rises and the part begins to form, a hydrostatic pressure across the area of the laminate would cause an increase in shear stress. However, most forming would occur at low pressures, as there would be no reaction force against the back side of the bottom diaphragm until it made contact with the tool, when inter-ply slip would have ceased. For press forming, the situation is more
186
A.M. Murtagh and P.J. Mallon
complex. Initially, the laminate is heated to forming temperature and is under no normal pressure, except for the parts of the surface directly beneath those parts of the forming die which first come into contact with the top ply. At a constant pull-out velocity, as pressure is increased further, the shearing force continues to increase, but the magnitude of further increases is much smaller than the initial "jump" when pressure was first reapplied. Thus, some regions may undergo inter-ply slip, under no normal pressure, until the forming die comes into direct contact. The pressure distribution across the laminate is much more difficult to predict than with diaphragm forming. Further tests were carried out to establish the shearing velocity/shear stress relationship as normal pressure was varied from 20 to 400 kPa. This involved preconsolidating (0, 90, 0)s APC-2 specimens at 1 MPa. Pull-out tests were then carried out on each specimen under a particular normal pressure, and the shear stress level was recorded as shearing velocity increased. Temperature during this programme of experiments was kept constant at 385~ Results are shown in fig. 5.29. As expected, shear stress rises as normal pressure is increased. Figure 5.30 shows a plot of normal pressure plotted against shear stress at two typical shearing velocities, 0.075 and 0.22 mm/s. This plot shows further evidence that significant increases in shear stress occur at lower values of pressure, especially so for the higher velocity. Once pressure increases beyond 100 kPa, the increase in shear stress is less pronounced as the pressure rises further. To examine the effect of varying fibre orientation, a series of experiments was carried out under standard conditions of pressure and temperature, for different lay30ee
(0, 90, 0)S APC-2 SPT 385~ CP 1 MPa
Normal
pressure (kPa) " " 20 eL ,=~
:':- -9- . , . e - - - -
v
w
f~
I
t-.
tll @
,=
I
0
0.0
i
0.1
,I
I
0.2
9
I
|
0.3
0.4
Shearing velocity (mmls) Fig. 5.29. Effect of normal pressure on inter-ply slip of APC-2.
9
0.5
40
-'-
100
i
200
----o---10
20
400
187
Shearing and frictional behaviour during sheet forming
20
A
cO
A.
Shear stress at velocity"
v
T= lo
.-e-
(4
-e=- 0.22mm/s
t,=
0.075mm/s
@
0
100
200
300
400
Normal pressure (kPa)
Fig. 5.30. Significant increase in shear stress at low normal pressure. up orientations. As it was not possible to change the orientation of the pull-out and exterior plies to any angle other than 0 ~ relative to the pull-out direction, the fibre angle of the free ply between the exterior and pull-out ply was varied. Thus the layup for a particular experiment may be expressed as (0, 0, 0)s, where 0 is any angle between 0 ~ and 90 ~ Figure 5.31 shows a plot of shear velocity versus shear stress for a number of lay-ups. Lay-ups where the fibres in the free ply lay at some angle other than 0 ~ to the pull-out direction gave similar results, showing that the resin layer in each case was fully existent and of similar proportions for all orientations. Some 30 (0, 0,0)S APC-2
01010
SPT 385~ =
013010
n_ 20
-=
0/60/0
m
=
0/90/0
CP 1 MPa NP 0.1 MPa
014510
A
(II
r, L_
Q
,c
10 "
,
0.0
1
i
i
illll
i
|
iii
0.1
i
i|1
i
illl
0.2
Shear velocity (mmls)
Fig. 5.31. Effect of lay-up variation on inter-ply slip of APC-2.
i
i
0.3
i
188
A.M. Murtagh and P.J. Mallon
fibre rotation was observed, especially in the 30 ~ and 45 ~ lay-ups. Fibres in the free ply tended to re-orientate themselves to become aligned with the fibres in the pull-out ply and the more grossly displaced specimens showed most evidence of this. In the (0, 0, 0)s lay-ups, shear stresses were much higher than that for angled layups: in this case, it might well be assumed that a distinct resin layer was not formed during consolidation, and shearing actually occurred through the thickness of the middle ply. Micrographs taken through a section of a (0)8 laminate indicate the absence of any distinct resin layer [12]. Further evidence of this inhibited form of shearing in (0, 0, 0)s lay-ups is shown in fig. 5.32, which illustrates the instability in shear stress as shearing occurred at a steady velocity. This is probably due to fibre interference and entanglement between plies. Observations on tested specimens showed that this was indeed the case, with some fibres being grossly distorted and buckled. As already mentioned, the effects of fibre interaction have also been observed by other researchers [11]. If we now consider a further analysis of the different slippage behaviour between the parallel-plied (0, 0, 0)s and other cross-plied lay-ups ((0, 90, 0)s (0, 45, 0)s, etc.), we can relate the different shearing rates that occur to determine an inter-ply slip viscosity. If we assume the cross-plied lay-ups to behave in the same fashion, i.e. a 6 pm resin layer being sheared between the plies, the associated shearing rate in this layer can be calculated and from this the viscosity can be found. For the parallelplied lay-up, shearing occurs throughout the thickness of the ply between the exterior 500
-
0.5 o
(0,0,0)S APC-2 SPT 385"C C P 1 MPa NP 0.1 M P a
400
z -"9 Io t~ o
0.4
300
0.3 Unstable
1
e=
G
200
0.2
#
l
100
0
0
200
IlL
,=.,.
i....
r @ ,c
I
0.1
--D- Shear load --
Velocity
400
o @ > ==
m
600
800
T i m e (sec.)
Fig. 5.32. Stress instability during (0, 0, I))s pullout.
1000
1200
0.0
r
Shear&g and frictional behaviour dur&g sheet form&g
189
and pull-out ply so the sheared layer thickness is equal to the ply thickness (125 ~tm). The viscosity is this case must be much higher due to the lower shear rates and this is shown in fig. 5.33. The viscosity of the parallel-plied lay-up ( > 20,000 Pa s) is almost two orders of magnitude greater than the viscosity of the cross-plied lay-up ( < 1,000 Pa s). The viscosity seen in the cross-plied lay-up is similar to the viscosity of neat PEEK resin as measured by Cogswell [5]. The slip viscosity for the (0, 0, 0)s lay-up may be related to the longitudinal intra-ply shearing mechanism, as mentioned in the previous section. Inter-ply slip of fabrics is affected by the tow straightening effect, as mentioned previously. Pull-out experiments on fabric samples were further complicated by having two directions of reinforcement tensile force could be applied to fibre tows lying in the pull-out direction, but the transverse tows at 90 ~ to the direction of pull-out tended to remain behind in the sample whenever deformations over a few millimetres occurred. This effect was not so pronounced in unconsolidated specimens and shear results shown here are for samples tested at lower pressures than would be expected during full consolidation. Figure 5.34 illustrates the difference observed in shear behaviour between unidirectional and fabric materials during a typical pull-out test. Both results shown are for a carbon fibre/PEI material (the fabric is 5-H satin weave), tested under similar conditions of heat and pressure. The 0~ pull-out ply in the (0, 90, 0)s allows direct transmission of the traction load to the sample and the measured shear load quickly rises to a steady level. For the fabric material, the shear load
100000
-
10000 A I0 v
s
/
F" : -
Parallel-plied layup
Cross-plied layup
looo
0 O
lOO
10
1000
i
a
i
J
t
a
i
iJ ........
I
I
10000 Shear stress (Pa)
Fig. 5.33. Inter-ply shearing viscosity for parallel-plied, cross-pSed lay-ups.
I
I
I
I I
190
A.M. Murtagh and P.J. Mallon
120
i-
lOO
t 7 b~176176 F ......... ,,
, ,,
,
Material:
CF/PEI
,
SPT 320 ~ NP 100 kPa
8O Z o
60
I
r a=
r
Unidirectional"
Transistion 55N
ID
,
A
i. , ..... i
_ L LI __.111 .
slip
(0,90, 0)~ Fabric"
40 Fibre straightening
20 0
([o/9o1, [o.9oi ),
0
50
100
150
Time (sec.)
Fig. 5.34. Direct comparison of unidirectional/fabric materials.
increases in a different f a s h i o n - initially, the load applied to the ply causes the fibre tows to straighten and this continues up to point A. Here, the nature of the load increase changes from an increasing to a decreasing rate. This point, at 55 N, can be regarded as the required load to cause inter-laminar shearing along the full length of the sample at the interface between two plies, rather than cause the tows to straighten any further. The stationary level of shear stress reached thereafter is composed of two parts the tensile force in the stretched fibres plus the viscous traction force required to shear the plies. The transition load is a function of the total amount of fibre stretching, in both the area of the sample under normal pressure and the pull-out ply length between the sample and ply clamp on the shearing apparatus. Any analysis of inter-ply slip in fabrics must take account of initial fibre stretching. Shear velocity/shear stress plots may be generated in a similar fashion to unidirectional materials by determining the steady shear load levels corresponding to various shear velocities. The effect of processing temperature on inter-ply slip of fabrics may be regarded as similar to that for unidirectional materials once fibre stretching has been taken into account. Figure 5.35 shows a plot of shear velocity versus shear stress for a Cetex 5-H satin fabric sample, sheared at temperatures between 300~ and 340~ Normal pressure remained constant at 100 kPa for each test. As with APC-2, an increase in temperature of 40~ reduces the level of shear stress occurring in inter-ply slip substantially, due a decrease in viscosity of the PEI matrix material. Localised inplane wrinkling of surface fibres that can occur at increased processing temperatures
Shearing and frictional behaviour during sheet forming
191
50
NP 100 kPa 3 plies 5-H Cetex 40
A t~
30
W
t= i_ tll Q
20 t
tn
300"C
10
,
_=
,
r. 0
I
0.0
|
!
0.1
I
320oc 3400C
,
0.2
I
0.3
Shearing velocity (ram/s)
Fig. 5.35. Effect of processing temperature on Cetex fabric slip. with unidirectional fibre-reinforced composites is not such a large problem with fabrics as the woven nature of the tows constrains any excessive movement of fibres. For variation of normal pressure, tests were performed between applied pressures of 20 kPa and 400 kPa. Figure 5.36 shows the obtained results; load cell limitations reduced the attainable shearing velocity to approximately 0.25 mm/s at higher normal pressures. As with APC-2, a significant increase in shear stress is seen as the pressure is increased. The magnitude of the recorded stresses are much higher compared with the unidirectional material. Even allowing for matrix and viscosity differences, there is a much greater resistance to inter-ply slip for the Cetex material, again probably due to the nature of the interaction between adjoining plies, with much more interference to sliding being caused by the uneven surface of the woven plies. Initial test results showed that inter-ply slip of laminates was not initiated until a certain yield point had been reached. Even at low shearing velocities, pull-out forces required to shear the laminate were substantial. In order to determine this value more accurately, the test set-up was changed. Rather than use the leadscrew to provide the pull-out force, a pulley system using dead-weights was installed at the front of the shearing frame. To measure the very small displacements at loads approaching the yield point, two miniature LVDTs were mounted behind the ply clamp (see fig. 5.37). This allows very small movements of the central ply to be recorded. Care was taken to exclude any movement of the ply clamp and any displacement necessary to take up slack in the pull-out ply. The yield load was defined
192
A.M. Murtagh and P.J. Mallon
50
40
Normal
pressure (kPa) lg a,. A
30
20
w
"-='-
20
--e--
50
=
100
--o-
200 400
m @
10 SPT 320~
3 plies 5-H Cetex 0
9 0.0
~ ,, 0.1
,
I 0.2
i
I 0.3
=
I
0.4
"
0.5
Shearing velocity (mm/s)
Fig. 5.36. Effect of normal pressure on Cetex fabric slip.
.
0 LVDT A
3
...
Pulley
Load cell
o I
LVDT B Dead
weights
Fig. 5.37. Positioning of LVDTs for yield measurements.
as the force required to cause an irrecoverable displacement of the pull-out ply. For unidirectional materials, this point was relatively easy to determine as pull-out force is transmitted directly through the straight fibres to the specimen, For fabrics, however, initial fibre straightening meant it was difficult to separate fabric stretching from true slippage displacement in the lay-up. Inter-ply shear tests were carried out under various conditions of temperature, pressure and lay-up to see their effect on the yield point. Figure 5.38 shows a typical yield plot of applied dead-weight loading and recorded displacement against time for a steel foil sheet being sheared from between two plies of APC-2 (i.e. friction of
Shear&g and frictional behaviour dur&g sheet forming
i
'F
40
8
30
6
193
-
20
(g
2
t~ 100 0
0 500
1000
1500
2000
Time(sec.) Fig. 5.38. Yielding situation for APC-2/steel foil.
composite against a smooth surface see section 5.5). This result shows the ideal s i t u a t i o n - no displacement is seen until the critical load is applied, when a definite yielding of the ply is observed, and seen to increase steadily as the load is kept constant. This sudden yield occurs only when there is no interaction between fibres from adjoining layers. Further increases in applied load causes the displacement to occur at a faster rate. In reality, most samples with two plies interacting did not show this behaviour. Instead, initial yielding showed a slight displacement (as in the initial part of fig. 5.39), followed by an almost complete cessation Of movement. This behaviour represents an elastic/plastic effect and cannot be regarded as a true yield in terms of continuous inter-ply slip. Instead, the yield point can be defined as the point at which irrecoverable, steady shear flow commences. A typical response in yielding is shown in fig. 5.39, for a (0, 0)s APC-2 sample. Here, as would be expected, application of an increasing dead-weight force causes movement of the pull-out ply, but without continuous sliding occurring. At point A, when 45 N of load has been applied, displacement of the pull-out ply has reached approximately 0.2 mm. Region B shows the response when the load is taken off; the ply displacement recovers back to practically a zero value, indicating an elastic effect. This effect may be due in some measure to the fact that as loading occurs, fibres from adjacent plies interact and cause an elastic resistance to shearing, and "spring-back" when the load is removed. Although this behaviour was observed in all lay-ups of APC-2, it was most evident in the (0, 0)s sample. This behaviour in the fibre direction may be related to a similar elastic "spring-back" effect observed in the through-
194
A.M. Murtagh and P.J. Mallon
100 ' " l Note: Noise in the load cell signal [ causes each dead-weight load to
-I 2.0 ,,--.~
80 I flickerto some degree
n
60 ~- (0,O)sAPC-2 [ SPT 385"C '~k~ 40 P I a CPr 1MPaNP100
I
I
0
A /-d
Load
ne
/
I
ii
]
1.5
/!
/I
fl 1"0 e=eue 0.5
i~
20
o0
250
500
750
1000
io.o
1250
Time (see.) Fig. 5.39. Yielding of a (0, 0)s sample.
thickness direction, in other research by Muzzi [29], where the fibre bundles are assumed to behave like coiled springs. Applying a compressive force to a bundle of stiff, slightly wavy fibres causes them to compress elastically, and when the compressive load is removed they can recover. In shear, adjoining fibre layers m a y impress u p o n one another under pressure and slippage of one layer across the other causes a slight elastic effect where some fibres m a y " s n a g " temporarily. Once shearing is halted, elastic recovery of any " s n a g g e d " fibres m a y occur. Increasing the load to the level shown at C caused complete yielding of the specimen. Table 5.2 summarises the results for various yield stress measurements carried out on unidirectional materials under different conditions. F o r APC-2, the yield stress TABLE 5.2 Yield stress values for APC-2 Material
Lay-up
Conditions
APC-2
(0, 90, 0)s
365-405~ NP 50,100 kPa NP 400 kPa 365-405~ NP 50,100 kPa NP 400 kPa
(0, 0)s
Yield stress (kPa) 1.1• 1.2-t-0.2 2.4• 2.6+0.2
Shearing and frictional behaviour during sheet forming
195
more than doubles (1.2 to 2.6 kPa) when the fibres are in an aligned state, i.e. (0, 0) s, compared with a (0,90) arrangement, or for any other lay-up where adjacent plies lie at an angle 0 to one another. To model the inter-ply slip behaviour of APC-2 and Cetex fabric, a modified form of the Herschel-Buckley power model was used as shown (v = shear velocity): (5.9)
Z" = "gyield + k(v) n
This allows a value for a yield s t r e s s (Z'yield) t o be inputted and then a power-law relation can be used to describe the viscous flow in the resin interlayer between plies. The values of the parameters Z'yield, k and n are dependent on the various process conditions (temperature, normal pressure, fibre orientation). Using the experimental data obtained, a curve-fitting technique was used to determine relationships between the process conditions and between ~'yield, k and n. For example, the effect of normal pressure on shear stress for APC-2 can be described by r(P, V) = (0.95 + 1.28 e-3(P)) + (-28.639 +
31.143(logP))(V~176176176 (5.10)
i.e.
"~yield - - 0.95 + 1.28 e-3(P) kPa
k = (-28.639 + 31.143(log P) n = 0.1635 + 0.3079(log P) valid for: velocity 0 < V < 0.5 mm/s, and pressure 20 kPa < P < 400 kPa. The curve-fits for the effect of normal pressure are shown in fig. 5.40. In order to combine these three different models into one master equation to calculate shear stress for any arbitrary combination of temperature, T, relative fibre angle, 0, and normal pressure P, in terms of shearing velocity, V, a factoring technique was applied. Using this method, a standard set of values was used to normalise all other points, which results in three factors fr, fp and fo, which, when multiplied by the original standard value, give the shear stress for any set of parameters. The standard conditions chosen for APC-2 were as follows: 9 Temperature 385~ 9 Normal pressure 100 kPa 9 Fibre orientation 90 ~ This results in the standard model: rs = 1.0 + 28.7083 V 0"8152
(5.11)
Then we can write
Apc-2(v, T, P, O) =iT "Up- Jb-
(5.12)
or "~APC_2(V, T, P, 0) =
r(v, T) r(V, e) r(v, 0) rs
rs
rs
(5.13)
A.M. Murtagh and P.J. Mallon
196 35
3O
20 kPa model ---2
40 kPa model
25
-"-= 100 kPa model
tl Q" 20
"-'-8 400 kPa model
__.4 200 kPa model
A
L..
o r
10
0 0.0
0.1
0.2
0.3
0.4
[]
20 kPa exp
0
40 kPa exp
&
100 kPa exp
X
200kPaexp
4-
400 kPa exp
0.5
Shear velocity (mints) Fig. 5.40. Model versus experimental values - - pressure variation.
A similar method was applied to modelling the behaviour of Cetex fabric, the exception being that the possibility of tow stretching has to be taken into account. From fig. 5.34, it is known that the transition between tow stretching and inter-ply slip occurs at a shear stress of 5.5 kPa. By equating shear stress in the interlayer to tensile stress in the ply at this point, we can write: ~'A s - - f i a t
(5.14)
where As is the sheared area and A t is the cross-sectional area of the ply under tension. Assuming constant width, we can than deduce that the tensile stress in a ply, that causes a shear stress of 5.5 kPa, can be represented by a-~
rLf
= 1.667 e7Lf
(5.15)
where r = 5.5 kPa, and tp = ply thickness = 0.33 mm Thus the effective "span" length, Lf, of the structure being formed, i.e. the length of the ply undergoing shear, is critical in determining the tensile stress. If the flange length Lr is sufficient that the total required amount of inter-ply slip (depending on geometry conditions) can be accommodated as pure fabric stretching, then no slippage between the plies need occur. Thus in a numerical simulation of press forming, a check should first be made to calculate the required amount of inter-ply slip from eq. (5.6). Using this value, and by calculating the required amount of longitudinal strain, the associated tensile stress in the deforming ply can be found from eq. (5.7).
Shearingandfrictional behaviourduringsheetforming
197
If the resulting tensile stress is sufficient to cause a shear stress of greater than 5.5 kPa in the interlayer, then inter-ply slip must occur and the power-law part of the model must be introduced. Similarly to APC-2, the effect of temperature and normal pressure on shear stress can be modelled using a curve-fitting/normalisation technique. The effect of fibre angle was ignored for the fabric. The standard set of conditions chosen for Cetex are as follows: 9 Temperature 320~ 9 N o r m a l pressure 100 kPa This results in the standard model: rs = 5.5 + 55.246(V) 0"4902
(5.16)
The master model thus obtained may be described by
r(v, 7") r(v, e)
"~Cetex(V, T, P) = ~
rs
~~'s rs
(5.17)
For both materials, the inter-ply slip model to predict inter-layer shear stresses may be condensed into an alternative general form as
T--(~TY(i)-~-kigni) "~s
(5.18)
The value of rs, yield stress ry(i) and of the power-law parameters k and n are given in tables 5.3 and 5.4. for APC-2 and Cetex respectively.
5.5. Friction during thermoforming In sheet forming of thermoplastic composite sheet, friction must occur due to the motion of the composite against the contacting surface which transmits the forming force. In the case of diaphragm forming, friction occurs between the composite and diaphragm, and between the diaphragm and tool surface. The study of this type of
TABLE 5.3 Inter-ply slip power-law model parameters for APC-2 Material: APC-2 rs = 1.0 + 28.708V~ i
ry
k
n
Conditions
1 2 3
1.0 0.95+(1.28 e-3)(P) 1.0 2.4
28.708 28.639+ 31.143 (log P) 28.708 79.14
-2.01 +(7.33 e-3)(T) 0.1635 + 0.3079 (log P) 0.8152 0.4471
360~
Velocity: 0 ~
198
A.M. Murtagh and P.J. Mallon
TABLE 5.4 Inter-ply slip power-law model parameters for Cetex fabric Material: Cetex 5-H satin fabric rs = 5.5 + 55.246 V 0"4902 i
ry
k
n
Conditions
1
5.5
-752.71 + 5.8641(T) -(1.043 e-2)(T)
-16.06 + 0.1019(T) -(1.5689 e-4)(T) 2
300~< T~<340~
2
5.5
11.52 + 0.5081 (P) -(7.086 e-4)(e) 2
0.5079(2.139 e-4)(e)
20 ~
3
5.5
55.246
0.4909
Velocity: 0 ~< V ~<0.5 mm/s
friction requires an understanding of the characteristics of the diaphragm and has been dealt with by Monaghan [30]. This research showed high coefficients of friction between the diaphragm (Upilex) and the steel tool, between values of 0.7 and 0.98 depending on temperature. This indicates the difficulty in allowing slippage of the diaphragm sheet over the mould surface. Diaphragm rupture occurred at some points where high shearing tractions were encountered. For friction of Upilex against APC-2 at melt temperatures, a Herschel-Buckley power-law model provided a good fit to the friction data, indicating the presence of a resin layer existing between composite and diaphragm film during forming. In press forming, matters are both more simplified and more complicated. Friction occurs between two mediums, the composite and tool surface. The anisothermal nature of press forming complicates matters as temperature has been shown to have a large effect on the friction of polymeric materials. In examining the friction that occurs during press forming, a number of parameters must be investigated, namely: 9 Interface temperature 9 Normal pressure 9 Fibre orientation 9 Mould surface/presence of a release agent As expected, these parameters are similar to those influencing other forming processes, for example as shown in section 5.4 for the inter-ply slip mechanism. The main difference is that friction is a surface characteristic of the material, and the other various shearing deformations occur internally within the composite. Two main types of friction have been identified for surfaces in contact: Coulomb friction and hydrodynamic friction [31]. Coulomb friction occurs between "dry" surfaces and in general, the frictional force is proportional to the applied normal force and independent of the sliding velocity. Hydrodynamic friction is a form of lubrication whereby a thin film of fluid exists between the two surfaces in question and viscous shearing of the film can occur in this region. In this case, sliding velocity may affect the frictional force. For polymeric composite friction, especially at high
Shear&g and frictional behaviour during sheet form&g
199
temperatures, hydrodynamic friction is dominant, due to the presence of a surface resin layer. However, a certain degree of Coulomb friction may occur wherever fibres come into direct contact with the hard surface. In terms of explaining the actual mechanisms of friction on a microscopic scale, two theories have been presented in the literature [32] - - the deformation theory and the adhesion theory. The deformation theory involves considering asperities from the hard (i.e. normally the tool) surface "ploughing" into the softer (polymeric matrix) material and the frictional force being related to the net loss in energy due to hysteresis as the material is strained elastically (see fig. 5.41). This net loss of energy can be related to the bulk visco-elastic properties of the polymer for a particular temperature, contact pressure and rate of deformation, rather than some special surface condition. With adhesion, it is thought that bonds are being continuously made, broken and re-made between contacting asperities from adjoining surfaces. The work needed to break these molecular bonds results in friction. The amount of adhesion, caused by mutually attractive van der Waals forces between the bodies, is dependent on the area of true contact which in turn is increased by van der Waals forces around the contacting regions. This effect is small when the normal load is large and increases in importance as the normal load is reduced. Strong adhesion is to be expected between "rubbery" materials, i.e. polymers above their softening temperature and other bodies. With rougher interfaces, adhesion is much reduced. Bartenev [33] treated rubber as a viscous fluid and introduced the concept of decrease in area of true contact (decrease in adhesion) as the speed increased, or as the temperature decreased. Bahadur and Ludema [34] lent further credence to the adhesion theory with this work relating sliding friction of polymers to visco-elastic properties and the strong connection between area of contact, adhesional shear strength and friction force. Previous work on friction of polymeric and fibre-reinforced materials have shown the influence of the parameters of temperature, normal load, sliding velocity and fibre orientation. Tanaka and Yamada [35] investigated the friction of polymeric materials, including PI, PES, PPS and PEEK, sliding against a smooth steel disc at various temperatures. Low coefficients of friction were observed at low temperatures,
Fig. 5.41. Ploughing of asperities in soft polymeric material.
200
A.M. Murtagh and P.J. Mallon
which increased rapidly as the temperature rose due to an increase in the deformation component at higher temperatures. At even higher temperatures, approaching or at the polymer melt temperature, a thick, transferred polymer layer was produced at the interface. This allowed direct comparison of frictional force with the shear strength of the material assuming that shearing occurs below the actual interface, in the polymeric material. Grosch [36] was able to describe the velocity and temperature effects on friction of various polymers, using a single master curve generated by the Williams-Landel-Ferry (WLF) transform to correlate velocity effects in terms of a universal temperature function, which is related to the glass transition temperature of the material. Friction tests carried out over a wide range of normal loads show that the coefficient of friction does not remain constant but increases as the load is reduced [37]. Thus, a simple linear relationship between frictional force and applied normal load does not exist. The type of relationship which can be used is of the form: F--
otW n
(5.19)
where c~ is a constant and n is some value less than 1. This is of similar form to the power-law model used to describe the shear velocity/shear stress relationship for inter-ply slip (see section 5.4). Over an extensive load range n is not a constant. If the adhesive mode of friction is assumed to be dominant, then the frictional behaviour of a material can be assumed to be dependent on the applied load/area of true contact relationship. This area of true contact has been shown to be difficult to measure in the past. It should not be considered to be the same as the sheared area undergoing inter-ply slip within the composite. The effect of surface fibre orientation on the friction of composites at elevated temperature has not been fully investigated. At room temperature, some researchers [38] have shown outer ply orientation to have little effect on friction. Friction coefficients were much more dependent on the resin surface characteristics. At higher temperatures, however, fibres may come to the surface and establish contact with the metal surface. In wear experiments using short glass-fibre-filled PPS [35], it was shown that at elevated temperatures, the fibres became exposed on the surface and were seen to be supporting the contact load. For hydrodynamic friction to occur during forming, a thin resin layer must be assumed to exist between the fibres and tool surface. Mould surface finish and the presence of surface coatings also need to be investigated to see how they may affect the frictional behaviour of the deforming composite. In most press forming operations, the mould surface in normally ground and polished. This is because regardless of any advantages that may result from being able to vary the surface roughness in certain circumstances [39] in the majority of cases the final surface finish is of paramount importance and for this reason, the mould surface must be as smooth as possible. Table 5.5 shows the results of surface roughness measurements taken from a typical mould surface used in thermoforming, a steel foil used during friction testing, and for comparison, measurements for APC2 prepreg in the longitudinal and transverse directions.
Shearing and frictional behaviour during sheet forming
201
TABLE 5.5 Surface roughness (Ra) measurements (in micrometres) Reading
Mould surface
Steel foil
APC-2 (long.)
APC-2 (trans.)
Steel foil (roughened)
1 2 3 4 5
0.6 0.7 0.8 0.7 0.7
0.22 0.18 0.25 0.22 0.22
1.3 1.2 1.25 1.5 1.2
6.0 6.5 5.8 6.0 6.1
0.9 0.85 0.8 0.82 0.8
Average
0.7
0.22
1.3
6.1
0.83
A 1-gm diamond paste was used to slightly roughen the surface of the foil and this consequently showed a similar value of surface roughness to the mould surface material. During actual friction testing [40], it was observed that there was little or no discernible difference in friction between the unroughened and roughened foil material, suggesting that any roughness differences at a level of less than 1 gm have little effect on friction of thermoforming composites. As expected, the surface roughness of the APC-2 prepreg is greater across the fibres than along them. The roughness value is similar in size to the diameter of a carbon fibre, which might be expected. Pre-consolidated laminates have a much smoother surface finish due to the presence of a surface resin layer. In many moulding operations, a surface release agent is applied to the surface of the mould which allows easy removal of any parts and can also improve the surface finish. Most of the agents work on the principle of depositing a solvent-soluble substance, e.g. a silicone, as a thin film a few molecules thick on the surface of the mould which acts as a barrier to prevent sticking and adhesion of the polymer to the moulding surface. In most cases, the release agent is applied using an aerosol spray or simply wiped on and allowed to dry. Multiple coatings may be applied. In terms of friction, release agents may also assist in reducing the degree of adhesion during sliding and so reduce the coefficient of friction. Two different experimental set-ups were used by Murtagh [40] to investigate the frictional properties of APC-2 under forming conditions. One method consisted of drawing a central specimen (composite or tool material) from between two sheets of the other material, all three being initially subject to a normal load between two heatable platens located between the jaws of a hydraulic press. Horizontal motion of the central specimen was achieved using the shearing apparatus already used for investigating the inter-ply slip mechanism explained previously in section 5.4. Using this set-up, isothermal conditions of between 25~ and 400~ were easily achievable and normal forces of between 0.1 kN and 100 kN could be applied. A shim placed at the rear of the pull-out sample maintained the alignment of the platens. Figure 5.42 shows a schematic of this apparatus. The other method used involved the development of a "friction sled', based on an ASTM standard [41], for obtaining frictional coefficients of plastic film and sheeting. This more accurately simulates the actual friction that occurs during press forming
202
A.M. Murtagh and P.J. Mallon
Fig. 5.42. Twin platen arrangement for friction testing.
as it mimics the anisothermal conditions and allows the concept of the two materials coming together suddenly, with the mould material being rapidly slid across the composite. It consists of an aluminium block with two embedded cartridge heaters, to which various test materials, in the form of plates, can be attached to the bottom surface (see fig. 5.43). The composite material is located on the surface of the consolidation apparatus already mentioned and is heated by conduction. Insulation is placed over the surface of the composite during initial heat-up and around the sled during testing to avoid excessive heat-loss. Weights may be applied on top of the sled to vary the normal load and the load range is lower than used with the twin platen arrangement. Again, the shearing apparatus is used to apply the sliding motion to the sled via a steel wire with a speed range of between 0.0125 mm/s and 4 mm/s. Friction force was measured using a 500 N load cell mounted at the front of the shearing apparatus and connected to the steel wire.
Fig. 5.43. Schematic of friction sled.
Shearing and frictional behaviour during sheet forming
203
Initial tests involved shearing a 50-mm wide sheet of 0~ APC-2 from between two sheets of steel foil, coated with Frekote release agent. The sample was placed between the platens of the consolidation apparatus and heated to testing temperature with 0.2 kN normal pressure being applied during heat-up. The results of this sample being sheared are shown in fig. 5.44. It was observed that due to the presence of this substantial normal load during heat-up that an adhesive bond tended to develop and this inhibited sliding from occurring until the shear strength of the bond had been exceeded at approximately 270 N shearing force. Following the breaking of this bond, the sliding force decreased to a steady level dependent on the shearing velocity characteristic of the assumed presence of a viscous resin layer between the composite and metal surface. This behaviour was typical of most tests carried out at high temperatures. In order to prevent the adhesive bond from being developed during heat-up, the test set-up was modified so that a slight gap was maintained between the top platen and the sample for most of the heating time. A brass plate, 6 mm in thickness, was placed over the sample which allowed heat-up via conduction through the bottom platen and when testing temperature had been reached, only then was the top platen brought into contact with test pressure being applied from the outset. Other research [37] has noted the effect of time of loading on friction due to the visco-elastic nature of the polymer material. In actual press forming, the laminate is not in contact with 350
300
.......
i
0.20
0.15
250
A A
z
200 0.10
_o
E E
_8
L_
m
v
150
t._
,00l
..2
t~
0.05
Shear load ] 50
Velocity
0
I
0
5
,
,,,
I
,
10
I
1 I
15
Displacement (ram) Fig. 5.44. Pull-out of 0 ~ APC-2 from between two sheets of steel foil.
20
0.00
204
A.M. Murtagh and P.J. Mallon
the mould until forming occurs so time of loading does not influence the frictional characteristics and it is important to eliminate this effect in friction testing. To examine the effects of surface temperature on the frictional characteristics of the composite/tool deforming system, the set-up using the twin platen/steel foil arrangement was used. In this case, isothermal conditions could be ensured throughout the sample, not just at the composite/tool interface. The temperature can be varied by adjusting the platen set-points and the appropriate effects observed. The friction sled set-up, although more accurately mimicking actual press forming conditions, as mentioned previously, is essentially a non-isothermal process, even in the event of both sled and composite having the same set-point temperature. The heat losses to the environment compared with the alternative platen set-up are quite large. In reality, the tool and composite are normally at different temperatures during actual press f o r m i n g - in the processing of APC-2 for example, the tool temperature may be as much as 150~ below the surface temperature of the composite just prior to forming. This allows more rapid cooling, and faster processing. Obviously, large thermal gradients are seen throughout the composite as it begins to deform after coming into contact with the mould and the frictional behaviour would be expected to vary accordingly. It has been not feasible to date to be able to cope with such anisothermal conditions. Instead, the effect at various isothermal temperature conditions has been observed and analysed. Figure 5.45 shows the influence of temperature on the friction of a 0 ~ ply of APC-2 as it was drawn from between two sheets of release-agent treated steel foil. The set 1.4
Normal load 0.2 kN
1.2
1.0
~,. ~ "6",~
0.8
~9 o
0.6
~
0.4
--o-
405 ~
--e~
385 ~ 3650
---e-:
0.0
I 0.0
0.5
1.0 Velocity (mmls)
Fig. 5.45. Effect of temperature on friction of 0~ APC-2.
1.5
,
t 2.0
345 ~ 25 ~
Shearing and frictional behaviour during sheet forming
205
point of both platens was varied for each test and a normal load of 0.2 kN was applied during shearing. The construction of the graph is similar to the method used for inter-ply s l i p - the behaviour is essentially viscous and various constant friction loads were measured as velocity increased. Rather than use shear stress as the measure of the ordinate, the coefficient of friction (It) was selected as it is the more traditional method of presenting friction results. Also, to present any result as a shear stress value would mean having to assume a value for the sheared area, which is to be considered invalid in any friction analysis. For the test performed at room temperature (25~ the frictional coefficient did not vary significantly with velocity and remains at a low value of approximately 0.16, similar to values obtained by Tanaka [35] at low temperatures. This non-variation with velocity change is indicative of simple Coulomb friction with a lubricating resin layer not existing at this point. As the temperature is increased to a value approaching the melt temperature, however, the friction behaviour begins to change significantly. The coefficient of friction increases and becomes more dependent on velocity. Tanaka [35] has shown an increase in friction for P E E K at lower temperatures than this (at the materials softening temperature, approximately at the glass transition temperature of 143~ The difference may be associated with different methods of testing (a steel sphere indenter compared with a flat plate interface) and the presence of a release agent in these tests. Above the melt temperature, in the temperature regions 365 to 405~ the friction coefficient rises rapidly from a common value of approximately 0.4 at low velocities up to a maximum of 1.2 at a velocity of 2 mm/s. This would seem to indicate a strong adhesive component as the P E E K matrix becomes molten and allows the real contact area (on a microscopic level) to increase substantially as temperature increases. It might have been expected that once a resin layer had become established at the interface that the friction force would be due totally to a shearing of the viscous fluid layer, and that increasing the temperature would cause shearing forces, i.e. frictional force, to reduce as the fluid viscosity decreased. This effect was observed previously with inter-ply slip behaviour. In this case, the opposite was observed, thus indicating that a strong adhesive bond develops between composite and tool surface as localised surface deformation occurs. Other factors such as a decrease in the resin layer thickness between the composite and steel foil as the temperature increases would also cause an increase in shearing force. Degradation of the release agent at higher temperatures may also be a factor. The influence of normal load was investigated in the range 0.1-0.75 kN. The lower limit was a function of the equipment available and the upper limit was considered a maximum that would occur on a laminate during any press forming operation, at least during the forming stage. Again, the results presented are for a 0 ~ ply of APC-2 sheared from between two steel foil sheets, coated with two layers of Frekote FRPNC release agent. Temperature was kept constant at 385~ for this series of tests. As shown in fig. 5.46, the coefficient of friction decreases as normal load is increased, a trend observed by other research [37]. The coefficient of friction varies from approximately 0.1 at low speeds (less than 0.25 mm/s) and a normal load of 750 N, to 1.4 at high speed (2 mm/s) and a low normal load (0.1 kN). It is to be observed that as the shear velocity increases, the coefficient of friction tends to level off.
206
A.M. Murtagh and P.J. Mallon
2.0
, NormalIo~ad:2NN 00100 --o-
1.5
Temperatu385~ re
500 N
_
n
A
Z
O
: ._u
1.0
0
[~ 0 "0
0.5 .
,
0.0 0.0
0.5 1
9 .
.
.
.
1.0 I
_
.
.
.
L+
.
- -
1.5 .1_
a
2.0
Sliding velocity (mmls)
Fig. 5.46. Effect of normal load on friction of 0~ APC-2.
Shear thinning (a viscosity decrease at higher rates) would explain a certain decrease in frictional/shearing force. However, another explanation for this phenomenon may be that as the speed and displacement of the sample is increased, the lubricating resin layer does not remain at a constant thickness, but increases as there is a build-up of transferred polymer layer to the steel foil surface. This is also evidenced by visual inspection of the foil after pull-out, where a substantial polymer layer had adhered to the foil. Once the part has formed, any transferred polymer is reconsolidated into the surface of the part. Figure 5.47 shows further evidence of this. Again, a typical steel foil specimen was sheared from between two 0 ~ APC-2 plies at a constant velocity of 2 mm/s. The frictional load increased to a maximum of 215 N, then levelled off and began to decrease. If displacement is terminated for a period of time (at point A), the shear load relaxes similar to visco-elastic relaxation behaviour. When shearing was recommenced (at point B), the force recovers to a value at point C, approximately 40 N greater than the point at which movement had ceased. This would be the expected behaviour if the resin layer had become thinner again before sliding had recommenced. A sample of steel foil with the resin residue still intact on the surface was taken from a specimen which had been sheared from between two 0 ~ sheets of APC-2 [12]. The specimen had been sheared at two distinct velocities (0.25 and 2 ram/s) and two bands of resin were evident on the steel foil. Three identically sized samples were taken: (i) one from an unsheared part of the steel foil, with no resin, (ii) a sample
Shear&g and frictional behaviour dur&g sheet form&g 300
.
.
.
.
.
.
.
.
200
~9
:,=
207
4
100
0
0
5
10
15
20
25
30
Time (sec.)
Fig. 5.47. Frictional force reduction at high velocities/resin layer build-up.
from the part of the foil that had been sheared at 0.25 mm/s, and (iii) a sample that had been sheared at 2 mm/s. By measuring the mass increase due to resin adhesion, it was possible to calculate the average resin layer thickness on the foil samples, assuming a resin density of 1.3 g/cm 3 [42]. Table 5.6 shows that as expected, an initial average resin layer thickness of approximately 10 ~tm on each side of the foil exists at a shearing velocity of 0.25 mm/s. As the T A B L E 5.6 Resin thickness measurements of steel foil Sample
Mass (g)
Resin mass (g)
Resin volume
(cm 3)
Resin (~tm)
(i) (ii) (iii)
0.6032 0.60482 0.60671 .
Resin density (1.318 g/cm) 3 Sample area 4.931 x 1.219 cm 2
1.62 e -3 3.51 e -3
1.229 e -3 2.663 e -3
10.22 22.12
thickness/2
208
A.M. Murtagh and P.J. Mallon
speed is increased to 2 mm/s, the resin layer thickness more than doubles in thickness to 22 ~m and this would have the effect of reducing the frictional forces. These resin layer thicknesses compare with a value of approximately 6 ~tm assumed to exist between individual plies in the inter-ply slip process. In the actual press forming process, this resin layer increase-in-thickness during forming may be of practical assistance, as it reduces frictional forces and any tendency that may exist for sticking on the forming surface. Even though there may be a local build-up of resin on the surface of those areas that move across the mould, it is reasonable to assume that during pressure application and consolidation of the part that any transferred polymer is re-percolated into the composite and accommodated amongst the fibres. To further investigate the presence of a surface resin layer, two micrographs were taken through a section of a tested specimen, one at 90 ~ and one at 0 ~ to the fibre direction, cooled under pressure to maintain an adhesive bond between the composite and steel foil, with no release agent [12]. Figure 5.48 shows that a thin layer of resin does exist at the boundary of the composite (APC-2 orientated at 90 ~ to the plane of observation) and steel foil/mould surface. Here, the resin layer seems to be of relatively constant thickness, compared with the interface in an actual laminate, which indicates the lack of fibre/fibre interference with frictional flow against the smooth foil surface. Shown inset is a section from a specimen with fibres running at 0 ~ At point A, a fibre can be seen to approach the steel surface, showing that fibre/ tool contact may be possible. It should again be emphasised here that the thickness shown in this micrograph is for a specimen which has interacted with the steel foil,
Fig. 5.48. Micrograph of tested specimen showing resin interlayer, fibres at 90 ~ (inset: fibres at 0~
Shearing and frictional behaviour during sheet forming
209
then cooled to room temperature. It does not conclusively illustrate the resin layer that is developed during processing. The effect of surface fibre orientation must also be taken into account in examining frictional characteristics of thermoplastic composites. Despite the fact that a thin resin layer is assumed to exist between the composite and tool, the presence and relative direction to flow of the fibres may affect the flow properties of the resin layer and indeed, the tool may come into contact with fibres in certain locations where resin "squeeze-out" has occurred. In terms of testing, the most suitable set-up to use in investigating the effect of fibre orientation was the friction sled [40]. With the twinplaten set-up, it proved too difficult to mount any specimen at the appropriate angle or to apply a traction force to any ply at an angle other than 0 ~ Care had to be taken that isothermal conditions existed for the sled tests. Set-point temperature of the sled and bottom platen was 385~ and 400~ respectively for all tests using APC-2. The higher temperature of the bottom platen was to allow for conduction through to the surface of the composite and to partly compensate for heat loss. Figure 5.49 shows a plot of sliding velocity versus coefficient of friction for a variety of different surface fibre orientations, relative to the direction of sliding. The highest resistance to sliding occurred when the fibres lay parallel to the sled. As the fibre angle changed, friction decreases, down to a minimum for a sample with fibres lying at 90 ~ or perpendicular to the sliding direction. Figure 5.49 shows that, even for an angular change from 0 ~ to 10~ the frictional force is significantly reduced. Therefore, it appears that although a resin layer is assumed to exist between the fibres and mould surface, the fibre orientation does have an effect on the 1.0
0.8 Fibre angle
~"
_
O.6
0~
T"~ 'i5
tJ
0
---4--
10 ~
--0--
90 ~
45 ~
0.4
0.2
0.0
,
0.0
I
0.5
,
....
9
,
1.0
.... I
1.5
Sliding velocity (mmls)
Fig. 5.49. Effect of surface layer fibre orientation on friction of APC-2.
,
.I
2.0
210
A.M. Murtagh and P.J. Mallon
frictional behaviour. The apparent higher resistance to movement for the 0~ tated fibres compared with 90~ is analogous to what was observed with the inter-ply slip behaviour of APC-2 (see section 5.4). However, fibre/fibre interactions between different layers cannot happen with a flat mould surface-composite interface. Therefore, as proposed by Groves [21], the flow behaviour and viscosity of the resin material is influenced by the fibre orientation ~ the longitudinal viscosity may be an order of magnitude above the transverse viscosity for measurements carried out on APC-2 using a torsional rheometer. In terms of the frictional behaviour of APC-2 against a mould surface above the melt temperature of the composite, the orientation of the fibres affects how replacement resin material is allowed percolate to the interface region, as the resin layer is being continuously sheared along the mould surface. Flow over 90 ~ fibres is potentially easier than resin flow along the fibre direction. Figure 5.50a and fig. 5.50b shows a possible mechanism for this; the flow of resin is less restricted when the fibres are perpendicular, or at some angle other than 0 ~ Various release agents were also investigated to determine their effect on the frictional behaviour of unidirectional APC-2 on stainless steel. It should be noted here that not all these agents (Frekote FRP-NC, Wurtz PAT 807-B, Wurtz PAT 808 and ChemTrend E274) were regarded as suitable for giving optimum results at such high temperatures (approaching 400~ Also, surface finish may be a more important quality in selecting the appropriate agent as opposed to reducing frictional forces to a minimum. For all these tests, two coats were applied to an acetonecleaned steel foil sample and allowed to dry. For the E274 agent, the foil had to be pre-heated to 400~ before the agent was applied using an aerosol. However, in practice, this agent has the advantage of not requiring as many "touch-ups" during repeated use compared with other agents. Normal load was 0.2 kN and temperature 385~ for all tested samples. As expected, fig. 5.51 shows that the highest coefficient of friction was recorded for an untreated foil sample. Frekote FRP-NC and E274 gave similar results. PAT-808
Fig. 5.50. Resin flow depending on fibre orientation.
Shearing and frictional behaviour during sheet forming 1.5
A Z
211
--
10
a-.O m 4..0 q.. O I:
05
No release agent Frekote NC
Temperature 385~ Normal load 0 2 kN ,
0.0
00
I
05
'
PAT 807B
A
PAT 808
",
I
,
E274 i
1.0
[
15
i
I
20
Sliding velocity (mmls) Fig. 5.51. Effect of surface release agent on friction of APC-2. (due to the fact that it dries to leave a thicker coat compared with the other agents) gave the best results. However, when compared with PAT-807B (slightly higher coefficient of friction), PAT-808 resulted in poor surface finish q u a l i t y - the surface had a matt rather than a glossy finish and tended to be of uneven quality, in typical, press-formed APC-2 parts [43]. Friction tests have been carried out on Cetex 5-H fabric (carbon-fibre-reinforced PEI) material to measure the material's frictional behaviour as a function of temperature, normal load and other process parameters [12]. Again, the twin-platen test set-up was used, with a sample of the composite being sheared from between two sheets of stainless steel foil, pre-treated with two coats of Frekote release agent. No significant normal load was applied to the sample prior to t e s t i n g - heat-up to test temperature was done with a slight gap between the platens. Normal load was applied only for two minutes prior to commencement of any test. Fibre straightening in the fabric sample was accounted for by allowing a known yield load to be developed in each sample before sliding would occur. This value was measured and defined to be 55 N for each test carried out. The significance of this was to exclude any internal stresses in the test sample, and to concentrate on surface phenomena only. Figure 5.52 shows results for a series of tests carried out at various test temperatures between 300~ and 340~ for Cetex. This shows a reduction in measured frictional forces as the temperature is increased. This is exactly the opposite trend as observed with APC-2 (fig. 5.45), where friction increased with temperature.
212
A.M. ~urtagh and P.J. Mallon
1.0
0.8 A
Z
0.6
-
c
._o u
o..
o
0.4
300~
C ~
o
,m
o
0.2
0.0
t
0.0
I
0.5
,
310oc 320~
o----
,
Normal load 2 5 0 N ,
... t, ..... =
340~ . . . . .
,
. . . .
I
,
,
,
1.0
I
1.5
,
~
,,
2.0
Sliding velocity (mmls)
Fig. 5.52. Effect of temperature on the friction of Cetex. Intuitively, one would expect friction to decrease if the viscosity in the intervening resin layer between the surfaces decreased with an increase in temperature [40]. The difference between the unidirectional material (APC-2) and the fabric may be explained by the possible development of the resin layer at the interface for APC2 or Cetex. In the unidirectional material, the volume fraction of the fibres is high, and compared with the under/over pattern of the fabric weave, offers a relatively "flat" surface to be offered to the mating steel surface. Consequently, the resin layer exists as a thin layer between the fibres and increasing the temperature may cause more fibre/steel surface contact if the resin were to flow back into the composite. For the fabric material, the amount of maximum fibre/steel contact is limited to the contact points between the highest points in the weave and the mould surface. At the same time, the amount of resin present at, or close to, the interface is higher than for unidirectional materials due to "pools" of resin lying in the interstices or low points on the surface of the weave. An increase in the temperature of the resin would lower the viscosity and reduce shearing forces in the interlayer. The adhesional properties of PEEK are known to be high, whereas PEI may not develop the same bond strength at such high temperatures. At the same time, the lower temperatures involved with the Cetex material may mean that the Frekote release coating may not degrade as rapidly as with typical APC-2 processing temperatures (,~400~ which would cause an increase in friction force. Using Frekote at such high temperatures is approaching the limit of the coating's effectiveness. Further tests were carried out to examine normal loading effects on the friction of Cetex. Similarly to APC-2 and other polymeric materials, the coefficient of friction
Shearing and frictional behaviour during sheet forming
213
decreases as normal load in increased. The lower bound for testing, 100 N, resulted in the highest level of friction. Figure 5.53 shows a plot of sliding velocity versus coefficient of friction for a variety of normal loading conditions, between 100 N and 500 N for Cetex. As with APC-2 (see fig. 5.46), a reduction in the coefficient of friction for Cetex is seen at higher sliding velocities, again possibly due to the development of a thicker resin interlayer as sliding progressed. Given the obtained experimental data, it should prove feasible to develop a predictive model for friction dependent on forming conditions. This has been achieved for APC-2 and Cetex 5-H satin fabric material [12]. A similar method was used to that for inter-ply slip since the power-law model form is again valid due to the viscous effects of the resin layer between tool and composite, which is analogous to the presence of a resin layer between plies for inter-ply slip. The relationship between friction coefficient, sliding velocity and the forming parameters (temperature, normal load and surface fibre orientation) can be described by the following:
air'i)
(5.20)
#s
~-i=l
/Zs
tz defines the coefficient of friction, V is the velocity in mm/s and/Zs is the coefficient of friction under standard conditions of temperature, normal load and fibre direction. The value of lZs and of the power-law parameters a and b are given in table 5.7 and table 5.8 for APC-2 and Cetex respectively.
1.4
i
Temperature3200C
......
1.2 ~"
1.0
cr
o
g
0.8
o
0.6
~
"O
~
100N
o.4
o
250N 0.2
0.0
0.0
500N
0.5
1.0
Sliding velocity (m mls) Fig. 5.53. Effect of normal load on the friction of Cetex.
1.5
2.0
214
A.M. Murtagh and P.J. Mallon
TABLE 5.7 Friction power-law model parameters for APC-2 Material: APC-2 /zs = 0.9276V~ i
a
b
Conditions
1
-34.136 + 0.172(T) -(2.1 e-4)(T) 2
(3.36 e-3)(T) - 0.874
345 ~
2
3.52- 1.127log(N)
0.686-0.116 log(N)
100~
3
0.9276 -- 0.0743(0)0.2606
0.4198 + (2.827 e-4)(0)
0~<0~<90~
Velocity: 0 ~
TABLE 5.8 Friction power-law model parameters for Cetex fabric Material: Cetex 5-H satin fabric /Zs = 0.868 V 0"4 i
a
b
Conditions
1
33.656 - 0.1917(T) +(2.79 e-4)(T) 2
-13.192 +(8.7525 e-2)(T) -(1.3375 e-4)(T) 2
300~< T~<340~
2
1.6687 - (4.026 e-3)(N) +3.293 e-6(N) 2
0.2348 + (1.229 e-3)(N) -2.2733 e-6(N) 2
100 ~
3
0.868
0.4
Velocity: 0 ~
References [1] Cogswell F.N, "The Processing Science of Continuous Fibre Reinforced Thermoplastic Composites", Intl. Polymer Processing, 1, 4, pp. 157-165, 1987. [2] Gutowski T.G., "A Resin Flow/Fiber Deformation Model for Composites", SAMPE Quarterly, 16, pp. 58-64, 1985. [3] Wheeler A.J., Ph.D. Thesis, University of Wales, Aberystwyth, 1990. [4] Lam R.C., Kardos J.L., "The Permeability of Aligned and Cross-Plied Fiber Beds during Processing of Continuous Fiber Composites", American Society for Composites, 3rd Annual Tech. Conf., pp. 3-11, 1988. [5] Cogswell F.N., from Thermoplastic Aromatic Polymer Composites, Butterworth-Heinemann, Oxford, 1992. [6] Barnes J.A., Cogswell F.N., "Transverse Flow Processes in Continuous Fibre Reinforced Thermoplastic Composites", Composites, 20, 1, pp. 38-42, 1989. [7] Mulholland A.J., Monaghan M.R., Mallon. P.J. "Characterisation of Consolidation Flow Processes in Continuous Fibre Reinforced Thermoplastic Composites", 13th Intl. Conf., European Chapter, SAMPE, Hamburg, 1992. [8] Spencer A.J.M., from Deformations of Fibre-reinforced Materials, Clarendon Press, Oxford, 1972.
Shearing and frictional behaviour during sheet forming
215
[9] Rogers T.G., "Rheological Characterisation of Anisotropic Materials", Composites, 20, 1, p. 21, 1989. [10] Groves D.J., Bellamy A.M., Stocks D.M., "Anisotropic Rheology of Continuous Fibre Thermoplastic Composites", Composites Manufacturing, 2, 2, 1992. [l l] Scobbo J.J., Nakajima N., "Dynamic Mechanical Analysis of Molten Analysis of Molten Thermoplastic/Continuous Graphite Fiber Composites in Simple Shear Deformation", 21st Intl. SAMPE Tech. Conf., pp. 730-743, Anaheim, California, 1989. [12] Murtagh A.M., "Characterisation of Shearing and Frictional Behaviour in Sheetforming of Thermoplastic Composites", Ph.D. Thesis, University of Limerick, May 1995. [13] Bergsma O.K., "Computer Simulation of 3-D Forming Processes of Fabric Reinforced Plastics", ICCM-9, Madrid, 1993. [14] Van West B.P.,"A Simulation of the Draping and a Model of the Consolidation of Comingled Fabrics", CCM Report 90-07, Center for Composite Materials, University of Delaware, 1990. [15] Johnson A.F., "Rheological Model for the Forming of Fabric Reinforced Thermoplastic Sheets", Composites Manufacturing, 6, 3-4, pp. 153-160, 1995. [16] Murtagh A.M., Mallon P.J., "Shear Characterisation of Unidirectional and Fabric-Reinforced Thermoplastic Composites for Pressforming Applications", ICCM-10, Whistler, Vancouver, 1995. [17] Blanlot R., Billoet J.L., Gachon H., "Study of Non-Polymerised Prepreg Fabrics in 'Off-Axes' Tests", ICCM-9, Madrid, 1993. [18] Scherer R., Friedrich K., "Experimental Background for Finite Element Analysis of the Interply-Slip Process During Thermoforming of Thermoplastic Composites", ECCM 1994, Stuttgart, 1990. [19] Scherer R., Zahlan N., Friedrich K., "Modelling the Interply-Slip Process During Thermoforming of Thermoplastic Composites Using Finite Element Analysis", Proc. CADCOMP 90, Brussels, 1990. [20] Xiang Wu, "Thermoforming of Thermoplastic C o m p o s i t e s - Interply Shear Flow Analysis", 21st International SAMPE Tech. Conf., p. 915, Anaheim, California, 1989. [21] Groves D.J., "A Characterisation of Shear Flow in Continuous Fibre Thermoplastic Laminates", Composites, 20, 1, p. 28, 1989. [22] Jones R.S., Oakley D., "An Interpretation of Rheological Data in Terms of Model Systems", Composites, 21, 5, p. 415, 1990. [23] Kaprielian P.V., O'Neill J.M., "Shearing Flow of Highly Anisotropic Composite Laminates", Composites, 20, 1, p. 43, 1989. [24] Soil W.E., "Behaviour of Advanced Thermoplastic Composite Parts", M.S. Thesis, Dept. of M.E., MIT, 1987. [25] Tam A.S., Gutowski T.G., "Ply-Slip During the Forming of Thermoplastic Composite Parts", J. of Composite Materials, 23, p. 587, 1989. [26] Morris S.R., Sun C.T., "An Investigation of Interply Slip Behaviour in AS4/PEEK at Forming Temperatures", Composites Manufacturing, 5, 4, p. 217, 1994. [27] Murtagh A.M., Monaghan M.R., Mallon P.J., "Investigation of the Interply Slip Process in Continuous Fibre Thermoplastic Composites", ICCM-9, Madrid, 1993. [28] Murtagh A.M., Monaghan M.R., Mallon P.J., "Development of a Shear Deformation Apparatus to Characterise the Interply Slip Mechanism of Advanced Thermoplastic Composites", IMF-8, University College Dublin, 1992. [29] Muzzi J., Norpoth L., Varughese B., "Characterisation of Thermoplastic Composites for Processing", SAMPE Journal, 25, 1, p. 23, 1989. [30] Monaghan M.R., Mallon P.J., "Study of Polymeric Diaphragm Behaviour in Autoclave Processing of Thermoplastic Composites", 14th Intl. Conf., European Chapter, SAMPE, Birmingham, 1993. [31] Barone M.R., Caulk D.A., "A Model for the Flow of a Chopped Fiber Reinforced Polymer Compound in Compression Moulding", J. of Applied Mechanics, 53, p. 361, 1986. [32] Tabor D., "Friction, Adhesion and Boundary Lubrication of Polymers", ACS International Symposium on Polymer Wear and its Control, Plenary Lecture, Los Angeles, 1974. [33] Bartenev G.M., El'kin E.I., "Friction Properties of High Elastic Material", Wear, 8, 8, 1965. [34] Bahadur S., Ludema K.C., "The Viscoelastic Nature of the Sliding Friction of Polyethylene, Polypropylene and Copolymers", Wear, 18, p. 109, 1971.
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[35] Tanaka K, Yamada Y., "Effect of Temperature on the Friction and Wear of Some Heat-Resistant Polymers", from Polymer Wear and its Control, American Chemical Society, Washington DC, 1985. [36] Grosch K.A., "The Relation between the Friction and Visco-elastic Properties of Rubber", Proc. Roy. Soc., A274, p. 21, London, 1963. [37] Bowden F.P., Tabor D., "The Friction and Deformation of Polymeric Materials", from The Friction and Lubrication of Solids, Part XIII, Clarendon Press, Oxford, 1964. [38] Herrington P.D., Sabbaghian M., "Factors Affecting the Friction Coefficents between Metallic Washers and Composite Surfaces", Composites, 22, 6, 1991. [39] Throne J.L., from Thermoforming, Hanser Publishers, Munich, 1987. [40] Murtagh A.M., Lennon J.J., Mallon P.J., "Surface Friction Effects Related to Pressforming of Continuous Fibre Thermoplastic Composites", Composites Manufacturing, 6, 3-4, p. 169-176, 1995. [41] ASTM Standard 1894-90,"Standard Test Method for Static and Kinetic Coefficents of Friction of Plastic Film and Sheeting", 1990. [42] ICI Thermoplastic Composite Handbook, 1992. [43] Maher E.J., "An Investigation of Isothermal Pressforming with Thermoplastic Composite Materials", M.Eng. Thesis, University of Limerick, Ireland, 1994.
Composite Sheet Forming edited by D. Bhattacharyya 9 Elsevier Science B.V. All rights reserved.
Chapter 6
Grid Strain Analysis and its Application in Composite Sheet Forming T.A. MARTIN, G.R. CHRISTIE and D. B H A T T A C H A R Y Y A Composites Research Group, Department of Mechanical Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand
Contents Abstract 217 6.1. Introduction 218 6.2. Large strain analysis 218 6.3. Method of least squares fitting 224 6.4. Forming a composite spherical dome 226 6.5. Forming a composite blister fairing 230 6.6. Draping theory of textile fabrics 234 6.7. Diagnostic applications 238 6.8. Concluding remarks 241 References 244 Abstract Sheet forming operations are often complex and induce large strains in the deformed materials. Grid strain analysis (GSA) provides a means of quantifying those strains by measuring the dimensions of the deformed grids and comparing them with the original grids printed on the material surface. In GSA macroscopic deformation of a sheet may be evaluated without any need to know the constitutive information of the deforming material. However, with additional information on the constitutive behaviour, it may also be possible to know the stress magnitudes from the strain distribution. GSA is extremely useful in the sheet forming industry for identifying problem areas and the design of tooling. This chapter introduces the large strain analysis technique using a continuum approach and shows its application in fibre-reinforced thermoplastic sheets. It appears that for bi-directional composite sheets, the surface strains adequately describe the deformation behaviour of the entire sheet. The results also match very well with those obtained from the kinematic analysis of draping bi-directional woven fabric. Material thinning and thickening (or compressive instability) zones can be very well identified and predicted, if necessary. For multi-directional 217
218
T.A. Martin et al.
reinforcement, the results need to be analysed with more careful judgement. Some examples are given to demonstrate the problem identifying and blank shape optimisation capabilities of GSA.
6.1. Introduction This chapter covers the fundamental mathematical concept behind grid strain analysis (GSA) and its application to components formed from continuous fibrereinforced thermoplastic (CFRT) sheets. While the theory may be formulated in a number of ways, this text deals with the recently adopted finite element strain analysis technique. Background information is also given regarding the development of some earlier techniques. Most readers will not necessarily wish to know the in-depth details of the formulation; however, a detailed outline of the theory is given for completeness in sections 6.2 and 6.3. Using specially developed software the analysis can be easily handled in a user-friendly graphical environment. Those who are merely interested in the concept and the practical applications of the method can quickly move on to sections 6.4 and 6.5, which contain the experimental strain analysis results for sheets formed into 3-D components. A clear description of the material behaviour for single-ply and multi-ply laminates follows from the analysis. The results are shown to be useful for quantifying the through-thickness deformation in most cases, even though only the surface strains are measured. In section 6.6, comparisons are made between the experimental findings and the theory of draping a network of inextensible fibres over a surface. Using this approach the deformed fibre paths are predicted on two particular surfaces, a hemispherical dome and a blister fairing, along with the associated strain distributions. Some experimental results are also given for sheets containing more than two families of fibres. Finally in section 6.7 some diagnostic applications of GSA are described and its usefulness in identifying problems related to tensile and compressive instabilities is described. Section 6.8 adds some concluding remarks with salient findings highlighted.
6.2. Large strain analysis Sheet forming operations are inherently complex and frequently induce substantial strains in the material being formed. In order to successfully deform a fiat sheet of material into a 3-D component, the appropriate die geometry and boundary constraints must be prescribed. However, the way in which these conditions affect the forming process is often not obvious to the designer. For many years the sheet metal industry has recognised the need to quantify the forming strains over an entire sheet, as this information is critical for designing new forming processes and/or altering existing ones to control the quality of finished products. In order to do this, the designer needs to interpret what has occurred during manufacturing using only two geometries; the initial blank shape and the finished part. The method of grid strain analysis, first devised by Sowerby et al. [1] and then extended by Schedin and Melander [2], provides a means to quantify the deformation in this situation. After the strains are known, draw beads and/or lubrication may be usefully applied
Grid strain analysis
219
to sheet metal forming operations to eliminate process failures like necking and wrinkling, which are associated with particular strain fields. With additional information about the constitutive relationship of the material, the stresses may also be evaluated as a consequence of the forming path. Grid strain analysis provides a mechanism by which the macroscopic deformation of a sheet may be evaluated, without needing to know any constitutive information about the particular material being formed. Using this method the surface of a sheet is marked with a series of circular or square grids prior to deformation and the altered shape of these grids is measured after forming. This may be done by measuring each element separately or by measuring the nodal coordinates of a deformed regular/irregular grid. One of the fundamental assumptions of the analysis is that the deformed component may be represented by a two-dimensional surface in 3-D space. The in-plane deformations of the sheet can then be quantified and the third principal strain direction is represented by the surface normal. It should also be noted that this technique is also often called large strain analysis, because it can be used reliably with processes which cause finite strains. The accuracy of the results depends on the accuracy of the nodal point measurement. When the strains are small, the positional error associated with each nodal coordinate can easily swamp the magnitude of the calculated strain; therefore, the strains need to be large so as to be physically measurable. In addition, the material is assumed to behave homogeneously and retain its geometric continuity throughout any deformation; i.e. the material does not rupture. Under these conditions the application of the grid strain analysis technique provides a means for establishing a mathematical relationship between a series of points marked on the surface of the sheet before and after deformation. Sowerby et al. and others [1,2] imposed the further constraint of pure homogeneous deformation within each element, in their development of the grid strain analysis method. This assumption confines the surface strain distribution to a discontinuous representation, which approaches the exact solution as the number of elements tends to infinity. While grid strain analysis has found favour in the sheet metal industry, it can also be successfully used to evaluate the strains in composite sheet materials. The observed strains may not always be interpreted as being characteristic of the deformation through the total thickness of the laminate, as the results correspond to the deformation in the surface layer only. However, the effect of the sub-surface layers on the surface layer can be demonstrated, even though it is impossible to determine the degree of inter-laminar shearing. Consequently, grid strain analysis provides a useful tool for observing the deformation behaviour of many different laminates. Some examples of the formation of spherical domes and a blister fairing are given later. Consider the deformation of a triangular element OAB, shown in fig. 6.1, which initially lies in the (X1, X2) plane. After forming the element has been translated to a new position in space and deformed into a new shape. If the x3 axis is directed along the surface normal, so that O'A 'B' lies in the (Xl, x2) plane, the relationship between the deformed and undeformed coordinates may be represented by a 2-D deformation gradient tensor, F. Ox i d x i = OXR
dXR -- FiR dXR
i, R - 1, 2
(6.1)
220
T.A. M a r t i n et al.
x3 ~ O'
X2
X3
A' x1
B
X, Fig. 6.1. Deformation of a triangular element.
Using the nodal coordinates of points A and B in their undeformed and deformed states respectively, two position vector matrices may be assembled. [ dX'IA dX'IB] dZ ~ [dX2 A dX.2Bj
and
dz
[ dxlA' = [dx2a,
dxl~,]
(6.2)
dXzB '
The deformation gradient tensor can then be constructed in terms of these two matrices by rearranging (6.1) to yield F = dz dZ -1
(6.3)
and the right Cauchy-Green strain tensor, C, is defined by C = FTF
(6.4)
The principal stretches may be determined from the eigenvalues of C and their directions from the eigenvectors of C, using the principal axis theorem (C - ZI)fi = 0
(6.5)
where
fi(1) ~ [COSo0 i_si n ] and ~(2)~ [ - sin01
[ cos 0 J
The principal stretches, X1 and )~2, are the ratios of the deformed element length to the undeformed element length along each principal direction and 0 represents the angle between the X1 axis and the direction of the first principal strain. In problems dealing with finite strains, the Lagrangian strain tensor is often used to express the degree of deformation. The principal Lagrangian strains on the surface are given by 1 Yi -- ~ (~2 _ 1)
(6.6)
Grid stra& analysis
221
Two alternative principal strain definitions may also be expressed in terms of the principal stretches, such as the engineering strain E i = Z i --
1
(6.7)
and the true strain Ei ~-- I n ( Z / )
(6.8)
If the material is incompressible, it must satisfy the incompressibility constraint equation El "+" e2 + e3 - - 0
(6.9)
which may be used to calculate the through thickness strains, e3. From the foregoing analysis it is clear that the grid strain analysis technique can be readily applied, if the deformed sheet is modelled as a polyhedral surface, by treating each element as a 2-D segment in space. This approach is reasonable when the surface contains a small degree of curvature. In regions of severe curvature, however, the calculated strains will be less accurate, unless a large number of elements is used to describe the geometry. Therefore, an improvement on the grid strain analysis technique has been developed by Duncan and Zhang [3,4]. Splines are fitted to the deformed geometry in their analysis. Each triangular element is then flattened out with the true lengths of each side equal to the arc length on the surface. A twodimensional analysis of the strains is then carried out in a similar manner to that just mentioned. While this approach has some merits, it distorts the element shapes and may adversely affect the results. The concept of discretising the surface into a finite number of elements is not new, but a better method for calculating the strains can be developed, if a convected curvilinear coordinate system is used to describe the deformation. When the surface geometry is adequately described by bicubic parametric elements, the basis functions provide a method for determining the continuous strain variation across the surface. This method of analysis, the finite element strain analysis technique, has been developed at the University of Auckland and implemented on a Windows platform by Christie [5]. The PC-based version provides an excellent graphical interface for preprocessing and post-processing the data. A complete mathematical formulation of the scheme follows. The large strain analysis procedure begins with two sets of raw data representing the undeformed and the deformed grid co-ordinates. To clarify the procedure, a one-dimensional data set is considered first, as this approach can be readily extended into more dimensions later. The collected raw data is assumed to be inherently smooth and continuous, like that shown in fig. 6.2. The potential problem associated with a multi-valued function is easily avoided, if two parametric functions, Xl (0 and x2(~), are introduced to approximate the data. ~ is allowed to vary from 0 to 1 along the arc length. In this way, an N th order polynomial expression in ~ can be used to exactly interpolate N - 1 data points. However, a high-order polynomial is needed to interpolate a large number of points and this may lead to an oscillating fitted curve, which is uncharacteristic of the actual data. Instead of attempting to match
222
T.A. M a r t i n et al.
X2
X1
Fig. 6.2. Graph of a multi-valued function.
the data at every point, a better solution can be obtained by dividing the domain up into a finite number of regions, within which low-order polynomials can be used to best fit the data. The fitted curves must also provide continuity from one element to the next. A suitable method for determining the least squares best fit will be discussed later. The parametric cubic equation is the lowest-order polynomial function which can be forced to meet four constraint conditions by an appropriate selection of its coefficients. Using cubic Hermite basis functions, the position and gradient vector at each boundary can be specified to ensure C 1 continuity from one element to the next. The formulae for these four basis functions are r
- 1 - 3~ 2 + 2~ 3,
~b2 - ~ ( ~ -
1) 2,
r
- ~2( 3 - 2~),
~4 -- ~3 _ ~2
(6.10)
The cubic Hermite basis functions are plotted in fig. 6.3. Functions r and r have zero slopes at both ends, a maximum value of one and a minimum value of zero at the endpoints. These two functions interpolate the nodal positions on the element boundaries. While r has a unit slope at ~ = 0 and a zero slope at ~ = 1, ~4 has a unit slope at ~ = 1 and a zero slope at ~ = 0. These two functions interpolate the gradients on the element boundaries. In the following development, bicubic Hermite elements are used to describe the surface geometry of both the undeformed and the deformed geometry. The bi-cubic Hermite element, shown in fig. 6.4, is topologically rectangular with four corner nodes. To construct a full bi-cubic basis over the element, four vector quantities are stored at each node: the nodal position, the slopes of the element sides along the ~l, ~2 directions, and a twist vector controlling the surface behaviour inside the element near the node. When these elements are joined together to form a finite element mesh, their adjoining elements share all of the properties of their common nodes, so that both strains and displacements are continuous across the entire geometry. With four vector quantities at each node and four nodes per element, 16 basis functions,
Grid stra& analysis
223
08 Q6
Q4 84
=,~
02-
Q1
Q 2 ~ = 0 4
Q5
Q6
Q7
Q8
~
1
.,02
Fig. 6.3. 1-D cubic Hermite basis function.
X4
Fig. 6.4. The bi-cubic Hermite element.
4~,(~1, ~2), are needed to interpolate the surface. These functions may be simply obtained from a cross-multiplication of the 1-D basis functions in eq. (6.10). The position vector of any point and its derivatives are defined at the surface coordinates (~1, ~2) by
Xi
16 Zn=l unq~n(~l' ~2)
OXi m
0~~
~ n=l
n O~n(~l, ~2) Ui -~~
i-1,2,3
or-l,2 (6.11)
where U i are the vector quantities at each node. When the assembled elements are all the same size, these interpolation equations are sufficient for continuity. However, if neighbouring elements are initially of different sizes, geometric continuity is preserved when the derivatives Oxi/OS~ are
224
T.A. M a r t & et al.
shared at a common node, where, S~ is the arc length along the ~t3 direction. The gradients, Oxi/O~ are then obtained by
~x____~i=
i - 1, 2, 3
~ X i ~OSfl
c~,/3 - 1, 2
(6.12)
The strains on either side of an element will differ slightly unless the nodal constants OS~/O~/3 are equal across the boundary. The data requirements for finite element strain analysis are identical to those required for grid strain analysis. All that is needed is a set of undeformed grid coordinates and their corresponding deformed co-ordinates. These may be entered in any order, provided there is a one-to-one correspondence between the sets. When the density of the data is increased at certain locations, more elements may be added to the mesh, allowing a more accurate solution. This is particularly important where the strain gradients vary markedly within a small region. The bottleneck in this process lies with the problem of accurately digitising the data points before and after deformation. The two predominant methods of acquiring this data involve using a threeaxis digitising table or photogrammetry [6]; the image processing of two or more camera views of the surface. The former method provides good accuracy at the expense of operator time, as opposed to the latter method, which reduces the data acquisition time but sacrifices accuracy. If a regular grid is used initially, this at least alleviates the problem of measuring the undeformed nodal coordinates. Pre-processing simply requires the user to lay out a mesh over the undeformed co-ordinates, with at least 16 data points within each element. The surface co-ordinates (~1, ~2) are then determined for each data point within its associated element. These are convected with the deformation, and this information is used together with the deformed nodal coordinates to find the best-fit surface. 6.3. Method of least squares fitting
Experimental data typically contains some measurement error, which may make nonsense out of any attempt to exactly interpolate it. Often it is more useful to find a solution which does not exactly match the data, but provides a better functional representation within each element. When polynomial basis functions are used to describe the surface geometry, their coefficients may be established by using a best-fit criterion. In this analysis the method of least squares is utilised. If the difference between the fitted surface approximation, Xg, and the actual data, gi, is called the error, Vg, a weighted squared sum of the error is defined by
'02 = E Wf(xP(~I, ~:2) -- /~p)2 p=l
=
E p=l
W pi
Un i ~ n p( ~ : l ,
~:2) --
U" pi
i = 1,2,3
\n=l (6.13)
where P is the total number of data points and W i is a weighting function. In order to optimise the fit, the squared error must be minimised. By differentiating (6.13) with respect to its coefficients and setting the result equal to zero it is found that
225
Grid strain analysis
0v2
OU----'fi =
2E p=l
n
u i ~ P ( ~ I , ~2) -- U i
W i
wiPc~Prn(~l, ~2)
p=l
r
(6.14)
' ~2) - - 0
\n=l r n=l
n
pap p W i u i ~ ) m ( ~ l , ~2)
, ~2) Ui - -
i -
1, 2, 3
p=l
(6.15) The portion of eq. (6.15) in square brackets represents the global system matrix. The right-hand side is the force vector and ui is the vector of unknowns. If the weighting function is identical for each of the xi data sets, the same global system matrix may be used to solve for all three co-ordinates. In any approach the fit may be improved by altering the weighting function at points which contain a larger degree of error. Sometimes the solution mesh may pass through the appropriate data points but exhibit severe oscillations in areas where the mesh has a large number of elements and few data points. Decreasing the element density and increasing the number of data points is the easiest way to alleviate this problem. Having obtained a solution for the deformed surface, a direct mathematical relationship has been established between the undeformed and the deformed surfaces. It is necessary to use this information to calculate the strains in a similar manner to that discussed previously. A deformation gradient tensor is formed by
Oxi
Oxi 0~o~
FiR -- O----~R= 0 ~ OXR
i - 1, 2, 3
c~, R -- 1, 2
(6.16)
The right Cauchy-Green tensor is calculated using eq. (6.4) and the principal strains are derived within each element across the surface using eqs. (6.5)-(6.8). It is clear from the foregoing theory, that this analysis method requires the collection of large amounts of data, which must be processed and presented in a useful format. It is not surprising that the availability and power of contemporary personal computers have put this analysis technique within the practical reach of many designers. Some appropriate graphical means of displaying this information will be considered next. There are essentially three useful methods for displaying the strain distribution over the surface of a component: (i) arrow diagrams, (ii) contour maps and (iii) strain space diagrams. Arrow diagrams indicate the magnitude and direction of the principal surface strains by displaying orthogonal pairs of two-headed arrows drawn on the undeformed/deformed geometry. The arrow sizes are proportional to the strain magnitudes. Inward-pointing arrows represent compressive strains and outward-pointing arrows represent tensile strains. Contour maps are useful for representing the distribution of a scalar function over the sheet surface. If the material is incompressible, a shaded strain contour map may be used to highlight thickness variations in the surface layer, by plotting the e3 strain distribution across the component. Strain space diagrams plot the principal surface strains against one another on a set of 2D Cartesian axes. These diagrams can be used to highlight various strain patterns such as biaxial strains, drawing strains and regions of plane strain. The strain space
226
T.A. M a r t i n et al.
diagram can be converted into a forming limit diagram, by specifying the degree of allowable deformation before failure in terms of the principal strains. This is often done for sheet metal materials, but has little relevance when considering the deformations of thermoplastic composite sheets. Because of the presence of kinematic constraints, it is impossible to obtain a full quantitative knowledge of the state of stress from the measured strain distribution, as the reaction stresses depend on the boundary conditions [7].
6.4. Forming a composite spherical dome Several examples will now be given which demonstrate the finite element strain analysis technique, using graphical methods to illustrate the strain distributions in components formed from continuous fibre reinforced thermoplastic sheets. The material used in these examples is called Plytron, a glass/polypropylene composite (35% fibre volume fraction) developed by ICI. One of the greatest problems with applying this analysis technique to thermoplastic composites is the need to keep the grid coherent on the surface. During forming the thermoplastic is in a molten state, but accurate data can be collected if care is taken not to smudge the surface. In the following examples the spherical domes were all formed using a single diaphragm on the underside of the laminate, to avoid interference with the grid pattern on the upper surface [8]. Figure 6.5 shows the true strain distribution on the deformed surface of a [0]4 Plytron laminate after being vacuum-formed into a dome shape at 190~ Only one quadrant of the geometry is illustrated because of the symmetry of the shell. The deformed fibre paths are shown by a series of solid lines on the surface. The fibres undergo little or no elongation during the forming process. Instead, the in-plane deformation is mostly accommodated by stretching transverse to the fibre direction. The small amount of simple shear along the fibres allows them to bend within the plane of the sheet in order to conform to the 3-D surface. The strain space diagram, shown in fig. 6.6, further emphasises the plane strain nature of the deformation of a uni-directional laminate, when it is formed into a
Fig. 6.5. A r r o w diagram for a [0]4 Plytron dome, elmax --" 18.2%, e2max -" 1.5%.
227
Grid strain analysis
0"16x~X 0.14:~j
0.06
0.04 0.02 I . . . . -0.15
t
I
-0.1
-0.06
C
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.....
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~ : 0.15
Fig. 6.6. Strain space diagram for [0]4 Plytron dome.
hemispherical dome. The magnitudes of the first principal strains reach over 18% in some elements in the body, while the magnitudes of the second principal strains remain small. Consequently, most of the data points remain close to the vertical axis. In the sheet metal forming industry, such a diagram is typically used to determine the acceptability of a forming operation based on the magnitudes of the principal surface strains. A failure envelope is determined from a number of tests which establish the forming limit of a metallic sheet under different strain conditions. Since failure is associated with thinning of the sheet, the lowest strain to failure point occurs when e2 is zero. The plane strain stretching of the sheet naturally leads to thinning, if the material behaves as an incompressible continuum. When this thinning is uniform there is no need to be concerned about the forming operation. However, if irregularities arise in the strain field, excessive stretching can occur in localised regions. The best means for reducing the occurrence of localised thinning in thermoplastic composites is provided by increasing the laminate thickness, adding layers with different fibre orientations and reducing the forming temperature. Figure 6.7 shows the strain distribution on the undeformed surface of a [0,9012s laminate, which was formed into a dome shape at 190~ Only one quadrant of the geometry is illustrated because of the symmetry of the problem. The undeformed surface fibres are illustrated by a series of solid lines, while the sub-surface fibres are illustrated by a series of dotted lines. In contrast to fig. 6.5, the strain magnitudes are greatest near the edge of the dome in the off-axis fibre direction, where a significant amount of shearing occurs. Large tensile radial strains are accompanied by large compressive circumferential strains in a pattern which is typical for sheet metal drawing processes. In CFRT materials the fibre lengths appear to remain unchanged as the sheet deforms by in-plane shearing. This deformation, shown in fig. 6.8, has been
228
T . A . M a r t i n et al. u I
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X2
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Fig. 6.8. Deformation of a trellis structure. described as trellis action by Cattanach et al. [9], because the extension of a trellis in one direction is accompanied by a contraction in the other direction. The kinematics of this problem have been discussed by several authors [10-12]. Using the geometry in fig. 6.8, the true strains, el and ez,can be expressed in terms of the extension ratios, 1 and ~'2 by E1
-
-
ln(L1) -- In \COS
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e 2 - ln(~'2)- ln(Sin~)ksin
(6.17)
When the fibres are initially orthogonal, i.e. 9 45 ~ the principal compressive strain always exceeds the principal tensile strain, so that an incompressible material thickens during forming. Another representation of the strain distribution, showing the theoretical and experimental results, is given in fig. 6.9, the strain space diagram. The scattered data corresponds to the experimentally determined strains and the solid line represents a theoretical solution for the deformation based on the trellis effect. It is clear that there is some measurement error involved in the large strain analysis method; however, the trend of the two results is quite similar. The experimental strains tend to be shifted upwards in the diagram, particularly in the regions where the second
Grid strain analysis
229
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Fig. 6.9. Strain space diagram for a [0, 9012s Plytron dome.
principal strains are small, because of the transverse spreading of the surface ply. In spite of this, the experimental data give support to the hypothesis that bi-directional laminates deform like networks of inextensible cords. This concept may be investigated further by considering the deformation of a bidirectional laminate, in which the fibres are not initially orthogonal. Figure 6.10 shows the principal strain distribution on the undeformed surface of a [+30, --3012 s laminate, which was formed into a dome shape at 190~ When forming ,,*'P
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Fig. 6.10. Arrow diagram for a [+30,-3012 s Plytron dome, elmax = 29%, e2max -- -28.5%.
230
T.A. M a r t i n et al.
a part like this, the initial angle between the fibres and the Xl axis may be considered to be either 9 = 30 ~ or 9 = 60 ~ depending on which region is being studied. This effect is clearly shown in fig. 6.10. In region 1, where large tensile strains are accompanied by smaller compressive strains, ~ - - 6 0 ~ and the fibres rotate from a high angle towards a low angle causing sheet thinning. In region 2, where large compressive strains are accompanied by very small tensile strains, ~ = 30 ~ and the fibres rotate from a low angle towards a high angle causing sheet thickening. The xl direction is considered to be aligned with the tensile strain direction in each region. The strain space diagram in fig. 6.11 shows how the theoretical principal strains for a trellis structure compare with the principal strains on the surface of a [+30,-3012 s laminate. The two solid lines represent the theoretical solutions for = 60 ~ and 9 = 30 ~ The experimental data is roughly divided into two clusters on the diagram which match the theory quite well. Under these circumstances, the thickness changes in the sheet can also be reliably predicted, since the finite element strain analysis method accurately quantifies the deformation of the whole laminate, as opposed to just the surface layer. Figure 6.12 shows the change in thickness in the laminate as a result of the deformation. This diagram clearly shows that the sheet thickens in region 2 and thins in region 1, as expected from the trellis model.
6.5. Forming a composite blister fairing In another illustrative forming example, the finite element strain analysis technique is demonstrated on an aerodynamic fairing. The shape of the flanged tear drop blister fairing can be seen in fig. 6.13, where the deformed grid points are superimposed on the undeformed grid points. The fairing was formed with [0, 9012s and [+45, -4512 s laminates to observe the effect of the fibre orientation on the deformation. The surface strains were slightly affected by the distortion of the diaphragm in ~=
60~
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0
+
'1
0.05
0.1
231
Grid stra& analysis
27.5% 20.6% 13.8% 6.9% 0% -6.9% -13.8% -20.6% -27.5% Fig. 6.12. Thickness strain contour map for a [+30,-3012s Plytron dome.
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contact with the upper surface, because each laminate was formed into a female mould using a double diaphragm thermo-forming process [13]. The first step in the analysis is to lay out a mesh over the undeformed grid. In this example there are three areas on the blank to consider: (a) the flat flange area, (b) the tear drop area and (c) the narrow bend region lying between them. As the curvature in the bend region is quite severe, a high concentration of elements is required there compared with the flange and the tear drop regions. The chosen mesh shown in fig. 6.14 is close to what could be considered optimal, given the final geometry. The next three figures show the main results for the analysis applied to the [0,9012s laminate. Figure 6.15 shows the principal true strains on the surface of the undeformed sheet. The surface fibres are represented by solid lines, while the sub-surface
T.A. M a r t i n et al.
232
%-'-+%%+ Fig. 6.14. Undeformed mesh for a [0, 9012 s laminate.
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fibres are represented by dotted lines. Note the particularly large strains in two elements on the right-hand side of the diagram, which seem inconsistent with the rest of the data. This is probably due to grid smudging or the inability of the fitted surface to match the actual surface in the narrow bend region. In components with a large degree of curvature it is sometimes more convenient to view the strains on the undeformed surface, to avoid losing sight of the elements curving away from view. The equivalent arrow diagram on the deformed surface is shown in fig. 6.16. These results must be interpreted somewhat differently from those in fig. 6.15. The principal strains are shown in their deformed orientations and the solid lines are the projections of the straight undeformed fibres onto the approximated surface. The dotted lines are naturally an estimate of the actual sub-surface fibre locations, as the grid strain analysis was performed on the surface ply only. Two important deformation processes may be identified in fig. 6.15 and fig. 6.16. The first is the trellis deformation of fibres at -t-45 ~ to the longitudinal axis in the part. In these regions the surface and the sub-surface layers behave similarly, so that the initially square fibre pairs deform into rhombi. The second process of unidirectional extension in the transverse fibre direction occurs in the centre of the tear drop region. This type of deformation is associated with thinning in the outer layer.
Grid strain analysis
233
......< .-!/-.
Fig. 6.16. Arrow diagram for a [0,
9012 s
blister fairing, elmax = 35.3%,
e2max --
-25.9%.
Another illustration of the principal strain results is given in fig. 6.17. In this figure the data is spread out on the drawing strain side of the diagram. The surface layer thinning is demonstrated by an accumulation of points along the plane strain axis. The magnitude of deformation associated with the trellis action is highlighted by the strains lying close to the solid line representing the ideal behaviour of an orthogonal network of inextensible cords. The results presented here have been derived from a successfully produced component. This demonstrates that CFRT sheets can sustain very large deformations while maintaining their structural integrity. In the following example a [+45,--4512 s laminate was utilised to form the same blister fairing part as that just discussed. The principal strains and the deformed fibre positions are shown on the final geometry in fig. 6.18. The solid lines are the projections of the straight undeformed fibres onto the approximated surface and the dotted r~ o35 03
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I
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234
T.A. M a r t & et al.
Fig. 6.18. Arrow diagram for a [+45,--4512s blister fairing. lines are an estimate of the sub-surface fibre locations. While the same deformation processes dominate this forming operation as those previously mentioned, the trellis deformation occurs along the transverse and longitudinal axes and the surface layer thins at an angle 45 ~ to the longitudinal direction. The difference between the deformed blank shape of fig. 6.16 and fig. 6.18 should also be noted. From these two figures it is evident that different initial blank shapes may be used to form the part, depending on the initial orientation of the fibres. This result has implications regarding the buckling stability of the laminate during forming [14].
6.6. Draping theory of textile fabrics When looking at the behaviour of a material as it is formed, it is often useful to compare the theoretical deformation of an ideal material with that of the one being studied. While the trellis model provides a good means to evaluate the relationship between el and e 2 in a bi-directional laminate, it does not provide any information about the maximum strain magnitudes and their locations in a manufactured component. This information is important when considering problems such as sheet thinning and buckling stability during forming. A more comprehensive approach is provided by considering the process of draping a textile cloth over a mould. Several authors [15,16] who have studied the problem of draping textile fabrics over arbitrary surfaces, have generated solutions by solving a series of non-linear simultaneous equations, which result from three simplifying assumptions: (a) the cloth behaves like an inextensible network of cords, (b) the yarn intersection points act like pinned joints and (c) the location of an intersecting warp and a weft yarn can be found by determining the intersection point between the surface and two spheres, whose radii are equal to the fibre arc length, ds. The third assumption leads to draping solutions which represent a polyhedral surface. In order to make the solution unique, a further kinematic constraint must be imposed. The fibre paths of one
235
Grid strain analysis
warp yarn and one weft yarn must be completely specified on the die surface. A simple case is shown in fig. 6.19, where the constrained warp and weft fibres lie in the x z and y z planes, respectively, on the surface of a hemispherical dome. Using the theory outlined by van West [15] draping solutions have been generated for a hemispherical dome and a blister fairing. The experimental results for these geometries have been discussed previously. Figure 6.20 shows the theoretical principal strain distribution on the undeformed surface of a [0,90] woven cloth, which is draped over a hemispherical dome. The constrained warp and weft yarns are highlighed by darker lines. This result compares favourably with the experimental result Drap~ start point
f
~x~
Fig. 6.19. Constrained
9
~
Co
d
warp and weft yarns on the surface of a hemispherical dome.
9
M
9
x
x
:c
9
9
9
X
x
x
x
9
9
9
X
X
x
x
x
x
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x
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x
x
,*
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X
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x
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9
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x
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x
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x
x
X
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X
X
9
.
Fig. 6.20. Arrow diagram for a woven cloth draped over a hemispherical dome, 82max = --12.2%.
elmax- 9.9%,
236
T.A. Martin et al.
shown in fig. 6.7. Since the deformed nodal coordinates were generated from the draping programme, there was no measurement error associated with determining their positions. The strains are consequently very uniform. From this arrow diagram it can be inferred that a balanced symmetric [0,90]s laminate deforms in a very similar manner to a bi-directional cloth material. The maximum compressive and tensile strain magnitudes also compare very well with the experimental evidence. Such an illustration demonstrates the usefulness of a strain analysis technique to identify what is happening in a composite material during forming. Unfortunately, it does not give any information about the inter-ply shear occurring as a result of the out-ofplane bending. In a similar manner, the strain distribution can be determined on the surface of a tear drop blister fairing. Figure 6.21 shows the arrow diagram for a textile cloth draped over one-half of a blister fairing with its weft yarn constrained along the longitudinal axis and its warp yarn constrained along the transverse direction. These yarns are highlighted by darker lines. In this case the results do not compare so well with the experimental results in fig. 6.16 because the real laminate exhibits some transverse spreading in the surface ply. However, the theoretical solution can be reliably used to determine the blank shape required to form the part. Consequently, the unnecessary material can be removed from the blank to help eliminate buckling instabilities during forming [14]. A final point to consider is the situation when there are more than two families of fibres in a laminate. In this case, the woven cloth analogy does not apply, since no inplane deformations are allowed by the kinematic constraints of the fibres. Only deformable surfaces can be produced from this type of cloth material and the grid strain analysis technique can provide no information about the deformation. However, a real laminate containing three or more fibre orientations deforms by a combination of in-plane and out-of-plane shear in order to take up a 3-D geometry.
Fig. 6.21. Arrow diagram for a woven cloth draped over a tear drop blister fairing, elmax-- 17.3%, e2max= --27.8%.
Grid strain
analysis
237
Figure 6.22 shows the strain space diagram for a [0, +60, -6012s laminate after being formed into a dome shape. From this diagram it is evident that the deformation tends to follow a trellis effect in some regions. Most of the data points fall within the limits of the two solid lines representing 9 = 30 ~ and 9 = 60 ~ for a bi-directional cloth material. Another illustration of the strain distribution is shown in fig. 6.23. In this case the solid lines represent the orientation of the undeformed grid points and the fibre directions are highlighted by darker lines on the arrow diagram. In the direction bisecting the first ply and the second ply ( ~ = 30 ~ the strains demonstrate a good agreement with the trellis model. However the strains along the third fibre direction
4> = 6 0 *
s,
0.3 0.25
0.2 x
x
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,,,=W,--..LX
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x X
-0.05
0
§
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0.05
o.1
Fig. 6.22. Strain space diagram for a [0, + 6 0 , - 6 0 1 2 s P l y t r o n dome.
Dome Pole
./
..
,,
Y,, ,,,,.
2nd fibre etionJ
~
-~, ~
_
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1st fibre direction Fig. 6.23. Arrow diagram for a [0, + 6 0 , - 6 0 1 2 s Plytron dome, elmax = 2 1 . 3 % , e2max = - - 2 9 . 6 % .
T.A. Martin et al.
238
do not agree so well with the woven cloth theory, as they are affected by the presence of the third fibre layer. In order for the third layer of fibres to play a minor role in affecting the surface strains, a significant amount of interply shear has occurred between the layers to accommodate the deformation. While the mechanical properties of the solid laminate are quasi-isotropic, the mode of deformation is clearly anisotropic.
6.7. Diagnostic applications Grid strain analysis can be very useful for studying the effects of different manufacturing variables such as forming temperature, speed of deformation and blank aspect ratios [8,17]. It may also be successfully utilised for early problem identification and diagnosis of instability problems in the form of sheet splitting or buckling. A simple but very good example of the former has been shown by Martin [8] while diaphragm forming a hemispherical dome from a [0]4 Plytron laminate. After forming to a certain depth a strain analysis was performed to reveal the distribution shown in fig. 6.24 that clearly shows irregular strain peaks in certain areas. This was the result of an uneven fibre distribution in the original composite laminate. Though only minor thinning was detected at this stage, when this dome was further deformed it revealed severe thinning and sheet splitting in the same area (fig. 6.25). The voids created on the surface spread from one edge to the other. However, once the problem was diagnosed, it was avoided rather easily by reducing the forming temperature slightly and effectively increasing the matrix viscosity. Similar bundle separation results have also been noticed by Martin et al. [17] by punch forming a dome from a [4512 s Plytron square (175 • 175 mm) laminate (fig. 6.26). A strain
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/
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p
B
,
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////\'
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\ X7 / 9
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Fig. 6.24. Tensile strain contour map for
a [0]4
Plytron dome, strain increment = 1%.
Grid strain analysis
239
Fig. 6.25. A Plytron dome showing matrix thinning as a result of unstable deformation, temperature > 190~
arrow diagram showing approximately one quarter of it is shown in fig. 6.27. Using a combination of coarse and fine grids in the flange and buckled areas respectively, a plane strain deformation transverse to the surface direction is evident under the punch nose, which indicates thinning and fibre bundle separation. This is not consistent with the deformation of a woven fabric and when such localised strain occurs in metallic sheets, it leads to tensile instability. This is a direct result of strain suppression along the length of the defect [18] and in this case the surface fibres provide an inextensible constraint which permits the defect to develop. For the same dome another interesting feature becomes apparent when out-ofplane sheet buckling is investigated. Figure 6.26 clearly shows that the extent of buckling increases with the forming speed. Furthermore, though the strain magnitudes are not very large compared to other buckle-free specimens studied [17], a large jump in compressive strain is evident where the buckling has occurred. Thus gross buckling can be linked to the strain gradient not the magnitude of strain alone. This is believed to be the case because the stress magnitude is dependent on strain rate which is reflected by the strain gradient. It becomes more obvious from fig. 6.28 where the strain contour map is shown for the same quarter. A similar but more
240
T.A. M a r t i n et al.
Fig. 6.26. 175-mm square Plytron blanks formed at varying speeds: (a) 2.5 mm/min, (b) 12.5 mm/min, (c) 50 mm/min; temperature = 175~ [17].
explicit example is shown by the same researchers while forming a dome in a carbon fibre/PEK laminate with long discontinuous fibres (LDF) of [4-45]4 s fibre architecture [14]. A digitised mesh plot of the surface of a 112.5 x 175 mm LDF sheet after forming is shown in fig. 6.29. The wrinkles in the direction off-axis to the fibres are evident in this diagram and the corresponding arrow diagram is shown in fig. 6.30. As expected, close to the hemispherical indentation large tensile strains directed towards the centre of the punch are accompanied by compressive strains. It is interesting to note that because the 45 ~ fibre direction does not represent a plane of geometric symmetry in this blank, the degree of deformation in short sides exceeds that in the longer side of the blank. Again the buckles coincide with the regions of high strain gradients. Although the strain magnitudes are much smaller near the indentation on the long side, the strain gradients are higher as is evident from the
Grid strain
analysis
241
surface fibre direction ._
Jr
,,,r
~ -~
§ ~
x
§
.+..~
+
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§
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o
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,
nnunn Fig. 6.27. Arrow diagram for one-quarter of 175-mm square Plytron sheet after forming at a speed of 12.5 mm/min, temperature = 175~ Elmax-- 17%, E 2 m a x - - - 1 6 . 5 % [17].
strain contour map (fig. 6.31). Interestingly the buckling is also reported to take place first in this region. 6.8. Concluding remarks 9 The grid (large) strain analysis technique provides a snapshot of the deformation process, which illustrates the characteristic surface strains generated during the forming of fibre-reinforced composite sheets. 9 Fibres behave as inextensible constraints and dominate the deformation process. A laminate with two directions of fibre reinforcement seems to behave like a bidirectional woven fabric when formed into a die. If this is the case, over the majority of the sheet the principal strain directions must change during forming as the fibres rotate. The knowledge of such deformation paths becomes important for calculating the resulting stress magnitudes but has no effect on the strain results.
"[17I] Do0L~ = o:mle~odtuol 'u!tu/tutu g'EI - poods ~u!ttuoj 'uotutoods tutu gLI x ~'EI I : I G ' ] m3 :toj tug~ge!p qsotu pos!l!~!G "6E'9 "g!~
"[L[] %1 = luotuoaou! u!13a~s 'DogLI = oan~aodtuo~ ' u ! t u / t u t u ~'EI jo poods 13 lg ~u!ttuo 3 ao1313~ooqs uoa~s oal3nbs tutu-g L I J o aola13nb ouo aoj d13tu ano~uoo u!13als OA!SsoadtuoD "9E'9 "~!d
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9 The results from sheets reinforced in more than two directions can be clustered into bi-directional results indicating the limits of strain magnitude. 9 Unlike metallic materials, the thermoplastic sheets can undergo a significant amount of compressive strain without buckling as thickening takes place. 9 Flange buckling generally takes place away from the fibres and is associated with a large compressive strain gradient rather than the strain magnitude alone. This is due to the strain rate dependency of the materials. 9 GSA identifies the regions of severe deformation and provides provide useful information regarding the optimisation of blank shape, and hence improving the quality of the product.
244
T.A. Martin et al.
LDF [ + ~ 5 ~ 1 7 6
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References [1] Sowerby, R., Chu, E. and Duncan, J.L., Determination of large strains in metal forming, J. Strain Analysis, 17 (1982)pp. 95-101. [2] Schedin, E. and Melander, A., The evaluation of large strains from industrial sheet metal stampings with a square grid, J. Applied Metalworking, 32 (1986) pp. 143-156. [3] Zhang, Z.T. and Duncan, J.L., Developments in nodal strain analysis of sheet forming, Int. J. Mech. Sci., 32 (1990) pp. 717-727. [4] Duncan, J.L. and Zhang, Z.T., Strain measurement and modelling of sheet metal forming, Materials Forum, 14 (1990) pp. 109-114. [5] Christie, G.R., Deformation modelling of fibre-reinforced thermoplastic sheets, PhD thesis to be submitted at Auckland University (1996).
Grid stra& analysis
245
[6] Vogel, J.H. and Lee, D., An automated top-view method for determining strain distributions on deformed surfaces, J. Materials Shaping Technology, 6 (1989) pp. 205-216. [7] Christie, G.R., Collins, I.F., Bhattacharyya, D., Out-of-plane buckling of fibre-reinforced thermoplastic sheets under homogeneous biaxial conditions, J. App. Mech., 62 (1996) pp. 834-840. [8] Martin, T.A., Forming fibre reinforced thermoplastic composite sheets, Ph.D. Thesis, University of Auckland, New Zealand (1993). [9] Cattanach, J.B., Cuff, G., Cogswell, F.N., The processing of thermoplastics containing high loadings of land and continuous reinforcing fibres, J. Polymer Engineering, 6 (1986) pp. 345-361. [10] Rivlin, R.S., Networks of inextensible cords, in Nonlinear Problems of Engineering, Academic Press, N.Y. (1964) pp. 51-64. [11] Green, A.E., Adkins, J.E., Large Elastic Deformations and Non-linear Continuum Mechanics, Clarendon Press, Oxford, 1960. [12] Spencer, A.J.M., Deformations of Fibre Reinforced Materials, Clarendon Press, Oxford, 1972. [13] Bradley, S., Pressure forming of continuous fibre reinforced thermoplastic using a double diaphragm, Project in Mechanical Engineering Report PME 94/12, University of Auckland, New Zealand, 1994. [14] Martin, T.A., Bhattacharyya, D., Pipes, R.B., Deformation characteristics and formability of fibrereinforced thermoplastic sheets, Composites Manufacturing, 3/3 (1992) pp. 165-172. [15] Van West, B.P., Pipes, R.B., Keefe, M. and Advani, S.G., The draping and consolidation of commingled fabrics, Composites Manufacturing, 2 (1991) pp. 10-22. [16] Heisley, F.L., Haller, K.D., Fitting woven fabric to surfaces in three dimensions, J. Textile Inst., 2 (1988) pp. 250-263. [17] Martin, T.A., Bhattacharyya, D. and Pipes, R.B., Computer aided grid strain analysis in fiber reinforced thermoplastic sheet forming, in Computer Aided Design in Composite Materials Technology, ed. S.G. Advani et al., Computational Mechanics Publications-Elsevier Applied Science (1992) pp. 143-162. [18] Hosford, W.F. and Caddell, R.M., Metal Forming: Mechanics and Metallurgy, Chapter 15, 1st edition, Prentice Hall, N.J., USA (1983) pp. 294-302.
This . Page Intentionally Left Blank
Composite Sheet Forming edited by D. Bhattacharyya 9 Elsevier Science B.V. All rights reserved.
Chapter 7
Implicit Finite Element Modelling of Composites Sheet Forming Processes C.M. O BR/i~DAIGH, G.B. McGUINNESS
a n d S.P. M c E N T E E
Composites Research Unit, University College, Galway, Ireland
Contents Abstract 248 7.1. Introduction 248 7.2. Modelling of composite sheets during forming 254 7.2.1. Review of published work 254 7.2.2. Ideal fibre-reinforced fluid 255 7.2.3. Rheological measurement of composite shear viscosities 258 7.3. Numerical solutions - - plane stress problems 258 7.3.1. Assumptions 258 7.3.2. Problem formulation and solution scheme 260 7.3.3. Finite element solution technique 261 7.4. Central indentation of a composite s h e e t - the shear-buckling problem 7.4.1. Punch experiments with circular uni-directional sheets 264 7.4.2. Uniform radial velocity case 265 7.4.3. Uniform radial pressure case 273 7.4.4. Parameter s t u d y - pressure loading case 275 7.4.5. Extension to multi-directional sheet analysis 282 7.4.6. Stability considerations 285 7.5. Experimental comparisons - - diaphragm forming 286 7.5.1. Motivation 286 7.5.2. Experimental procedure 286 7.5.3. Experimental results 288 7.5.4. Numerical analysis and models 290 7.5.5. Mesh sensitivity and design 291 7.5.6. Numerical results and comparisons 293 7.6. Conclusions of plane stress analysis 303 7.7. Numerical solutions - - plane deformation problems 305 7.7.1. Problem s t a t e m e n t - single-curvature forming 305 7.7.2. Plane deformation modelling 307 7.7.3. Problem formulation 308 7.7.4. Finite element solution technique 308 7.7.5. Computational details and results 310 7.7.5. Ply contact formulation and results 311 7.8. Conclusions of plane deformation analysis 315 247
263
248
C.M. O Brddaigh et al.
Acknowledgements 318 Nomenclature 318 References 319
Abstract
This chapter discusses implicit finite element methods of simulating composite sheet forming problems. Each ply is assumed to behave as a transversely isotropic, incompressible Newtonian fluid at forming temperature. The presence of high volume fractions of continuous elastic reinforcing fibres in the molten polymer leads to the kinematic constraint of inextensibility in the fibre direction, and associated arbitrary tension stresses. A mixed penalty numerical formulation is constructed by discretizing the weak forms of the constraint and governing equations for creeping flow, using independent interpolation of the velocity and tension stress fields. Numerical solutions are given for two types of planar problems, plane stress which is used to simulate the problem of diaphragm forming a small indentation in the centre of a large composite sheet, and plane deformation which is used to simulate single-curvature forming situations. The plane stress analysis calculates the stress and deformation patterns which are responsible for shear-buckling under rapid forming conditions, by considering uniform radial velocity or pressure boundary conditions applied at the inner radius of an annular sheet. Experimental results are presented which correspond with the numerical predictions. For multi-ply lay-ups, each ply is analysed individually, and average stress predictions for the laminate are obtained on this basis. A detailed comparison between numerical stress predictions and experimental buckling patterns is presented for central indentation of circular uni-directional, cross-ply and quasiisotropic preforms. Parameters influencing the magnitude and location of peak tangential stresses include tangential fibre lengths and diaphragm/composite viscosity ratios. The effect of sheet width and shape on the instability patterns is investigated for quasi-isotropic laminates using both numerical and experimental techniques. The plane strain finite element model presented can model isothermal shearing and plane transverse flows encountered in forming composite laminates into singlecurvature shapes. These flows are the dominant mechanisms in the forming of important industrial shapes such as J- and U-beams for aerospace applications. The finite element formulation uses a mixed penalty approach with independent interpolation of the velocity, pressure and fibre tension stress fields. Results are shown which agree well with available analytical studies for both single-ply and multi-ply deformations. Experimental characterizations of the inter-ply slip behaviour are used to develop a general-purpose contact-friction algorithm for forming situations. The results shown are an important step towards the development of a simulation tool for single-curvature composite forming. 7.1. Introduction
Sheet forming manufacturing processes are a means by which composite laminates can be formed into a variety of complex curvilinear shapes [1]. These processes offer
Implicit finite element modelling
249
the composite manufacturer the possibility of automating the placement of laminates onto complex curvature mould surfaces, thereby producing cost-efficient, highperformance structures with predictable and repeatable properties. Thermoplastic composite forming, in particular diaphragm forming and press forming, can provide an effective means of producing finished structural shapes in a single moulding process involving heating, forming, and cooling [2]. Sheet forming of thermoset composites is also an emerging technique for placement of prepregs onto mould shapes prior to bagging and autoclave curing [3]. The sheet forming of thermoset composites is widely known in industry as the "hot-draping" process. In both cases the forming parameters are similar: lay-up and mould geometry; method of application of forming loads (i.e. rubber pads, metal dies, hydrostatic pressure applied across diaphragms); forming cycle and material parameters. The only important difference is that the thermoplastic material is heated to melt temperature prior to forming and then cooled, whereas the thermoset material is formed at room temperature and then cured. Forming of a continuous fibre composite laminate, whether in a thermoplastic or thermoset matrix, can present considerable difficulties, as the material is highly constrained in the deformations it can follow [4]. The processing window in each case is dictated by limiting or failure conditions resulting in a lack of material conformability with the mould surface. Fibre-buckling, fibre-washing, excessive thickness build-ups, ply-splitting, void-formation and warping are some of the problems that must be avoided if a component is to be formed successfully. The quality of a sheet-formed component will depend on many factors such as mould design, laminate lay-up, forming rate and temperature etc. It is not the purpose of this chapter to describe the effect of the various forming parameters on the process, as these will be dealt with elsewhere in the volume. It is necessary, however, to introduce the basic deformation mechanisms of the material which are to be modelled, and to describe various experiments which have been carried out to assess the validity of the models developed. This chapter outlines the development of a finite element software tool that can simulate aspects of the sheet forming process, and thus assist the designer by allowing a prediction of the final part quality, helping to avoid the traditional (and expensive) trial-and-error design procedure. Finite element modelling of composite sheet forming processes has taken two routes - - the implicit and explicit approaches. The code presented here is based on an implicit approach, which has been applied to planar problems in sheet forming, or to problems that can be approximated as planar. In particular, the shear-buckling problem [5], which occurs when a small indentation or bead is drawn in the centre of a large composite sheet under rapid forming conditions, has been analysed using a plane stress variation of the analysis. A plane strain, or plane deformation analysis, has also been developed in order to simulate the forming of single-curvature components. Both approaches are based on the same model of the material, which is assumed to act as a viscous polymer melt, reinforced by high volume fractions of high-stiffness fibres. In each case, the finite element solution technique is an implicit one, i.e. it involves the formation of a tangent
250
C.M. 0 Brdtdaigh et al.
stiffness matrix which is inverted at each step of the problem in order to find the solution for that time step. The explicit finite element method has been applied to composite forming [6] as part of an overall programme in the simulation of composite press forming [7]. This method may have certain advantages over the implicit method in the area of tool and inter-ply contact, particularly at fast forming speeds, whereas the implicit method may be more suitable for slower forming methods such as diaphragm forming. More information on the implicit formulation presented here can be found in references [8-121. A uni-directional continuous fibre composite consists, during the forming process, of very stiff, continuous, highly collimated fibres embedded in a relatively compliant matrix. The matrix may be a thermoplastic, in which case the forming takes place above the melt temperature of the polymer. It may also be an uncured thermoset matrix, in which case the forming is carried out at temperatures between 50~ and 100~ where the matrix is soft but not quite molten. The presence of up to 60% volume fraction of continuous reinforcing fibres in the molten polymer matrix is the predominant factor in composite sheet forming. The mismatch in properties between the axial and transverse directions is so large that, in effect, the fibres may be regarded as inextensible. In practice, this results in a very high viscosity in the fibre direction compared to other unreinforced directions. The result is a material which deforms primarily in shear, along and transverse to the fibre direction. This situation is further complicated in reality as all practical laminates will have plies of more than one fibre orientation, all with different inextensible directions. Figures 7.1(a) and 7.1(b) illustrate the forming mechanisms observed in these processes. The two main shearing mechanisms are the intra-ply mode, where fibres within each ply move past each other in the plane of the ply; and the inter-ply mode, where plies that cannot stretch in a particular direction slide over each other in order to achieve a laminate curvature. Secondary mechanisms include transverse flow of fibres and matrix material in response to pressure gradients, and resin percolation, more often seen with thermoset matrix materials than with thermoplastics. Fabricreinforced composites have other forming mechanisms which will not be dealt with in this chapter. Figure 7.2 shows the hierarchy of mechanisms that is needed to form particular shapes in order of increasing importance. For consolidation of flat sheets with a compliant diaphragm, only resin percolation through the fibre bed is necessary. However, if matched dies are used, transverse flow of fibres and matrix will also be required. Forming of single-curvature shapes requires the inter-ply shear deformation mechanism, whereas all mechanisms are needed, in particular the intra-ply shear mechanism, in order to form double-curvature shapes. Diaphragm forming and matched-metal die forming are the forming processes which have received most attention from manufacturers and researchers. Diaphragm forming of thermoplastic composite laminates [13,5] has been developed on the principles of vacuum forming of thermoplastic sheets. The process involves placing the composite lay-up between two thin deformable diaphragms which are
Implicit finite element modelling
251
1, Intraply Shearing 2. Resin Percolation
1:
3. Transverse Squeeze Flow
~~
P
~ ~ ~ ~
Po o Oo O o o ~ t , 1oo6oooo
~5 o..ol
o~176176176 o~"ol 'I ~ o Q o o o o 9 "1
P
Iooo~O o~
.
o ao~oodoo %00,
oo
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Fig. 7.1 (a). Intra-ply forming mechanisms.
4. Interlaminar Slip
F
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F
M
F
M Fig. 7.1(b). Inter-ply forming mechanisms.
clamped around the edges, while the composite itself is unclamped. A positive hydrostatic pressure is then applied to form the heated assembly onto a suitable tool shape (fig. 7.3). The diaphragms maintain tension on the laminate during forming, while allowing the inextensible fibre ends to move inwards. Research into failure mechanisms has concentrated on the occurrence of instabilities during forming of simple shapes such as hemispherical section components. The term shear-buckling was used to describe the symmetric out-of-plane wrinkles which occur at 45 ~ to the reinforcement directions during central indentation of a large composite sheet [5]. The buckling has been observed to depend on the forming rate
252
C.M. O Br6daigh et al.
REQUIREMENT
MODE Consolidation: Compliant diaphragm
~NXX \XXXNX \XX~
Resin percolation +
Matched Die
~\\\\\\\\\\\\~, ~\\\\\\\\\\\\'x~
Shaping Single Curvature
Interply Slip
9
Double Curvature
Transverse Flow
~J'~-
Intraply Shear + Interply Rotation
Fig. 7.2. Hierarchy of forming mechanisms. Upper Diaphragm
P+
Mould /
/
Patm Lower Diaphragm
Composite Layup Free To Move Between Diaphragms
[~
!1
To Exhaust
Fig. 7.3. Schematic of diaphragm forming in pressurized autoclave.
and to increase in severity as the ratio of sheet external diameter/radius of internal formed area increases. Uni-directional and cross-ply laminates exhibit four symmetric wrinkles at the i 4 5 ~ locations, as shown in fig. 7.4. Quasi-isotropic [0~ 4 5 ~ 45~176 laminates exhibit eight symmetric buckling sites, as shown in fig. 7.5. The effect of increasing diaphragm stiffness is to eliminate the shearbuckling phenomenon completely. Researchers have also noted the dependence of buckling on forming rate and temperature during the matched-die forming of composite sheets [14,15]. In one case [14], the attachment of spring-loaded tensioning devices to the ends of a laminate was seen to alleviate fibre wrinkling in a 90 ~ bend.
Implicit finite element modelling
Fig. 7.4. Buckled female cross-ply laminate. Lay-up: [0~176 lines.
s.
253
Laminate exhibits shear-buckling on 4-45~
Fig. 7.5. Effect of diaphragm stiffness on shear-buckling of quasi-isotropic [0~ 4 5 ~ 45~176 female laminates. Left: part formed with Upilex-R diaphragms. Right: part formed with stiffer Upilex-S diaphragms.
254
C.M. 0 Brhdaigh et al.
Diaphragm failure during forming is a commonly encountered failure mechanism in composite sheet forming. The degree of coupling between the diaphragm and the composite depends on the stiffness of the diaphragm with respect to the laminate being formed, as well as the surface treatment applied to the diaphragms. Polymeric diaphragms tend to exhibit the same deformation pattern as the surface plies of the lay-up, thus resulting in localized stress concentrations at the ends of the fibres which have moved inwards most. Failure due to tearing is often encountered at these regions. Summaries of sheet forming techniques and research are included in references [1,16].
7.2. Modelling of composite sheets during forming 7.2.1. Review of published work The behaviour of composite sheets at forming temperature is termed "highly anisotropic" and is often modelled kinematically [17-20] with the restriction that the prediction of stresses is not possible. As the buckling phenomena have been shown to depend on the forming rate, temperature and externally applied tension forces, purely kinematic analysis cannot predict these instabilities. Constitutive approaches to the problem have considered the composite sheet either as a continuum, or as layers of discrete materials. Tam and Gutowski [21] modelled the three-point bending of a laminate using discrete layers of elastic and viscous materials. This approach was extended by Talbot and Miller [22] to an array of point forces, though the viscous layers were omitted. This approach is, however, limited to small deformation beam bending in one plane only, but the results may be compared to the numerical results of section 7.7.5 below. A continuum mechanics analysis of composite forming may be carried out by assuming the material to be inextensible in the fibre direction [24-26]. Some closed-form and numerical solutions for forming of simple shapes have been developed [27,28], but these techniques tend to be specialized and geometrically limited. A numerical analysis technique which incorporates the strong kinematics and ratedependence of the problem, while allowing for complete geometric generality is clearly needed. A thorough review of the modelling of the composite sheet forming process has been published by Tucker [29]. This chapter presents a general-purpose finite element simulation code which predicts the stresses and deformations in a composite sheet subject to predominately planar forming forces, either in the plane of the laminate, or in the through-thickness direction. The mechanics-based formulation treats each composite ply as a fluid reinforced with inextensible directions, and the diaphragms as unreinforced fluids. Though implicit and explicit finite element modelling of metal and polymer sheet forming processes are well developed [30], the methods have only recently been used for analysis of continuous fibre-reinforced sheet forming. Scherer et al. [31] modelled the forming of single-curvature beams of material using discrete elastic/plastic elements. Beaussart [32] developed an implicit thin shell element for the transversely isotropic material with an elongational viscosity of approximately twenty times the
Implicit finite element modelling
255
shear and transverse viscosities. However, the formulation which employed is an irreducible one, which has been shown to exhibit element locking [33] at the large elongational viscosity ratios associated with highly anisotropic materials. The mixedpenalty finite element system presented here has been developed by the author in earlier publications in order to avoid these problems of element locking [8-12] and was later extended to a shell formulation by Simacek [34].
7.2.2. Ideal fibre-reinforced fluid (IFRF) The theory of ideal fibre-reinforced materials has been developed for treatment of highly anisotropic elastic and plastic materials [35,24]. In this analysis, each ply of the composite sheet is modelled as a transversely isotropic Newtonian fluid, which obeys the twin constraints of inextensibility in the fibre direction and material incompressibility. The constraint of incompressibility is a well-documented condition, but the consequences of constraining the extensional deformation in one or more directions are not widely understood. In general, with a constraint there is associated an arbitrariness in the stress, which is determined only to within a stress which does no work in a motion satisfying the constraint. For example, in an incompressible material the stress is only determined to within an arbitrary hydrostatic pressure. With each constraint, an arbitrary scalar
function describing a stress becomes available for satisfying the equations of motion or equilibrium. In the case of inextensibility, the arbitrary stress that results is a tension stress in the fibre direction. The kinematic constraints restrict the range of possible deformations of the material, and are given [25] for a fluid as: Incompressibility dii ~" 0
(7.1)
lnextensibility in a fibre direction aiajd O. - - 0
(7.2)
where a is a unit vector representing the local fibre direction and d is the Eulerian rate-of-strain tensor: l (Ov i Ovj~ do. -- ~ IkOXj + OXi/I
(7.3)
where v is the velocity vector, and xi are the components of the Cartesian axes. In general, the local fibre orientation vector is a function of both space and time. a = a(x,t)
(7.4)
Assuming that the fibre directions rotate as material lines during deformation, Rogers [25] gives the following expression for rotation of the local fibre orientation vector:
Fibre rotation Da OV i D t = ak kOX~
(7.5)
256
C.M. 0 Brfdaigh et al.
where the derivative denotes differentiation with respect to time following a material particle, known as the total time derivative. The constitutive relations for an ideal fibre reinforced Newtonian fluid have been developed by Rogers [25]. The material is assumed to behave as a transversely isotropic linear viscous fluid, with a single family of reinforcement. Furthermore, the fibres are assumed to lie in a plane and be continuously distributed throughout, acting only as strong or preferred directions. If the fluid is incompressible and inextensible in the fibre direction, we may write the constitutive equation in indicial notation as follows: (7.6)
cro = - P ~ i j + Taiaj + 20Tdij + 2(r/L -- rlT)(aiakdkj + ajakdki)
where 3ij is the Kronecker delta, 80. = 1 if i - - j ; otherwise 60 = 0; and tr is the Cauchy stress tensor, 0L is the longitudinal shear viscosity, r/T is the transverse shear viscosity, d is the rate of deformation tensor, P is an arbitrary hydrostatic pressure, and T is an arbitrary tension in the fibre direction. The constraint equations have caused two arbitrary stress terms to appear in the constitutive equation (7.6). Physically, these terms are an arbitrary hydrostatic pressure (P) and an arbitrary tension stress in the fibre direction (T). We further note that the stress may be divided into two components, as follows: (7.7)
trij = rij q- r O.
where r is known as the reaction stress due to the kinematic constraints, and r is known as the extra stress due to the deformation, defined by r ij = - P ~ ij -[- Za i aj
(7.8)
~ij = 20Tdo" + 2(r/L -- OT)(aiakdkj + ajakdki)
(7.9)
If we assume the fibres to be initially straight and parallel in a Cartesian co-ordinate system (fig. 7.6) with axes Xl, x2, x3 and to lie in the 1-2 plane at an initial angle, 0 to 3 axis b
Fibre Direction
0
f
1 axis
Fig. 7.6. Lamina configuration with co-ordinate system and fibre direction.
~
I
~
h < < a,b
2 axis
257
Implicit finite element modelling
the xl-axis, then the components of the local fibre orientation unit vector at any time instant, t become: (7.10)
(al, a2, a3) - (cos O(t), sin O(t), O) then, if we let m fill o-22 ~ 023 O'31 O'12
cos 0(t) and n - sin 0(t), eq. (7.6) may be written:
Oil 0 0 D22 0 0 m 0 0 0 0 _D16 D26
0 0 D33 0 0 0
0 0 0 D44 D45 0
0 0 0 D45 D55 0
D16 026 0 0 0 D66_
dll d22 d33 2d23 2d31 2d12
Tm 2 - PTn 2 _ p
--F
-P 0 0 mnT
(7.11) _
where, for convenience, the tensorial shear strain rates are replaced by the engineering shear strain rates, 2d23, 2d31, 2d12. The terms of the viscous constitutive matrix are given as follows: Dll -- 2r/T(1 -- 2m 2) + 4rlL m2 D12 -- 2r/r(1 - 2n 2) + 4rlL n2 D33 = 2r/T
D44 - r/T(1 -- n 2) + OLn 2
(7.12)
D55 -- 0r(1 - m 2) + r/zm 2
D66 -- r/L D45 = OIL - ~lT)mn
D16 = D26 = 2(r/L- OT)mn The incompressibility constraint, eq. (7.1) may be written in Cartesian co-ordinates as follows: Or1 ~- OV2 + OV3 -- 0
(7 13)
Similarly, the inextensibility condition, eq. (7.2) may also be written in Cartesian coordinates" c~
+ OXl] + sin20(t)-~x2
+ c~
(7.14)
For an arbitrary fibre orientation at any instant of time, t, eqs. (7.11), (7.13) and (7.14) completely describe the kinematic and constitutive behaviour of the highly anisotropic, incompressible Newtonian fluid. Finally, the equation for rotation of fibre angle in Lagrangian co-ordinates is given as: 80
8V2
8V2
a t = Ox----~+ tan O(t) ~x2
(7.15)
258
C.M. O Brddaigh et al.
The ideal fibre-reinforced fluid model represented by these equations will now be used as the basis for numerical solution of planar forming problems, under conditions of plane stress in the 1-2 plane, and under conditions of plane deformation, or strain, in the 1-3 plane.
7.2.3. Rheological measurement of composite shear viscosities Rheological studies of molten composite sheets mechanisms have been mainly carried out using torsional rheometers with centred and off-centred composite specimens. The material most investigated is APC-2 [36], a carbon-fibre-reinforced PEEK thermoplastic material, which has been examined by several researchers using dynamic oscillatory rheometers [37,38]. Though such experiments involve both axial and transverse intra-ply shearing modes, a derivation for separating these components is given by Kaprielian and Rogers [39]. Cogswell [2] surveys this work and reports average shear viscosity values of r / L - 6,000 Pa s and r/T-4,000 Pa s. Recent work using a newly devised picture-frame shear apparatus [40] for in-plane loading of APC-2 and other composites suggests that the true shear viscosities may be several orders of magnitude higher than those found by torsional rheometry [41]. For the purposes of this chapter, however, the values reported by Cogswell will be used unless it is indicated otherwise. 7.3. Numerical solutions w plane stress problems
7.3.1. Assumptions A number of assumptions will be employed in order to make the solution of composite diaphragm forming problems more tractable: 1. Each composite ply is assumed to behave as a transversely isotropic incompressible Newtonian fluid which is inextensible in a single reinforcement direction (ideal fibre-reinforced fluid) 2. The diaphragm material is assumed to behave as an isotropic incompressible Newtonian fluid. 3. The diaphragms and composite plies are thin in the direction perpendicular to the plane of the fibres (see fig. 7.6) and are subjected to stresses in this plane only. 4. The reinforcing fibres act only as inextensible directions, denoted by the unit vector a, i.e. the material is a continuum. 5. The fibres lie in the 1-2 plane at an angle 0 to the xl axis. 6. The fibres are assumed to rotate as material lines during deformation. 7. Each ply in the composite laminate is assumed to deform independently of adjacent plies. 8. The diaphragm is only present between the clamping points and the extremities of the laminate, to which it is rigidly connected. 9. Acceleration effects may be neglected, yielding what is known as a creeping flow problem.
259
Implicit finite element modelling
10. Body forces, such as gravity, may also be neglected. 11. The problem time domain may be subdivided into a series of small independent time steps, during which steady-state conditions are said to exist. This is known as the quasi-steady-state assumption. 12. Changes in sheet properties, such as thickness, deformed shape and local fibre orientations may be updated at the end of each independent time step. The assumptions above obviously limit the scope of the analysis, particularly the planar restriction on the deformation. It will be shown later, however, that certain forming problems may be modelled adequately as planar deformations. The assumption of a purely viscous response is also limiting, as elastic effects have been observed in both composite sheets and polymeric diaphragm materials at typical forming temperatures [37,42,43]. However, the authors believe that the extra effort in carrying out a full viscoelastic analysis is not, at this stage, justified. The assumptions that the plies behave independently of each other, and are therefore in a state of plane stress, also represent first-order approaches to the problem. In reality, as shown in fig. 7.7, traction between plies is transferred via inter-laminar shear deformation in a thin lubricating resin-rich layer [2]. The planar assumption also necessitates that the diaphragm tension be transferred into the laminate at the extremities, rather than along the surfaces. Neglecting acceleration effects should be reasonable for diaphragm forming, which tends to take place in minutes rather than seconds, but may have to be revised for faster matched-die forming.
Top Diaphragm ~, Traction
I
....
,
_
Resin-Ricl Layers, exaggerat~
/
Bottom Diaphragm
Tracti,
t
Diaph ragm
Composite
Fig. 7.7. Boundary conditions between diaphragms and composite during forming. Top: traction is transmitted from diaphragms into composite via inter-ply shearing of resin-rich interlayers. Bottom: plane stress analysis assumes diaphragm connected to outside of composite.
260
C.M. O Brddaigh et al.
7.3.2. Problem formulation and solution scheme For conditions of plane stress in the 1-2 plane, eq. (7.12) becomes
Elll [011+033~ 0261E 111[m2J 0"22 0"12
--
033 D16
022 --I-D33 D26
026 066
d22 2d12
+
Tr/2
(7.16)
mnT
where m = cos0(t) and n = sin O(t). The components of the plane stress constitutive matrix, D are given in terms of m, n, r/L and r/r as follows: Dll = 2r/T(1 -- 2m 2) + 4r/Lm2 D22 = 2r/L(1 -- 2n 2) nt- 4OLn2 D16 = D26 -- 2(r/L- ~r)mn
(7.17)
D33 = 2r/T D66 = r/L Note that the hydrostatic pressure does not explicitly appear in these equations, as the incompressibility condition (7.13) is directly satisified by adjusting the deformation rate in the through-thickness direction [8]: dll --1-d22 + d33 = 0 :=:} d33 -- - ( d l l -1- d22)
(7.18)
The plane stress assumption, in turn, leads to a direct relationship between the hydrostatic pressure and the in-plane extensional strain rates: (7.19)
P -- -D33(dll -+- d22)
Equations (7.14) and (7.15) for fibre inextensibility and fibre rotation respectively are unchanged by the assumption of plane stress. The diaphragm material is to be modelled as a thin isotropic incompressible Newtonian fluid. Such a material is characterized by a single fluid viscosity 17 with the following constitutive equation [44]:
0"0 = -Pgij + 2odij
(7.20)
Assumption of plane stress conditions in the diaphragm leads to the following equation set:
[11] 0"22 0"12
--
20 0
40 0
01[ll1
0 r/
d22 2d12
(7.21)
Equation (7.18) can also be employed to calculate the through-thickness deformation rate of the diaphragm sheets at any time instant. The problem is divided into a series of independent problems in a quasi-steadystate fashion. Following the solution for a single time step, the configuration and properties of the body are updated and used as input for the next time step. Let us
Implicit finite element modelling
261
assume that at time t, the flow domain is given by f2(t) with a fibre orientation vector a(t). The equilibrium and boundary traction equations are given as:
aO.J = 0 in ~2(t)
(7.22)
tj -----niaij on rt(t)
(7.23)
where rt(t ) is the boundary at time t on which the tractions t are prescribed, and ni are the components of the unit normal vector to the boundary. By solving the constitutive equations (7.16) and (7.21) and the inextensibility equation (7.14) with the equilibrium and boundary traction equations, one obtains a velocity field v(t). At time (t + Atn) the new domain S2(t+ Atn) is updated by moving all its points by means of the following explicit algorithm: (Xl, X2)n+ 1 --" (Xl, X2) n + (At)n(V 1 +/32) n
(7.24)
Furthermore, the sheet thickness, h, is also updated at each point from eq. (7.18): h ( x 1, X2)n+ 1 -- h(Xl, X2)n{1 -- (dll + d22)n}(At)n
For the composite body, the orientation vector is updated after the time step from eq. (7.15):
O(Xl, XZ)n+l -- O(Xl, XZ)n + (AOn FXl + tan O(Xl, x2)~ ~x2
(7.25)
(At)n (7.26)
The solution procedure may be summarized as follows:
Step 1 The boundary value problem is solved at time t. Solution variables include velocity, stress and strain rate fields.
Step 2 The body coordinates, thickness and fibre orientation field are updated from
Step 3
the velocity field solution. The terms of the constitutive matrices (7.16) and (7.21) are then updated to take account of changed fibre orientation and thickness. Return to step 1.
7.3.3. Finite element solution technique The weak form of the equilibrium and inextensibility equations are found [8] by introducing arbitrary velocity and tension stress vectors, 6v and 8T and integrating eq. (7.14), (7.22) and (7.23) over the problem domain, f2 as follows:
I ~k~Dk d~2 + 16kraT dr2 - 0 f2
(7.27)
~2
I(ST) TaTSv dr2 - 0
(7.28)
r
where the components of the stress, a, the virtual strain rate, e, and the fibre orientation vector, a, have been arranged in column matrices, and S is the plane stress
C.M. 0 Br6daigh et al.
262
derivative operator matrix. Equation (7.27) is known as the virtual work equation, and eq. (7.28) is the weak form of the inextensibility condition. The next step is to introduce suitable shape functions to discretize the velocity and tension fields at nodal points throughout the domain. Here, the tension and velocity fields will be discretized independently, in the same manner that incompressible flow problems are solved numerically by discretizing the velocity and hydrostatic pressure stress separately. Two sets of interpolation functions are therefore needed: v ~ "~-- NVav
(7.29)
T ~ T-
(7.30)
N tat
where N ~ and N t are the velocity and tension interpolation functions, and a ~ and a t are the listings of nodal velocities and tensions. Applying the above interpolations to eqs. (7.27) and (7.28) leads to the following finite element system:
K
Kt
av
f
For any two nodes i and j, the matrix components are given by K , y - I(B~.)T DB)' dff2 f2
K~. - I(B~')r aN~ d a
(7.32)
f2
f-
-J(N~') T t" dF F
where the matrix B v represents the first derivative of the velocity shape functions. Further details on the derivation of this equation system are included in references [8] and [9]. It may be shown [8] that a formal analogy exists between the plane stress formula-
tion for a Newtonian fluid reinforced with inextensible fibres and the equations of linear isotropie incompressible fluid mechanics (Stokes flow). In both cases the velocity is the primary variable. The secondary or constraint variable is an arbitrary stress, in one case a hydrostatic pressure, in the other a tension stress in the fibre direction. By analogy with Stokes flow, we may write a mixed penalty formulation for the current problem, the first step being the construction of an approximation for the tension field, as follows: T -- oraTe = otaTSv
as ot --> oo
(7.33)
Where ot is a large number, known as the inextensibility penalty number. Discretization of eq. (7.33) is followed by substitution into eq. (7.31), leading to the following penalized system: (Kt) r
-(1/o0M t
at
+
0
-0
asa~ee
(7.34)
Implicit finite element modelling
263
where M t is known as the tension m a s s m a t r i x , defined as follows:
Mb -- I N it'N j d ~
(7.35)
~2
As the tension field may be discontinuous between elements, a v is eliminated, yielding {K + o t K t ( M t ) - l ( K t ) T } a v -- f
(7.36)
After solution of the velocity field from eq. (7.36), the tension field may be recovered, as follows: a t = ot(Mt) -1 (Kt) rav
(7.37)
The finite element equation system for an incompressible, isotropic Newtonian fluid under plane stress conditions is given as follows: {K}av = f
(7.38)
where, for any two nodes, i and j, K / j - I(B}')rDB~d~2
(7.39)
This simple linear set of equations does not require any special solution technique, and together with eq. (7.36) yields a solution for the velocity field in the composite and diaphragms at any time.
7.4. Central indentation of a composite s h e e t -
the shear-buckling problem
A specialised finite element program has been developed for composite sheet forming problems, using the mixed penalty formulation of eq. (7.36) for composite elements and the simple irreducible system (7.38) for the diaphragm elements. The program, known as FEFORM, is based on a general-purpose finite element code called PCFEAP [45]. For more details on the element and program construction, the reader should consult references [1] and [46]. The inputs to the program include initial mesh and node locations, the initial sheet dimensions, thickness and fibre orientation, the composite material viscosities 0L and 0T, the diaphragm viscosity, ~, the total time and number of time increments and finally, the loading and boundary conditions. According to the solution scheme outlined earlier, the boundary value problem is solved in F E F O R M for each time step, using the mesh configuration, element thickness and fibre orientation resulting from the previous time step. As the equation system is linear, no special solution technique is required, the only extra effort compared to irreducible problems being the evaluation of the constraint matrices.
264
C.M. 0 Br6daigh et al.
7.4.1. Punch experiments with circular uni-directional sheets
These experiments [42] have been carried out using a heated circular punch and mould, as shown in fig. 7.8. The clamped diameter of the mould is 280 mm and the punch diameter is 25.4 mm. A circular uni-directional APC-2 laminate of 216 mm diameter is shown in fig. 7.9, after being formed by the punch to a depth of 15.5 mm. A polar grid had been inscribed on the surface of the laminate prior to forming. The deformed grid shown in fig. 7.9 illustrates that only a thin band of fibres at the centreline of the laminate have been forced to move inwards by the punch. The deformation is propagated to the outside of the sheet in the fibre direction, as this direction is inextensible. The occurrence of shear-buckling at the 4-45~ lines, emanating from the rim of the sheet, is depicted clearly in fig. 7.10, with some secondary buckling occurring further out in the sheet. This problem is now analysed using the FEFORM code. The objectives of this analysis are (i) to investigate the stress and deformation states of the composite sheet during this type of forming process, with particular consideration given to the conditions at the rim of the mould and (ii) to draw conclusions concerning the sensitivity of the shear buckling phenomenon to the different process parameters which may be varied. The limitations of this analysis are governed by the plane stress assumption in the finite element formulation. The punch deformation experiment is modelled as the in-plane loading of a large circular composite sheet. For the case of only one direction of fibre reinforcement, the analysis may be further simplified by observing that the disk is symmetric in four quadrants. Symmetry conditions are applied to the edges at x = 0 and y = 0 as shown in fig. 7.11. The diaphragm material is represented as an outer ring of
Fig. 7.8. Photograph of punch and mould of punch deformation apparatus.
Implicit finite element modelling
265
Fig. 7.9. Photograph of 8-ply uni-directional laminate formed in punch deformation apparatus. Crosshead speed = 12.7 mm/min. Central deflection = 15.5 mm.
isotropic viscous material attached to the outside edge of the composite sheet. The diaphragm is clamped at its outer radius, as is the case in real forming situations. Since the central forming region of the sheet is not modelled, assumptions must be made concerning the conditions at the rim of the mould. Two simple cases are considered, those of an applied uniform inward radial velocity and of a uniform normal pressure at the inner radius of the model. The composite material viscosities used in these analyses are, unless otherwise stated: r/L =6,000 Pa s and ~T =4,000 Pa s. The diaphragm viscosity is taken to be 108 Pa s. The quadrant is meshed using Q9/4 biquadratic velocity, bilinear tension elements. A simple polar mesh, such as that shown in fig. 7.12a, is used to analyse this problem in all the cases described here.
7.4.2. Uniform radial velocity case The first case we consider is that of a uniform inward radial velocity applied at the inner radius. This assumption is considered suitable for the case of the material being deformed into the mould at a constant rate by a punch. In reality, the velocity conditions at the rim of the mould will be influenced by the anisotropy of the central forming region of the disc. Since we are not in a position to predict these conditions, the uniform radial velocity condition will serve as an approximation to the real case. The simulation presented is for forming at a rate of 0.51 mm/min. The initial polar
266
C.M. 0 Brdtdaigh et al.
Fig. 7.10. Close-up photograph of shear-buckling emanating from inner radius, also showing secondary buckling.
Fig. 7.11. Symmetry and boundary conditions used to simulate punch deformation experiments. Outer ring of isotropic diaphragm elements are clamped at outer radius.
Implicit finite element modelling
267
FibredirectionI
b
d
c
e
Fig. 7.12. (a) 1,927-node, 460-element mesh of composite elements only. Outer radius is free, inner radius is subject to a uniform inwards velocity of 0.51 mm/min. Fibres lie in direction of y-axis. (b) Deformed mesh after 200 seconds, 4 time steps. (c) Deformed mesh after 400 seconds, 8 time steps. (d) Deformed mesh after 600 seconds, 12 time steps. (e) Deformed mesh after 900 seconds, 18 time steps.
mesh is depicted in fig. 7.12a. It contains 1927 nodes and 460 elements. In this case, none of the diaphragm elements were included, although the effect that diaphragm viscosity has on the model's behaviour will be examined later. Figure 7.12b shows the deformed mesh after 200 seconds, taken in 4 time steps of 50 seconds each, with only a small amount of deformation noticeable. As the simulation continues to 400, 600 and 900 seconds (fig. 7.12c--e), the deformation becomes much more marked. Figure 7.13 shows the deformation at 600 seconds, superimposed on the original mesh for a clearer illustration of the mechanism involved. There are two distinct regions of behaviour evident in this deformation, divided by the fibre which intersects the inner radius at the x-axis. The first is dominated by the shearing of the inextensible fibres to accommodate the applied velocity condition, while the second is typical of flow of a fibre-filled fluid transverse to the reinforcement direction. This discontinuity is better exposed by the use of a rectilinear mesh with a series of nodes positioned on the intersecting fibre [47]. Finally, note that the inner radius remains circular due to the uniform radial velocity applied. The deformed polar meshes of figs. 7.12d and 7.12e show a strong resemblance to the experimental grid displacements of the punched laminate of fig. 7.9. In order to investigate the correlation in more detail a laminate was deformed at a rate of 0.51 mm/min in the punch deformation apparatus for a total time of 10 minutes (600 seconds), to compare directly with the 600-second result of fig. 7.12d. Care was taken to ensure that the circles inscribed on the laminate coincided exactly with the
C.M. 0 Br6daigh et al.
268
Shear of Inextensible Fibres
A
I
I
Fibre Direction
Transverse Flow
Fig. 7.13. Deformed mesh after 600 seconds superimposed on original mesh.
radius of nodal points in the simulation [46]. A series of deformed circles on a quadrant of the punched laminate were digitized and are compared directly to the simulation results in fig. 7.14. The agreement is excellent in the fibre direction, but not so exact in the transverse direction. This discrepancy is probably due to the assumption that all points along the inner radius move radially inwards with the same velocity, which may not be the case. Other possible reasons for this discrepancy could be the arbitrary transverse viscosity value of 4,000 Pa s used, and the fact that the analysis does not include the diaphragm. Nevertheless, the excellent quantitative agreement between experimental and predicted deformations in the fibre direction are a major source of confidence in this approach. The stress results for the simulation will now be presented. In all cases shown below, an extra ring of clamped isotropic diaphragm elements have been added at the outer diameter of the composite quadrant, as illustrated in fig. 7.11. Shear stress O'xy contour results are shown in fig. 7.15 for the first time step of velocity loading.
Implicit finite element modelling
12011,
,
,~ 1
0
E
80
~"
20
,
i-,
0
,
,
i
,
,
~
or,,,~,,,it,,il~ 0
20
40
X - C o o r d i n a t e
,
i
,
,
,
,
,
,
,
i,,,
,,,,
A Experimental - 'Numerical
60
269
,
A,~,l ~t~,;:t, 80
100 120
( m m )
Fig. 7.14. Experimental versus numerical comparison of deformed grids. Experimental: 8-ply APC-2 laminate formed at 0.51 mm/min, after 10 min deflection. Numerical: deformation after 600 seconds, 12 time steps, 0.51 mm/min.
Fig. 7.15. trxy shear stress contours, radial velocity = 0.51 mm/min, diaphragm viscosity = 108 Pa s.
270
C.M. 0 Brdtdaigh et al.
The input radial velocity is 0.02 in/min and a diaphragm viscosity of 108 Pa s is used. The region of highest shear stress is located along the singular fibre which intersects the inner radius at the x-axis. There is a shear stress discontinuity between this region and the rest of the sheet, which is a feature of the behaviour of materials with directions of inextensibility [35]. The shear stress at the inner radius peaks at about 15~ to the x-axis, as shown in fig. 7.16. The boundary conditions of zero shear stress on the x- and y-axes are clearly satisfied. The diaphragm viscosity has no effect on the shear stresses predicted here, since the applied velocity boundary conditions completely specify the shear deformations which occur at the inner radius. Furthermore, due to the strong kinematics of the composite, the shear stresses at any point in the composite are completely independent of the diaphragm properties. The value of shear stress at any point is also directly proportional to the input velocity (fig. 7.17), and this holds for all stresses. This result is to be expected as the constitutive relationships employed are Newtonian. The same effect is noted for the axx stresses (in the transverse direction to the fibres) shown in fig. 7.18. The Cryy stress contours (in the fibre axial direction) are shown in fig. 7.19. These are dominated by tensile stresses in the region between the y-axis and the singular fibre. This is explained by considering the small area of diaphragm material at the outer diameter. This material is clamped at the outside (i.e. zero displacement) and attached to the fibre ends at its inner diameter. As these fibre ends are moving inwards at the same rate as the applied radial velocity, the diaphragm is under tensile stress in the y-direction. This tension is reacted into the fibre ends, causing the tensile stresses shown. Increasing the diaphragm viscosity therefore increases the tensile stresses in the fibres between 30 ~ and 90 ~ to the x-axis, as shown in fig. 7.20. These high tensile stresses in the diaphragm elements at the fibre ends are also the cause of diaphragm splitting and tearing during forming experiments.
25
mm
B
20
a,,
m
m
I
m
g
~ 1 5 -
!
e~
dlap v i s c = 1E5
visc
•
diap
+
diap v i s c = lEO
visc
=
=
1E7
1E3
I
u|nanlmiulmi|m!
i
5
dlap
o
m It
g
mlO L_
~
mwmm
-m
mm
mBD~
O
_L
0
~
L
,
,
I
~
,
J
,..
P..L
~
J
I
,
15 30 45 60 75 Angle from X-axis, Theta (deg)
Fig. 7.16. Effect of diaphragm viscosity 0.51 mm/min.
on
,
90
O'xy shear stress at inner radius for imposed radial velocity of
Implicit finite element modelling 600 500
m 400 300
I,,,,
~'~ L_
t'
'~ 'oc. b
-
"~
,.-
,-,
o'
I''
t
~'
,~
. . . .
/, v=O.02 intmin
o
i o,S
i
t
ax v=0.05 in/min v=0.1 in/rnin + v=0.2 in/min o v=0.5 in/min
oo
-8 Z-~ --
or. 200
--'7
100
]~
(o
0
-100
++ + + + +
WO0000000000Oo00
0
+
++
l -
0O +~- + + + r - + - - + +
+ + ++~
~+XrIc._~3UCDmr X X x x X X X X X X X X X X x X
~- - ~ ~ ~~ ~ ' ~
+_ L
'~'~ U 0
9 ,=++:
~ ~ ' ~ ~ ~ ~ ~
F ,_-i,, 0
I
15
,
30
I
,
45
271
I , ,
~
60
,
_
-t
-1
'~Q~
,
75
~
90
Angle from X-axis, Theta (deg) Fig. 7.17. Effect of imposed radial velocity Pa s.
200
~-
~oo
~r~" ,.~oOC.ol., I
~Fo ~~
9
w 9
....
-,
i .....
, + + + .+ +. +
!
....
OQOOOo -+~-+,
0
1
~ ~in/rnin I ~- I ~ v=0.05 in/min I
r
~ I = V:0.2 in/min I- I o v=0.5 in/m]n E i , "
F I • v=o., in/min
L .....
.....
i
~-100
0
,
~xy shear stress at inner radius, diaphragm viscosity = 108
oo
._r++ +~
/300
on
2
15 Angle
....
I . . . . .
30 from
o'++-,_+
~o
I
~176
~176 -i
I I I !
+++++-~ ~
. . . . .
45 X-axis,
I . . . .
60 Theta
' .....
75 (deg)
] %0,~k -
90
Fig. 7.18. Effect of imposed radial velocity on axx transverse stress at inner radius, diaphragm viscosity = 108 Pa s.
Quite a strong axial compressive stress is shown at the fibre which intersects the xaxis, the absolute value of which increases as the diaphragm viscosity increases. This could be a cause of fibre in-plane buckling, but experiments have shown a decrease in buckling as the diaphragm stiffness is increased. The stress components may also be usefully transformed into radial and tangential stresses. The radial stresses are shown in fig. 7.21, and are dominated by the fibre tensile stresses at 90 ~ to the x-axis. The total forces involved in achieving a particular deformation rate for different diaphragm viscosities can be represented as the areas under the curves in fig. 7.21. From this figure, forming composite sheets with higher-viscosity diaphragms will require much higher hydrostatic pressures. This has been experimentally observed [48] and
C.M. 0 Br6daigh et al.
272
Fig. 7.19. ~ryy axial stress contours, radial velocity = 0.51 m m / m i n , d i a p h r a g m viscosity = 108 Pa s.
6000
k .....
~ .....
i. . . . . . . . . . .
1
4000
AZX AAAA
2000
~_
0
-4000
~ 7
&A
.,
~O0~COOCCOCO~EDOP, D]O~
_..
~
diap vise = 1E6
~
-~ diap visc = 1E5
!-~
x
diap visc = lEO
+
no
p
-6000
[-t
0
_
zx
&
-2000
! ....
& A A z~ A 6, & / X / X A s A & A ZX A Z~ 'X A ~ _
.
:
9 :
i
I
15
,
,
j
i
= [
30
/
,
i
. . . .
45
diaphragm
_
i
60
. . . . .
I
. . . .
75
] 7 :
90
Angle from X-axis, Theta (deg) Fig. 7.20. Effect of d i a p h r a g m viscosity 0.51 m m / m i n .
on
tTyy axial stress at inner radius for imposed radial velocity of
Implicit finite element modelling
5000L
,,
~_ 4 0 0 0
== 3 0 0 0
~
t-
, .... ,
,
,
,
i
.
~ diap visc = 1E61 [] diap visc = 1E51 x diap visc = 1E411
+
no
diaphragm
. LX~
. 1 0 0 0 L ~ _ _ _ . ~ _ ~ _ _
0
~
I
. . . .
....
A
n,j.30
,,
~
,
,
273
_]
..zxt,/~c------I
,~A
--.' J
D D E C D E D C 0 FIO rn r"l D 0
:
! , .
:
,
~
15 30 45 60 75 90 Angle from X-axis, Theta (deg)
Fig. 7.21. Effect of diaphragm viscosity on arr radial stress at inner radius for imposed radial velocity of 0.51 mm/min.
the potential of finite element analysis to quantify this aspect of composite sheet forming is noted. Although the deformed shape predicted closely matches the experimental results, the stress results do not suggest that buckling should be expected at the 45 ~ point. The reason for this is believed to be the unrealistic nature of the applied velocity boundary condition at the inner radius. This assumption stipulates that the inner radius remain circular, which is not supported by experiments. However, the capacity of the approach to predict the overall sheet deformation, away from the inner radius, has been demonstrated. 7.4.3. Uniform radial pressure case
The second loading case considered involves the application of a uniform radial pressure on the inner radius. This is an approximation to the loading which would be experienced by a composite sheet during pressure forming in an autoclave, for example. The deformed and original meshes are shown in fig. 7.22 for the first time step results. In this case, a very high diaphragm viscosity is used (108 Pa s), which essentially restrains the fibre directions from significant inwards movement. The result is predominately transverse deformation, yielding an elliptical inner radius which is closer to that observed during forming, even with the punch apparatus. It is difficult to compare detailed mesh displacements with experimental deformations for this case as the inputs are in the form of prescribed loads, rather than displacements. Figures 7.23 and 7.24 illustrate the effect of the diaphragm viscosity on the predicted shear stresses, shown as normalized by the applied pressure. A zero diaphragm viscosity produces the stress pattern shown in fig. 7.23, with a peak at the inner radius approximately at 45 ~ to the x-axis, but with significant shears
274
C.M. O Brfdaigh et al.
Fig. 7.22. Deformed mesh after first time step for uniform radial pressure boundary conditions, shown superimposed on original mesh. Outer two rings are isotropic diaphragm elements.
throughout the sheet. Increasing the diaphragm viscosity to 108 Pa s produces the contours shown in fig. 7.24, again with a peak shear stress around 45 ~ but with zero stresses elsewhere. Therefore, the effect of increasing the diaphragm viscosity is to limit shear deformations to the immediate inner radius area. Experimentally, the composite is always permitted to propagate its shear deformations to the outer radius, via the inter-ply slip boundary condition with the diaphragm, shown in fig. 7.7. Diaphragm viscosity has a negligible effect on the distribution of shear stress at the inner radius (at least for this ratio of outer to inner diameter; see fig. 7.35), as shown in fig. 7.25. It is this stress which is believed to cause the shear-buckling phenomenon observed in composite sheet forming. The normalized transverse stress distribution (fig. 7.26) exhibits the same pattern as for the applied velocity case. The axial (fibre direction) stress contours, shown in fig. 7.27, are also similar to previous results, exhibiting a tensile region between 45 ~ and 90 ~ to the x-axis, and a very small compressive stress value where the fibres intersect the x-axis. However, fig. 7.28 shows that the effect of increasing the diaphragm viscosity in this case is to decrease the value of this compressive stress at this point. The importance of the diaphragm in composite sheet forming is to transfer tensile stresses into the lay-up during forming, as shown in this example. As shear-buckling always occurs in the radial direction during forming, it is useful to look at the tangential stresses, aoo at the inner radius (fig. 7.29). The surprising feature of fig. 7.29 is that, apart from the large (axial) compressive stress at 0 ~ which
Implicit finite element modelling
275
Fig. 7.23. Oxy shear stress contours, normalized by applied pressure, no diaphragm used.
is greatly reduced by increasing diaphragm viscosity, the stresses reach a maximum compressive value at 90 ~. This is due to the compressive transverse stresses at this point. However, no transverse buckling is observed in forming of uni-directional composite sheets under any conditions, perhaps due to the fact that the fibres at this point are oriented at 90 ~ to the transverse stress and are also subject to tensile axial stresses.
7.4.4. Parameter s t u d y -
pressure loading case
The effect of varying the material viscosities, r/L and Or, as well as the inner and outer diameters of the sheet will now be examined, assuming a constant diaphragm viscosity of 108 Pa s. Previous results have been for a material viscosity ratio of r/L/0r = 1.5. Figure 7.30 shows that the peak shear stress value at the inner radius increases significantly as this ratio is increased to a value of 4.0. Furthermore, the location of the peak stress moves from 65 ~ to approximately 45 ~ as the ratio is varied from 0.5 to 4.0. The effect of the viscosity ratio on the axial (fibre) stress is shown in fig. 7.31. The value of compressive stress at the x-axis can be decreased and changed to a tensile
C.M. 0 Br6daigh et al.
276
Fig. 7.24. O'xy shear stress contours, normalized by applied pressure, diaphragm viscosity = 108 Pa s.
1 , 0
"r
:3 m
'
'
I
'
'
'I
0.4 -
OX+ OX+
o•247 oX+ ~+~+x++
'
'
I
'"
r
II
I
~ diap visc = lEO o diap visc = 1E2 x diap visc = 1E4 + diap visc = 1E6 diap visc = 1E8
02 "
~,
L.
m ~e-
I
O o
-
*-' ffl
'
0.8 06
~
~
0.0
I
-0.2
0
,
I
,
,
L
,
,
I
~
~
I
,
~
l
,
90 15 30 45 60 75 Angle from X-axis, Theta (deg)
Fig. 7.25. Effect of diaphragm viscosity 203.2 mm; inner diameter = 25.4 mm.
on
tTxy normalized shear stress at inner radius. Outer diameter =
Implicit finite element modelling
277
Fig. 7.26. Crxx transverse stress contours, normalized by applied pressure, diaphragm viscosity = 108 Pa s.
stress by decreasing the viscosity ratio. The tangential stresses (fig. 7.32) show that decreasing the ratio below 1.5 can cause the 0 ~ to 45 ~ section of the inner radius to experience tensile tangential stresses, but that in all cases, the remainder of the edge is in compression. The maximum compressive stress increases in value as the viscosity ratio decreases. Given that the measurement of shear viscosities for a transversely isotropic composite sheet at melt temperature is still the subject of considerable research [40,41], this parameter study is of more than academic interest. The second effect to be studied is the variation of inner and outer diameters. This is a practical problem for composite manufacturers which has been seen to influence the occurrence of shear buckling in components [5]. Figure 7.33 shows that decreasing the outer diameter from 203 mm to 50.8 mm for an inner diameter of 25.4 mm causes a decrease in maximum shear stress of about 30% at the inner radius. A similar decrease is shown in fig. 7.34 for an increase in inner diameter from 50.8 to 102 mm. In fact, the results for similar ratios of outer diameter to inner diameter Do/Di are identical. The maximum shear stress is plotted against this ratio in fig. 7.35, for a series of diaphragm viscosities. For diaphragm viscosities of 104 Pa s and above, the effect of increasing the ratio is to increase the maximum shear stress experienced at the inner radius of the
C.M. 6 Brfdaigh et al.
278
Fig. 7.27. O'yy axial stress contours, normalized by applied pressure, diaphragm viscosity = 108 Pa s.
2.0
00
:3 U} .~ -2.0 n t._.
"~
9
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.
.
.
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.
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15
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+
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o dlap vlsc=lE8 I
45
60
75
t
1
90
Angle from X-axis, Theta (deg) Fig. 7.28. Effect of diaphragm viscosity 203.2 mm; inner diameter = 25.4 ram.
on
Oyy normalized axial stress at inner radius. Outer diameter =
Implicit finite element modelling .m
2"01
'
'
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'
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xx ~ :~
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279
,
,
15
,
Angle
I
,
30
,
from
,
,
45
,
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I
1E4 1E6 1E8 ,
60
,
75
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90
(deg)
Fig. 7.29. Effect of diaphragm viscosity on ~r00normalized tangential stress at inner radius. Outer diameter = 203.2 mm; inner diameter = 25.4 mm.
1.0
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,
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~
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= 0.5
5fi
-
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= 1.5
= 3 = 4
I t. J I , 45 6O X-axis, Theta
Fig. 7.30. Effect of composite shear viscosity ratio
,
_
I J , 75 90 (deg)
r/L/Or on Oxy normalized shear stress at inner radius.
composite. The opposite effect, however, is seen with the diaphragm viscosity of 103 Pa s, with much higher peak shear stresses being experienced. This plot shows the necessity of using a diaphragm with at least the same order of magnitude viscosity as the shear viscosities of the composite (in this case: 6,000 and 4,000 Pa s), if shearbuckling is to be avoided. The plot also graphically shows that peak shear stresses increase with diameter ratio for the higher-viscosity diaphragms, with a converging value of shear stress as the outer diameter of the composite sheet approaches that of the clamped diaphragm. Note also that the relative effect of diaphragm viscosity becomes more
280
C.M. O Br6daigh et al.
1.5 L-
=
t~ t~
1.0
n.
0.5
Ik=
0.0
. ~^ ~ ~ ~ _
.
~
-
AL~/~A L~/~/~ Z~/~/~/~/~ Z~/~~ a/~/~ A L~A ~ . ~ ; ~
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0
.
.
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.
.
o
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o~176 /
x
Ratio = 3.0
[
o
Ratio = 4.0
(D(3
X XX
xxXX x x x x
I I
4
o oo~
15 30 45 60 75 90 Angle from X-axis, Theta (deg)
Fig. 7.31. Effect of composite shear viscosity ratio r/L/r/T o n Oyy normalized axial stress at inner radius.
1.0 =
0.5
!.-.
,
~A66~6666~a~a ,~ O O ~ u u . . .~. u U n ~
~ -0.5
-
~A
'XXXXXXXXXXXxx~
O00
XXXXXXXxx
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---. Ratio = 0.5 I Ratio = 1.5 i
C
x
Ratio = 3.0
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0
Ratio = 4.0 I
-1.5
m -2.0
m !--
- 2 5"
00
v~
0
d
6
15
30
45
A
OOn A
oooooooooooooooooooooo68~~o~9ooooo~
~- -1 0 =i
~
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....
q
xxxxxx~
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~
~
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60
75
90
Angle from X-axis, Theta (deg) Fig. 7.32. Effect of composite shear viscosity ratio OL/riT o n aoo normalized tangential stress at inner radius.
pronounced at lower diameter ratios, as more area of diaphragm is involved in the analysis. Experimental observations have shown the onset of shear-buckling at diameter ratios varying from 1.5 upwards, depending on the forming rate and diaphragm properties. The shear stress results shown in fig. 7.35 will increase proportionally with the forming rate, as the materials are assumed to behave in a Newtonian manner. Increasing the diaphragm viscosity will also have the effect of increasing the critical buckling load of the diaphragm, a factor which is not treated here. In the absence of a stability analysis, the higher diaphragm viscosity results of fig. 7.35 could be regarded as a type of master plot for shear-buckling of uni-directional
Implicit finite element modelling
281
o8 m ~....
0.6
L-
~ L
0.4
~^
~XooO~ ~x~
~
E
0.2
"C~ oo
~x ~ x "
.0.2
_
ooO~
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~o~
o
I , , I . . . . 0 15 30 Angle from
_ _
-o~ _~
0 ooO~ ............ ~ [] x
-
-~Oooxx~
~ko ~
I- ~ o ~ L,~ ~~ T ~
0,0
~o ~
x~
oOOOOOo•
Outer Dia. = 8 inches Outer D i a . = 6 inches OuterDia.=4inches Outer Dla. = 2 inches
, . . . . . . 45 60 X-axis, Theta
/
II 5 7 I ~ I I
'', , I 75 90 (deg)
Fig. 7.33. Effect of inner diameter on Crxynormalized shear stress at inner radius, outer diameter - 203.2 mm; diaphragm viscosity = 108 Pa s.
0.8
z~A~DDODOo~A
m =(n 0 . 6
_
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x
.~Xo~
,, [] x o
6
-
0
15
_O
oX~
r,u O"
30
~X^ -I~ OX OL~
.... Inner Dia. Inner Dia. Inner Dia. Inner Dia.
45
X
O
= = = =
60
1 2 3 4
inch inches inches inches
75
O o
9O
Angle from X-axis, Theta (deg)
Fig. 7.34. Effect of outer diameter o n Orxy normalized shear stress at inner radius. Inner diameter = 25.4 mm; diaphragm viscosity = 108 Pa s.
sheets, as it is unlikely that a diaphragm with lower viscosity than the composite would be used. Finally, the effect of varying the inside and outside diameters of the composite on the axial stress distribution at the inner radius is shown in figs. 7.36 and 7.37. The maximum compressive stress, at 0 ~ to the x-axis, is shown to increase as the outside diameter is decreased, or the inner diameter increased. This is the opposite effect to that shown with the shear stresses, and suggests that the use of a maximum diameter sheet will alleviate the tendency for columnar buckling of fibres to occur at this point. Indeed, experiments with the forming of full hemispheres with small diameter
C.M. 0 Brddaigh et al.
282
I.uu
.... ~ .... I .... I .... I ....
_
:3
0 9. 0
---
r! - . l--
co
~
0
9
•
~
~
x
-' t
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~
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, ....
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co
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,
Diameter of Clamped Diaphragm
!.._
A
Diap Vise = 1E8 I1
[]
Diap Vise = 1E5 I ~
x
Diap Vise = 1E4 I 1
o
Diap,,, Vise = 1E3
,l~l,,,ll~,,,t,,,:l,,,~l,t,,I 1
2
3
4
.... 5
Diameter
6
I,,,,I
7
Ratio,
8
9
Do/Di
Fig. 7.35. Effect of diameter ratio (outer/inner) diaphragm viscosities.
O'xy normalized shear stress at inner radius, for various
on
1.0 !_.
=
u~ u~ l_
n
0.5
_
xxxxxxxxxXXo~" x ~_ o o o o D . o o ~ 2 ~ : ' "
vmOD.^
~,z~;~,, " ~ ' ' ~ "
0.0
X
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x I=,.
X
-0 5
~ z ~
o
_
o
-
._~
x
-1.0
.. Dia.
....
--- 1 inch
•
Inner
o
Inner Dia. = 2 Inches
x
I n n e r Dia. = 3 i n c h e s
o
I n n e r Dia. = 4 i n c h e s
O
i
So
. . . .
0
15 Angle
~
30 from
,
,
i
,
45 X-axis,
,
t
,_,
60 Theta
I
,
,
75 90 (deg)
Fig. 7.36. Effect of inner diameter o n O'yy normalized axial stress at inner radius. Outer diameter = 203.2 mm; diaphragm viscosity = 108 Pa s.
ratios has shown the tendency for localized areas of columnar buckling, or "fibrewash" to occur around the lip of the formed hemispheres [46]. 7.4.5. Extension to multi-directional sheet analysis
The analysis is now extended to investigate the behaviour of laminates with different shapes and more than one direction of reinforcement. For the purposes of analysing multi-directional laminates, some consideration had to be given to the manner in which the different plies interact with each other during forming. This
Implicit finite element modelling
283
1.0 w s_.
0.0
i " i
m -1.0 L_
=.
-2.0
0
0
m -3.0 G) t_
r
0
-4.0
"~ -5.0 <1: -6.0
O
-
Outer
Dia. = 8
inches
[]
Outer
Dia. = 6
inches
x
Outer
Dia. = 4
inches
0
Outer
Dia. = 2
inches
IIII
II
- O -O
-
0
,
~
I
15
,
~
I
L
,
30
L.
45
~
~
1
60
~
,
I
75
,
Angle from X-axis, Theta (deg)
Fig. 7.37. Effect of outer diameter on mm; diaphragm viscosity = 108 Pa s.
1 1
90
ayy normalized axial stress at inner radius. Inner diameter = 25.4
matter is currently the subject of research, and ultimately physical laws will be used to represent the real behaviour in numerical simulations. Experimental investigation has shown that a resin-rich layer exists between two adjacent plies during forming, which facilitates sliding between plies [2,49]. However, as a simple approximation to this behaviour, it was assumed that individual plies deform independently of each other. This means that each ply was analysed independently, and an average condition for the laminate was calculated. The stress quantity of most interest is the tangential stress, which acts in the direction of buckling at all of the buckling sites. The effect of introducing additional directions of reinforcement on the tangential stress distribution at the inner radius of the model is shown in fig. 7.38. Direct comparison between the values of tangential stress at the buckling locations for different lay-ups is not valid, since these stresses will act on elements of material with different resistances to out-of-plane buckling. It is of interest to compare the tangential stress predictions at the various buckling locations for different preform shapes, with the same lay-up, since the laminate properties at these points should be the same. Figure 7.39 shows a comparison between the predictions of the model for tangential stress at the inner radius for a quasi-isotropic circle and square. Since the tangential stresses at the 0 ~ 45 ~ and 90 ~ sites cause the out-of-plane shear buckling, the prediction was made that the square and circle could be expected to exhibit the same level of buckling at the 0 ~ and 90 ~ sites, but that the square would be significantly less prone to buckling at the 45 ~ point than the circle. A discussion of the consideration given to the buckling behaviour of quasi-isotropic laminates is given in the next section. The effect of the ratio of longitudinal shear viscosity to transverse shear viscosity on the tangential stress distribution at the inner radius of a quasi-isotropic circular sheet is shown in fig. 7.40.
r~ 0
~
=.
00
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I
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'
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--
b
q
Implicit finite element modelling 0.0
'
'
I
'
..... '
285
L
-0.2
~,
'
'
t
-~
-0.4
-0.6 -0.8
-1.0
j
-1,2 0
30
Angle (Degrees)
60
90
Fig. 7.40. Effect of viscosity ratio on tangential stress distribution for circular quasi-isotropic laminates.
7.4.6. Stability considerations A complete analysis of the buckling phenomena discussed in this chapter would require a stability analysis of the composite laminate as it deforms into the mould [57]. The different components of stress predicted for the laminate would then have to be considered in the light of the manner in which the material might be expected to respond. In the case of quasi-isotropic laminates, experiments reported here and elsewhere [5] suggest that buckling will always initiate in a direction at 0~176176 ~ to the fibre directions, and, for a female hemispherical mould, will always begin at the rim. Without attempting a proper stability analysis at this stage, these observations are used to consider the implications of the analysis results. The tangential stress predictions along the inner radius of our model are examined, because this stress acts in the direction of buckling at the likely buckling sites. Note that, though the solution values for tangential stress at all points along the inner radius are shown, these are not all of the same significance, because they act along material directions which have different susceptibilities to buckling. However, comparison between the tangential stress values at the different buckling points (i.e. 0 ~ 4 5 ~ 45~ ~ does offer a useful insight into the material behaviour, since the material reinforcements are at the same orientations at each point. Comments on the numerical results are made on this basis. It should be remembered, however, that other geometrical factors may influence the stability of the laminate.
286
C.M. (9 Brfdaigh et al.
7.5. Experimental comparisons- diaphragm forming 7.5.1. Motivation
A series of diaphragm forming experiments using a female hemispherical mould was carried out to investigate the effect of preform shape on the buckling patterns observed during forming with quasi-isotropic laminates. In particular, the intention was to test the hypothesis that, in this kind of experiment, a square laminate is less prone to buckling at the 45 ~ point around the inner rim than at the 0 ~ and 90 ~ points, as predicted above. Further experiments involving rectangular and other laminate shapes are also reported, together with conclusions drawn. 7.5.2. Experimental procedure
The diaphragm forming experiments described in this paper were performed in a computer-controlled thermoforming autoclave [50]. This autoclave operates at pressures of up to 2 MPa, and at temperatures of up to 450~ An LVDT displacement transducer offers the possibility of monitoring the deflection of the composite lay-up during forming, or of controlling the deflection rate through the use of control software. Heating and cooling of the autoclave may also be controlled by a personal computer. Full details of the experimental equipment used may be found in reference [50]. The mould used is a shallow hemispherical mould, known as the Female-A mould (fig. 7.41), which has a depth of 19 mm. This shallow mould is chosen to minimize the effect of elastic bending stresses due to fibres deforming into the mould. The diameter of the cavity is 89 mm, while the overall diameter of the mould is 216 mm. The composite material involved in the present study is APC-2 and the diaphragms used are Upilex-R polymeric diaphragm sheets produced by UBE Industries [51]. An initial series of experiments was conducted by pressurizing the autoclave to form the composite lay-ups into the evacuated mould cavities. The rate at which the shape is formed is limited by the maximum pressurization rate possible with the autoclave, so that maximum forming rates of approximately 2.5 mm/s can be achieved using this approach. Many of the parts formed in this way were of good
Clamping Ring
24.13cm
V
Vacuum Ring Mould Base Fig. 7.41. Female-A mould.
Cavity Vent
Layup
Diaphragms
Implicit finite element modelling
287
quality, exhibiting little buckling. It has been demonstrated previously that out-ofplane buckling in composites sheet forming is dependent on forming rate [5,48], so faster forming rates should be expected to promote buckling. These conditions were achieved by pressurizing both the autoclave and the cavity of the mould to 0.4 MPa at 380~ The laminate was then allowed to consolidate before forming (fig. 7.42(a)). The pressure in the cavity of the mould is then released, allowing the full 0.4 MPa autoclave pressure to form the laminate into the mould (fig. 7.42(b)). The displacement of the centre of the top diaphragm in the lay-up is recorded using the LVDT. An example of the displacement versus time relationship for a typical experiment carried out in this manner is given in fig. 7.43. The laminates form at a rate in the region of at least 5 mm/s.
P
BC~pon~~rraP/msree To Move
Upper Diaphragm
![
~
cVl~l
Fig. 7.42(a). Laminate consolidating before forming. p
I
I Mould /
I
/
I
V / ~,
,
,
Upper Diaphragm
~
V
i
~
!
I
Patm Lower Diaphragm
Composite Layup Free To Move Between Diaphragms
Fig. 7.42(b). Laminate formed as cavity pressure released.
Valve open
C.M. O Br6daigh et al.
288
70.0 65.0 .-. E E
"-" t,'-
E
60.0
55 0
O
50.0
-~a
45.0 40.0
-'127mm (5 inch) rectangle | ----o--Truncated 152 mm (6inch) square A 203mm (8 inch) circle
I
I
35.0 300
305
310
315
320
325
330
335
340
Time (Seconds) Fig. 7.43. Displacement versus time for the centre of the deforming laminate.
7.5.3. Experimental results The purpose of the experimental programme undertaken was to investigate the basic geometrical factors which influence if and where a quasi-isotropic sheet will buckle if it is formed into a double-curvature shape. In particular, the tests were designed to investigate the relationship between buckling and the length of the fibres which run tangent to the rim of the mould at the potential buckling site. The initial experiment involved a square preform, with reinforcement in the directions parallel to its edges and in the diagonal directions. The relationship between the geometry of this sheet and the circular cavity into which it is formed is shown in fig. 7.44, along with schematic representations of the tangent fibre lengths of interest. The tangent fibres which run parallel to the edges are, of course, 152 mm long, but those which run parallel to the diagonals are only approximately 127 mm in length. The formed shape for this experiment is shown in fig. 7.45. The laminate exhibits obvious tangential buckling when formed at a speed in the region of 10 mm/s. On the 152-mm square, eight buckles form symmetrically around the rim of the mould. Four of these are very severe and four are smaller. The four severe buckles form, at the rim of the mould, at 0 ~ and 90 ~ to the X-axis, as marked in fig. 7.11. The smaller buckles form at + 45 ~ and - 4 5 ~ to the X-axis, where the shorter fibres are tangential. In each case, the sheet has buckled along a fibre direction, transverse to a fibre direction and at + 45 ~ and - 4 5 ~ to a fibre direction. If the resistance to buckling of a laminate depends only on these local properties, then it is fair to directly
Implicit finite element modelling 152,4
289
Mm
\ 152.4
\ \
turn
,q
\ \
\ 126.49
Fig. 7.44. Geometry of the 152 mm square experiment.
associate the stresses experienced at each of these buckling points with the occurrence or size of the buckles observed. Previous experiments for circular quasi-isotropic preforms resulted in eight equalsized buckles at the same buckling locations as for the square [5]. For a circle, of course, all of the tangential fibres are the same length. Two experiments involving rectangular quasi-isotropic laminates are now reported. A 127 x 152 mm rectangle is first considered. Clearly, the fibres parallel to the 127-mm edge of the rectangle have been shortened, and so, to a lesser extent, have the diagonal tangent fibres (108 mm). The laminate geometry for the experiment is shown in fig. 7.46 and the result is shown in fig. 7.47. The larger buckles are clearly evident again in the same positions, but the secondary buckles, at + 45 ~ and - 4 5 ~ to the X-axis, are now either smaller or non-existent, coinciding with the shortest tangential length. The trend continues with the 102 x 152 mm rectangle, the geometry for which is shown in fig. 7.48. The result is shown in fig. 7.49. Large buckles form at 0 ~ to the X-axis, where the tangential fibre length is still 152 mm, with much smaller buckles at 90 ~ to the X-axis (102 mm), and none at the +45 ~ and - 4 5 ~ points (91 mm).
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Fig. 7.45. Formed 152 mm square laminate.
A further experiment was carried out to assess the effect of cutting across two corners from a square preform. This modification to the lay-up geometry, shown in fig. 7.50, has the effect of slightly shortening the 0 ~ and 90 ~ tangent fibres (from 152 mm to 125 mm) while, as close inspection of the geometry will reveal, leaving the diagonal tangent fibres the same length (again approximately 125 mm). The expected result of this experiment, in view of the hypothesis that tangential fibre length controls the phenomenon, would be buckles of roughly equal size at each of the eight buckling sites. The actual result of the experiment is shown in fig. 7.51, and the major discrepancy is that buckling does not occur at the two diagonal sites which have not had corners cut off. This interesting observation is discussed in the following section, in the light of further numerical analysis results.
7.5.4. Numerical analysis and models The analysis models which are used to investigate the experiments are similar to those described above. The symmetry conditions for these problems are different, however, and must be considered in the context of the assumption that the plies behave independently of each other, and are therefore analysed separately. The consequences of this for the symmetry of the model are as follows: If a quasi-isotropic rectangular laminate is to be analysed, this will include some +45 ~ plies. Even though the laminate as a whole will exhibit symmetry about the
Implicit finite element modelling 127
\
291
mm
152,4 m m
\
\
\
108,46
Fig. 7.46. Geometry of the 127 • 152 mm rectangular laminate.
X- and Y-axes, the solution fields for these particular individual plies will not. Therefore simply analysing a quadrant of the problem will not yield a correct solution, and the full model should be used. A schematic of a typical model used is shown in fig. 7.52.
7.5.5. Mesh sensitivity and design The issue of the sensitivity of axial stress predictions to the density of the mesh and to distortion of the elements in the region of the tension stress singularity must be addressed. In section 7.4, when F E F O R M was used to model circular shapes, and above for square preform shapes, it was possible to use meshes of uniform density with regularly shaped elements. When modelling the rectangular shapes described, it is no longer easy to construct a mesh which is completely uniform. In order to assess the mesh density needed to reasonably represent the true stress conditions in problems of this kind, a convergence study was carried out using a simple mesh of a quadrant of a circle (fig. 7.53). The inner radius of the model is 12.7 mm, while the outer radius of the composite material region is 50.8 mm. The outer radius of the diaphragm region is 107.95 mm. This mesh is typical of those used in
292
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Fig. 7.47. Formed laminate: 127 x 152 mm rectangle.
the analyses of section 7.4 and the same symmetry and boundary conditions apply. A uniform normal pressure is applied to the inner radius. Three different meshes were used to solve this problem. The first and coarsest mesh has 143 nodes and 30 elements, as shown in fig. 7.53. The axial stress predictions for this mesh are shown in fig. 7.54. The ratio of the axial stress singularity to the applied pressure is predicted as -246. Figure 7.55 shows the solution for a much denser mesh (2,009 nodes, 540 elements), and it is qualitatively similar to that of the coarse mesh. The value of the tension stress singularity is now -319. Inspection of the two results suggests that the coarse mesh works well, and is only significantly inaccurate at the point of the singularity. The results of a study of the convergence of the stress at the singular node are presented in fig. 7.56. The second factor we wish to investigate is the accuracy of predictions in the region of elements with poor aspect ratios. The same problem is considered, but this time elongated elements are used. A sample mesh, consisting of 539 nodes and 120 elements, is shown in fig. 7.57, with the axial stress predictions depicted in fig. 7.58. The value of the tension stress singularity ratio is significantly under predicted, at only -113. In the light of these results, it is clear that element distortions have a potentially more serious effect on the calculation of tension stress singularities than mesh density does. The studies described above were intended to guide the mesh design for the analysis of the forming of rectangular laminates. Each of the meshes used to model the
Implicit finite element modelling
293
101,6 M~
152,4 MM \
Fig. 7.48. Geometry of the 102 • 152 mm rectangular laminate.
forming of rectangular quasi-isotropic laminate (for example fig. 7.59) has a varying mesh density around the inner radius. This was done in order to reduce element distortion in this area. Similar meshes representing laminates of 101.6 mm, 114.3 mm, 139.7 mm and 152.4 mm widths have also been constructed, and are used in the analyses that follow. In order to further minimize the possibility of mesh irregularities distorting the relationship between analysis results for different shapes, the central square region of the mesh is identical for each case. Only the outer region of each mesh varies.
7.5.6. Numerical results and comparisons The results of the finite element parameter study to investigate the effect of varying sheet width are now discussed. Example results are shown for uni-directional and multi-directional cases, and then comparisons between analysis and experimental trends are made. Initially the results for the uni-directional case will be considered. In order to illustrate the contributions of the various different plies to the averaged stress state that are calculated for the laminate, the case of the 114.3-mm rectangle is
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Fig. 7.49. Formed laminate: 102 x 152 mm rectangle. I01,6 M m
50,8 Mrn
\
\\\\
I
50,8
Inm
33,48 rnrn
101,6 ~M
I
9
,
. . . . .
Fig. 7.50. Geometry of the truncated square experiment.
Implicit finite element modelling
Fig. 7.51. Formed laminate: truncated 152 mm square.
",,,]L
Fibre directions
Diaphragm elements
Composite elements
'L
Uniform radial pressure Outer edge clamped
Fig. 7.52. Schematic of planar forming model for rectangular laminate with boundary conditions.
295
296
C.M. 0 Br6daigh et al.
Fig. 7.53. 143-node, 30-element mesh for forming with a uni-directional circular sheet.
Fig. 7.54. Axial stress predictions for 143-node, 30-element mesh.
Implicit finite element modelling
297
Fig. 7.55. Axial stress predictions for 2,009-node, 480-element mesh.
-240.0
~
-260.0
~
"o
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,
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~-
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t
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e-
._. "
-340.0 -360.0
L ....
0
L.___L__~_
..... ~ .....
100
.1
200
L.--_J.
300
N u m b e r of E l e m e n t s in M e s h
Fig. 7.56. Convergence study for axial stress.
.....
400
500
C.M. O Brddaigh et al.
298
,,
i
x
x
\
", x
\
-
\ \
,'
\
\
\
\
\
\
\
\
iiii
Fig. 7.57. 539-node, 120-element mesh for forming with uni-directional circular sheet.
Fig. 7.58. Axial stress predictions for a mesh with high aspect ratio elements.
Implicit finite element modelling
299
X
Fig. 7.59. Mesh used to analyse 127 x 152 mm quasi-isotropic laminate.
now considered. The tangential stress distributions at the inner radius of the model for the case of each of the four fibre directions are shown in fig. 7.60. The mesh used to analyse the 139.7 mm case is shown in fig. 7.61, and the average tangential stress distribution for the laminate is displayed in fig. 7.62. The tangential stress pattern is clearly influenced by the bands of compressive axial stress which occur where fibre directions run tangent to the inner radius of the model. These bands are similar in nature to those discussed in the mesh sensitivity study. Referring back to the experimental results above, it is now possible to obtain numerical results which correspond to these experiments. A comparison between the different sheet widths is shown in fig. 7.63, which gives the tangential stress distributions at the inner radii of the models, in the 0 ~ to 90 ~ quadrant. Firstly, consider the result for a 152.4-mm square, which corresponds geometrically with an experiment reported in section 7.5.3. The tangential stress at the 0 ~ and 90 ~ points is the same, as would be expected, given the symmetry of the shape. These stresses are considerably in excess of those reported at the 45 ~ point, which has a shorter tangential fibre. This offers an explanation for the severe buckles which form at 0 ~ and 90 ~ in the experiments, and the slight buckles which appear at the 45 ~ point. Looking at the predictions for the rectangular sheets, a clear trend emerges. As the width of the laminates is reduced, the tangential stress which could cause buckling increases at both the 0 ~ point and the 90 ~ point, but in each case the 0 ~ site bears the greater stress. This is also in agreement with the experiments reported in section
C.M. 0 Br6daigh et al.
300
5.0
.
.
.
.
. . . . . . . . . . . .
I
.
.
.
.
,
.
I
,
,
!
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-s.o
== I-
-10.0
15.0
-20.0
i
,
i
,,
[
30
i
i
i
i
1
60
Angle from X Axis (Degrees) Fig. 7.60. Tangential stress distributions for individual plies (127 x 152 mm).
E Fig. 7.61. Mesh used to analyse 139.7 x 152.4 mm quasi-isotropic laminate.
90
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301
Fig. 7.62. Contour plot for averaged tangential stress, quasi-isotropic sheet.
1.0
~
t
,
t
,
,
i
'
'
~ ......
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'
t
'
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'
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~
... ra ._~
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-.
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r
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=
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5 Inch Width
-3.0
-4.0
- --o- -4.5 Inch Width III II
-5.0 ~
0
II IIII I
' 30
60
Angle from X Axis (Degrees)
Fig. 7.63. Tangential stress distribution at inner radius, different laminate widths.
90
302
C.M. O Brddaigh et al.
7.5.3. where buckles form more severely at the 0 ~ site. The relationship between sheet width and predicted tangential stresses responsible for buckling is approximately linear, as demonstrated in fig. 7.64. Finally, consider the case of the 152.4 m m square laminate with two opposite corners cut away, the experimental result for which was seen in fig. 7.50. The mesh used for this analysis is given in fig. 7.65, and the results will be given in tabular form, for ease of comparison with the predictions already reported for the 152.4 m m square. Tangential stress results at each buckling site for both of these cases are presented in table 7.1, with the angle measured in degrees from the positive X-axis. The most obvious difference between the two cases is at the 0 ~ 90 ~ and 180 ~ points. The value of the stress singularity at these points is considerably lower for the case of the laminate with the truncated corners. This coincides with the points around the inner radius of the model where the tangential fibres have been shortened. The stress concentrations reported where these fibres are tangent to the rim of the mould in the 0 ~ and 90 ~ plies are substantially reduced. There is only a small effect on the stresses predicted for the + 45 ~ and - 4 5 ~ plies, which is related to the fact that the singular fibres in these cases have not been shortened at all. It must be noted, however, that this result would suggest that, for the new truncated case, one would expect buckles to form at the 135 ~ point with the same severity as the 45 ~ point. This is not in agreement with the experimental results. The reason for this may be that the amount of surrounding material at the 135 ~ point inhibits buckling to a greater degree than at the 45 ~ position. A stability analysis which accounts for this effect, such as that carried out for elastic composite sheets by
-3.4
-
,
,
,
"~ . . . . '
i
J
i
~
...... 1--
1
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i
j
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- ....
:
i
i
i
,
~
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-3.6 o~
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-4.0
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-4.2
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,-" t~
-4.4 "--...... 90 Degrees from X axis !
-4.6 -4.8
__•
4
....
L ..... _ L _ _ _ i _ _ _ L _ _ _ L
4.5
....
L . . . . . a ........ L....... _ L . _ _ _ _ t _ _ . _ . t . . . .
5
L__-___A___.a-
. . . . _a . . . . .
5.5
Laminate Width (Inches)
Fig. 7.64. Variation of tangential stress at 0~ and 90~ with sheet width.
L.__~__2
....
6
-___~
....... ~ . . . . . . = _ . . t
6.5
Implicit finite element modelling
303
Fig. 7.65. M e s h u s e d to a n a l y s e t r u n c a t e d 101.6 m m s q u a r e l a m i n a t e .
T A B L E 7.1
Angle
0~
152.4 m m s q u a r e
-3.435
152.4 m m s q u a r e w i t h corners cut
- 1.242
45 ~
90 ~
135 ~
180 ~
-0.547
-3.435
-0.574
-3.435
-0.508
- 1.259
-0.696
- 1.242
Coffin [57], will be necessary in order to further interpret the experimental and numerical results presented here. 7.6. Conclusions of plane stress analysis
The application of the plane-stress finite element analysis to the problem of central indentation of a composite sheet has been demonstrated for uni-directional and multi-directional composite sheets. This is an important practical problem in sheet forming of composite materials, where shear-buckling of the laminate can occur at faster forming speeds, particularly where a small indentation, such as a beaded stiffener or rib, is being made in a large sheet. Two different boundary conditions are used at the inner radius of the sheet in order to approximate the problem as a planar one, uniform radial velocities and uniform radial pressures. The velocity boundary conditions enabled a direct
304
C.M. O Br6daigh et al.
comparison between the deformed grid predicted by the finite element analysis and the experimentally deformed grids obtained by a punch-deformation apparatus. Excellent agreement was seen in the fibre direction, with somewhat poorer results in the transverse direction. The pressure boundary conditions were seen to provide a more accurate representation of the shape of the inner radius after forming and are thought to be more appropriate for diaphragm forming conditions. Furthermore, shear stresses in uni-directional sheets were seen to peak at roughly 45 ~ to the fibre direction, depending on the ratio of composite shear viscosities employed. These stresses cause the shear-buckling phenomenon in composite sheet forming. The effect of increasing the diaphragm stiffness was seen to reduce the peak shear stress encountered in the buckling region, especially where small ratios of outer to inner diameter are used. In general, the value of peak shear stress can also be reduced by reducing this diameter ratio. The necessity of utilizing a diaphragm material with a viscosity of the order of the composite shear viscosities was also demonstrated, in order to avoid the possibility of bucking occurring during forming. Finally, the stiffening effect of using the diaphragm material to reduce the value of axial compressive stress encountered was also demonstrated. It should be noted that for unidirectional laminates, the knowledge of the stress pattern during forming is not, in itself, sufficient to predict where buckling may occur, as the laminate has locally higher resistance to buckling in the fibre direction, and essentially behaves as a filled fluid in the transverse direction. A methodology has also been presented for calculating average tangential and radial stresses in multi-directional laminates. A separate analysis is carried out for each fibre direction in the laminate and the stress results averaged through the thickness. For cross-ply laminates, the difference in local buckling resistance means that buckling prediction is still difficult. For quasi-isotropic laminates, however, the average tangential stress results may be used to predict buckling at each of the 0 ~ 4 5 ~ 450/90 ~ sites in the laminate, as the local properties at each point will be the same. The multi-directional analysis has been used to assess the effect of laminate shape on the formation of buckles in a quasi-isotropic sheets, and to qualitatively predict the buckling pattern for circular, square, rectangular and truncated-square laminates. In all cases, the axial compressive stress in the fibres which run tangent to the buckling site, and the shear stress at this point, are found to be the major contributing factors to the tangential stress which causes instabilities. The magnitude of the tangential stress depends on the lengths of these tangential fibres, as well as other geometric features of the sheet. The numerical results predict correctly the location and relative magnitude of instabilities in the circular and square laminates, as well as predicting the effect on the buckling patterns observed when the width of the square sheet is reduced to a rectangular strip. The need for a complete laminate stability analysis is highlighted by the comparison between the numerical and experimental results for the truncated square experiment, where the stress predictions followed the expected pattern (reductions
305
Implicit finite element modelling
associated with shorter tangential fibre lengths) but could not be used to fully explain the instability patterns of the experiment. Overall, quite good agreement has been established between the results of the finite element model and experiments. The issue of the true nature of the boundary conditions between the diaphragm and the composite is somewhat more complex than that utilized here, but it is believed that the essential features of the stress and deformation patterns have been uncovered, leading to an increased understanding of the phenomena of buckling in composite sheet forming. 7.7. Numerical s o l u t i o n s - plane deformation problems
7.7.1. Problem statement
single-curvature forming
Single-curvature structures are an important class of sheet-formed products, including J-beams, U-beams, sine-wave spars, and Z-stiffeners for aerospace applications, some of which are illustrated in fig. 7.66. The key deformation mechanisms here are the slip between the plies and the transverse flow within each ply in response to pressure gradients as the material contacts the tool. In particular, the assumption of plane strain, or plane deformation conditions, in the plane transverse to the axis of the component is employed (i.e. the 1-3 plane). This assumption is appropriate for regions near the centre of a long prismatic shape, as shown denoted by section A - A in fig. 7.66, rather than at the ends (sections B-B and C-C), where flows along the component axis (in the 2-direction) may also occur. Typical photomicrographs of inter-ply slip at the edge of a single-curvature component (in this case a 90 ~ bend), and of thickness gradients in the direction transverse to the mould axis, caused by transverse flow of fibres and matrix, are shown in figs. 7.67 and 7.68 respectively. The formulation developed in this section is capable of modelling both of these phenomena, but does not model resin percolation, or transverse flow in the mould axis direction. It should be noted that the formulation does not allow for transverse flows of plies whose angles are other than 90 ~. However, it is felt that, as most thickness changes in the 1-3 plane are due to transverse flows in the 90 ~ plies, rather than in the angle plies, the model is not unduly restricted.
B
.
.
.
.
.
B
3 B A-
~ B A
Fig. 7.66. Typical single-curvature components formed from flat composite sheets.
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Fig. 7.67. Inter-ply slip in a thermoplastic composite laminate formed over a 90 ~ bend.
Fig. 7.68. Transverse flow in a thermoplastic composite laminate formed over a male hemisphere. Note thickening of centre 90 ~ plies towards bottom of photograph.
Implicit finite element modelling
307
Previous approaches towards modelling single-curvature forming of composite parts have used mainly closed-form analytical approaches. Tam and Gutowski [21] derived a model for small deformations of a linear viscoelastic beam, composed of alternating layers of stiff elastic and soft viscous materials. Similarly, Talbott and Miller [22] generalized this approach for an array of point forces, using alternating stiff and soft elastic layers. The general conclusions of these approaches was the high axial forces in the upper and lower skin layers, and the importance of the resin-rich layers in reducing forming forces. Scherer et al. [31] used a finite element approach to the problem, again modelling the stack of plies as alternating stiff and soft elastic layers, but incorporated a yield stress into the soft layers. Finally, Martin et al. [52] modelled the 3-point bending of glass fibre/polypropylene sheets using a continuum approach and noticed significant viscoelastic effects in the experiments.
7.7.2. Plane deformation modelling The individual plies of the composite laminate will be considered here to behave as transversely isotropic Newtonian fluids, which are incompressible, and also inextensible in a single fibre direction. This formulation may be extended to a power-law, or other non-Newtonian forms without much difficulty. The ideal fibre-reinforced fluid model of sections 7.2 and 7.3 above will again be employed, but in a plane deformation, instead of plane stress formulation. The particular form of the rheological constants will depend on experimental investigations being developed at present [40,41]. Likewise, the assumption of isothermal conditions during forming may be somewhat simplistic and may be improved in the future as the full forming simulation is developed. An important consideration in modelling the forming process is the mechanism of inter-ply shearing. It is generally understood that the inter-ply shearing mechanism occurs in a thin resin-rich layer [2] which evolves between adjacent plies of different fibre orientations, during the consolidation process. A typical movement of one ply over another during forming of a quasi-isotropic 10 cm diameter hemisphere, for example, would be of the order of 3-5 cm. If this shear takes place over a resin-rich layer thickness of about 10 ~trn, then we may estimate the overall value of inter-ply shear as (see fig. 7.69): Y13
-
'
-
30-50/0.01 - 3,000-5,000 units of shear
3-5 cm
BoHom
71
Composite Ply
I Resin-Rich Layer
Fig. 7.69. Schematic diagram of the inter-ply shear mechanism.
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C.M. 0 Br6daigh et al.
This level of deformation poses a practical problem for numerical modelling of the process, namely the difficulty of refining and revising our mesh sufficiently during the process to model these large shears. Another approach is to consider the inter-ply shear to be a lubricated friction mechanism, and to use contact elements between adjacent plies. The advantage of this approach is not only the avoidance of high shear gradients, but also that good experimental data is available for thermoplastic composite inter-ply slip, which appears to follow this type of relationship [49,53]. In this section we will illustrate both approaches, the use of continuum isotropic layers to represent the resin-rich inter-ply regions, and the development of contact elements to match experimental results for thermoplastic composites. The first approach is shown to match the numerical approach with some available analytical solutions, whereas the second is the more practical in developing a forming simulation.
7.7.3. Problem formulation The behaviour of the composite ply during forming is again modelled as an ideal fibre-reinforced Newtonian fluid (IFRF), represented in Cartesian co-ordinates by eq. (7.11) with material constants given in terms of the two shear viscosities in eq. (7.12). The constitutive relationship for this model also involves the two kinematic constraints of incompressibility (eq. (7.13)) and inextensibility in the fibre direction (eq. 7.14)). If the ply can be assumed to be infinitely long in the 2-direction (see fig. 7.6 above), then plane deformation can be assumed in the 1-3 plane. The conditions for plane strain in the 1-3 plane are d22 = d23 = dl2 = 0. When these are substituted into eq. (7.11), the constitutive relationship reduces to (in matrix form):
Elll [ol 0 01[ llj 0"33
--
o'13
0
D33
0
d33
0
0
D55
2d13
+
-p 0
j
(7.40)
where again, m = cos(0), 0 being the fibre angle in the 1-2 plane. The constitutive matrix coefficients are as follows: Dll = 2r/T(1 -- 2m 2) -+-4r/Lm2 D33 -- 2r/T
(7.41)
D55 = r/T(1 -- m 2) + r/Lm2 Dynamic terms are omitted from the equilibrium equations, as with the plane stress formulation of section 7.2 above, and the problem is treated as being quasi-static over the flow domain f2. The equilibrium and boundary traction equations are the same as before, i.e. eqs. (7.22) and (7.23).
7.7.4. Finite element solution technique The finite element formulation for the plane strain ideal fibre-reinforced Newtonian fluid is now presented along with a basic outline of its derivation. The
Implicit finite element modelling
309
formulation is based on the formulation for plane stress deformation presented in section 7.3 above. A detailed derivation of the formulation is currently being prepared for publication elsewhere. The weak form of the equilibrium equations can be stated as: J(,k)TDk dr2+ l(Sk)raT dff2+ J(,k)rmp d r 2 - J ( , v ) r b dr2 n r n [(su)Tt "dF = 0 -
n
n
(7.42)
-
d
F
8k and 8v are vectors of virtual strain rates and velocities respectively. They form the relationship 8k = SSv. The column matrices a and m are given by: I1 = (COS2 0
0
m = (1
0)T
1 0)T
(7.43)
The next step involves discretizing the governing integral equations of the problem at nodes distributed throughout the domain f2. The solution parameters at each node are the velocity vector v, the scalar fibre tension stress T, and the scalar hydrostatic pressure p. The velocity, fibre tension, and pressure fields are descretized independently: v ~ ~3- NVq~V;T ~ T - Ntdpt; p ,~ ~ - NPdpP; and k - e - SNV~bV;- BVq~~ (7.44) where N v, N t, and N p are matrices of the interpolation functions for velocity, fibre tension, and pressure respectively; and 4~~, 4/, and 4~p are vectors of nodal velocities, fibre tensions, and pressures respectively. This leads to the discretized form of eq. (7.42):
n
n
(7.45)
--J(NV)Tf~ b d~2- I(NV)Tr i d F } - 0 Weak forms of the inextensibility constraint equation (7.14) and the incompressibility constraint equation (7.13) can be found. Their discretized forms are, for inextensibility and incompressibility respectively: (~dpt) T [ ( N t ) T aTBVqf d n - 0
(7,46)
o
f~ (8~~ T [(NP) T mTB~ck v dr2 -- 0 J
f2
(7.47)
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C . M . 0 Brfdaigh et al.
When eqs. (7.45-7.47) are combined a mixed system of equations is formed. This system of equations can be solved by using the mixed penalty method. This leads to the final formulation" { K v + o t t K t ( M t ) - l K t + otpKP(MP)-IKp}~v -- f
(7.48)
where the matrices are given by: K v - I(BV) T DB v dr2
K t-
f2 K p --
I(BV) T m N
](BY) T a N t d~ fl
p
dr2
(7.49)
f2
~2 M t-
J'(Nt) T Nt d~
M p --
I(NP) T N p
F d~
f~
at and c~p are scalar penalty numbers for fibre tension and pressure respectively. The constraints of fibre inextensibility and incompressibility are enforced exactly as at ~ ~ and ap ~ c~. Note that N v and N t must now consist of discontinuous interpolation functions. Once the velocity field has been obtained from eq. (7.48), the fibre tension and pressure fields can be obtained from the following relationships: ckt
-
-
ott(Mt) -1 (Kt) T~bv
q~P - - _ O t p ( M p ) - l ( K p ) T ~ v
as c~t ~ c~
(7.50)
as ap ~ oo
(7.51)
7.7.5. Computational details and results
The mixed penalty formulation is implemented in F E F O R M , using a Q9/4/4 element. This element utilizes a biquadratic velocity interpolation function, and discontinuous bilinear interpolation functions for the fibre tension and pressure. The fibre tension and pressure nodes are located inside the element at the 2 • 2 Gaussian integration points. All the element matrices are calculated using full 3 • 3 Gaussian integration. Equation (7.48) is solved for the velocity field, and the fibre tension and pressure fields are then obtained from eqs. (7.50) and (7.51) respectively. All the simulations in this section used values for the penalty numbers of Ott -- 1012 and % - 10 l~ The viscosities used were 0c = 6,000 Pa s and tiT = 4,000 Pa s, unless otherwise stated. The I F R F plane strain element formulation was assessed by using it to model flows for which analytical solutions are available. All the results shown can be regarded as the first time step of a quasi-static system. Kaprielian and O'Neill [54] analysed shear flow in composite laminates, modelling the plies as IFRFs, and including a Newtonian resin-rich layer between the plies (fig. 7.70). As this analysis involved three-dimensional flow it cannot be directly compared to the present formulation. However, if the component of velocity in the 2-direction in fig. 7.70 is zero
Implicit finite element modelling
311
Fixed rigid plate resin
]~.
'
Fibre Angles (degrees)
31
[upper ply, lower ply]
IFRF ply
X finite element results
'"r~ ~
resin hI
IFRF ply resin Moving rigid plate
' [;0,90]
. ,z 0
13
,
[90,0]
~flN~~~,~
~ Force o
'"r~
= Force/area
1
;\
2
3
4
5
11 V
-o 13 h IFRF
1
Fig. 7.70. Shear flow in a two-ply laminate. Comparison of plane Couette flow with finite element results.
then the problem becomes a case of plane Couette flow, which can be modelled using the plane strain formulation. Hull et al. [26] provide an analytical solution for Coutte flow in an IFRF, and this is used to find the flow field for the problem. Figure 7.70 also shows the comparison between the analytical and the finite element models with very good agreement. Figure 7.71 shows results for a I F R F cantilever beam with 0 ~ fibres that is being sheared. Rogers and Pipkin [55] provide a solution for this problem, and the results in fig. 7.71 are in good agreement with this. The shear stress field in fig. 7.71a shows a small variation because the finite penalty numbers do not enforce incompressibility and fibre inextensibility exactly, thus allowing a small, but insignificant, amount of "bending". The stress field shown in fig. 7.71c is dominated by the fibre tensions. The stress concentrations at the top and bottom, with almost zero stress in between, is characteristic. 7.7.6. Ply contact formulation and results
When a multi-ply laminate is consolidated a "resin-rich" layer consisting of the matrix material forms between the plies. This layer is thin, for instance typically 10 ~tm thick between thermoplastic plies of 0.125mm. Shear deformation in the inter-ply layer is an important flow mechanism during the forming of curved components. Experiments have been carried out to investigate the nature of this inter-ply shear [49,53]. These indicate that the inter-ply layer has a yield stress and that inter-ply shear has a normal pressure-dependency, thereby suggesting that inter-ply shear could be regarded as being frictional in nature. Therefore, modelling the inter-ply shear using frictional contact elements appears to be the logical way to proceed. The
i
_
i
i
i
:
i
.
,
,
1
i
|
,
,
;
i
|
|
,
~--~
'
'
l~v
I
'
F
~
.
I
i
.
1
:
ix? . . . . . . ,
-j
.
.
,
,
,
.~
.
,
|
,
"
-"
i
i
.
.
,
,
|
i
i
.
,
|
~
_-
.
Implicit finite element modelling
313
Fig. 7.71. Beam with 0 ~ fibres being sheared. (a) The 0"13 shear stress field (Pa). The beam is completely constrained on the left side, and a shearing force is applied on the right side. (b) The velocity field (mm/s). (c) The Cql stress field (Pa).
contact formulation is implemented using a penalty approach the basic details of which are outlined here. A more detailed description of a penalty contact-friction formulation is given by Laursen and Simo [56]. The weak form of the governing equations for a laminate with p I F R F plies is:
C'(O', ~ ' ) +
[-i~v. aO~v- i~. 8 ~ ] dr'
- 0
(7.52)
re where the function G i represents the weak form of the equilibrium equations for each IFRF ply i (eq. (7.42)); [~r is the contact traction on ply i acting in the direction normal to the line of the contact interface; similarly, [~r is the contact traction on ply i acting in the direction tangential to the line of the contact interface. At a point on the interface between two plies the normal contact stress t N is calculated by penalizing the relative normal velocity: tN = eNg
(7.53)
314
C.M. 0 Brfdaigh et al.
where g is the "gap" function which defines the relative interfacial velocity in the normal direction between two contacting plies, e N is a penalty number. As Eu ~ O0 then g--+ 0 and thus there is no penetration or separation between plies at the contact interface. The manner in which the tangential contact stress tr is calculated depends on how the inter-ply shear mechanism is characterized. If it has a yield stress and the inter-ply shear stress is below the yield stress then the situation can be regarded as a case of "stick" friction. To deal with this a trial solution must first be calculated to test whether or not the tangential contact stress is below the yield stress. This is done by penalizing the relative tangential velocity between the two contacting plies (the "slip" velocity s): t~"ial -
s
(7.54)
S
where e r is another penalty number. If
t trial
is below the yield stress then"
(7.55)
t T - - tt~ ial
The condition of stick friction is enforced as 8 T ~ 0 0 . If t ~ ial is above the yield stress then t T is calculated from the model used to characterize the slip mechanism. In a simple model inter-ply slip could be assumed to be hydrodynamic in nature, based on the viscosity of the matrix material ~: (7.56)
tT = rls/h
where h is the average thickness of the inter-ply resin layer. The contact tractions are calculated by integrating the contact stresses over the contact interface, these are then assembled together to form the contact residual vector. The components of the contact stiffness matrix are found by linearizing the contact residuals. The contact stiffness matrix and residual vector are then assembled into the global equations which are solved using the Newton-Raphson method. The contact penalty numbers used in the following examples are e T - - e N - - 1012. Figure 7.72a shows a four-ply lay-up subjected to a uniform distributed load on top. The inter-ply slip in this example is modelled as hydrodynamic friction with no yield stress, 0 - 3,000 Pa s and h = 10 ~tm. The results for axial and shear stresses are shown in figs. 7.72c and 7.72d. The important feature of the multi-ply result is that individual plies experience tensile stresses on their top surface and compression on their bottom surfaces, with shear traction in between adjacent plies. This is in agreement with analytical results published elsewhere [21,22]. The inter-ply shearing mechanism has been investigated experimentally for APC-2 by Murtagh [49,53]. These experiments found that inter-ply shear stress is a function of the inter-ply slip velocity, the forming temperature, the normal pressure, and the ply angles; and also that inter-ply shear has a yield stress. The following comprehensive model for the inter-ply shear mechanism was derived: tAPc-2
-
(s, T , P , 0) -
r(s, T) r(s, P) r(s, 0) r~
rs
r~
r~
(7.57)
Implicit finite element modelling
315
where r(s, T) -- 1.0 + 28.7083 s (-2"0082+7"3335•
T)
r(s, P) - 0.95 + (1.28 • 10 -3) P + (-28.639 + 31.143 log(P)) (S0"1635+0"30791~
r(S, 0) --
0 > 5 ~ =~ 1.0 + 28.7083 s (-2"0082+7"3335• T) / 0 < 5 ~ =~ 2.0-+- 79 14 S(-0"88194+3"452• T)
/
rs - 1.0 + 28.7083 s ~ Conditions: 360~
< T < 400~
20 kPa < P < 400 kPa 0~ < 0 < 9 0 ~ 0 mm/s < s < 0.5 mm/s where s is the slip velocity, P is the normal pressure, T is the forming temperature, and 0 is the difference in fibre angles between contacting plies. The normal pressure is equivalent to t u . The finite element implementation of the model was evaluated as follows. The model lay-up consisted of two flat plies (see fig. 7.69), the lower one was fully constrained on the bottom, the upper one was subjected to uniform normal and shearing stresses on top. The resulting flows were simple shear flows. The nodal slip velocity along the inter-ply contact interface was compared to the applied shear stress. The slip velocity s was found by subtracting the velocity at the contact nodes on the lower ply from the velocity at the corresponding contact nodes on the upper ply. The cr13 stress distribution was not completely uniform throughout the model because of the tendency of the plies to "bend" slightly, a consequence of the fact that it is not possible to implement perfectly the incompressiblilty and fibre inextensibility constraints. Because of this, the cr33 stress distribution along the inter-ply contact interface resulted in negative values for P. It was necessary to average the normal pressure over the contact interface in order to calculate the contact tangential stress, on account of the log(P) terms in eq. (7.57). This was possible in this case because of the simple geometry of the problem. In a more complex problem eq. (7.57) would have to be recast in order to calculate the contact tangential stress from the normal pressure at each individual contact node. Figure 7.73 shows a comparison between the results from the finite element formulation and the analytical curves given by eq. (7.57). The agreement is excellent.
7.8. Conclusions of plane deformation analysis A plane strain finite element formulation has been developed to model sheet forming of single-curvature composite shapes. Numerical analysis of single-ply and multi-ply forming was demonstrated for the initial time step. Each composite
316
C.M. (J Brddaigh et al.
uniform pressure 12 kPa
(a)
I
(b)
i
I
I
I.
I i
Implicit finite element modelling
317
Fig. 7.72. F o u r - p l y laminate with applied uniform load. Inter-ply slip is h y d r o d y n a m i c with no yield stress, = 3,000 Pa s and resin layer thickness o f 10 lxm. E a c h ply has dimensions 10 x 0.5 m m . (a) M o d e l and b o u n d a r y conditions. (b) Velocity field. (c) T h e all stress field. (d) T h e crl3 shear stress field.
Normal pressure
Layup
shear stress
Temperature
shear stress
shear stress
25
60
20
20
50 40
15
30
!0
(kPa)
360 C 380
15 !()
20 5 0
5
10
0
O.1 0.2 0.3 0.4 slip velocity (mm/s)
0.5
0
0
O.1
0.2 0.3 0.4 slip velocity (mm/s)
0.5
0
0
O.1
0.2 0.3 0.4 slip velocity (mm/s)
0.5
Fig. 7.73. Inter-ply slip curves from eq. (7.57) [49]. The marks indicate c o m p a r i s o n with results from the finite element m o d e l o f inter-ply slip.
ply was modelled as a transversely isotropic incompressible Newtonian fluid, with a single direction of inextensibility. These restrictive kinematic conditions were implemented correctly using a mixed penalty method. An experimentally determined lubricated friction law, incorporating rate effects and yielding behaviour, was implemented between the plies and used to model the
318
C.M. 0 BrSdaigh et al.
first step in forming a multi-ply laminate. Future work will include development of geometry updating and tool contact algorithms, in order to fully simulate the composite sheet forming process. Acknowledgements
The authors would like to acknowledge the European Commission, who supported much of this work through Brite-EuRam Contract No. BE-93-5092, as well as the collaboration of our industrial and academic partners and the Center for Composite Materials, University of Delaware, USA. Also, our appreciation is due to the staff of the Department of Mechanical Engineering at University College Galway for their assistance with the experimental work, and to Eoin Simon and Gary Fahey for their help with the manuscript. Nomenclature a
at av (Aa)n
B~ d D f
g Gi
K Kt m
m M t Mp n n
nt nv Nt
NP N ~ P
s S
i tN tT
Fibre orientation vector List of nodal tensions List of nodal velocities Incremental change in fibre orientation during nth time step Velocity shape function derivative matrix Deformation rate tensor Constitutive matrix Residual (right-hand side) vector Contact gap function Weak form of equilibrium function for IFRF ply Stiffness matrix Tension constraint matrix Cosine 0 Divergence forming vector for incompressibility constraint Tension mass matrix Pressure mass matrix Sine 0 Unit normal to boundary Number of tension unknowns in element Number of velocity unknowns in element Tension shape function vector Pressure shape function vector Velocity shape function vector Arbitrary hydrostatic pressure Contact slip function Partial derivative matrix Traction vector Normal contact traction Tangential contact traction
Implicit finite element modelling
(At)n T 7~
u dv xi ol Olt
% 8 13N ET
C # Op Ft d~ rl rlL rlT 0 o r, r s f2
319
nth time step A r b i t r a r y tension stress Discretized tension stress Velocity vector Discretized velocity vector Virtual velocity vector C o m p o n e n t s o f fixed Cartesian axes Penalty n u m b e r Fibre tension penalty n u m b e r Pressure penalty n u m b e r K r o n e c k e r delta function N o r m a l contact penalty n u m b e r Tangential contact penalty n u m b e r Velocity vector Fibre tension vector Pressure vector Traction boundary Strain rate vector Virtual strain rate vector Isotropic d i a p h r a g m viscosity C o m p o s i t e longitudinal shear viscosity C o m p o s i t e transverse shear viscosity Fibre orientation angle Stress tensor Material p r o p e r t y function Problem domain
References [1] 6 Brfidaigh, C.M., "Sheet Forming of Composite Materials", Chapter 13, Flow and Rheology in Polymer Composites Manufacturing, Volume 10 of Book Series: Composite Materials. Editor: S.G. Advani, Elsevier Science Publishers, Amsterdam, 1994. [2] Cogswell, F.N., Thermoplastic Aromatic Polymer Composites, Butterworth-Heinemann Ltd., Oxford, 1992. [3] Klenner, J. and Brandis, H., "The Production of Non-Developable Composite Aircraft Structures", Proceedings of Third International Conference on Automated Composites, ICAC'91, Amsterdam, October 1991. [4] Gysin, H.J., "Mechanical Parts Made of Thermoplastic Matrix Composites", Proceedings of the 3rd International Conference on "Flow Processes in Composite Materials", University College Galway, July 1994. [5] O Brfidaigh, C.M., Pipes, R.B and Mallon, P.J., "Issues in Diaphragm Forming of Advanced Composites", Polymer Composites, Vol. 12, No. 4, August 1991, pp. 246-256 [6] Pickett, A.K., Queckb6rner, T., De Luca, P., and Hang, E., "An Explicit Finite Element Solution for the Forming Prediction of Continuous Fibre Reinforced Thermoplastic Sheets", Composites Manufacturing, Vol. 6, No. 3-4, pp. 237, 1995. [7] European Commisssion Brite-EuRam Contract BE-5092, 1992, "Industrial Press Forming of Continuous Fibre Reinforced Thermoplastic Sheets and the Development of Numerical Simulation Tools".
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[8] 6 Brfidaigh, C.M. and Pipes, R.B., "A Finite Element Formulation for Highly-Anisotropic Incompressible Elastic Solids", International Journal for Numerical Methods in Engineering, Vol. 33, 1992, pp. 1573-1596. [9] 6 Br/tdaigh, C.M. and Pipes, R.B., "Finite Element Analysis of Composite Sheet-Forming Processes", Composites Manufacturing, Vol. 2, Nos. 3 & 4, 1991, pp. 161-170. [10] 6 Brfidaigh, C.M., McGuinness, G.B. and Pipes, R.B., "Numerical Analysis of Stresses and Deformations in Composite Materials Sheet Forming: Central Indentation of a Circular Sheet", Composites Manufacturing, Vol. 4, No. 2, 1993, pp. 67-83. [11] McGuinness, G.B. and 6 Brfidaigh, C.M., "Effect of Preform Shape on Buckling of Quasi-Isotropic Thermoplastic Composite Laminates during Sheet Forming", Composites Manufacturing, Vol. 6, Nos. 3-4, 1995, pp. 269-280. [12] McEntee, S.P. and 6 Brfidaigh, C.M., "Numerical Modelling of Single-Curvature Composites SheetForming", Proceedings of the ASME Materials Division, MD-Vol. 69-2, IMECE, pp. 1119-1132, 1995. [13] Cattanach, J.B. and Cogswell, F.N., "Processing with Aromatic Polymer Composites", Developments in Reinforced Plastics, Editor: G. Pritchard, Applied Science Publishers, 1986, pp. 1-38. [14] Soil, W.E. and Gutowski, T.G., "Forming Thermoplastic Composite Parts", SAMPE Journal, Vol. 24, No. 3, pp. 15-19, 1988. [15] Cakmak, M. and Dutta, A., "Instrumented Thermoforming of Advanced Thermoplastic Composites. II: Dynamics of Double Curvature Part Formation from PEEK/Carbon Fibre Prepreg Tapes", Polymer Composites, Vol. 12, No. 5, pp. 338-353, 1991. [16] Okine, R.K., "Analysis of Forming Parts from Advanced Thermoplastic Composite Sheet Materials", Journal of Thermoplastic Composite Materials, Vol. 2, No.I, pp. 50-76, 1989. [17] VanWest, B.P., Pipes, R.B., Keefe, M. and Advani, S.G., "The Draping and Consolidation of Commingled Fabrics", Composites Manufacturing, Vol. 2, No. 1, pp. 10-22, 1991. [18] Bergsma, O.K. and Huisman, J., "Deep Drawing of Fabric Reinforced Thermoplastics", Proceedings of CADCOMP '88, Editor: C.A. Brebbia, Computational Mechanics Publications, Springer-Verlag, Berlin, 1988. [19] Gutowski, T.G., Hoult, D., Dillon, G. and Gonzalez-Zugasti, J., "Differential Geometry and the Forming of Aligned Fibre Composites", Composites Manufacturing, Vol. 2, No. 3/4, pp. 147-152, 1991. [20] Gutowski, T.G., Dillon, G., Chey, S. and Li, H., "Laminate Wrinkling Scaling Laws for Ideal Composites", Composites Manufacturing, Vol. 6, Nos. 3/4, pp. 123-135, 1995. [21] Tam, A.S. and Gutowski, T.G., "Ply-Slip during the Forming of Thermoplastic Composite Parts", Journal of Composite Materials, Vol. 23, pp. 587-605, 1989. [22] Talbot, M.F. and Miller, A.K., "A Model for Laminate Bending under Arbitrary Curvature Distribution and Extensive Interlaminar Sliding", Polymer Composites Vol. 11, No. 6, pp. 387397, 1990. [24] Spencer, A.J.M., Deformation of Fibre-Reinforced Materials, Clarendon Press, Oxford, 1972. [25] Rogers, T.G., 1989, "Rheological Characterisation of Anisotropic Materials", Composites, Vol. 20, No. 1, pp. 21-27. [26] Hull, B.D., Rogers, T.G., and Spencer, A.J.M., "Theoretical Analysis of Forming Flows of Continuous-Fibre-Resin Systems", Flow and Rheology in Polymer Composites Manufacturing, Editor: S.G. Advani, Elsevier Science Publishers, Amsterdam, pp. 203-256, 1994. [27] Rogers, T.G. and O'Neill, J.M., "Theoretical Analysis of Forming Flows of Fibre Reinforced Composites", Composites Manufacturing,, Vol. 2, No. 3/4, pp.153-160, 1991. [28] Golden, K., Rogers, T.G. and Spencer, A.J.M., "Forming Kinematics of Continuous Fibre Reinforced Laminates", Composites Manufacturing, Vol. 2, No. 3/4, pp. 267-278, 1991. [29] Tucker, C.L., "Sheet Forming of Composite Materials", Advanced Composites Manufacturing, Editor: T.G. Gutowski, John Wiley, New York, 1997. [30] Thompson, E.G., Wood, R.D., Zienkiewicz O.C. and Samuelsson, A. (Eds), Numerical Methods in Industrial Forming Processes, Numiform 89, Fort Collins, CO, June 1989, A.A. Balkema, Rotterdam, 1989.
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[31] Scherer, R., Zahlan, N., and Friedrich, K., 1990, "Modelling the Interply Slip Process during Thermoforming of Thermoplastic Composites Using Finite Element Analysis", Proceedings of CADCOMP '90, Brussels, Belgium, pp. 39-52 [32] Beaussart, A., Pipes, R.B. and Okine, R.K., "Modeling of Sheet Forming Processes for Thermoplastic Composites", Proceedings of CADCOMP '92, Editor: S.G. Advani et al., Computational Mechanics Publications, Southampton, 1992. [33] Simacek, P., Kaliakin, V.N. and Pipes, R.B., "Pathologies Associated with the Numerical Analysis of Hyper-Anisotropic Materials", International Journal for Numerical Methods in Engineering, Vol. 36, pp. 3487-4508, 1993. [34] Simacek, P., "Numerical Modelling of the Sheet-Forming Process", PhD Dissertation, Department of Mechanical Engineering, University of Delaware, 1994. [35] Pipkin, A.C. and Rogers, T.G.,"Plane Deformations of Incompressible Fiber-Reinforced Materials", Journal of Applied Mechanics, Vol. 38, pp. 634-640, 1971. [36] ICI Fibreite Data Sheet 5, Fabricating with APC-2 Composites. Fiberite Corporation, 1986 [37] Groves, D.J. and Stocks, D.M., "Rheology of Thermoplastic-Carbon Fibre Composites in the Elastic and Viscoelastic States", Composites Manufacturing, Vol. 2, No. 3/4, pp. 179-184, 1991. [38] Scobbo, J.J. and Nakajima, N., "Dynamic Mechanical Analysis of Thermoplastic Composites and Resins", Polymer Composites, Vol. 12, No. 2, pp. 102-107, 1991. [39] Kaprielian, P.V. and Rogers, T.G., "Determination of the Shear Modulii of Fibre-Reinforced Materials by Centred and Off-Centred Torsion", Proceedings of the 34th SAMPE International Symposium,1989. [40] McGuinness, G.B., Canavan, R.A., Nestor, T.A. and 6 Brfidaigh, C.M., "A Picture-Frame Intraply Shearing Test for Sheet-Forming of Composite Materials", Proceedings of the ASME Materials Division, MD-Vol. 69-2, IMECE, pp. 1107-1118, 1995. [41] McGuinness, G.B. and 6 Brfidaigh, C.M., "Characterisation of the Processing Behaviour of Unidirectional Continuous Fibre Reinforced Thermoplastic Sheets", Proceedings of 4th International Conference on "Flow Processes in Composite Materials', University of Wales, Aberystwyth, September 1996. [42] 6 Brfidaigh, C.M. and Pipes, R.B., "A Punch Deformation Experiment for Sheet Forming of Thermoplastic Composites", Proceedings of International Conference on Automated Composites (ICAC '91), The Hague, Netherlands, 1991. [43] Monaghan, M.R. and Mallon, P.J., "Study of the Mechanical Behaviour of Diaphragm Films", Composites Manufacturing, Vol. 2, No. 3/4, pp.197-202, 1991. [44] Tadmor, Z. and Gogos, C.G., Principles of Polymer Processing, John Wiley and Sons, New York, 1979. [45] Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method, 4th Edition, Vol. 1, McGraw-Hill, London, 1989. [46] 6 Brfidaigh, C.M., "Analysis and Experiments in Diaphragm Forming of Continuous Fibre Reinforced Thermoplastics", PhD Dissertation, Department of Mechanical Engineering, University of Delaware, 1991. [47] McGuinness, G.B. and 6 Brfidaigh, C.M., "Finite Element Analysis of a Thermoplastic Composite Sheet Forming Process", Proceedings of the Irish Materials Forum Conference (IMF-8), University College Dublin, Sept. 1992. [48] Monaghan, M.R., 6 Brfidaigh, C.M., Mallon, P.J. and Pipes, R.B., "The Effect of Diaphragm Stiffness on the Quality of Diaphragm Formed Thermoplastic Composite Components", Journal of Thermoplastic Composite Materials, Vol. 3, July 1990, pp. 202-215. [49] Murtagh, A.M., "Characterisation of Shearing and Frictional Behaviour in Sheetforming of Thermoplastic Composites", Ph.D. Dissertation, Department of Mechanical and Aeronautical Engineering, University of Limerick, Ireland, 1995. [50] Monaghan, M.R. and Mallon, P.J., "Development of a Computer Controlled Autoclave for Forming Thermoplastic Composites", Composites Manufacturing, Vol. 1, No. 1, pp. 8-14. [51] Upilex Product Data Sheet. UBE Industries, Tokyo, Japan, 1987. [52] Martin, T.A., Bhattacharyya, D., and Collins, I.F., 1995, "Bending of Fibre-Reinforced Thermoplastic Sheets", Composites Manufacturing, Vol. 6, No. 3/4, pp. 177-187, 1995.
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[53] Murtagh, A.M., Monaghan, M.R., Mallon, P.J., "Investigation of the Interply Slip Process in Continuous Fibre Thermoplastic Composites", Proceedings of the Ninth International Conference on Composite Materials (ICCM-9), Madrid, July 1993. [54] Kaprielian, P.V., and O'Neill, J.M., "Shearing Flow of Highly Anisotropic Laminated Composites", Composites, Vol. 20, No. l, pp. 43-47, 1989. [55] Rogers, T.G., and Pipkin, A.C., "Small Deflections of Fibre-Reinforced Beams or Slabs", Journal of Applied Mechanics, Vol. 38, pp. 1047-1048, 1971. [56] Laursen, T.A., and Simo, J.C., "A Continuum-Based Finite Element Formulation for the Implicit Solution of Multibody, Large Deformation Frictional Contact Problems", International Journal for Numerical Methods in Engineering, Vol.36, pp. 3451-3485, 1993. [57] Coffin, D. W., "Flange-Wrinkling and the Deep-Drawing of Thermoplastic Composite Sheets", PhD Dissertation, Department of Mechanical Engineering, University of Delaware, 1993.
Composite Sheet Forming edited by D. Bhattacharyya 01997 Elsevier Science B.V. All rights reserved.
Chapter 8
Rheology of Long Fiber-Reinforced Composites in Sheet Forming S.G. ADVANI,
T.S. C R E A S Y *
and S.F. S H U L E R t
Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA
Contents Abstract 324 8.1. Introduction 324 8.1.1. Sheet forming philosophy 325 8.1.2. Processing methods 327 8.1.3. Materials 327 8.2. Rheological properties 329 8.2.1. Unfilled polymer melts 329 8.2.1.1. Viscosity and relaxation phenomena in shear flows 329 8.2.1.2. Viscosity in elongational flows 336 8.2.2. Filled viscous fluids 338 8.2.2.1. Viscosity and relaxation in shear flows 339 8.2.2.2. Viscosity in elongational flows 343 8.3. Rheological measurement techniques 348 8.3.1. Standard techniques 348 8.3.1.1. Rotational viscometers 348 8.3.1.2. Capillary flow 350 8.3.1.3. Use with filled polymers 351 8.3.2. Non-conventional apparatus 352 8.3.2.1. Shear viscosity 353 8.3.2.2. Elongational viscosity 354 8.4. Why the rheological properties are important and how to use them in sheet forming 8.4.1. Mechanisms 356 8.4.2. Important properties 357 8.4.3. Resin percolation 357 8.4.4. Transverse squeezing flow 358 8.4.5. Axial intra-ply shear deformation 361 8.4.6. Inter-ply shear deformation 362 8.4.7. Extensional flow 364 8.4.8. The importance of rheological parameters to thermoforming 364 8.5. Outlook 366 References 367 * Currently at University of Southern California, Los Angeles, CA, USA. t Currently at General Electric Co., Pittsfield, MA, USA. 323
356
324
S.G. Advani et al.
Abstract
This chapter introduces the reader to sheet forming philosophy, processing methods, and the materials applied to produce thermoformed components. Rheological properties that must be understood are introduced first for unfilled polymer melts by discussing viscosity and relaxation phenomena in shear and elongational flows. With this foundation we present the same properties for filled viscous fluids and we show the sometimes surprising results such as shear thickening. Filler aspect ratio affects shear and elongation phenomena; this presentation shows these effects and lists the theories developed to account for the filler/matrix interactions. Following the background material, the section on rheological measurement techniques discusses standard techniques for unfilled materials (rotational and capillary flow viscometers) and their limited use with filled polymers. The presented nonconventional apparatus (linear, squeeze flow and elongational viscometers) reveal the scale issues involved in measuring rheological properties of thermoforming systems. The chapter concludes by considering why the rheological properties are important and how to use them in sheet forming. Mechanisms that affect forming flows (resin percolation, transverse squeezing flow, axial intra-ply shear deformation, inter-ply shear deformation extensional flow) show the complexity of the process. Finally, we deliberate the outlook for application of thermoforming and improvement of the knowledge base of system behavior. 8.1. Introduction
Although the enhanced performance characteristics of collimated fiber composite materials are widely acclaimed, an obstacle limiting the utilization of advanced composites has been the historic manufacturing difficulty. Traditional composite manufacturing methods are often labor-intensive, slow and difficult to control precisely. One basic strategy in composite processing is to pre-impregnate the fiber tape or tow preform, assemble it into a multiaxial laminate, conform it to a curvilinear tool surface and then subject it to thermal and pressure cycles sufficient for crosslinking cure of the thermoset polymeric matrix. This is typical of thermoset filament winding, matched die molding and autoclave processing techniques. Another composite processing strategy is to impregnate, shape and consolidate the fibers, and cure the resin in one step. This occurs in pultrusion and the liquid composite molding processes of resin transfer molding (RTM) [1] and reaction injection molding (RIM). The processing difficulty and rate-limiting step associated with these processing techniques surrounds the need to control the cross-linking chemistry of the thermoset matrix. The use of high-temperature thermoplastic polymers in the composite preforms presents an opportunity to develop faster, lower-cost manufacturing methods. Heated to forming temperatures, thermoplastic polymers undergo a reversible phase change from solid to liquid allowing them to be formed in ways similar to conventional materials. This offers potential automated manufacturing using processes such as tape laying, pultrusion and sheet forming [2] and eliminates the need
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for expensive autoclaving equipment. The thermoplastic matrix can be rapidly hardened after forming simply by cooling. Reduced forming cycle times are possible due to the absence of a time-consuming cross-linking step. Sheet stamping is a well established and widely used process in the sheet metal forming industry. Its counterpart in composite sheet forming is a promising processing technique for thermoplastic matrix composite part manufacture [3]. The basic material form used in thermoplastic composite sheet forming consists of individual plies of either randomly placed short fibers or long aligned fibers imbedded in a thermoplastic matrix. The individual lamina can be stacked to form multiaxial laminates ranging from uniaxial to quasi-isotropic form. With such material precursors increased forming cycles are possible since the thermoplastic, which controls the forming characteristics, can be heated and cooled rapidly. One obstacle faced in forming these advanced thermoplastic composite material systems is the lack of extendibility of the reinforcing fibers. While random short fiber systems offer excellent formability they lack the superior directional properties desired in advanced composites. The enhanced mechanical properties achieved by using continuous aligned fiber reinforcement comes at the expense of material inextendibility in the fiber direction during forming. Extendibility in the fiber direction can be obtained by introducing breaks along the individual fiber lengths. The resulting aligned long discontinuous fiber reinforcement provides enhanced formability in multiaxial sheets with little loss in final mechanical properties [4]. The objective of this chapter is the description of important non-linear rheological effects associated with the sheet forming of thermoplastic-matrix fiber-reinforced sheets near the melt temperature of the polymer. The particular material forms considered are multiaxial thermoplastic matrix laminates containing high concentrations of aligned, collimated continuous and discontinuous fiber reinforcement. We begin with a summary of properties basic to sheet-formable composites and the various basic sheet forming methods. Next is a review of the rheological properties of both viscous fluids and viscous fluids filled with long fibers. What follows is a review of various standard rheological measurement techniques and the problems encountered when using conventional measurement techniques with these composite materials. Finally, there is a discussion of how the rheological properties are important in fiber-reinforced thermoplastic composite sheet forming. Insight into the forming mechanics and material flow behavior is necessary to predict the geometry and fiber structure of finished parts. Knowledge concerning the material flow behavior will ultimately assist in the prediction of properties and performance of formed parts and facilitate the development of low-cost conventional forming methods vital to increasing the market for advanced composites.
8.1.1. Sheet forming philosophy Composite sheet forming is a process well suited for the forming and shaping of thermoplastic matrix short and long fiber-reinforced composites. The material preform is non-reinforced or reinforced with uni-directional or multiaxial fibers in a sheet and can be either in stacked or pre-consolidated form. Sheet forming starts
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with heating the preform to its forming temperature, defined as the temperature where the viscosity of the resin is low enough to allow the fibers to slide relative to one another and permit easy shaping of the sheet. Then either mechanical or hydraulic pressure forms the sheet over a curvilinear tool surface. The forming step is analogous to several common sheet metal bending and forming operations and includes deformation of the sheet both in and out of the plane. This material flow is highly viscous (the viscosity may be 7 to 12 orders of magnitude greater than that of water) and is characterized by the flow of both resin and fibers together. This chapter will focus on this aspect of the process and discuss methods of measuring some transient rheological properties. After the forming step is complete consolidation pressure remains on the part until it cools. Once sufficiently cooled, the part is removed from the tool surface and, if required, a final edge-trimming step is performed. If needed the reversible solid-liquid phase change characteristic of thermoplastics enables the once formed part to be transferred to another tool surface and repeatedly reformed or incrementally formed until the final desired geometry results. The oldest form of sheet forming, also known as thermoforming, exists with the processing of non-reinforced thermoplastic sheets. Isotropic in nature, the sheets are usually held along their edges over a tool surface and brought up to their material softening temperature. This is usually somewhere slightly below the actual melt temperature to work the material while in a compliant but not liquid state. Common forming methods are "hot stamping" the sheet between matching dies and "vacuum forming" where a vacuum drawn through small holes in the tool face pulls the sheet down and spreads the sheet over the surface. The processing science for long and short fiber-reinforced thermoplastic sheets grew out of a need for large parts with strength and stiffness higher than nonreinforced sheets. Faster processing cycle times than those for thermoset-matrix composites were required also. Besides faster forming cycles, thermoplastics have more flexible processing parameters since the viscosity is a function of temperature not of cross-linking as in thermosets. As previously mentioned, the ability to remold and reshape thermoplastics can be an advantage to using thermoplastic sheet formed parts over similar thermoset parts. Sheet formed parts also have the potential to reduce the total part count in a structure by molding in and incorporating reinforced areas. Common reinforced thermoplastic processing methods are injection molding, pultrusion and tape laying. The sheet forming process holds several unique advantages over each of these methods. The nature of injection molding combined with the high viscosity of the thermoplastic precludes the use of high aspect ratio fibers (> 1,000) that provide the necessary mechanical properties desired in high-performance applications. Also, it is difficult to make large parts by injection molding. Sheet forming provides greater control over and ability to predict the final fiber architecture. Thermoplastic pultrusion solves this by providing continuous fiber reinforcement in one direction. Still, it would be difficult to pultrude parts of any great width or net shape and multiaxial laminates would not be possible. Finally, thermoplastic tape laying permits precise long fiber placement within large multiaxial and even axisymmetric parts, however, the process cycle times are much longer than for sheet formed
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parts. Why has sheet forming not become the manufacturing method of choice? The main hurdle may be the combination of the directional properties and the inextendibility of the fibers that may lead to instabilities in the process. Better understanding of the material deformation behavior during the forming process can help manufacturing engineers design the tool and processing conditions to avoid the instabilities. 8.1.2. Processing methods
The major composite sheet forming processing methods can be broadly classified as hot stamping, diaphragm forming, and incremental processing. Composite sheet stamping or matched-die press forming imitates the stamping methods employed in the field of sheet metal forming as a high-volume, low-cost manufacturing process. The heated composite blank is pressed against the tool surface. A variation on this is rubber tool stamping where the sides of the die are compliant. In the case of tool misalignment this helps maintain an even consolidation pressure across the part. In diaphragm forming two disposable plastically deformable diaphragms of either superplastic aluminum or polyimide polymer hold the blank between them. During the forming cycle the diaphragms have their edges clamped, they heat up with the blank and the sandwich is deformed by air pressure toward the tool surface. The diaphragms serve to hold the blank in tension and prevent fiber buckling that can occur under compressive stresses. When forming parts containing continuous fiber reinforcement the diaphragms are clamped but the blank cannot be. This is due to the inextendibility of the fiber reinforcement. Hydroforming is a process similar to diaphragm forming where hydraulic fluid provides the pressure behind a permanent rubber diaphragm. Incremental processing enables the forming of large structures using smaller, lower-cost fabricating equipment. After the usual heating and forming steps there is an additional part transfer step added to the forming cycle. Incremental forming produces final shapes that would be difficult otherwise. Stock shapes can be produced and later incrementally formed into custom configurations. One promising incremental forming method is stretch forming. Stretch forming makes exclusive use of aligned long discontinuous fiber-reinforcement technology. As the name implies an extensional mode of deformation in the fiber direction is enlisted during the forming. Since the fibers are discontinuous the composite sheets may be locally heated and deformed. This allows certain incremental forming techniques not possible with continuous fiber-reinforced materials. For example, linear beams can be stretch formed into curved sections with favorable mechanical properties since the fibers follow the curvature of the beam [5]. The key to successful stretch forming is precise control over the final fiber placement. This is achieved by clamping both ends of the uniformed part, heating up the portion between the clamps and then carefully forming the stock shape to the desired curvature [6]. 8.1.3. Materials
The basic materials consist of the polymer matrix and the reinforcing fibers, which may be continuous or discontinuous. Thermoplastic matrix polymers are the
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common constituent in all sheet-formable composite structures. The difference between thermoplastic and thermoset polymers is the lack of cross-links in thermoplastics. This results in the ability of thermoplastics to be heated and reshaped repeatedly. Their higher initial viscosity makes thermoplastic polymers more difficult to process than thermosets. But the multistep processing and cross-linking necessary in thermosets can be time-consuming. An additional processing advantage associated with the use of thermoplastic matrix polymers is their infinite shelf life at room temperature. This significantly eases composite preform storage in contrast to thermoset matrix polymers that require constant refrigeration. Thermoplastic polymers also offer the mechanical properties of toughness, good adhesion to the fibers, wear resistance and environmental resistance. The high-temperature thermal properties of thermoplastics are inferior to those of thermosets; however, advanced thermoplastics such as polyetheretherketone (PEEK) and polyetherketoneketone (PEKK) have working temperature ranges than can make them competitive. Other thermoplastic matrix polymers of interest are polyphenylenenesulphide (PPS), polyethersulphone (PES), polyamide (PA), polyetherimide (PEI) and polypropylene (PP) [7]. The reinforcing fibers provide the high strength and stiffness to the composite. The desired fiber properties in a composite are stiffness, strength, low density, environmental resistance and a resistance to temperatures encountered during the processing cycles. The reinforcing fibers utilized in sheet-formable composites include organic fibers (Kevlar), inorganic fibers (glass) and carbon fibers (AS4). Besides determining the final mechanical properties of the composite, the fibers, particularly long fibers, serve to dictate the overall flow of the sheet blank during forming. Although random short fiber-reinforced composites have good forming characteristics they lack the high directional properties of aligned long fiber reinforcement. This tradeoff goes both ways since part complexity is limited by the movement capability of the fiber reinforcement during forming. Aligned long discontinuous fiber reinforcement offers improved forming behavior since it allows for an extensional mode of deformation in the fiber direction. It is important to combine the selected matrix and fiber components into material preforms that are easy to handle and are suitable for forming into structures. The most common preforms exist with the fiber reinforcement preimpregnated with the matrix to form a "prepreg". Any forming process must occur above the melt temperature of the matrix polymer. At these elevated temperatures oxygen may interact with the matrix and fiber surfaces. Oxidation reactions can affect the matrix/fiber surface adhesion and in turn affect the mechanical properties of the composite. By preimpregnating the fiber reinforcement before the actual forming process takes place extended time at high temperature can be avoided. The general classifications of composite sheet preforms are woven fabrics, unidirectional plies or tapes, and random short fiber sheets. Additionally, individual plies may be stacked to form multi-ply and multi-angle laminates. One widely used perform is prepreg uniaxial continuous fiber-reinforced tape. This preform's advantages include simple storage and handling, easy cutting and placement to construct multi-angle laminates with specific fiber orientations, and rapid forming ability. Forming is rapid since the tape only needs to be melted and fused at a low pressure
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to form the final structure. The reinforcing fiber geometry and orientation within the preform determine both the forming behavior of the sheet and its final mechanical properties.
8.2. Rheological properties The deformation behavior or the fluid dynamics of polymeric melts can be described by a set of rheological properties called material functions. These functions describe the manner in which the fluid response changes as a function of the flow conditions imposed upon the melt. These conditions can be the applied strain rate (y for shear, k for elongation) or the applied stress ('gyx for shear, rxx for normal stress causing elongational flow). Material functions can be defined under both steady state conditions and as transient functions with time as an additional parameter. The transient functions should approach the steady functions for large times under steady applied strain rate or stress. Deformation behavior of Newtonian materials can be defined by one material function called the viscosity. For non-Newtonian materials more than one material function is necessary as the melt may be shear thinning and visco-elastic, and may exhibit time-dependent behavior. For more details see chapter 3 of Bird et al. [8]. In this section we present important material functions to describe some steady and transient rheological properties. These functions can then be compared with the flow phenomena of filled melts to show the change from unfilled melt behavior. This will show that, in addition to an expected increase in the viscosity, fillers also can perturb local flows with unexpected results.
8.2.1. Unfilled polymer melts The material functions for unfilled polymer melts or neat fluids provide a good basis for understanding complex melts. The two most important flow fields investigated are shear flow, and elongational flow, as most deformations can be expressed as a combination of these two flow fields.
8.2.1.1. Viscosity and relaxation phenomena in shear flows Steady-state shearing viscosity, ~/s, is the most basic material function used to quantify the flow of a fluid. The flow condition used to define r/s is conceptually simple and called planar Couette flow [9]. Figure 8.1 illustrates the flow velocity profile for a layer of fluid in simple shear between two parallel plates. The top plate moves with a constant velocity, U, while the bottom plate remains fixed. The analysis assumes a "no-slip" boundary condition is at each plate. Thus the fluid elements touching the surface of the plate remain fixed to the plate and keep the same position with respect to the plate always. The resulting velocity profile is linear and is proportional to the height above the fixed lower plate as given by the relation OVx Vx - -~y y
(8.1)
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Fig. 8.1. Simple flow of a fluid layer between two plates. (a) Couette flow geometry. (b) Stress on a fluid element.
where the velocity gradient OVx/Oy - ~)yx, the shear strain rate in the fluid. The shear stress, ryx, is proportional to the applied f/yx as expressed by the relevant relationship "gyx = OSf/yx
(8.2)
Thus the steady shear viscosity relates the shear stress to the shear strain rate. Note that r/s is not necessarily a constant. The nature of dependence of r/s on i/yx classifies the fluid as either Newtonian (no dependence) or non-Newtonian. If nonNewtonian, r/s may be function of strain rate for inelastic fluids and may exhibit time-dependent behavior for a visco-elastic melt. Stress relaxation following shear flow is the decay of the steady ryx that occurs when the applied ~ stops. This material function is noted as ry-~,where the minus sign indicates a decreasing function of time. This function is by definition a transient response of the fluid and it will be discussed with other transients. We will start by describing the Newtonian fluid. For Newtonian fluids the shear stress in the fluid is proportional to the shear strain rate through the relationship [9] -
.;,yx
(8.3)
where/z, defined as the Newtonian viscosity, replaces the more general r/s. For a Newtonian fluid,/z is a single constant value for any ~'yx. Newtonian fluids have a simple molecular structure with interactions between molecules over short distances. The fluids of interest in sheet forming are usually non-Newtonian polymer melts, which may be neat or particle-filled. For these complex fluids the entanglement of the polymer macromolecules produces a rate dependence of the steady viscosity [8]. Figure 8.2 illustrates the steady-state shear viscosity as Yyx increases for Newtonian, shear thinning (viscosity decreases) and shear thickening (viscosity increases) fluids. To differentiate the Newtonian from the non-Newtonian liquids /z is replaced by ~ s - ~s(fJyx) so that "ryx -~ 17S(f/yx)f/yx
(8.4)
As shown by fig. 8.2, polymer melts behave as Newtonian fluids in creeping flows, ~)yx~ 1.Osec -1, but then demonstrate "shear thinning" for large y. At decreasing
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Shear Thickening
J Inll s
Shear Thinning
Fig. 8.2. Steady shear viscosity as a function of shear strain rate.
i/yx, Os
approaches a constant value called rl0. This is the "zero shear rate viscosity" the viscosity of the fluid would have when deformed infinitely slowly. Shear thinning materials are common among polymer solutions and melts [10]. However, these melts also may exhibit visco-elasticity. Visco-elasticity can be characterized by measuring material functions such as first normal stress difference, second normal stress difference and relaxation parameters. Usually Deborah or Weissenberg number characterizes the elasticity effect in the melt [8]. The shear thinning behavior exhibited by polymer melts under steady-state conditions, rls(i/yx), can be modeled by either the power-law [11,12] or Carreau [13] relations as illustrated in fig. 8.3. In many engineering processes e.g., injection molding i/yx is high everywhere and the transition to steady-state flow occurs rapidly. For these conditions the decreasing rls region of fig. 8.2 is the only range of interest. Usually an empirical power-law relation of the form
,('-~) ~?S(~'yx) -- mryx
(8 5)
10 5
"~ _
"................. "'" ......... u ~-,~,~
I 0 Huppler 1967 IE] Dodd 1966 ] ........ Power L a w
InT1 s
_
10 0 10-4
I
I
........
!
I
I
10 2
I.t. Fig. 8.3. Steady shear viscosity as modeled by the empirical power law and Carreau relations.
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is sufficient, where m and n are the parameters used to obtain the best least-squares fit of the data on a ln-ln scale. Note that for shear thinning fluids n < 1 so that the power-law slope is negative. Once m and n are determined for a given fluid rls can be quickly calculated for any i'yx in the power-law range. Note that significant error appears when applying the power-law relation to low strain rates. Should the application need shear viscosity values for a broad range of f/yx the Carreau relation should be used. The Carreau relation provides a good description of the steady viscosity from Newtonian creeping flows to high power-law strain rates. The basic relation is r t s - 00[1 + (~.?'yx)2] (n-l)/2
(8.6)
where the new parameters are rl0 and ~. Again, 00 is the zero shear rate viscosity while ~ (with units of time) is a constant for the material. ~ adjusts the position of the "knee" in the ~s-~ curve. Here n has the same definition as in the power-law relation; it still controls the slope of the curve in the power-law region. When a Newtonian fluid at rest is set into motion in shear the shear stress moves from zero to ryx in a step function as shown in fig. 8.4. Also, as shown in the figure, when the flow stops the stress immediately drops to zero. This immediate response means that Newtonian liquids do not exhibit any memory effects. In contrast, the response of a polymeric liquid does not follow immediately the applied shear rate. This is characteristic of visco-elastic behavior. Such fluids exhibit a retardation or a stress growth phenomenon. Similarly, when the flow stops the stresses do not go back to zero instantaneously. This is relaxation. The relaxation and retardation times are characteristic of the material under consideration and are important material functions to characterize transient response. Another important transient phenomena is shear recovery. If the shear stress is removed after reaching steady state, a Newtonian fluid will not deform any further. A visco-elastic polymer will recoil before coming to a complete stop and the negative deformation is shear recovery. A purely elastic body in these conditions will exhibit a complete recovery. Hence the partial recovery can be used as a characterization transient material function. These
0 Simple Fluid D Linear Viscoelastic 9 Non-linear Viscoelastic
~
+
,
0
!:
FlowStart up -. ~ I ~
Stress R e l a x a t i o n ~
I ~~-"'-"~
Time
Fig. 8.4. Transient effects in simple shear. ?'yx is applied at t = 0. ~'yx= 0 at t = tl.
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transient material functions are important to thermoforming since the forming process involves rapid deformation and short processing times. When polymeric liquid is set into shear flow from rest it takes finite time for the fluid to reach steady state. This is the result of the visco-elasticity of the fluid. Since "gyx and r/s are now functions of both f/yx and time they usually are denoted by + 9 ryx(Yyx, t) and r/~(i/yx, t). The plus sign represents a flow begun from rest at time "zero" with " ryx + or r/s+ increasing into positive time. Given sufficient time ry+~--~ ryx and r/~ ~ r/s. Visco-elastic fluids may be classed as linear or non-linear depending on the way that ry+xrises to ryx. Figure 8.4 demonstrates stress growth for both visco-elastic fluid types. For linear visco-elastic fluids the stress grows up to the steady stress level without rising above it. A time constant relevant to the flow, usually denoted by )~1 and called the "relaxation time", is introduced as a measure of the time required to reach the steady state response of the system. The relaxation time changes the arithmetic equation (8.4) into a differential equation on stress such as the familiar Maxwell [14] equation (constant r/s) or White-Metzner [15] equation r/s(f/yx)
ry+x+ Z, 4:y+x= r/s(f/yx))yx
(8.7)
Note that this equation uses the steady shear thinning viscosity r/s(f/yx). The transient viscosity is defined by the solution to eq. (8.7): r/-~(i/yx, t)= ryx(Yyx, + " t)/i/yx. For a linear viscoelastic fluid each r/~ curve falls below the curves found at lower strain rates. The slowest strain rate curve will have the highest viscosity. The relaxation response of the linear viscoelastic fluid is different from the Newtonian liquid too. Just as ry+xrequires a finite time to reach 72yx, now ryx takes a finite time to decay to zero stress. When the flow stops the stress relaxes to zero with exponential decay of the form [10] "Cyx -- ry x e [-t/~'l]
(8.8)
where ryx represents the decaying stress, rex is the steady-state stress present when the flow stopped and Z l is the same relaxation time defined above. The above discussion has been simplified from the situation of polymer melts with a single-valued relaxation parameter ~1. Actually, the distribution of molecular weights and the variety of molecule to molecule interactions available in a polymer melt make the material respond with a distribution or spectrum of relaxation times. However, ~'1 can be used to designate the largest relaxation value present in the distribution and thereby simplifying the discussion. Some polymers also may exhibit non-linear visco-elastic response. The material functions still carry the same notation, e.g., Tyx(Yyx, + " t), it is the way in which r;+x approaches ryx that characterizes the non-linearity. At small 1) usually for ~'yx < 1/).1 - the non-linear visco-elastic fluid will act just like a linear one [16]. When f/yx grows large enough it stimulates non-linear stress growth. As fig. 8.4 shows, ry+ grows larger than ryx and then falls to reach the steady value. Also, as f/yx rises further the ratio of the peak ry+xto the steady ryx increases. The peak value of ry+xis the yield stress rYx.
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1.0
~
looo~o
0.1
+co
10,000 ;/0
0.01 0.01
Time Fig. 8.5. Transient viscosity function for a non-linear visco-elastic fluid in shear.
A set of r/~ functions for a non-linear visco-elastic fluid is presented in fig. 8.5. The slowest i/yx data forms an envelope curve that is a maximum 0~ function for the fluid. As i/yx grows each r/~ curve falls under this envelope. The steady r/s values reached at each strain rate are equivalent to the steady-state shear thinning property shown in figs. 8.2 and 8.3. The appearance of this non-linear stress growth relates to the change in the conformation of the polymer melt. At rest and without applied stress the long chain molecules reach a random entangled network, which is the preferred rest conformation. This rest conformation is depicted by the characteristic relaxation constant ,kl. When exposed to small i/yx small enough to produce linear viscoelastic response stress growth and relaxation are linear with ,k1 as the relevant parameter. That is, the rate is small enough that the melt can adjust to the flow by the same diffusion processes that allow attainment and maintenance of the rest conformation. Higher i/yx stimulates a rearrangement of the conformation. The fluid cannot accommodate the strain rate with the diffusion processes and it undergoes a conformation adjustment. The stress grows until sufficiently large to change the mechanism by which the molecules adjust to the stress. This creates a catastrophic realignment. Stress falls to ryx and becomes stable [16]. During this conformation change the relaxation constant decreases. That is, )~1 falls from its initial value to a new value called ~.'. Further, ,k' decreases as i/yx and y increase, thus, ~.' = ,k'(~, t). This impacts the stress relaxation of the fluid. Once the flow stops ryx falls initially with a time constant ~'(~, t). However the fluid can regain its original "structure" or conformation as the stress falls. Therefore ~' approaches Z l as the relaxation progresses. Figure 8.4 demonstrates the effect of this change. Initially the stress relaxes faster than that of the linear visco-elastic fluid. As ~' increases the stress relaxation is delayed such that a longer time is needed to remove the stress than taken by the linear fluid. These fluids require the use of nonlinear constitutive relations and the introduction of additional material parameters [17,181.
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The simplest constitutive equation that has a qualitative description of non-linear fluid response is the corotational Jeffreys fluid. As presented in Bird et al. [8] the from the relation is 1: -'1- ~.1~1 -'~-1/~.1 {71 "'1: .qt_.,1:,71 /
- -
r/0171 +/~'272 +
kzl't1"~'l }]
(8.9a)
where ~ is the stress tensor, 7 is the strain tensor, rl0 is the zero shear rate viscosity, k l is the relaxation time and )'2 is the retardation time. One constraint of the model is that )~2 is always less than k l. The subscripts (1) and (2) on the stress and strain tensors are Bird's notation for the zeroth and first convected derivatives of the tensor respectively. The strain tensor and its derivatives have the following definitions: 71 - ~ / - Vv + (Vv) T 72
D -
72
~-~
-
(8.9b)
{(Vv)T'71 + 71 .(Vv)}
(8.9c)
The derivative of the stress tensor comes from 1:1
~
D ~-~1;
{(Vv)T.~ + ~.(Vv)}
(8.9d)
For situations where k2 "~,k] the model can exhibit reasonable response to simulated flows. However, as k2 approaches k l the stress-time curve begins to oscillate and the model becomes less suitable (see fig. 8.6). In this example the rate of strain tensor is
'~1-
[0,01 l
0
0 0
0 0
~,
(8.9e)
with kl = 7.85, k2 = 0 and y = 0.172 for the solid line. With k2 ~ ~.1 = 11.274 the model peaks at the same time and with the same ratio of peak stress to steady stress for ~ = 0.315. The dashed line shows the excessive oscillations that can occur for large k2. The Giesekus model provides realistic behavior for non-linear visco-elastic fluids. The form of the model used is from Bird et al. [8]:
L"~~
.
' ' ".
.
'll .. . . . .
.
.
.
~ -o .
.
.
.
t (seo) Fig. 8.6. Jeffreys fluid in simple shear.
.
.
~,2 __- ~1
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(~,2)2 zl [ 17-+- ~.1171 --O~m{ 17"17}-O~2{71''~-]'17"71} -- 170 71 - ~ - ~ ' 2 7 2 - - C t ~ {]r ~ r/0 )~1
1 (8.9f)
where t~ is the parameter that determines the degree of non-linear behavior, the retardation time is ~'2 ~'~ )vl/l,000 and the other terms are defined in the Jeffreys equation. When done it shows the non-linear behavior shown in figs. 8.4 and 8.5 without the oscillations of the Jeffreys fluid.
8.2.1.2. Viscosity & elongational flows Shear viscosity is the primary material function of use in most fluid flow applications. When one turns to thermoforming, however, the elongational viscosity is also of great importance in analyzing the flow. The volume of the fluid remains constant under the assumption of the incompressibility of a fluid. The incompressibility effect renders elongation a three-dimensional flow. Therefore strain rate is expressed best in tensor notation as
~t~t-
0.5
0
0 )
o
o.5
o
0
0
-1
o oo)
o -o.5 0
~
0
Biaxial elongation
Uniaxial elongation
(8.10)
-0.5
where k is the elongational rate. Note that only the terms along the diagonal are nonzero. Thus these elongational flows are shear-free flows. Figure 8.7 shows a thermoforming sheet clamped at the edges with a greater pressure applied to its lower surface. The sheet has inflated like a bubble. An element of the sheet, also shown in fig. 8.7, deforms in biaxial elongation. The sheet stretches in x and y as it thins in z. Collecting material functions in biaxial flow is difficult.
(a) (b) Fig. 8.7. Thermoforming of fluid sheets with clamped edges in a biaxial elongation flow. (a) Inflated sheet. (b) Fluid element.
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Most elongational tests are conducted in uni-directional elongation to simplify the procedure. Uni-directional elongation is the drawing of a fluid in one direction, say in direction x, as one would extend a solid at the strain rate k in a tensile test. Figure 8.8 presents a fluid bar extended in the x direction. Since the substance is a fluid the total possible strain is much greater than for solids in tension. For most solids the possible strains are so small that assumed constant L - L0 applies. Thus a constant k test may be conducted at a constant velocity v such that [19] 1 dL
v
---- ~ = - L dt L0
(8.11)
For fluid systems with large strains we have L - L(t). With constant velocity applied to the fluid we immediately know the form of L(t): (8.12)
L(t) - Lo + vt
Rearranging eqs. (8.11) and (8.12) yields k-
+t
and k is not constant. After integrating and rearranging eq. (8.11) one obtains k t L(t) - Lo eet
(8.13)
l n ( L / L o ) or
(8.14)
Thus, for constant k, both L and v are exponentially increasing functions of time. For constant applied v, k is a decreasing function of time for large strains. For most forming processes the strain rate will vary during the procedure. By analogy with the shear stress/shear strain rate relation in eq. (8.3) the elongational stress is
......y" ...........i"""
Fig. 8.8. Uniaxial elongation of a rectangular fluid element.
338
S.G. Advani et al.
r~x = r/Ek
(8.15)
where OE is the elongational viscosity. A Newtonian fluid has a single shear viscosity defined as/z. It can be shown that there is a single value for the elongational viscosity of Newtonian fluid given by the Trouton relation 0E - 3/z
(8.16)
This equation also holds for unfilled polymer melts when ~ is small enough (creeping flow) to keep the response within the Newtonian range. In this case r/+ - 3 r / ~ . However, such strain rates are too small to be of use in a commercial process. From the discussion of transient behavior above one could easily presume that in elongation a shear thinning fluid would behave like a Newtonian liquid in creeping flows and then show greater deviation as k increases. This is correct except that the direction of the deviation is reversed. Where the shear thinning fluid has a reduced Os with increasing y, the same fluid will have an r/+ that rises above the Newtonian curve as ~ increases. Figure 8.9 shows this for a polystyrene melt [20] at 170 ~ For - 0 . 0 0 6 3 s -1 r/+ follows the Newtonian curve up to 100 s elapsed time. Then the non-Newtonian rise in ~+ occurs. As k increases the non-Newtonian performance occurs at earlier times. This phenomenon is shear hardening. In elongation the slowest ~ test forms an envelope curve, but for ~+ it is a lower bound envelope.
8.2.2. Filled viscous fluids The material functions and unfilled melt behavior discussed above are the starting point for understanding the rheology of filled polymers. To obtain the properties required of thermoformed systems the polymers must be filled with a reinforcement of particles or fibers. Volume fractions of fillers can reach 60% and the possible effect on the rheology under some flow conditions can be substantial. Filler interactions by direct contact, change in orientation or relative motion modifies the flow field. Local flows may change the material response from that expected applied flow condition. 1010
......
10 9
.
+111
0 0.0063 Q 0.02 ~ 0.063 ,o.,
108 10 7
~ " 10 6
lllll
10 5
lO-I
lOo
I I I IIIIII
lO+
lO2
I II IIIII lO,+
Seconds Fig. 8.9. Transient elongational velocity of polystyrene at 170~ [20].
Rheology of longfiber-reinforcedcomposites
339
8.2.2.1. Viscosity and relaxation in shear flows The effect of fillers on the shear viscosity can be surprising. A shear thinning fluid can be converted into a shear thickening mixture. Non-linear transients may be introduced at shear rates where linear visco-elasticity applied in the unfilled melt. These and other effects will be summarized here. Shear thickening. One qualitative change that fillers can introduce is that the suspension may become shear thickening from shear thinning. The first change in the behavior of shear thickening fluids is shown in fig. 8.2. At low shear rates the fluid behaves like a Newtonian liquid, but, as ~ reaches a critical speed yr the viscosity increases with rising shear rate. Finally, at sufficiently high rates the fluid may become shear thinning [21]. The strain rate at the peak viscosity is Ym, the maximum thickening rate. In the region between ~r and y~, the power-law relation, eq. (85), may be applied with values of n greater than 1. The factor contributing to thickening is the lack of interaction of the filler particles to one another. Particles that form an adhering network within the fluid will not shear thicken [22]. Adhering particles increase 00 but demonstrate decreasing viscosity under flow as the network breaks. Thus, shear thickening requires particles that are either neutral or repulsive. Barnes summarized the characteristics that control shear thickening [23]; they are: particle volume fraction, particle size, particle size distribution, particle shape. Increasing the filler volume fraction decreases ~. For low volume fractions the material may still shear thicken but ~ may be outside the range of interest or the limits of the viscometer. Increasing the particle size also decreases y~. Spreading the filler over a range of sizes reduces shear thickening. A clay with a broad particle size distribution had much lower viscosity at 0.7 volume fraction than one with a narrow size distribution at 0.6% loading [24]. Finally, anisotropic particles enhance shear thickening. Most polymers of industrial use are shear thinning. Effect of filler aspect ratio on shear phenomena. One measure used to describe a filler is the aspect ratio, ar. This is the ratio of the particle's greatest length to its shortest. Nearly spherical particles have ar of nearly 1. Reinforcing fibers can have an ar from 5 to well over 1,000. For small particles and short fiber fillers, at < 50 and it is possible to measure the response to shear strain using conventional equipment as described later if the diameter of the fibers is a few microns. The effect of aspect ratio on shear stress in a filled polyamide was measured by Laun [25]. Figure 8.10 shows the results for a fixed ~ (0.1 s-l), filler volume fraction (30%) and three at of the glass filler: 1 (beads), 7.3 and 25.1 (short fibers 13.5 ~tm diameter). The neat melt shows a rapid rise to steady ryx (28.4 Pa) at small strain. The glass beads extend the total strain required to reach steady ryx to 24 and they increase the apparent viscosity. The randomly oriented fibers increase 0s and add a stress overshoot effect. Table 8.1 summarizes the viscosity changes from that experiment. The stress overshoot is the result of the initial random orientation of the short fibers [25]. When flow begins those fibers that are perpendicular to the flow (or nearly so) inhibit the deformation and the stress rises. As the flow continues these fibers align with the flow direction and the stress falls to the steady stress. Therefore, this
340
S.G. Advani
150
Aspect Ratio O 27 E] 7.3 9 1 9 Neat
100
§
e t al.
,50
OF 0
50
100
150
Fig. 8.10. The effect of aspect ratio on the shear response of a polyamide with 30 vol. % filler [25] ) = 0.1 s.
TABLE 8.1 Effects of a r on shear viscosity of a polyamide melt at ~ = 0.1 s -1 and f = 0.30 (Pa s)
ar
rls
Neat 1 7.3 27
284 420 419 794
Peak r/~ (Pa s)
849 1421
stress overshoot should not be confused with the neat non-linear visco-elastic fluid stress overshoot discussed above. To compare the two overshoots, fig. 8.11 presents Laun's [25] for an ar of 27 glass-filled polyamide and a schematic of a neat non-linear visco-elastic fluid. After the stress has relaxed the neat fluid will repeat the initial stress growth response with overshoot. However, the filled polyamide resumes the flow without an overshoot because the initial random fiber orientation does not return. 4
00 (a)
5
10
15
(b)
Fig. 8.11. Stress overshoot of (a) non-linear visco-elastic fluid and (b) a short fiber-filled fluid [25].
20
Rheology of longfiber-reinforced composites
341
Although both ar of 1 and 7.3 show the same increase in apparent viscosity the a r o f 7.3 filler produces an overshoot. The a r of 25.1 filler demonstrates that there is a viscosity increase for larger ratios. This may only be due to the increased entanglement of the longer fibers (at the same volume fraction) as they attempt to rotate into the flow planes of the Couette apparatus. For a given f/yx there is only a certain amount of stress energy available to remove the tangles and align the fibers. Once the fibers are in equilibrium with the resulting ryx no further improvement occurs in the array. Mutel and Kamal [26] showed this with the experiment in fig. 8.12. When a sample of glass fiber-filled polypropylene shears at 0.04 s-~ it shows a slight stress overshoot before approaching ryx. When f/yx suddenly increases to 0.06 there is another overshoot before the approach to ryx at the new rate. Next a new sample sheared at 0.06 s-1 approaches the same ryx as the first sample deformed at the same rate. Both samples had the same stress energy available to align the fiber array since both deformed at 0.06 s-]. Finally, a sample initially sheared at 0.20 s-1 has its shear rate reduced to 0.06. The resulting ryx is lower than for the first two samples. The higher stress level provided the extra energy needed to improve the orientation of the fibers. This shows that the effect of increasing a r on the change in apparent viscosity may be different for different initial and final fiber orientations. Note that we cannot directly calculate r/s for the filled experiments and must be satisfied with apparent viscosity values. As discussed below, there are theories that suggest the relative motion of the fillers increases the local y and thus the effective f/yx is uncertain. However, by analogy to engineering stress and strain one can calculate an apparent viscosity for the systems using the applied global shear rate. Thus the addition of fillers increases the apparent viscosity of the system and greater a r have a larger effect. Details on short fiber rheology and flow can be found in Milliken and Powell [27] and Tucker and Advani [28]. Theories f o r filled s y s t e m s in shear. Many theories have been presented for predicting the viscosity of a filled fluid based upon the character of the filler and the Os of the fluid. Einstein presented a relation for the viscosity of a suspension [29]. r/-- r/o(1 + 2.5~b) 2000
eL
+~
(8.17) ,
1500
,
1/sec
1000 500
[ 0
0
250
I
0.04 l/sec
-
500
!
750
I
1000
Seconds Fig. 8.12. Effect of short fiber entanglement on apparent viscosity [26].
342
S.G. Advani et al.
where 4~ is the volume fraction of the filler. This linear relationship is suitable only under several restrictions. The filler must be spherical or nearly so. The filler particles must not interact or form a structure within the fluid, which usually is the case only for dilute suspensions. As filler concentration grows the interaction of the particles lead to deviations from the linear growth depicted and another model must be employed. For concentrated packing of spheres Frankel and Acrivos [30] developed a relation for the filled fluid viscosity r/' recast by Christensen [31] as rl'
r/s@)
--+
27Cm
1011
-
-
(C/r
as c ~ Cm
(8.18)
where r/s is the viscosity of the neat fluid at the applied y and Cm is the maximum packing fraction for uniform spheres. Pipes [32] and Christensen [31] have developed relations for the transverse (rlyz) and longitudinal (Oyx) shear viscosity of highly aligned concentrated fiber arrays. Figure 8.13 displays the geometry of these shear flows for highly aligned arrays. Pipes applies a micromechanics approach to obtain: rlyz = rlyx = XOs(i/)
(8.19)
with 7 the applied longitudinal shear rate as shown in fig. 8.13. Here the transverse and longitudinal shear viscosities are the same. Note that eq. (8.19) contains no a r effect. The effective shear viscosities are related to the matrix viscosity through the fiber volume fraction parameter, x: ,c =
(8.20)
1 -v/7/F
Fig. 8.13. Geometry of shear modes.
Rheology of longfiber-reinforced composites
343
where f is the fiber volume fraction and F is the maximum possible packing fraction for a given geometric fiber array. The parameter x depends solely upon the fiber packing geometry and concentration. Since eq. (8.19) contains no effect of ar the fibers could be short or continuous in this micromechanics approach. However, the experiment shown in fig. 8.10 demonstrates that for fixed applied y and f , the 27 ar filler reached a steady viscosity 1.9 times greater than that of the 7.3 at melt. Christensen [31] used the fluid mechanics approach developed by Frankel and Acrivos [30] to obtain expressions for the longitudinal and transverse shearing viscosities. For a hexagonal packing array of fibers in a Newtonian fluid he developed the following semi-empirical expressions for the longitudinal shearing viscosity:
1 + aft~F)
)
x/,(1 _ tiff~F))(1 - (f/F)) rl;
r/12 --
(
)3
(8.21)
a = 0.8730, t3 = 0.8815
and for the transverse shearing viscosity:
023 -
v/(1
-
1 + a(f/F) fl(f /F))(1 (f /F))
O;
-
a = -0.1930,/3 = 0.5952
(8.22)
8.2.2.2. Viscosity in elongational flows The elongational viscosity of filled systems shows a dramatic change from the neat fluid response. The phenomenon discussed next demonstrates that the change in local flow within the filled melt shifts the dominant mechanism. Elongation phenomena for low aspect particles. As shown in fig. 8.9, typical neat fluid r/e demonstrates rising viscosity with increasing k. Particle-filled fluids can manifest the shift from this reaction to a more shear-similar response. That is, creeping flows produce the maximum r/E envelope and further increases in decreases r/~:. This is demonstrated by the carbon-black particle-filled polystyrene [20] in fig. 8.14. The particles were sub-micron size. At 20 vol. % there is no strain hardening and increasing k reduces the r/+ curve. The ~ = 0.0063 envelope curve moves to higher r/+ than for the neat melt. At 25 vol. % the rise in the envelope 1010
1010
10 g
10 9
~o 7 "~
o o.,~ [] o.,~
w lO e F
10-I
.
-
100
a o.oes
10 S~ond8
102
,o 7
~
o o.**a 0.,2
/*~ /
103
lO 0 I -
10-1
_ _
100
I a o.os.~
10 ,~ond8
Fig. 8.14. Carbon-black-filled polystyrene in elongation. Compare with fig. 8.10 [20].
102
10'I
344
S.G. Advani et al.
curve is even more pronounced. The change in the magnitude of r/+ with the particles added is best demonstrated by comparing the lowest k envelope curves in fig. 8.15. Table 8.2 summarizes the numerical results of the changes in filler content and k. First the effect of the filler at k - 0.0063 shows that r/+ (10) increases by 22 times with 20 vol. % carbon black added. The 25% filler melt has an r/~ (10) 51 times as great as the neat polystyrene. The change from elongation to shear-dominated behaviour appears in the change in 0+ as k increases for each material. Neat polystyrene shows little change in 0+ (10) until k reaches 0.2 s-1. Then the strain hardening becomes apparent with the elongational viscosity increasing 263%. The filled melts have a drop in r/+ (10) with each k increase. The percentage change from one rate to the next is similar for both filled fluids. The higher packing of low aspect ratio fillers has little effect on additional local shear flow. Elongation phenomena with fiber-filled melts. Data for short fiber-filled melts in elongation is scarce. Several studies have been performed with fiber spinning of
10 10 10 9
aI +m ~"
10 8 10 7 10 6 10 5
!L ~
~
-----
10-1
Filler 25% [] 20% /X Neat g - 0.0063 0
100
101
102
103
Seconds Fig. 8.15. Envelope r/+ curves for polystyrene melts in elongation [20]. TABLE 8.2 Change in 17+ at 10 s elapsed time for polystyrene melts (s -1) 0.0063 0.02 0.063 0.2 (S-1) 0.0063 0.02 0.063 0.2
Neat melt r/+ (IO)(MPa s)
Change from prior k (%)
13.3 " " 35.1
0 0 + 263
20% particles 0+ (1 O) (MPa s)
Change from prior k (%)
25% particles r/+ (1 O) (MPa s)
Change from prior k (%)
289 132 76 4
--54 -42 -45
679 270 173 98
-60 -36 -43
Rheology of longfiber-reinforced composites
345
Newtonian oils with low concentrations of short fibers [33-35]. Weinberger and Goddard [35] showed that the short fibers increased shear viscosity by 8% but increased elongational viscosity by one order of magnitude. Mewis and Metzner showed elongational viscosity increased by 260 times for longer aspect ratios [33]. Thus the effect of short fibers in elongation is much greater than in shear. A recent study of long discontinuous fiber (LDF) filled polyether-ketone-ketone (PEKK) shows shear dominated reaction to simple elongation [36]. The "global" elongation causes "local" shearing between the fibers. The material is extreme in both fiber content (60 vol. %) and aspect ratio, which is around 8,000. Figure 8.16 shows both the stress and equivalent r/e +. When compared to fig. 8.5, the performance in elongation is similar to a non-linear visco-elastic fluid flow in simple shear. Theories for filled systems in elongation. Several models have been proposed for the elongational response of filled melts. Batchelor proposed a solution for suspensions of large ar fibers in a fluid [37]. The length of the fibers is L, the diameter of the fiber is D and the center to center fiber spacing is S, as shown in fig. 8.17. If L ~>S~> D, the fiber spacing is close but without them contacting one another. The elongation velocity gradient applied in the x-direction (the fiber direction) moves each fiber with the velocity of its centroid: u~ = x ~
(8.23)
where u~ is the velocity of fiber c~, x~ is the distance of c~'s centroid from the origin. Under a no-slip condition the velocity of the fluid at the fiber surface is also u~. The fibers at the edge of the cell shown by the dashed lines have zero velocity relative to the top fiber. Now the fiber array is replaced by the cell model with zero velocity at the cell edge and velocity u -- u~ at the fiber surface. Batchelor [37] solved Laplace's equation for u = (L/2)k at r = D/2 and u = 0 at r = S. The solution is
u-
kL ln(r/S) 2 ln(D/2S)
(8.24)
250 200
~
-
~ ~ 10"1
150
~-:" 100 ~ lo-4
50
l 0 F.
I 0.02
* 10-5 ...... ] 0.06 0.08
I 0.04
~:| a) ~
growth function.
Fig. 8.16. Elongation of LDF/PEKK [36].
§ ~
~"
0.10
t -" lo-4
E
_.~ 10 0
I 100
I 200
J_O_ 10-3 I 300 400
Seconds b) ElongMIonld vi~oslty function.
346
S.G. Advani et al.
Fig. 8.17. Fibre spacing for shear and elongation models.
where D / 2 <~r<<,S. F r o m this one obtains the "local" shear strain rate in the cell: ~L 1 i ~ - 2r l n ( 2 S / D )
(8.25)
with the same limits on r as in eq. (8.25). We define the ratio i//k to be the strain rate magnification factor. Figure 8.18 shows the u and ~ distribution across the cell model. A brief example will illustrate the shear magnification effect for fiber-fluids in elongation. Select the following data: L=5.6• S-
-2
L/100-
D=7.0•
m
5.6 • 10 -4
10 -6
m
m
L / D = ar -- 8,000
Fig. 8.18. Batchelor cell model for fiber volume concentration 0.0078%.
Rheology of longfiber-reinforcedcomposites
347
For hexagonal packing we find the fiber volume fraction is 0.0078%. The fiber and cell in fig. 8.18 scale to the values of S and D selected here. The strain rate magnification is ~)/k = 1,580 and the strain rate tensor for this filled fluid becomes: 1 1,580 1,580-0.5 1,580 0
~,--
1,580] 0 k -0.5
(8.26)
For the large ar selected it seems reasonable that the response of the filled fluid to elongation should be dominated by local shear behavior rather than the elongational components along the diagonal of ~/. Mewis and Metzner confirmed the Batchelor relation [37] in the spinning of polybutene containing glass fibers [33] for fiber concentrations to 0.93% and a r to 586. The same principle could be used for concentrated aligned fiber systems to show the infinitesimal "global" elongation can cause substantial "local" shearing of the melt between the fibres. Batchelor's work [37] resulted in a predictive model for the effective extensional viscosities for dilute and semi-dilute suspensions of collimated fibers in a Newtonian fluid. In the semi-concentrated regime Batchelor's cell model simulated the effect of hydrodynamic screening. As summarized by Metzner [38], the Batchelor equation for extensional viscosity may be expressed as 011-77 3 + 3
449(L/D)2 ]
In(n/C)]
(8.27)
where r/is the Newtonian suspending fluid viscosity, r is the fiber volume fraction and L / D is the average fiber aspect ratio. Goddard [39,40] later extended Batchelor's analysis to include shear thinning power-law fluid effects. If the power-law viscosity of the suspending fluid is expressed as (8.28)
g/ = A ~ ' ( n - l )
the Goddard analysis may be expressed as 2ok A B n ( L / D ) (n+l) ,~(n+l)
(2 + n)
/711 =
(8.29)
where (1 - n)
B -- [1
-
((])/7r)(1-n)/2n]/n
(8.30)
and k is the extensional strain rate of the suspension. Shaqfeh and Fredrickson [41] used a different approach to develop the extensional viscosities of fiber suspensions. Through an examination of the momentum transport process of suspensions they determined the contributions of the fibers to the transport properties of the suspension. For semi-concentrated aligned fiber suspensions they developed the following relation: r/ll r/
8n'n L 3 3[ln(1/r + In In(1/r + O.1585]
(8.31)
348
S.G. Advani et al.
where n is the fiber number density, L is the fiber half length and 4) is the fiber volume fraction. Pipes and coworkers [42,43] have developed an expression for the extensional viscosity of a highly concentrated aligned hyperanisotropic fiber suspension, as depicted in fig. 8.13. The relative sliding motion between adjacent fibers generates a shearing stress on the fiber surfaces. Through a force equilibrium argument on the fibers the following expression is obtained for the case of a Newtonian suspending fluid: f (x - 1)
(8.32)
where 0 is the fluid viscosity, f is the fiber volume fraction, L / D is the fiber aspect ratio and x is the array geometry parameter listed previously. Using the same arguments expressions for the cases of power-law and Carreau model suspending fluids have also been developed [42,43]. Pipes and co-workers have also examined the effects of fiber orientation and fiber length distribution on the extensional viscosity [44,45]. These results are summarized in Pipes et al. [42]. The next questions are: how can we measure the transient behavior of filled systems and how is it pertinent to sheet forming? In the next section we briefly discuss the traditional rheological measurement techniques. Also, since they are not suitable for measurement of filled systems with large aspect ratio particles, we introduce the non-traditional measurement techniques.
8.3. Rheological measurement techniques Rheometry of polymer melts is a complex task even for neat fluids. The rheologist must use good technique to generate accurate and reproducible values of the material functions. The standard devices discussed below have had many years of development and use with both neat and low a r filled fluids. Non-traditional devices have been proposed for the more complex filled liquids. The goal is to measure the rheological properties in shear and elongational flow. 8.3.1. Standard techniques
Several standard techniques have evolved for collection of the material functions. They are developed to commercial availability and standard operation techniques. Many are now computer-controlled and provide immediate data reduction and display as well as storage of information for further analysis. The basic physical characteristics of these devices are described below. An in-depth discussion of the flow analysis of each instrument may be found in Bird et al. [8]. Practical considerations and interpretation of results is presented in detail by Cogswell [46]. 8.3.1.1. Rotational viscometers
The most widely used rheometer for distinguishing the properties of viscous fluids is perhaps the rotational device shown in fig. 8.19. The top rotor is conical. The angle
Rheology of longfiber-reinforced composites
349
Fig. 8.19. Rotational rheometer.
of the cone corrects for the variation in shear strain rate that would occur if flat rotors were at both the top and bottom. Sometimes both rotors are flat; this requires additional analysis and testing to correct for the variation in strain rate along the radius of the rotors. The device is used as follows. The polymer sits between the conic upper rotor and the flat lower rotor. After the polymer heats to its melting point the cone and plate move together. Excess polymer flows over the edge of the rotors and the operator removes the excess prior to the experiment. The temperature and atmosphere of the sample may be controlled. Inert gas protects polymers that react with oxygen to avoid material degradation. The experiment begins after the sample is stable at temperature for a sufficient time to assume equilibrium. The upper rotor spins under one of three control schemes: constant stress (ryx), constant strain rate (~'yx) or dynamic strain rate (Ymax at w). Constant stress tests apply to constant torque to the upper platen. This happens with either fixed weights or microprocessor-controlled motors. Since torque is fixed the only measured quantity is the resulting rotation rate of the upper rotor. This converts to the resulting ~'yx through the data analysis system. Constant strain rate tests occur when the top rotor spins at a fixed rate that corresponds to the required ~'yx. The resulting torque on the lower rotor provides the data that becomes the stress (r+x(t)) values. Dynamic strain rate testing drives the upper rotor with a sinusoidal function generator. A maximum strain, y, is reached at peak and trough of the waveform supplied at the frequency ~o. The resulting torque on the lower rotor is sinusoidal also with the same w as the input. However, the stress trails the response of strain input due to viscous dissipation of the polymer. The level of this phase angle shift at various frequencies and temperatures provides an insight to the polymer's characteristic relaxation spectrum [10]. Table 8.3 summarizes the properties available from rotational rheometry and fig. 8.20 shows the type of data produced.
S.G. Advani et al.
350
TABLE 8.3 Properties measured by rotational rheometry Control
Measure
;,yx
~;x(;.yx. t) --. ~.(;,.) N+ (~'yx, t) --> N] (~'yx)
~x
;,y+x(~yx.t) ~ f..yx(~yx) N+(ryx, t) ~ Nl(ryx)
(1), Yyx
ryx(O), ~/yx)
Measure
CalculateTransients '
r
i
t
y
-1
I
~yx t
t
o m"
_
t
......
~ ~ ~ / G'
r9
ExtractSteady
t
IIII~
Cox-MertzRule Laun'sRule In
Fig. 8.20. Typical rotational rheometer data.
During any of the schemes discussed the normal force on the rotors can be recorded. Visco-elastic fluids in shear generate a normal force, i.e. perpendicular to the direction of/,, that tries to push the rotors apart. A transducer collects the information and the data system processes it simultaneously with the shearing results.
8.3.1.2. Capillary flow Figure 8.21 shows the basis of the capillary rheometer. A charge of polymer in the reservoir reaches the test temperature and equilibrates. Load applied to the piston forces the fluid flow through the capillary. A transducer near the inlet measures the
Rheology of longfiber-reinforced composites
351
Fig. 8.21. Capillary rheometer.
resulting inlet pressure. The pressure loss corresponds to the shear strain rate at the capillary wall. Some devices measure the load applied to the piston, but the pressure drop along the reservoir may be significant and the accuracy of the results are affected. Mass flow rate measured by monitoring the displacement of the piston or by weighing the material extruded in a fixed length of time provides the other component of the data. There are several sources of error in the instrument. Corrections adjust the viscosity derived from the capillary instrument. At least two lengths of capillary should be used at the same flow rate. This data corrects the entrance and exit effects as the flow enters and exits the capillary. Additional data can correct for the non-parabolic velocity profile of the polymer. Cogswell [46] discussed these at length. 8.3.1.3. Use with filled polymers
The standard devices just discussed are well established and their effectiveness understood. However, they are only useful for filled polymers with a scale that permits a charge to be treated as a continuum on the physical scale of the instrument. For example rotational devices are suited to filled polymers when the fillers are small (micron scale) and either spherical or of low aspect ratio. Mutel showed that for 50 a r glass fibers in polypropylene [47] the stress peak disappeared if the gap height reduced to 0.4 mm. This occurred because the small gap precluded fibers from standing perpendicular to the rotors. Closing the rotors pre-aligned the sample and the results differed from those expected in an application. One unusual application of a standard rotational rheometer was for shear flow in continuous fiber melts [48,49], A symmetric 00-90 ~ laminate placed between two fiat circular platens received dynamic stimulation. Small disconnected pieces of laminate placed between the plates provided the longitudinal and transverse shear behavior of
352
S.G. Advani et al.
the continuous fiber-reinforced fluid. Such use of the rotational device is limited to small cyclic deformations minimize change in sample configuration and fiber orientation. During extended testing the fibers may drift away from the desired orientation. Post-experiment analysis must be careful and thorough to assure that the data suits the flow problem of interest. Capillary rheometry has been used to measure the shear and extensional viscosities of fiber-filled polymer melts [50,51]. The difficulty with capillary measurements arises from the high volume fraction of long fibers present in the suspensions. Because of this fibers tend to entangle as a result of the contraction flow into the capillary die entrance. Pressures rise when fibers obstruct or block the die entrance. As these fibers break and pass through there is a rapid pressure variation. These breaks will change the final length of the fibers passing through the capillary. Care must be taken to establish the actual average fiber lengths during the testing. This phenomenon has been documented and measured by Binding [50]. When testing melt suspensions at room temperatures high fiber volume fractions may raise the suspension viscosity to a level where viscous heating of the fluid in the capillary changes its temperature and thus its viscosity. This may be reduced by using a shorter capillary, however the tube must be sufficiently long (at least several fiber lengths) to obtain fully developed flow. There also is evidence that the viscosity of fiber filled melts is a strong function of the capillary diameter and thus suspect [47]. Sectioned samples of extrudate show the fiber orientation varies through the cross-section [25]. Along the capillary walls, where the shear stress is highest, the fibers were highly aligned. In the center of the specimen the fibers were much less aligned. This variation may change the velocity profile of the flow. Capillary flow, however, is dominated by the effects at the wall. Therefore, the experimental results will indicate the shearing behavior of a moderately aligned system. Another researcher [52] showed that capillary extrudates of 10-40 wt. % ar 40 melts have significant distortion of the extrudate surface at low flow rates. Variation in fiber alignment could easily create viscosity variations of ten-fold or more. As the extrudate leaves the capillary the velocity profile changes from parabolic to fiat within two diameters of the exit. This velocity rearrangement causes the less aligned fibers to rotate and deform the surface. High shear rates increased the quality of fiber alignment and improved the surface quality of the extrudate.
8.3.2. Non-conventional apparatus
The increasing interest in highly filled fluids has lead to the development of specialized rheometers. These devices aim at producing the viscometric information for fluids that cannot work with the scale of the standard devices. Additionally, the standard instruments produce data of use with flow processes such as injection molding. When sheet forming processes involving long or continuous fiber fillers are of interest new techniques must be developed. The forces required to deform the system may be orders of magnitude higher than conventional rheometers provided.
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8.3.2.1. Shear viscosity Shear rheometers for filled materials fall into two classes. Linear devices attempt to create the Couette flow geometry shown in fig. 8.1. Squeeze flow instruments close the plates and force the fluid to flow out of them. Unlike rotational or capillary machines, these rheometers are easy to scale up to the size required for the filled fluid. The flow conditions are similar to those of thermoforming. Linear viscometers. A number of researchers have been building linear shear viscometers. Figure 8.22 shows a schematic of a linear viscometer. Two parallel plates surround the fluid with one plate set in motion with respect to the other. As for the rotational device the normal force can be measured by a suitable transducer. One researcher placed a pressure transducer in the center of one plate to do this [53]. Displacement of each plate is recorded by suitable transducers. The upper plate is controlled while the lower plate responds to the displacement. The displacement of the lower plate is small due to the stiff springs and it represents the resulting ryx of the sample. These devices can be operated in either steady or dynamic (sinusoidal) modes. Conserving the conformation of the specimen limits both modes to small total strains. However, these devices easily scale to the configuration of the filler so that accurate results may be obtained. Squeeze flow viscometers. Figure 8.23 shows three types of squeeze flow rheometers. In the first a fluid sample sits between plates that are much larger. A fixed load acts on the upper plate once the fluid has reached the melt temperature. The displacement of the top plate is recorded versus elapsed time until the flow ends. With this device the applied pressure drops as the surface area of the fluid on the plate surface increases. The flow ends when either the reduced pressure is insufficient to overcome the viscosity of the fluid or the filler particles interlock and stop the flow. The second device is a constant-pressure squeeze flow rheometer. The initial sizes of the fluid and the platens are the same. The platens may be any suitable shape. Circular platens are used for isotropic fluids. Continuous fiber-filled fluids can be tested with square platens. As the flow continues the fluid leaves the platens keeping the contact area constant. This device works in either load (i.e. pressure) or displacement control
Fig. 8.22. Linear viscometer.
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Fig. 8.23. Squeeze flow rheometers.
with the other parameter recorded as the dependent value. For continuous fiber-filled fluids this type of device allowed a video imaging system to record the material flow during testing [54]. A grid was silk screened on the sides of the specimen with hightemperature paint. Following the grid deformation as the flow progressed allowed the actual flow field to be measured. Although the squeeze flow of a neat fluid is biaxial shear, filled fluids may not perform in the same manner. For example, continuous fiber-filled materials are nonextendible in the fiber direction. Thus the flow becomes transverse shear only. As the fluid becomes sufficiently thin the fibers may contact and lock. Then resin percolation flow may be noted in the fiber direction. The third device is a channel flow instrument. Here the channel top and sides restrict the flow to one dimension. Control modes are the same as the second device. By controlling the slip condition at the interface this device can be used for transverse shear flow (no slip) or elongational flows (slip) of fiber-filled liquids.
8.3.2.2. Elongational viscosity Elongational viscosity is of great importance to thermoforming. Unfortunately it is also a difficult property to measure. Many devices have been suggested. Figure 8.24 presents three devices. The first is a constant strain rate (k) device. The fluid and the driving wheels remain at the melt temperature. The fluid may be suspended in a silicone oil so that it does not sag during the test. Constant strain rate results from
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Fig. 8.24. Elongational rheometers.
the fixed distance between each pair of driving wheels. As noted in eq. (8.11), if the gage length L0 is fixed constant v produces a constant k. The gage length under extension does not vary during the experiment and thus the constant rotation speed of the wheels creates a constant strain rate. This instrument is limited to fluids that may be suspended in silicone oil and that can be gripped and pulled by the wheels at the melt temperature. The use of a suspending fluid limits the temperature range available for testing. The second device is a constant stress (Vxx) mechanism. Here the gage length increases and thereby the cross-section of the fluid is decreased under the condition of incompressibility. Maintaining a constant stress requires reducing the load as the sample elongates. The machine shown here achieves this by a shaped weight. The portion of the weight entering the reservoir displaces enough liquid to reduce the load appropriately. The third device is a general use instrument based upon the availability of realtime computing and control. This allows the specimen to be tested under constant or varying k or Vxx or some combination of the two modes. Only a section of the fluid melts; the extremes remain cold so that the grips can hold them firmly. The microprocessor produces constant ~ tests by exponentially increasing the displacement of one end of the specimen to correct for the growing gage length. As shown by eq. (8.14), L(t) - Lo eet. Constant Vx~ is achieved by decreasing the load as a function of displacement using the incompressibility assumption. This machine allows filled polymers to be characterized without suspending fluids or specially designed weights. However, it is limited to fluids with sufficiently high r/e to support their own weight in the vertical orientation. For highly filled fluids this may not be a problem.
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8.4. Why the rheological properties are important and how to use them in sheet forming
The viscous deformation of reinforced thermoplastic sheets at their melt temperature involves the flow of resin and solid reinforcement together. The four basic variables that describe the viscosity of a filled suspension are the nature of the matrix fluid, the geometry of the reinforcing particles, the concentration of the suspended particles and the shearing motion of the fluid between the particles. In composite sheet forming the presence of fibers, particularly long fibers, dictates the overall flow of the sheet blank during forming. The complexity of the part to be created is, therefore, limited by the movement capability of the fiber reinforcement. A sheet reinforced with randomly oriented short fibers will have a similar flow behavior to that of an unreinforced sheet. This permits the creation of moderately complex parts with stiffness and strength somewhat greater than the unreinforced resin. Composite structures that require greater directional stiffness and strength, however, require fiber reinforcement that is longer, more directionally oriented and more densely packed. With this type of fiber reinforcement the viscous flow behavior will exhibit strong directional effects. The most extreme example is that of a polymer sheet reinforced with 60 vol.% aligned, collimated, continuous fibers. The flow characteristics become even more complex with the sheets stacked to form multiply, multi-angle laminates. Understanding the limits and mechanisms of flow for these highly anisotropic materials will enable the creation of quality, complex and structurally efficient parts. The initial ply lay-up and orientation combined with the viscous deformation during the forming process determines the final fiber position. Final fiber position, in turn, determines the composite's overall mechanical properties. The prediction of material flow behavior and the resulting final fiber position is, therefore, an essential aspect of sheet forming technology. This can be approached through modeling and experimental study of the rheology of these directionally reinforced viscous material systems. Understanding the deformation limits and fundamental mechanisms of flow for these highly anisotropic materials will enable the creation of quality complex and structurally efficient parts. 8.4.1. Mechanisms
Consider the ideal aligned long fiber array as shown in fig. 8.13. Continuous fiberreinforced materials have a direction of inextendibility in the fiber direction. All changes in shape are, therefore, achieved through shear deformation. Within a single continuous fiber-reinforced lamina there are three main deformation mechanisms. They are resin percolation through and along the fibers, in-plane shear deformation, and transverse squeezing flow, see figs. 8.13 and 8.25 through 8.27. With long discontinuous fiber reinforcement extensional modes of deformation are possible (see figs. 8.7 and 8.8). Within a laminate composed of many lamina of varying orientations the deformation may be further classified as inter-ply and intra-ply deformations. Intra-ply deformation is the motion of fibers within a single layer and includes
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the three main deformation mechanisms mentioned above. Inter-ply deformation is the motion of one layer of fibers relative to another. These deformation modes are illustrated in fig. 8.27. Depending on the deformation taking place one of these modes may be dominant but in actual forming operations many of these modes can act simultaneously.
8.4.2. Important properties The important material properties to be considered depend on the deformation modes or combinations of modes employed during the forming operation. If resin percolation occurs the fiber bed permeability transverse and parallel to the fibers dominates. Matrix resin flowing through and along an essentially fixed fiber bed characterizes this deformation mode. Important properties to consider for this flow mechanism are the resin viscosity and the fiber bed permeability. But if the bed permeability is small (small pore size) the high viscosity of thermoplastics makes impregnation of the resin into these pores highly improbable with the fiber bed stationary. The remaining deformation modes of in-plane shear deformation, inter-ply deformation, transverse squeezing flow and extensional deformation involve the flow of both resin and fibers together. This two-phase flow of fibers and resin make these deformation mechanisms inherently different from percolation flow. The important properties to consider, therefore, are the resin viscosity and the effective in-plane shear, transverse shear and extensional viscosities of the fiber-filled suspensions. The effective bulk viscosities of the suspension will, in turn, set the forming rates, deformations and forces that will occur in the forming process. The goal in studying the rheological properties of molten thermoplastic composites is to gain an understanding of the flow behavior. This may be approached through a combination of experimental studies with analytic and numerical modeling. The knowledge gained assists in finding the best methods for shaping and forming to achieve the desired final part structure and geometry. The following is a summary of the main deformation mechanisms and the various modeling approaches used to describe them.
8.4.3. Resin percolation Resin percolation is the flow of resin either through the fiber bed or along the fiber lengths (fig. 8.25). It is the main mechanism for healing by locally redistributing resin and allowing the bonding of adjacent plies. Flow through porous media has traditionally been described by the empirical Darcy's law [55] that relates the fluid flow rate through porous media to the pressure gradient, fluid viscosity and permeability of the porous media. In multiple dimensions it can be stated as t7 =
K VP 0 L
(8.33)
where u is the macroscopic average velocity of the fluid flow, ~ is the viscosity of the fluid, P/L is the pressure gradient in the direction of flow over a characteristic
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m
9~ 9 i~1~~ Fig. 8.25. Resin percolation.
dimension L, and K is the permeability tensor of the preform. The permeability is a measure of the ease of flow thorough the medium and is a function of both the structure and porosity of the preform. There have been several studies concerning percolation flow through fiber beds of various orientations and configurations [56-60]. The general approach to obtain the permeability of a preform is to develop a relationship between the pressure drop and the flow rate and compare it to Darcy's law. Most quantify the permeability in terms of the Darcy permeability coefficient experimentally obtained by measuring the resistance to flow through various fiber beds. Analytical and experimental studies showed that the resistance to flow is less along the fibers than perpendicular to them [59]. Percolation flow in thermoplastic sheet forming is most likely encountered during the consolidation processing phase. If the edges of the composite are left unconstrained or compliant diaphragms are used the resin may percolate along or through the fiber bed. This would result in resin-rich surface and edge areas. Percolation flow is usually limited due to the combination of high resin viscosity and the dense packing of the fiber reinforcement, which results in low permeabilities. In the most general case the sheet preforms will consist of multiple layers through the thickness of anistropic, non-uniform layers. One method of accounting for the different permeabilities of the preform layers is to average the permeabilities of each layer through the thickness of the preform [1]. However, during forming the fibers and resin move together and hence the flow mechanism is squeeze flow of an anisotropic suspension rather than percolation flow.
8.4.4. Transverse squeezing flow Transverse squeeze flow is the resulting deformation with pressure applied normal to the laminate surface, as shown in fig. 8.26. There have been several experimental
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Fig. 8.26. Transverse squeeze flow.
and analytic studies of squeeze flow deformation for both filled and unfilled fluids [61-63]. Due to the high extensional viscosities in the fiber direction infinite for continuous reinforcement the resulting deformation is strictly perpendicular to the fiber direction. This has been observed experimentally [63,64]. If there is no resin percolation the composite may be viewed as an incompressible anisotropic fluid with an effective shear viscosity. Transverse squeezing flows have the characteristic features of compression molding and depending on the geometry and flow type can be modeled similarly to hydrodynamic lubrication systems. The modeling assumptions made are that (1) the material is incompressible, (2) no body force acts on the material, (3) the squeeze motion is very slow, (4) there is no flow in the fiber direction, and (5) the material height is much smaller than its width so that the velocity component in the perpendicular direction is negligible. From these assumptions the Navier-Stokes equation reduces to -
OP Orxz = Ox Oz
(8.34)
-
Applying a Newtonian fluid model (8.35)
rxz-- lz Oz
to the material and the boundary conditions of flow symmetry about the x and z plane and no slip at the upper and lower surfaces the velocity distribution may be obtained: 2m Ox z 2 -
(8.36)
From the incompressibility assumption the expression that relates the flow rate at any cross-section to the platen closure speed dh/dt can be derived: dh Q - -x~-~
(8.37)
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Integrating the velocity distribution and setting it equal to the expression for flow rate the pressure distribution across the platen may be obtained: P(x, t) -
6m dh [x2 _ L2] + Patm h(t) 3 dt
(8.38)
Note that this pressure distribution is parabolic in both the x- and y-directions. Finally, the pressure distribution may be integrated over the platen surfaces to obtain a relation between the force on the platen, F, and the platen closure rate, dh/dt: F(t) - -8tz W ~
h-~
(8.39)
A similar result has been derived by Barnes and Cogswell [64]. A comparable approach was taken by Balasubramanyam [63] who also investigated the influence of slip on the platen surfaces. At higher platen closure speeds the shear thinning nature of the matrix resin may become apparent. A better approximation may be to employ the power-law fluid model for the bulk viscosity: ~(~,) = m yn-1
(8.40)
The velocity distribution for the power-law model may then be obtained as
z
-
1
-z
- m -~x
(1/n + 1)
~1/n+1)
1
(8.41)
and the force-closure rate relation as
(dh)n(2+l/n)n2(2+n) F(t) -- Wm --d-[
(2 + n)
L (2+n)
[h{,+Zn)_ h0{l+Zn)]
(8.42)
The major limitation of the power-law fluid model is the over-prediction of viscosity at low shear rates. The Carreau [13] fluid model takes this into account and describes the viscosity as Newtonian behavior at low shear rates followed by shear thinning power-law behavior at higher shear rates. Employing the Carreau model leads to the following constitutive relation: r x z - 7011 + (~.~)2]{,-1)/2 aVx Oz
(8.43)
Substituting this constitutive relation into the x-direction equation of motion, the following expression results:
.011 +
n-1)/2
- -
Oz
= ~at,z Ox
(8.44)
Due to the non-linearity of this expression, explicit solutions for the velocity and force are not possible. The velocity distribution and forces may, however, be found numerically.
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There have been several analytic attempts at describing the transverse shear viscosity, r/23. Pipes and co-workers [42] have developed simple relations for the effective transverse shearing viscosity (see section 8.3). The modeling assumptions are that the fibers exhibit rigid body displacements with a no-slip condition at the fluid-fiber interface. Christensen derived an alternate method for determining the longitudinal and transverse shearing viscosities by equating the rate of dissipation energy for the heterogeneous fiber-filled fluid to an effective transversely isotropic homogeneous fluid [31]. For a hexagonal packing array of fibers in a Newtonian fluid he developed the semi-empirical expression in eq. (8.22) above for the transverse shear viscosity for the full range of fiber volume fractions. These developed relations are in general agreement at the high volume fractions of fibers usually encountered in thermoforming composites. They conclude that the effective transverse shear viscosity is a function of the fiber volume fraction and packing geometry and is greater than the shear viscosity of the neat matrix polymer. With several cross-plies stacked together each ply exhibits its own independent squeeze flow response. Since there is no flow in the fiber direction the plies effectively are decoupled, each interface acting as an independent boundary on the flow. If resin percolates out of the fiber bed during the squeeze flow the incompressible fluid assumption is no longer valid as the fiber volume fraction will change during the flow. Gutowski and co-workers [65,66] have shown that in this case the fiber bed elasticity becomes important during the consolidation and squeeze flows. Transverse squeeze flow occurs during the consolidation phase of sheet forming and determines the final thickness of the part. With matched-die forming there is likely to be a no-slip composite/die surface interface. In this case all the deformation will be transverse intra-ply shear. But if a diaphragm is used there may be slip at the laminate surface. 8.4.5. Axial intra-ply shear deformation
Axial intra-ply shear deformation is the mechanism by which in-plane shear deformation occurs. The axial intra-ply shear deformation mode is shown in fig. 8.13. Many such models have already been introduced in section 8.3. An effective in-plane or longitudinal shearing viscosity can be defined as /'/12 = KT/
(8.45)
The factor x, described previously, amplifies the effective shearing viscosity of the assembly as compared to the viscosity of the neat fluid. This effect is due to a magnification of the fluid deformation rate caused by the presence of fibers. This equation is equally valid for the case of long discontinuous fiber systems since in plane shearing deformation imposed upon the unit cell occurs without differential movement of collinear fibers. Christensen [31] also derived an expression for the inplane longitudinal shearing viscosity (eq. 8.21). All noted models for both the dilute and concentrated suspensions predict an increased viscosity over that of the neat polymer. The studies also conclude that
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the composite in-plane shear viscosity is a function of the viscosity of the matrix polymer, the fiber volume fraction and the fiber array packing geometry [31,43]. Note that in these models the in-plane shear viscosity is independent of fiber aspect ratio. This was observed experimentally by Miles, Murty and Molden [67] using capillary rheometry. Axial intra-ply shear deformation is a dominant flow mechanism when deforming planar sheets within the plane. It is also a common deformation mode when the sheet deforms out of the plane.
8.4.6. Inter-ply shear deformation Inter-ply shear deformation is the method by which individual plies can move past one another. The inter-ply shear deformation mode is depicted in fig. 8.27. In order for multi-ply laminates of varying fiber orientations to deform out of the plane, individual plies must be allowed to move relative to each other. Upon consolidation a thin resin layer (-~ 7 ~tm) forms between successive plies [67]. Studies suggest that inter-ply shear deformation occurs only within this resin rich layer [69]. During forming, the relative ply translation or rotation results in a Couette shear flow of the matrix resin between the plies. The inter-ply shear deformation, therefore, is dependent only on the shear viscosity of the matrix fluid. Consider the relative motion of two fiber lamina as illustrated in fig. 8.27. The resin-rich layer is in simple shear between the lamina. Applying a "no-slip" boundary on the upper and lower surfaces creates a linear velocity profile proportional to the height above the lower lamina. The velocity may be expressed as:
Vx =
~v~
Oy y
(8.46)
where the velocity gradient, OVx/Oy, is the shear strain rate, Yxy, in the matrix layer. Experimental and theoretical studies have shown that during forming stresses build up in each fiber layer until a certain yield stress occurs in the resin-rich layer and inter-ply slip begins [68,70,71]. The inter-ply slip behavior, therefore, may be
Fig. 8.27. Modes of inter-ply shear.
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modeled as behaving as a Bingham plastic fluid. Bingham plastic fluid behavior may be described as follows: r = ry + r/y12
(8.47)
where ry is the yield stress required to initiate flow. Cogswell [68] suggests that the yield stress is due to adjacent plies initially touching locally. Once flow begins the plies move apart to form the resin-rich lubrication layer. During flow the inter-ply shear stress, r, is proportional to the resin viscosity, r/, and interfacial strain rate, ~'12
(8.48)
r = ~'12
Then, the bulk strain rate should be proportional to the interfacial strain rate:
hi ~/- (hp- hi) ~'12
(8.49)
where h i is the interface thickness and hp is a single ply thickness. If the resin viscosity is Newtonian the apparent bulk viscosity can be expressed as [72]: OBulk
-'-
(hp - hi) hi 0
(8.50)
In order to measure the yield stress and the effect of the lubricating resin layer lamina "pull-out" tests have been studied both analytically [70] and experimentally [71]. Kaprielian and O'Neill [70] examined the pull-out configuration as depicted in fig. 8.28, modeling the resin layers as Newtonian viscous fluids. Their model covers an arbitrary number of plies in an arbitrary stacking sequence. Their specific examples demonstrate the occurrence of a non-zero cross-flow velocity in the x2 direction even when the applied force is in the x l direction. Bersee and Robroek [71] performed experimental studies using pull-out tests on the middle lamina of a five-layer heated and pressurized laminate. They found that the shear yield stress, ry, increased when the applied normal stress increased and decreased when either the temperature or the lubricating resin layer thickness increased. Inter-ply shear deformation is a dominant flow mechanism when forming any type of out of plane curvature. This "deck of cards" type deformation is necessary to form any type of multi-angled laminate. The inter-ply shear mode relieves the build-up of
Fig. 8.28. Lamina pull-out test.
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compressive stresses that could result in fiber buckling when forming over sharply curved tool surfaces.
8.4.7. Extensional flow Extensional deformation of long fiber-reinforced laminates is unique to long discontinuous fiber-reinforced composites. The use of aligned, collimated discontinuous reinforcement allows the fibers to translate in the fiber direction. The deformation mechanism is depicted in fig. 8.8. The extendibility of these composites in the fiber direction provides improved formability while maintaining up to 90% of the mechanical properties of similar continuous fiber-reinforced materials [6,73]. The relative fiber sliding imparts a lubrication-type shear flow on the matrix polymer similarly to intra-ply shear deformation. There is general agreement in all the theories that the extensional viscosity of a highly filled suspension is a function of the viscosity of the neat matrix resin and the reinforcing fiber volume fraction and packing geometry. Additionally, unlike the shear viscosities, the extensional viscosity is a strong function of the fiber aspect ratio. All the theories above predict the extensional viscosity to be several orders of magnitude greater than the viscosity of the neat matrix resin. There have also been several experimental measurements of the extensional viscosity by capillary rheometry [50,51] and by direct pulling methods [36]. Because the extensional viscosity of these systems is so high, the preferred deformation modes during forming are still inter-ply and intra-ply shear. The extensional deformation mode performs most effectively in localized heating and deformation processes. As demonstrated [6], clamping both ends of a section of a long discontinuous fiber-reinforced beam and then locally heating it allows it to bend a forming process not possible with continuous fiber reinforcement.
8.4.8. The importance of rheological parameters to thermoforming In thermoforming a sheet of polymer (that may be filled with particles or fibers) forms by stamping it between matched dies (controlled strain rate) or by applying differential pressure to it (controlled stress level) (see fig. 8.29). Initially at rest, the sheet flows as it makes contact with the mold. A distribution of strain rates and total strain occurs over the area of the sheet. Steady flow may only be achieved in limited regions if at all. The primary deformation mechanism during the process is transient biaxial elongation. However, various shear modes (longitudinal, inter-ply, etc.) may be stimulated as the melt contacts the die. Once formed the part must be held in the mold until forming stresses decay sufficiently to reduce warping of the final shape. Then the melt cools and solidifies so the finished piece can be separated from the mold. At each stage of the process specific rheological characteristics of the sheet must be known to predict the forming behavior. Controlled strain rate forming occurs in matched dies. The fluid sheet becomes trapped at several points between the dies and stretches at a rate proportional to the die closure speed. Figure 8.30 shows schematically three stages of matched die
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Fig. 8.30. Stages in matched die forming.
thermoforming. Controlled elongation rate tests, ideally collected at several constant strain rates, provide transient elongational viscosity data, r/+ - r ~ : / ~ . This data describes the loads that will be encountered in forming a useful component at the high elongation rates desired for mass production by allowing a prediction of the stresses generated. With the parameters for a constitutive model obtained, strain rate tensor results from a finite element analysis provide stress data for any complex flow. Figure 8.30 shows the sheet wrapping around the sinusoidal protrusions of the dies. Die contact algorithms can track the changing surface contact with the sheet and then modify the strain tensors accordingly. The stress decay data shows the effect of strain rate on the relaxation process of the melt. As a practical consideration this data describes the time required to hold a part within a mold in order to reduce the stresses generated by the forming process to an acceptable level. Note that stress relaxation data shows that high k reduces
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processing time in two ways. First the flow step is shorter so that the part forms in less time. Second, the flow stress, although higher with greater k, decays faster. This second effect reduces the process time dramatically. One can note that a residual stress distribution can occur that is the inverse of the distributed strain rate. Areas near the edge of the sheet may have smaller k and r + but will have less stress relaxation for a short process cycle. Residual stress in the slowly extended regions could be higher than that of the high flow regions unless the process allows longer relax times.
8.5. Outlook
The benefit of utilizing fiber-reinforced thermoplastic matrix composite materials is demonstrated in many different applications and great possibilities exist for expanded use in the automotive and aerospace industries. Effective manufacturing methods for parts constructed of these materials have been developed but still require further refinement and optimization. Due to the unique composition of these materials, any growth in manufacturing technology will require a parallel growth in the field of rheology of these materials. This has been reflected in the recently increased volume of literature regarding the rheology of fiber-filled suspensions. Composite materials offer the benefits of tailored properties and the combination of increased strength with decreasing weight. With respect to the potential automotive and aerospace markets these characteristics offer attractive savings coupled with increased performance. Further acceptance of these materials will mean that future fiber-reinforced thermoplastic components will be of increased size and complexity and will result in increased manufacturing difficulties. Better understanding of the forming characteristics and rheological properties of these materials will assist in addressing these issues. Understanding the flow and rheology of such fiber-filled suspensions will help cut manufacturing costs, improve final part finish and material properties and predict final fiber positions. Additionally, this knowledge will help predict the required forces and pressures necessary to form components and assist in the materials selection process by allowing a designer to consider materials not only for their final properties but also for their formability. This will all make it possible to form net shape parts of complex geometry once we can characterize how the material deforms under various loads and rates at which the loads are applied. Growth in the use of composite material components will certainly involve the development of new material components and types of reinforcement. In order to achieve their full potential it will be necessary to keep processing technology in pace with these new material developments. Determining the rheological properties of these unique material forms will continue to present some unique challenges to be offset through insightful modeling efforts and the imaginative development of novel test techniques.
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References [1] Advani, S.G., M.V. Bruschke and R.S. Parnas, "Resin Transfer Molding Flow Phenomena in Polymeric Composites," Chapter 12, p. 465-511 in Flow and Rheology in Polymer Composites Manufacturing, edited by S.G. Advani (Elsevier, Amsterdam 1994). [2] Advani, S.G., Chapter 1 in Flow and Rheology in Polymer Composites Manufacturing, edited by S.G. Advani (Elsevier, Amsterdam 1994). [3] Okine, R.K., "Analysis of Forming Parts from Advanced Thermoplastic Composite Sheet Materials," SAMPE Journal 25 (3), 9-19 (1989). [4] Chang, I.Y. and J.F. Pratte, "LDF Thermoplastic Composites Technology," J. Thermoplastic Comp. Mat., 4, 227-252 (1991). [5] O'Toole, B.J., "Modelling the Effects of Heterogeneity in Curved Composite Beams," Ph.D. Dissertation, University of Delaware (1992). [6] Medwin, S.J. "Long Discontinuous Ordered Fiber Structural Parts," Proceedings of the 34th International SAMPE Symposium, Reno, May 8-11, 171-177 (1989). [7] Engineered Materials H a n d b o o k - Engineering Plastics. ASM International, Metals Park, OH, 1988. [8] Bird, R.B., R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Volume 1 Fluid Mechanics (John Wiley & Sons, New York 1987). [9] Panton, R.L., Incompressible Flow (John Wiley & Sons, New York 1984). [10] Rodriguez, F., Principles of Polymer Systems (Hemisphere, Washington, D.C. 1982). [11] Ostwald, W., Kolloid-Z., 36, 99-117 (1925). [12] de Waele, A., Oil Color Chem. Assoc. J. 6, 33-88 (1923). [13] Carreau, P.J., "Rheological Equations from Molecular Network Theories," Trans. Soc. Rheol., 16 (1), 99-127 (1972). [14] Maxwell, J.C., Phil. Trans. Roy. Soc. A157, 49-88 (1867). [15] White, J.L. and A.B. Metzner, J. Appl. Polym. Sci., 7, 1867-1889 (1963). [16] Matsuoka, S., Relaxation Phenomena in Polymers (Hanser, Munich 1992). [17] Jeffreys, H., The Earth (Cambridge University Press, London 1929). [18] Giesekus, H., "A Simple Constitutive Equation for Polymer Fluids Based on the Concept of Deformation-Dependent Tensoral Mobility," Journal of Non-Newtonian Fluid Mechanics 11, 69109 (1982). [19] Petrie, C.J.S., Elongational Flows: Aspects of the Behaviour of Model Elasticoviscous Fluids (Pitman, London 1979). [20] Lobe, V.M. and J.L. White, "An Experimental Study of the Influence of Carbon Black on the Rheological Properties of a Polystyrene Melt," Polym. Eng. Sci. 19, 617-624 (1979). [21] Otsubo, Yasufumi, "Normal Stress Behavior of Highly Elastic Suspensions," J. Colloid and Interface Science 163, 507-511 (1994). [22] Freundlich, H. and A.D. Jones, "Sedimentation Volume, Dilatancy, Thixotropic and Plastic Properties of Concentrated Suspensions," Journal of Physical Chemistry 40, 1217-1236 (1936). [23] Barnes, H.A., "Shear-Thickening ('Dilatancy') in Suspensions of Nonaggregating Solid Particles Dispersed in Newtonian Liquids," J. Rheol. 33, 329-366 (1989). [24] Price, C.R., Southern Pulp Paper 40, 13 (1977). [25] Laun, H.M., "Orientation Effects and Rheology of Short Glass Fiber-Reinforced Thermoplastics," Colloid & Polymer Science 262, 257-269 (1984). [26] Mutel, A.T. and M.R. Kamal, "Characterization of the Rheological Behavior of Fiber Filled Polypropylene Melts under Steady and Oscillatory Shear Using Cone and Plate and Rotational Parallel Plate Rheometry," Polym. Compos. 7, 283-294 (1986). [27] Milliken, W.L. and R.L. Powell, "Short-Fiber Suspensions," Chapter 3, p. 53-80 in Flow and Rheology in Polymer Composites Manufacturing, edited by S.G. Advani (Elsevier, Amsterdam 1994). [28] Tucker, C.L. and S.G. Advani, "Processing of Short-Fiber Systems," Chapter 6, p. 147-197 in Flow and Rheology in Polymer Composites Manufacturing, edited by S.G. Advani (Elsevier, Amsterdam 1994).
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[29] Einstein, A., "Eine neue Bestimmung der Molekuldimension," Ann. Physik 19, 289-306 (1906). [30] Frankel, N.A. and A. Acrivos, "On the Viscosity of a Concentrated Suspension of Solid Spheres," Chemical Engineering Science 22, 847-853 (1967). [31] Christensen, R.M., "Effective Viscous Flow Properties for Fiber Suspensions under Concentrated Conditions," Journal of Rheology 37, 103-121 (1993). [32] Pipes, R.B., "Anisotropic Viscosities of an Oriented Fiber Assembly with a Power Law Matrix Fluid," J. Comp. Mat. 26, 1536-1552 (1992). [33] Mewis, J. and A.B. Metzner, "The Rheological Properties of Suspensions of Fibers in Newtonian Fluids Subjected to Extensional Deformations," Journal of Fluid Mechanics 62, 593-600 (1974). [34] Kizior, T.E. and F.A. Seyer, "Axial Stress in Elongational Flow of Fiber Suspension," Transactions of the Society of Rheology 18, 271-285 (1974). [35] Weinberger, C.B. and J.D. Goddard, "Extensional Flow Behavior of Polymer Solutions and Particle Suspensions in a Spinning Motion," International Journal of Multiphase Flow 1,465-486 (1974). [36] Creasy, T.S., S.G. Advani, R.K. Okine, "Transient Rheological Behavior of a Long Discontinuous Fiber Fluid System," Work in progress (1995). [37] Batchelor, G.K., "The Stress Generated in a Non-dilute Suspension of Elongated Particles by Pure Straining Motion," Journal of Fluid Mechanics 46, 813-829 (1971). [38] Metzner, A.B., "Rheology of Suspensions in Polymeric Liquids," Journal of Rheology 29 (6) 739-775 (1985). [39] Goddard, J.D., "Tensile Stress Contribution of Flow-Oriented Slender Particles in Non-Newtonian Fluids," Journal of Non-Newtonian Fluid Mechanics 1, 1-17 (1976). [40] Goddard, J.D., "The Stress Field of Slender Particles Oriented by a Non-Newtonian Extension Flow," J. Fluid Mech. 78, Part 1, 177-206 (1976). [41] Shaqfeh, E.S.G. and G.H. Fredrickson, "The Hydrodynamic Stress in a Suspension of Rods," Phys. Fluids, A 2 (1), 7-24 (1990). [42] Pipes, R.B., D.W. Coffin, P. Simacek, S.F. Shuler and R.K. Okine, "Rheological Behavior of Collimated Fiber Thermoplastic Composite Materials," Chapter 4, p. 85-125 in Flow and Rheology in Polymer Composites Manufacturing, edited by S.G. Advani (Elsevier, Amsterdam 1994). [43] Coffin, D.W. and R.B. Pipes, "Constitutive Relationships for Aligned Discontinuous Fiber Composites," Composites Manufacturing 2, No. 3-4 (1991). [44] Pipes, R.B., J.W.S. Hearle, R.K. Okine, A.J. Beaussart, and A.M. Sastry, "A Constitutive Relation for the Viscous Flow of an Oriented Fiber Assembly," J. Comp. Mat. 25, 1204-1217 (1991). [45] Pipes, R.B., J.W.S. Hearle, A.J. Beaussart, and R.K. Okine, "Influence of Fiber Length on the Viscous Flow of an Oriented Fiber Assembly," J. Comp. Mat. 25, 1379-1390 (1990). [46] Cogswell, F.N., Polymer Melt Rheology (John Wiley & Sons, New York 1981). [47] Mutel, A.T. and M.R. Kamal, "Rheological Properties of Fiber-Reinforced Polymer Melts," in TwoPhase Polymer Systems (Hanser, Munich 1991), pp. 305-331. [48] Groves, D.J., "A Characterization of Shear Flow in Continuous Fiber Thermoplastic Laminates," Composites 20, 28-32 (1989). [49] Groves, D.J. and D.M. Stocks, "Rheology of Thermoplastic-Carbon Fiber Composite in the Elastic and Viscoelastic States," Composites Manufacturing 2, 179-184 (1991). [50] Binding, D.M., "Capillary and Contraction Flow of Long (Glass) Fiber Filled Polypropylene," Composites Manufacturing 2, 243-252 (1991). [51] Shuler, S.F., Binding D.M. and Pipes, R.B. "Rheological Behavior of Two and Three Phase Fiber Suspensions," Polymer Composites 15, No. 6, 427-435 (1994). [52] Becraft, M.L., "The Rheology of Concentrated Fiber Suspensions," Ph.D. Thesis, University of Delaware (1989). [53] Oakley, J.G. and A.J. Giacomin, "Sliding Plate Normal Thrust Rheometer for Molten Plastics," Polymer Engineering and Science 34, 580-584 (1994). [54] Shuler, S.F., S.G. Advani, "Squeeze Flow Behavior of Laminates Composed of Long Aligned Fibers within a Thermoplastic Matrix," 66th Annual Society of Rheology Mtg, Philadelphia, PA, October 6 (1994). [55] Darcy, H., Les Fontaines Publiques de la Ville de Dijon (Delmont, Paris 1856).
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[56] Cai, Z., "A Generalized Model of Flow of Polymer Fluids through Fibrous Media," J. Advanced Materials, 529-539, October (1993). [57] ,~str6m, B.T. and R.B. Pipes and S.G. Advani, "On Flow through Aligned Fiber Beds and its Application to Composites Processing," J. Comp. Mat. 26, 1351-1373 (1992). [58] Ranganathan, S., F.R. Phelan and S.G. Advani, "A Generalized Model for the Transverse Permeability of Unidirectional Fibrous Media," Submitted to Polymer Composites, 1995. [59] Bruschke, M.V. and S.G. Advani, "Flow of Generalized Newtonian Fluids across a Periodic Array of Cylinders," J. Rheol., 37, 479-498 (1993). [60] Bafna, S.S. and D.G. Baird, "An Impregnation Model for the Preparation of Thermoplastic Prepregs," J. Comp. Mat., 26, No. 5, 683-707 (1992). [61] Tadmor, Z. and C.G.Gogos, in Principles of Polymer Processing (John Wiley & Sons, New York). [62] Barone, M.R. and D.A. Caulk, "A Model for the Flow of a Chopped Reinforced Polymer Compound in Compression Molding," Journal of Applied Mechanics 53, 361-371 (1986). [63] Balasubramanyam, R., R.S. Jones and A.B. Wheeler, "Modeling Transverse Flows of Reinforced Thermoplastic Materials," Composites 20, No. 1 (1989). [64] Barnes, J.A. and F.N. Cogswell, "Transverse Flow Processes in Continuous Fibre-Reinforced Thermoplastic Composites," Composites 20, No. 1 38-42 (1989). [65] Gutowski, T.G., "A Resin Flow/Fiber Deformation Model for Composites," SAMPE Quarterly, 16, No. 4 (1985). [66] Gutowski, T.G., T. Morigaki and Z. Cai, "The Consolidation of Laminate Composites," The Journal of Composite Materials 21 (2), 172-188 (1987). [67] Miles, J.N., N.K. Murty and G.F. Molden, "The Viscosity of Fiber Suspensions at Low Fiber Volume Fractions," Polymer Engineering and Science 21, 1171-1172 (1981). [68] Cogswell, F.N., Thermoplastic Aromatic Polymer Composites (Butterworth-Heinemann, Oxford, 1992). [69] Cogswell, F.N., "The Processing Science of Thermoplastic Structural Composites," International Polymer Processing 1, No. 4, 157-165 (1987). [70] Kaprielian, P.V. and J.M. O'Neill, "Shearing Flow of Highly Anisotropic Laminated Composites," Composites 20, No. 1, 43-47 (1989). [71] Bersee, H.E.N. and L.M.J. Robroek, "The Role of the Thermoplastic Matrix in Forming Processes of Composite Materials," Composites Manufacturing 2, No. 3/4, 217-222 (1991). [72] Muzzy, J.D., X. Wu and J.S. Colton, "Thermoforming of High Performance Thermoplastic Composites," ANTEC 1989. [73] Pratte, J.F., W.H. Krueger, and I.Y. Chang, "High Performance Thermoplastic Composites with Poly(ether ketone ketone) Matrix," Proceedings of the 34th International SAMPE Symposium, Reno, 2229-2242 (1989).
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Composite Sheet Forming edited by D. Bhattacharyya 9 Elsevier Science B. V. All rights reserved.
Chapter 9
Bending of Continuous Fibre-Reinforced Thermoplastic Sheets T.A. M A R T I N , S.J. M A N D E R , * R.J. D Y K E S and D. B H A T T A C H A R Y Y A Composites Research Group, Department of Mechanical Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand
Contents Abstract 371 9.1. Introduction 372 9.2. Development of an idealised viscous bending model 374 9.2.1. Constraint conditions and kinematics 374 9.2.2. Stress in a constrained material 376 9.2.3. Constitutive equation 377 9.2.4. Stress equilibrium 377 9.2.5. Kinematic model for a vee-bend 378 9.2.6. Admissible stress fields 379 9.3. Experimental procedures 380 9.4. Results and discussion 382 9.5. Modified constant shear rate tests 392 9.5.1. Determination of transverse shear behaviour from vee-bending 393 9.5.2. Transverse shear viscosity tests 395 9.6. Conclusions 399 Acknowledgements 399 References 400
Abstract This chapter is primarily concerned with the rheological behaviour of continuous fibre-reinforced thermoplastic ( C F R T ) materials in shear. The analysis presented in this chapter centres a r o u n d a novel piece of testing equipment which establishes veebending as a means of determining both the longitudinal and transverse shear viscosities of such materials. In the analysis presented here, no distinction is drawn between the constituents of the composite. Instead an idealised c o n t i n u u m model subject to the kinematic constraints of incompressibility and fibre inextensibility has been adopted. The first part of the chapter deals with various concepts related to the deformation and related flow mechanisms which p r e d o m i n a t e in C F R T materials. This is followed by the development of an idealised material model for an *Currently at McKinsey & Company, Sydney, NSW, Australia. 371
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incompressible viscous fluid reinforced with a single family of inextensible fibres. The analytical model leads to a straightforward interpretation of the effects of forming speed and geometry on the bending stresses expected in a real sheet during forming. An experimental programme is then outlined which details the forming rates and temperatures over which the vee-bending experiments are conducted. The results of these tests are discussed in two parts. Firstly, the quality of the samples is assessed with regards to fibre instability and the spring-back/forward phenomenon. The second part of the discussion centres around the interpretation of the material's longitudinal shear behaviour. A further modification to the bending mechanism is then introduced which allows the tests to be carried out at constant shear rates. This is then followed by the development of a method for predicting both the longitudinal and transverse shear viscosities of CFRTs. The experimentally obtained viscosity ratios from these final tests are compared to a number of alternative models which relate the longitudinal and transverse viscosities to the fibre volume fraction and the viscosity of the matrix. 9.1. Introduction
The formability of a continuous fibre-reinforced thermoplastic is governed by four basis flow mechanisms: resin percolation, transverse fibre flow, inter-ply shear and intra-ply shear [1]. These mechanisms play important roles in different forming operations. Resin percolation and transverse squeeze flow are normally associated with consolidation, welding and compression moulding processes, as they enable gaps to be closed or "healed," thereby ensuring a good bond between adjacent layers [2]. Whereas, shaping processes which induce single curvature and double curvature require inter-ply and intra-ply shear deformations. Cogswell and Leach [3] have suggested that various flow processes are commonly present in combination according to the hierarchy shown in fig. 9.1. This chapter investigates out-of-plane bending of CFRT sheets. Since the fibres resist in-plane deformations along their lengths [4], a single-curvature operation necessitates inter-ply shear. Consequently, it might be misleading to refer to this type of deformation as bending in the classical sense. The usual assumptions, that plane sections remain plane and straight lines remain straight are not valid for reinforced thermoplastics with high loadings of continuous fibres. The entire deformation is accommodated by shear. Nevertheless, it seems appropriate to use the word bending in the current context, since it correctly implies the introduction of curvature into a sheet. In the light of the mode of deformation it also seems logical to use a bend test to investigate the inter-ply shear properties of CFRT laminates. The following text examines the subject of forming CFRT composites into vee-bends at elevated temperatures in order to observe the deformation mechanisms in a relatively simple single-axis bend. The three main objectives are: (a) to observe the quality of 90 ~ Plytron vee-bends formed under isothermal conditions in a novel bending device, (b) to measure the experimental loads required to form the strips, and (c) to compare these results with an idealised beam bending model for an incompressible viscous fluid reinforced with a single family of inextensible fibres.
Bending of continuousfibre-reinforced thermoplastic sheets DEFORMATION MODE
Consolidation
Matched Die
Single Curvature
Ff~
~///'////~
373
REQUIRED FLOW MECHANISM
Resin Percolation.
plus Transverse Flow plus Interply Shear
plus Interply Rotation
o
%
[ O|174174174] |174174174174174 --~ oOO|174174174 --~ *,414 IOOOOO6OOOl ~!ooooooooo] IOOOOO6O'6ol lOOOOOOOOoI
t00oooooooi IOOOOOO•OOl
Double Curvature plus Intraply Shear
Fig. 9.1. Hierarchy of deformation processes.
The novel vee-bending test method outlined in this chapter allows the unidirectional laminates to be bent into shape without compressing them between contacting surfaces and constrains them to a known geometry. The experiments lead to a precise study of the fibre placement within the bend region, the fibre wrinkling instability, the spring-forward phenomenon, and the longitudinal and transverse shear viscosities as functions of the forming speed and the forming temperature. Two main methods of analysis have been used to study the inter-ply shear properties of CFRT laminates" (a) oscillatory shear experiments [5] with a conventional rotating disc rheometer and (b) ply pull-out tests [6], as a variation on the standard sliding rheometer test. Oscillatory shear experiments are useful for determining the visco-elastic properties of CFRT laminates for small amplitudes of deformation. However, the results of this linear approach are not really applicable to real forming operations with very large deformations and high strain rates. On the other hand, ply pull-out tests tend to cause a gross redistribution of the fibres during testing which greatly influences the behaviour of the laminates. Even for a Newtonian fluid matrix, the experimentally determined shear viscosities obtained from this test method can be unreliable [7]. Several researchers have manufactured vee-bends using both cold and hot matched faced dies in order to understand the practical problems with forming such sections [8-10]. In Tam's paper [9], a model for bending a laminate of elastic layers separated by thin viscous fluid layers is developed; however, no attempt is made to establish an analytical load/displacement relationship based on a theoretically derived strip deflection curve. One way to simplify the analysis of CFRT
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materials is to assume an idealised behaviour by imposing suitable kinematic constraints on a continuum model. Some authors have recently published theoretical solutions for bending fibre-reinforced elastic, plastic and linearly visco-elastic cantilever beams subjected to two kinematic constraints: fibre inextensibility and incompressibility [11-13]. Rogers and O'Neill [14] deal specifically with a viscous fluid model using these constraints. An idealised material model for an incompressible viscous fluid reinforced with inextensible fibres is utilised in this chapter. In addition, a plane strain constraint is imposed. A purely kinematic approach is taken to generate the forming solution, which is made statically admissible, by satisfying the boundary conditions. The analytical model yields a straightforward interpretation of the effects of forming speed and geometry on the bending stresses in a real sheet during forming. It also provides an excellent basis for further work with non-linear visco-elastic material models. 9.2. Development of an idealised viscous bending model Thermoplastic polymers are known to exhibit viscous behaviour when formed at temperatures within or above their melting ranges. The matrix material in a CFRT composite sheet may therefore be idealised as an incompressible Newtonian fluid in its molten state. In addition, the fibres can be treated as thin homogeneously distributed inextensible cords, since they severely limit the deformation along their lengths. In the current section we consider plane strain bending of an initially fiat plate with uniaxial fibre reinforcement in its plane. The fibres lie perpendicular to the bend axis when the sheet is subjected to vee-bending, as shown in fig. 9.2. Spencer [15] has derived many of the mathematical expressions presented in the following theory. 9.2.1. Constraint conditions and kinematics
The first two considerations regarding this idealised material concern the kinematic constraints imposed on any deformation by the assumptions of incompressibility and
!
I[
Fig. 9.2. Vee-bending of an incompressible inextensible beam.
I
Bending of continuous fibre-reinforced thermoplastic sheets
375
fibre inextensibility. In the following analysis, capital letters indicate vector quantities referred to the undeformed configuration, whereas small letters indicate vector quantities referred to the deformed state. In a Cartesian reference frame the incompressibility constraint may be expressed mathematically as Ovi Oxi
=dgi-O
(9.1)
where x and v refer to the deformed co-ordinate vector and the velocity vector respectively, and d O9is the rate of deformation tensor defined by 1 "-(OUi nt- OVj~ dij -- -2 ~OXj OXi,]
(9.2)
In a continuum containing a family of fibres, each fibre has its own path, which may be characterised by a field of unit vectors. In the current configuration these may be represented by a unit vector field a(XR, t). The trajectories of a represent the fibres themselves and the components of a are denoted by ai. During a deformation the fibres will be convected with the continuum and the same particles will lie on a given fibre at any time. Using this definition the fibre inextensibility condition may be written as OVg __ aiajdi j a~aJ-~x j
(9.3)
_ 0
The only other constraint imposed on the deformation is that of plane strain. At this point it is necessary to consider a series of n-lines, which represent orthogonal trajectories to the a-lines. These lines are not material curves in general, because particles lying on a normal line before deformation will not necessarily lie on the same normal line after deformation. Pipkin and Rogers [16] have derived the governing equations for plane strain deformations of incompressible, inextensible materials. In the present analysis a plate of uniform thickness is considered, in which the fibres all lie in parallel surfaces in the plane of deformation. Two significant kinematic results follow from the analysis of Pipkin and Rogers: (a) If the a-lines are initially parallel, the n-lines are initially straight and must remain straight throughout any plane strain deformation. Thus, the fibres remain in parallel surfaces. (b) The normal distance between any two adjacent fibres must be constant at all points along that pair. Therefore, the thickness of the sheet cannot change during plane strain deformation. These two requirements permit only simple shear deformations along the fibres and the amount of shear is conveniently expressed by the change in angle between two adjacent fibres, y. Consider fig. 9.3, in which an initially flat plate is deformed by shear. Rigid body rotations and translations are ignored. In the (Xl, x2) plane, the a vector represents the deformed fibres as a family of a-curves and the n-vector represents a family of curves normal to the a-curves. These two families have the vector components a = (cos r sin r 0),
n = ( - sin r cos r O)
(9.4)
T.A. Martin et al.
376
X2
a
Xl
(a)
~
xI
(b)
Fig. 9.3. (a) Undeformed element. (b) Element after deformation.
where 4~ represents the angle between the tangent fibre direction and the X1 axis. When the fibres are embedded at one end, or a line of symmetry exists along which there is no shear deformation, the shear angle, y, may be expressed simply as y = 4~
(9.5)
and the shear rate is given by ~' = q~
(9.6)
9.2.2. Stress in a constrained material The next important step in this analysis involves the introduction of a stress tensor which divides the stress into two distinct parts. The total stress in a constrained material can be thought of as the sum of a reaction stress, rij, and an extra stress, SO.. aij -~- rij %- S O" -- -p(Sij - aiaj) + Taiaj + Sij
(9.7)
where S 0. satisfies the constraints aiajSij - - 0 and ninjS #. = O. In other words, S/j involves no normal stress component on surface elements normal to the a-direction or the n-direction. The reaction stress does no work in a deformation and the reactions p and T arise as a result of the incompressibility and fibre inextensibility constraints respectively. T is the total tension on elements normal to the fibre direction and p represents the total pressure on elements normal to the ndirection. These scalar terms must be determined by solving the equilibrium equations. The deviatoric stress tensor, SO., needs to be specified by an appropriate constitutive relationship. If the material has reflectional symmetry in the x3 plane and the deformation is homogeneous, under plane strain conditions the only non-zero components of a/j are frO" --- --P(SiJ -- aiaj) "q- Taiaj -q- S(ainj + ajni) + S33kikj
(9.8)
Bending of continuous fibre-reinforced thermoplastic sheets
377
where k is the vector normal to the plane of deformation. A constitutive relationship is required to define $33 and S. 9.2.3. Constitutive equation In the present analysis, the deformation of an incompressible Newtonian fluid reinforced with a single family of inextensible fibres in the (Xl, x2) plane is considered. According to Rogers [17], the constitutive relationship for a viscous fluid subjected to these two constraints is given by Sij = 2lzTd O + 2(/zL --/zv) (aiakdkj + ajakdki )
(9.9)
where r/L and r/7- are the respective viscosities of the continuum along and transverse to the fibres. In a plane strain deformation, v3 = d33 = 0. Therefore, $33 = 0 and cr33 = - p . Hence, a stress must be applied normal to the plane of deformation to maintain the plane strain condition. Using eqs. (9.8) and (9.9) it may be shown that [18] S-
ainjcro. - ainjSij - 21zLainjd(i -- l l ~ L f / - lZL(b
(9.10)
where S is the shear stress associated with simple shear along the fibre direction. For a viscous fluid model, S depends only on the shear rate. In this particular example the only material property which can be determined in the constitutive equation is the longitudinal shear viscosity,/zL. By including fibre layers in the laminate which are not aligned with the longitudinal direction, the transverse shear viscosity may also be determined as will be shown later [19]. 9.2.4. Stress equilibrium By considering the equilibrium of an elementary cross section of the deformed plate, shown in fig. 9.4, it is a simple matter to deduce that 8T*
S* = 0
(9.11)
OS* T* + - - ~ -- roPo
(9.12)
when the only boundary traction is the pressure exerted on the inner surface. The force resultants, S* and T *, are the resultant shear and tensile forces per unit length in the k direction acting across a normal line 4~- constant. h
S*-ISd ~ o
h and
T*-ITd o
(9.13)
o
Using eqs. (9.10)-(9.12), the shear stress and the fibre tension can be determined throughout the entire sheet, when the deformed geometry is prescribed as a continuous function of time.
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T.A. M a r t i n et al.
~176176176176176176176176 o~176176 Oo i
ro(~)
:
S* .~
Fig. 9.4. Quasi-polar co-ordinate system.
9.2.5. Kinematic model for a vee-bend
Martin et al. [20] obtained a solution for three-point bending of an ideal viscous beam using the theory just outlined. However, the solution did not compare well with the experimentally observed behaviour of Plytron laminates. A possible explanation for this is that the elastic flexural rigidity of the fibre bundles caused them to deform by flexure, so that the deformed geometry did not match the theory. The selfweight of the samples was also ignored in the model. In the analysis presented here the deformation of the strip is constrained by using a specially designed jig, which allows the true shear response of the material to be measured. The kinematics and dynamics of this solution are detailed as follows. The following kinematic solution is proposed for the deformation of a flat plate subjected to the novel method of vee-bending shown in fig. 9.5. Because the bending is symmetrical about the radius bar, a geometric construction for only one half of the deformed strip is illustrated. The specimen sits on a pair of rectangular platens which are hinged at the centre of the radius bar. This mechanism causes the sheet to wrap around the radius bar and shear along its length. The platens are supported on a pair of pivot wheels and the central hinge pin is connected to a load cell. With progressive deformation, fan regions grow beneath the radius bar and at the free end, while the region between the radius bar and the free end remains straight. The half-strip, shown in fig. 9.5, has been divided into three main sections: [ABEF] represents the fan region beneath the radius bar, [CC'D] represents the developing fan region at the free end and [BC'DE] represents the straight section between the two fan regions. Using the geometry in fig. 9.5 the angular rotation rate of the platens, 4~, can be readily related to the punch velocity, w.
Bending of continuousfibre-reinforced thermoplastic sheets
L '" 0
379
~i
~X~
W h
...........
"1,t
B~ A~
RS FiE Fig. 9.5. Vee-bend deformation model.
cos2~
)
q~- 4v L - (Rs + b)sin 4~
(9.14)
The kinematic model leads to the result that the shear rate in sections [ABEF] and [CC'D] is zero. Consequently, these fan regions move downwards like rigid bodies and S* is zero there. In region [BC'DE] the strip remains straight and the shear strain rate, ), is equal to the rate of rotation of the strip, ~, defined in (9.14). The shear rate varies with time, when the punch speed remains constant. 9.2.6. Admissible stress fields
The foregoing kinematically admissible solution leads to an admissible stress field solution, which admits stress discontinuities across a-lines and n-lines in order to satisfy the boundary conditions. A finite tensile force is carried by the lower surface and a finite compressive force is carried by the upper surface; however, the surface fibre layers have an infinitesimal thickness. Consequently, the upper and lower surfaces of the sheet carry infinite stresses. The tensile and compressive forces increase from zero at points D and C' in fig. 9.5 up to their maximum values at points E and B and remain constant in region [ABEF]. Such a result is characteristic for solutions involving inextensible fibres [16]. In reality, most of the stress is carried near the surface of the sheet during forming. This result is supported by Tam and Gutowski's research [9] on the forming of thermoplastic composite beams. In addition, region
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T.A. Martin et al.
[ABEF] carries a hydrostatic pressure, which leads to transverse fibre spreading, if the sides of the sheet are not constrained during deformation. In the experimental study a radius bar collar was used to enforce the plane strain condition. Figure 9.6 shows the variation in the resultant shear force as a function of the arc length along an a-line in the sheet. Clearly, the resultant shear force is discontinuous across n-lines: BE and C'D. This is possible, provided point loads are applied on the surface at points B and D with force magnitudes equal to the jump in shear stress across each line of discontinuity. The shear force resultant acting on the sheet in region [BC'DE] can be obtained from eqs. (9.13) and (9.14). h
S*-J" tZL i/ d o -
COS2 t~ tZ L h 'iv L - ( R ~ + b ) s i n
) 4~
(9.15)
0
The net downward load on the radius bar can be calculated by considering the equilibrium of the vee-bending mechanism in fig. 9.7. When the sample is bent to an angle 4~, the moment and force equilibrium equations enable the forming load per unit width, P, to be expressed in terms of the longitudinal shear viscosity of the material. p = 21ZLh fv(ls - ( R r "l- h ) f l p ) c o s 4 ~b
(9.16)
(L - (Rr + h + tp + Rs)sin t~) 2
where ls is the half-length of the sheet. This concludes the development of the theoretical model.
9.3. Experimentalprocedures A series of vee-bend experiments were carried out with Plytron [0]8 preconsolidated laminates using the mechanism previously described. Plytron is a polypropylene/glass composite with a nominal 35% fibre volume fraction. The laminates were cut into sheets 140-mm long and 40-mm wide. The tests were carried out under S*
AF
BE
C'D
CD
Fig. 9.6. Resultant shear force along the length of the beam.
~:
Bending of continuousfibre-reinforced thermoplastic sheets
381
P
L .....
,,
,N
I
0
Q+
RS
Q
Fig. 9.7. Force equilibrium of the vee-bending mechanism.
isothermal test conditions over a range of forming speeds and f o r m i n g temperatures, outlined in table 9.1. A schematic d i a g r a m of the experimental set-up is shown in fig. 9.8. D u r i n g testing the forming speed was kept c o n s t a n t and the gross load was measured using a 50-N load cell in an I n s t r o n 1185 testing machine. Later the tare load of the platens and the weight of the specimen were subtracted f r o m the gross load to obtain the experimental forming load. After being f o r m e d into 90 ~ bends, the
TABLE 9.1 An outline of the test parameters Crosshead speed
140~
150~
160~
170~
180~
50 mm/min 100 mm/min 200 mm/min 500 mm/min
v' r v' r
r v' r r
r r r r
r r r r
r r r v'
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T.A. Mart& et al.
50 N Load Cell
Load Cell Amplifier ~
\
t
/z/z////////////
I
71o, Controller
-
' &
c o n v e r s i o n A to
load D board
Fig. 9.8. Schematic representation of the experimental set-up.
specimens were allowed to cool to 60~ before being removed from the test rig. At this point the final angle of each bend was measured and some samples were prepared for microscopic investigation under a scanning electron microscope. 9.4. Results and discussion
In this type of bending operation two factors are of fundamental interest: (a) the quality of the final part and (b) the shear viscosity characterising the behaviour of the material during forming. The part quality will be considered first and the effects of temperature and shear rate on the material response will be discussed later. One of the most notable features about folding uni-directional Plytron sheets into vee-bends is the routine shape taken up by the strips as a result of the deformation. Figure 9.9 illustrates a number of typical sections after complete forming and solidification. It is clear from these samples that the deformation has been accommodated by shear and that the strip width has remained constant. The specimens exhibit a smooth shiny surface in the bend region, where they have been compressed against the radius bar during forming. One problem which can undermine the structural integrity of CFRT components is the migration of the reinforcing fibres towards the inner radius of a bend during forming. Cogswell [1] has discussed how the mobility of the resin affects this fibre migration. A matrix with a low viscosity can lead to resin-rich and resin-starved regions, as demonstrated by Martin et al. [21] when forming Plytron sheets at temperatures above 180~ whereas a matrix with a high viscosity can cause fibre wrinkling. In this study no evidence of fibre migration was observed. A micrograph of a
Bending of continuousfibre-reinforced thermoplastic sheets
Fig. 9.9. Vee-bend sections at various forming temperatures: (i) 140~ speed = 50 mm/min.
(ii) 160~
383
(iii) 180~
forming
typical specimen in fig. 9.10 shows how the fibres are evenly distributed through the sheet thickness. This lack of fibre movement can be attributed to the novel bending mechanism used to fold the specimen. Another material defect which can cause concern in structural applications is the presence of fibre buckles. This phenomenon occurs in regions where the fibres are subjected to compressive loading causing instability. In the detailed theoretical analysis presented by Martin et al. [19], the model predicts a maximum compressive stress at the inside of the bend radius where the sample meets the radius bar. This stress stays at its maximum value in the fan region [ABEF], which is where fibre wrinkling is most likely to occur. On all samples formed at 150~ and above there was no evidence of fibre wrinkling. However, the samples formed at 140~ did exhibit fibre wrinkling beneath the radius bar which continued 2-5 mm into the straight section of the sample (see fig. 9.11). Furthermore, the wrinkle amplitudes were greater in the samples formed at 50 mm/min as opposed to 500 mm/min. This may be explained by considering the post-buckling behaviour of the fibres. Once the fibres reached their critical buckling load they wrinkled. The slower crosshead speeds allowed more time for these defects to grow as the sheet continued to deform. The preceding results indicate that fibre wrinkling is more likely to occur in a matrix resin with a higher viscosity. This finding is in agreement with Cogswell's comments, but contradicts research by Martin et al. [20] on deep drawing hemispherical shells. Martin's results demonstrated that at forming temperatures greater than 180~ the fibres in Plytron sheets wrinkled in the plane of the sheet during deformation, whereas below 180~ the sheets buckled out-of-plane. The key difference in this
384
T.A. M a r t i n et al.
Fig. 9.10. Micrograph of the bend radius region in a typical vee-bend. Temperature = 180~ speed = 50 mm/min.
forming
Fig. 9.11. Vee-bend section showing fibre wrinkling.
study is that the sheet itself was not subjected to an in-plane compressive load, so gross buckling did not occur. Consider the micrograph shown in fig. 9.12. At forming temperatures greater than or equal to 150~ the deformed samples exhibit interply slip. The degree of shear deformation is indicated by the angle of slip between the
Bending of continuous fibre-reinforced thermoplastic sheets
385
Fig. 9.12. Micrograph of free end showing inter-ply and intra-ply deformation. Temperature = 150~ forming speed -- 500 mm/min.
fibre layers at the free end. This behaviour is consistent with the continuum theory presented earlier in this chapter. In contrast fig. 9.13 shows a sample formed at 140~ which exhibits predominant slip in the resin-rich layers between the lamina, characterised by distinct steps at the free end. Deformation of this type leads to severe fibre wrinkling and is probably associated with the lack of shear within each prepreg layer. The low forming temperature and forming speed allowed the partial recrystallisation of the material before it was completely moulded; thereby inhibiting a uniform shear deformation across the thickness of the sheet. The fibre wrinkling occurs in the surface layer underneath the radius bar, as predicted by the theory. These findings indicate that fibre wrinkling occurs when the shear deformation is restricted in the laminate. Another interesting feature of the micrographs in fig. 9.12 and fig. 9.13 is the lack of curvature at the free end of the sheet. There is no fan region there as predicted by the theoretical model. This is understandable, since the elastic flexural stiffness of the fibre bundles is several orders of magnitude greater than the shear viscosity of the matrix. In reality the flexural rigidity of the fibres counters the formation of a fan region at the free end, so that the fibres completely re-straighten during the relaxation phase of the forming process. This type of response could be accounted for by including an elastic bending stiffness term in the constitutive equation, but it would greatly complicate the kinematic solution of the bending problem, while hardly affecting the outcome. It is presumed that the violation of the theoretical model in the small region near the end of the sheet does not represent a gross error.
386
T.A. M a r t & et al.
Fig. 9.13. Micrographof a free end showingmostlyinter-ply shear. Temperature = 140~ forming speed - 500 mm/min. A final point to consider regarding the quality of the formed parts is the included angle of the bends after solidification. A well-documented phenomenon occurring during the processing of CFRT sheets is the spring-forward effect. Upon cooling from its melt temperature a moulded thermoplastic composite laminate decreases its included angle in the final part. This effect is primarily due to the anisotropic thermal contraction of the material. Zahlan and O'Neill [22] have investigated this phenomenon and derived a simple theoretical expression for the change in angle of an L-section with differential in-plane to thickness contraction. A0 -- (c~R -- or0)0
(9.17)
where an and c~0 are the respective radial and circumferential coefficients of thermal expansion for a linear thermoelastic material. 0 is the mould angle and A0 is the difference between the forming temperature and the ambient temperature. Their experimental results agree favourably with eq. (9.17). Hou et al. [10] have demonstrated the effect of the forming speed on the magnitude of the spring-forward in a series of matched die stamp forming tests. Their results indicate a marked increase in the degree of spring-forward as the forming rate is increased. However, this anomaly has been attributed to fibre migration and squeeze flow as a result of contact between the dies. Table 9.2 shows the effect of the forming speed and the forming temperature on the degree of spring-forward in Plytron vee-bends in the current study. There appears to be little correlation between the forming parameters and the springforward. Considering the temperature range over which the samples were deformed (140~ T~< 180~ and the linear relationship between A0 and AT in (9.17), these
Bending of continuous fibre-reinforced thermoplastic sheets TABLE
387
9.2
Experimental spring-forward for 90 ~ Plytron vee-bends Crosshead speed
140~
50 m m / m i n
7~
5.5 ~
6.5 ~
6.5 ~
200 mm/min
7.5 ~
6.5 ~
500 m m / m i n
6.5 ~
Average
6.8 ~
100
mm/min
150~
160~
170~
180~
5.5 ~
5~
5.5 ~
5.5 ~
5.5 ~
5.5 ~
5~
5~
5.5 ~
6~
5.5 ~
5.5 ~
5.5 ~
6.1 ~
5.4 ~
5.3 ~
5.5 ~
results appear to contradict Zahlan's theory, but it should be remembered that PP does not behave like a linear thermoelastic material when cooling from its melt state. Polypropylene remains molten until cooling to 125~ when rapid recrystallisation occurs. At this point the polymer dramatically decreases its volume causing the spring-forward effect. Such a result would be expected for any crystalline or semicrystalline thermoplastic composite and is supported by the observations made while the experimental samples were being cooled to room temperature. There was no observable spring-forward effect until the sample temperature reached 120-130~ whereupon the bends rapidly changed their shape by a few degrees. Samples which were cooled rapidly from their forming temperature also developed a considerable amount of curvature across the width of the strip, due to thermal gradients through the sheet thickness. The experiments suggest that moderate controlled cooling leads to the best quality parts. Figure 9.14 shows the first set of experimental results for the loads required to form Plytron sheets at various temperatures and a constant crosshead speed. In this --
~
140~
--0--
150~
5 -
---0-- 160~
4
- - O - - 180~
w
Z .~
,,,
170~
O
~3
1
0...
I
I
.-, I
t
0
5
10
15
20
......
I
I
25
30
Time (sec) F i g . 9.14.
Graph of forming load versus time for Plytron vee-bends. Forming speed
=
500
mm/min.
388
T.A. M a r t i n et al.
case the load is plotted against time so that the relaxation effects in the material may be seen. It is not surprising that the highest peak load of ~6 N occurs at the lowest forming temperature, while the lowest peak load of ~ I N occurs at the highest forming temperature. In all cases the forming loads are small. This demonstrates the ease with which thermoplastic composites may be moulded when molten. The load profile can be described by three distinct stages: (b) a rapid rise in the forming load as the sheet starts to deform, (c) a quasi-steady-state forming load which decreases as forming proceeds, (d) a rapid decay in the forming load at the cessation of forming. The first stage corresponds to the transient load response at the initiation of flow. This lasts for a very short time before the load reaches its maximum value and thereafter slowly declines. The gradual decrease in load after reaching a peak value corresponds to a decrease in the shear rate with the increasing punch depth. A rapid drop in the load follows at the completion of forming, as would be expected for a viscous fluid. This is followed by a gradual relaxation in the load which demonstrates some visco-elasticity in the material. The load tends towards zero at a very slow rate. A note of caution is given here regarding the interpretation of the results obtained from this type of test. Given the visco-elastic nature of the response, it is unwise to draw conclusions regarding the elongational behaviour from the shear behaviour. Lodge [23] has demonstrated how the mode of deformation substantially affects the theoretical viscosity for the simplest rubberlike liquid constitutive equation. In steady shear flow the rubberlike liquid exhibits a viscosity which is independent of the shear rate. In steady elongational flow it has an indefinitely increasing viscosity. Therefore, shear type experiments are not sufficient on their own to universally validate a constitutive model. The results make it tempting to apply linear time-dependent material models to characterise the observed visco-elastic behaviour, but the large strains and strain rates encountered during forming make this a pointless curve-fitting exercise. Instead, a generalised non-linear visco-elastic model is needed to account for the finite strains and high strain rates, if mathematical sense is to be made of the rheological behaviour exhibited here. The instantaneous drop in the load at the cessation of forming shown in fig. 9.14 is in contrast to the results obtained from a series of three-point bend tests performed by Martin et al. [19]. Under similar test conditions, the Plytron sheets showed a much smoother load decay at the completion of forming, due to the elastic recovery of the curved fibres between the load supports. It is interesting to note the effect of forming speed and temperature on the load profile in fig. 9.15. When the platen angle reaches 45 ~ the forming stops and thereafter the load curves are plotted against time. At a forming temperature of 140~ there is a marked change in the shape of the load curve for the two different forming speeds. This effect is not evident in the specimens formed at 180~ This result clearly shows the effect of recrystallisation on the material behaviour as the temperature is lowered. At 140~ and 50 mm/min as the material recrystallises, the matrix behaves more like a visco-elastic solid than a visco-elastic liquid. Under these conditions the
Bending of continuous fibre-reinforced thermoplastic sheets
389
-9- 0 - - ! 80~ 50 ham/min ---0-- 180"(2 500 mm/min 140~ 50 mm/min X
140~ 500 mm/min
~4
.~3
o~ 0o
I
[
15~
30~
[t,.o~
"'
t, = 4 0 sl
45 ~
Fig. 9.15. Graph of load versus platen angle for Plytron vee-bends.
constitutive response of the material becomes more dependent on the shear magnitude than the shear rate. The relaxation response is also altered by the change. Using the aforementioned experimental load profiles and eq. (9.16), the apparent longitudinal shear viscosity, /zL, was calculated as a function of temperature and forming speed. During forming the sheets did not experience a steady shear flow, since the shear rate was not constant; therefore, the apparent shear viscosity is illustrated with the shear rate in fig. 9.16 as a function of the platen angle. The actual variation in strain rate over the duration of the forming period is quite 9000
0.2200
8000 o rr~ t~
000000000O 0
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7000
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6000
AA
0.2000
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00 AAAAA/~&AA&
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0
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. ~ " 5000
0.1800 o
0O 0
AAA A
0.1600 "~
AAAAAAAAAAA
4000
0.1400 o
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~o~176176176176176176176176176176176176176 0.1200 "N
..~
2000
o o oO o o oo Oo oo
0.1000
1000 ~ o o ~ 0-'~-
I
I
I,
I
1
I
0
5
10
15
20
25
30
"
0.0800
I
35
40
Platen Angle ~ (degrees) []
50mm/min
•
100mm/min
0
200mm/min
O
500mm/min
Strain r a t e
Fig. 9.16. Longitudinal shear viscosity and shear rate versus platen angle,f.
390
T . A . M a r t i n et al.
small at around 0.02 rad/s. The profile of/ZL is seen to rise and fall in a similar manner to the shear rate. It is also evident from this figure that the samples exhibit a shear thinning phenomenon: a decrease in the apparent viscosity with increasing shear rate. Clearly the material response is non-Newtonian. In the particular case of simple shear flow of an incompressible viscous fluid, the only non-zero strain rate invariant can be expressed as a function of the shear rate [24]. Given the nature of the results, the shear stress may be expressed as a function of the shear rate using the "power-law" relationship described by r = m~ n
(9.18)
where r is the shear stress and m is the consistency index. When n < 1 the material is said to be shear thinning and when n > 1 the material is said to be shear thickening. A Newtonian response is obtained by setting n - 1, so that r is a linear function of the shear rate. The apparent (Newtonian) viscosity of the material,/ZL is defined by "~ I~ L - - -
= my
(9.19)
n-1
This model has been commonly applied to molten polymer solutions. Using the forming load curves in the range 15 ~ < 4~ < 30 ~ an average /ZL was calculated for each average shear rate. The averaged data is shown on a log-log plot in fig. 9.17 for sheets formed at 180~ Figure 9.17 represents a fairly good fit between the power-law model and the experimental data. A linear regression analysis was performed to determine the m and n coefficients for the Plytron samples at various test temperatures. The numerical results from the linear regression are given in table 9.3 along with the residual squared error of the best fit. The n values increase with increasing temperature in accordance with the general trend for polymer melts. A typical n value for PP in the temperature range 200-230~ is 0.3-0.4. This compares favourably with the result for the sample formed at 180~ On the other hand, the consistency index shows a wide variation with increasing temperature. This finding
-~ 10000
~o o,-, t~ O O
\ N~
xx,
>
<
1000 0.01
0.1
Log Shear Strain Rate dr
1
(tad/s)
Fig. 9.17. Log-log plot of/z L versus shear rate for Plytron samples.
391
Bending of continuous fibre-reinforced thermoplastic sheets
TABLE 9.3 Power-law parameters for the longitudinal shear viscosity of Plytron laminates Temperature (~
m (Pa sn)
n (dimensionless)
R2
140 150 160 170 180
165 510 395 285 445
-0.335 0.172 0.198 0.144 0.353
0.9993 0.9697 0.995 0.9897 0.9904
is contrary to the rapid decrease which would be expected for a polymer. Further experimental work is needed to verify this result. A further comparison is made between the power-law model and the experimental results in fig. 9.18. By determining the instantaneous apparent shear viscosity at each platen angle, based on the current shear rate and the power-law coefficients from table 9.3, the theoretical load was calculated using eq. (9.16). The theoretical load curve compares well with the measured load profile. This result indicates that a power-law relationship can be used with confidence to predict the shear stresses in C F R T sheets at various strain rates. Using table 9.3 and eq. (9.16), the apparent longitudinal shear viscosity of the unidirectional Plytron sheets was calculated in the temperature range 140~ T ~< 180~ The results showed a variation in /ZL from 5,000-55,000 P a s depending on the forming rate. The shear viscosity increased with decreasing forming temperature, as expected. At 140~ and 50 mm/min the shear viscosity reached its highest value, which was far greater than most of the calculated values at higher forming temperatures and faster forming speeds. The increased viscosity associated with
Theoretical load Experimental
z
-~ 2 t~ et0
O
z
Ol 0
i ................ 5
I 10
I 15
i 20
Time (sec) Fig. 9.18. Theoretical and experimental forming loads versus time. Temperature = 170~ forming speed = 200 mm/min.
392
T.A. M a r t & et al.
recrystallisation in crystalline and semicrystalline CFRTs should be considered in general sheet forming processes. 9.5. Modified constant shear rate tests
This part of the chapter introduces a slight modification to the vee-bending mechanism detailed earlier as well as establishing the method as a means of determining the transverse shear behaviour of CFRTs. The vee-bending mechanism described earlier demonstrated itself to be very useful in terms of constraining the deformation of the strip and allowing the longitudinal shear response of the material to be isolated and studied. Unfortunately, one of the drawbacks of the set-up was that the rate of angular rotation of the platens and consequently the shear rate of the sample, y, were not constant. While the actual variation in strain rate in the sample was very small compared with the strain magnitude, it would seem preferable to test at constant rates of shear. An effective way of doing this is to replace the circular pivot wheels, upon which the platens sit, with a support profile which manipulates the rate of angular rotation of the platens to be constant and proportional to the speed of the punch. This modification, illustrated in fig. 9.19, has no effect other than X2
L
,, 0
.
.
.
.
.
X~
j
~1
i ]
W
p B- N
A-N
Fig. 9.19. Kinematic model of the modified vee-bending mechanism.
Bending of continuous fibre-reinforced thermoplastic sheets
393
to alter the relationship between the speed of the punch and the rate of rotation of the platens which after some manipulation reduces simply to ~i,_ s -- L 4' cos,/,
(9.20)
where ep is the orthogonal distance between the centre of the radius bar and the point of contact between the platen and the support, ep and cos 4~vary with time but the ratio of the two remains constant and equal to L for all 4~.
9.5.1. Determination of transverse shear behaviour from vee-bending The transverse shear viscosity of CFRTs is often very difficult to determine using methods such as oscillatory shear and ply-pull-out tests. An alternative technique for gaining some insight into the transverse shear response of such materials is to subject a laminated strip, which possesses both longitudinal and transverse layers, to the same type of single-axis bending operation as described earlier. The introduction of transverse layers, in which the fibres are aligned with the x3 axis, poses somewhat of a dilemma as they are not subject to the same kinematic restrictions which govern the plane strain deformation of the longitudinal layers. In fact, the only kinematic effect these layers have on the strip is to enforce the plane strain condition already assumed. It therefore becomes necessary to make a further assumption regarding the deformation of the transverse layers so that useful solutions may be obtained. It is assumed that the thickness of the transverse layers in a laminated strip remains constant during any plane strain deformation. In other words, they exhibit the same kinematic behaviour as the longitudinal layers. By way of an example, consider the shear deformation of an initially flat laminated plate consisting of three layers, as shown in fig. 9.20. The outer two (longitudinal) layers possess fibres which lie in the plane of deformation while the reinforcement in the central (transverse) layer is aligned in the direction of the x3 axis. In this example, the unit vector b represents the shearing direction such that a-b = 1 in the longitudinal
x~
X2
b
iiiiiliiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii] iiiiii!!i!!ii!ii v
Xl
(a) Fig. 9.20. (a) Undeformed plate. (b) Deformed plate.
v
(b)
Xl
394
T.A. Mart& et al.
layers, and a-b = 0 in the transverse layer. Again a represents the fibre direction in the material and the unit vector n is defined to be orthogonal to both the shearing direction and the x3 axis. The shear strain and strain rate of the entire plate can then be specified as before using eqs. (9.5) and (9.6). By once again applying the same constitutive relationship (eq. (9.10)) as that adopted earlier leads to the result: S L -- binjo'ij - binjSij = 21ZLbinjd(i - t X L Y - - tXL~
(9.21)
S T -- binjo~i = binjSij = 21ZTbinjdij - - ~ T Y - - ~ T ~
(9.22)
where S L and S T a r e the shear stresses associated with the shear deformation in the longitudinal and transverse layers respectively. According to eqs. (9.21) and (9.22) the shear stress through a laminated strip, possessing both longitudinal and transverse layers, varies discontinuously at the boundary surfaces and interfacial surfaces between the differently orientated layers. Initially it would appear as though equilibrium between such layers is therefore unachievable. However, there exists the possibility that a sheet of fibres can support a finite force and hence an infinite stress across the layers. In this analysis it is necessary to accommodate discontinuities in S at the outer surfaces of the strip and also at the interfacial surfaces. We may achieve this by using step and delta functions that allow T to take infinite values in the fibres in the material adjacent to the boundary surfaces, and in the fibres adjacent to, and on either side of, interfacial surfaces. The effect of this is to introduce simple shear stress jump discontinuities. For further clarification of this stress solution the readers are referred to Rogers and Pipkin [25] where the discontinuous stress condition is used to satisfy the shear traction boundary conditions in plane strain bending problems. Spencer [26] has also shown how the same property can be used to admit shear stress discontinuities for the more general case in which the individual layers of an elastic laminated beam can assume any orientation oblique to the plane of deformation. By considering the equilibrium of the entire modified bending mechanism shown in fig. 9.21, it becomes a straightforward task to determine the net downward bending load on the punch. When the sample is bent to an angle 4~, the moment and force equilibrium equations enable the forming load per unit width, P, to be expressed in terms of both the longitudinal shear and transverse shear viscosities of the material. p = 2r
- qb(Rr + h))(tZLhL + tZThT)
L2
(9.23)
where h L and hr are the combined thicknesses of the longitudinal and transverse layers respectively. For the case in which the strip possesses no transverse layers (i.e. hr = 0), it becomes a simple matter of rearranging eq. (9.23) to yield an expression for the longitudinal viscosity. Once/xL has been established for a particular forming condition, the subsequent introduction of transverse layers into the laminated beam may then be used to determine/Zr.
Bending of continuous fibre-reinforced thermoplastic sheets
395
s*, . . . .
Q+S*
Q
Fig. 9.21. Equilibrium of the modified vee-bending mechanism. Note the introduction of transverse layers into the laminate.
9.5.2. Transverse shear viscosity tests
A series of experiments were performed using the modified vee-bending mechanism in an attempt to establish its usefulness as a means of determining both the longitudinal and transverse shear viscosities of CFRTs. These tests were performed using exactly the same procedures as those outlined for the earlier case. However for these tests, two different preconsolidated laminates were constructed from Plytron; [0~ and [0~176176176 s. The temperature and shear rate dependency observed for the longitudinal viscosity in the earlier experiments were also noticeable in the results of the tests performed on the modified apparatus. One apparent discrepancy though, that would seem to be of concern were the magnitudes of the viscosity results which are shown in fig. 9.22. The longitudinal viscosity results obtained from the modified testing jig were found to be significantly greater than those observed earlier on. This discrepancy can, however, be somewhat accounted for by considering the effect of the modifications on the shear rate of the strip for a given forming speed. It is to be noted that, although these set of tests were performed at identical punch speeds to the previous experiments, the actual forming rate, under the modified conditions, was slightly higher than the average rate encountered using the earlier set-up. In other words, for the same punch speed, the entire forming operation was completed in a shorter period of time. Bearing in mind the shear thinning behaviour of the material, and the slightly
396 b ,--,
T.A. M a r t & et al.
40 35
,.d
30 0~
25
+
50mm/min
~
;~
lOOmm/min
20
200mm/min "~
15
0
~
~
10
~
5
I
.
0
f
t
I
I
0.2
0.4
0.6
0.8
500mm/min
Platen Angle, ~ (rads) Fig. 9.22. Apparent longitudinal viscosity of Plytron versus platen angle. Forming temperature = 180~
different shear rates in this set of experiments, it would be inappropriate to make direct comparisons between these results and those obtained earlier. Using the method outlined in the previous section, a further series of tests were performed under identical forming conditions on bi-directional laminates which possessed both longitudinal and transverse layers. The results of these tests were used in conjunction with the longitudinal viscosity results, shown in fig. 9.22, to determine the transverse shear response of the material. The transverse shear viscosity results are shown in fig. 9.23. It is interesting to note that the same shear thinning effect observed for the longitudinal response is also apparent in the 3O t~
25
::3.
.~ 20
lOOmm/min --/l--- 200mm/min
0
0
500mm/min
~> 10
<
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Platen Angle, ~ (fads) Fig. 9.23. Apparent transverse viscosity of Plytron versus platen angle. Forming temperature = 180~
Bending of continuousfibre-reinforced thermoplastic sheets
397
transverse viscosity. Another interesting feature of the curves presented in fig. 9.23 is how flat each of the viscosity curves remain throughout the deformation. The results obtained from these experiments allow a direct comparison to be made between the transverse and longitudinal shear viscosities. This provides an important opportunity to verify a number of theoretical models which have been proposed in an attempt to relate #T and IzL to the fibre volume fraction f and the matrix viscosity #M. A summary of the models that have been proposed by Pipes [26], Christensen [27], and Binding [28] is given in table 9.4. These models are based around geometric arguments and assume somewhat of an idealised behaviour. The models of Pipes and Christensen predict that IZT > IZL for all fibre volume fractions, while the model of Binding predicts only IzL as a function o f f and #M. Furthermore, the longitudinal viscosity, as predicted by Binding, is larger than either of the transverse viscosity terms predicted by Pipes and Christensen. It should be noted that the viscosity results obtained for Plytron indicate that tXT < /XL for all the temperatures and forming speeds investigated, which is clearly not in agreement with the models of Pipes and Christensen. However, it is interesting to note that, by combining the model of Binding with those of Pipes or Christensen to eliminate/XM, expressions for the viscosity ratio #T/IZL may be readily obtained which are in line with the experimentally observed results: Pipes/Binding /s /zL
1 -V~ 1 --f
Christensen / B in din g (1 - 0.193f)3(1 _ f ) 2 /XL
(1 - 0.5952f)3/2(1 -f)3/2(1 - f )
Assuming that Plytron has a hexagonally packed arrangement of fibres and a nominal fibre volume fraction of 35 %, the above expressions would suggest that the ratio TABLE 9.4 Theoretical models relating ]s and ].LL to the fibre volume fractionf and ]s l.t T / l~tM
Pipes [27]
I~tL / l~tM
1
2(1 - v/f) Christensen [28] Binding [29]
(1 - 0.193f) 3 (1 - 0.5952f) 3/2 (1 _f) 3/2
1 + 0.873f (1 - 0.8815f) 1/2 (1 -f),/2 1-f (1 V~)2 -
T.A. Martin et al.
398
lZT/i~L is approximately 0.58 using the combined Pipes/Binding theory, and 0.54 using the alternative Christensen/Binding theory. These values seem to slightly underestimate the experimentally obtained results shown in fig. 9.24. The experimental results would tend to indicate that at 180~ the ratio of the two viscosity terms remains within a narrow band for the various deformation rates studied in this investigation. This would appear to be in agreement with the idealised theory presented above, in which no rate-dependent terms arise. The results would also tend to suggest that the shear thinning, or rate-dependent, behaviour observed in both the longitudinal and transverse directions, is largely attributable to the non-linear behaviour of the molten matrix material. It should also be noted that the idealised models presented in table 9.4 fail to adequately take account of the resin-rich layers that form between the individual plies. It is the authors' belief that the presence of these thin layers, coupled with a slight amount of fibre misalignment, accounts for the discrepancy between the theoretical and actual results. Earlier on in the chapter, it was shown how a power-law expression could be used to give a reasonably good description of the shear thinning behaviour observed for the longitudinal shear viscosity. Given a similar type of trend for the transverse shear response, it would not seem unreasonable to assume the same sort of general relationship for the transverse shear viscosity. We therefore assume that both the longitudinal and transverse shear viscosities can be approximated by power-law expressions which take the same form as eq. (9.19). Using this these types of relationships enables the ratio of the two to be written as [A T
m T f/n T
= ~
#L
(9.24)
m L ~ 'nc
where the constants m and n are the same as those defined earlier. If eq. (9.24) is set to a constant, as the results shown in fig. 9.24 would suggest, then it can be readily 2
A
1.8
A
---0--- 50mm/min
7
----/t-- 200mm/min
1.6
~
1.4
lOOmm/min
1.2
0.8 0.6
- ....
r
.\
_.
o, t / x '~
ol 0
.... .
ding
o.2
. 0.1
.
.
. 0.2
.
.
.
. 0.3
.
.
. 0.4
.
.
. 0.5
,
,
,
0.6
0.7
0.8
Platen a n g l e , ~ (rads) Fig. 9.24. T r a n s v e r s e to l o n g i t u d i n a l viscosity ratios for v a r i o u s f o r m i n g rates. F o r m i n g t e m p e r a t u r e = 180~
Bending of continuous fibre-reinforced thermoplastic sheets
399
shown that n L = n T = n. From this result it becomes obvious that the shear thinning, or rate-dependency, of both viscosity terms is almost certainly attributable to the non-linear matrix material behaviour. Upon reflection, this might seem unsurprising as the shear flow of such materials is largely dominated by the thin resin-rich layers which form between the individual plies.
9.6. Conclusions The novel vee-bending mechanism discussed in this chapter allows the longitudinal and the transverse shear viscosity of CFRT sheets to be isolated and measured. The simple shear deformations between layers of fibres in CFRT laminates are characterised particularly well by the bending model. By reducing the forming temperature from 180~ while remaining above the recrystallisation temperature, the degree of elasticity in the Plytron laminates is increased. However, this leads to additional fibre loads and increases the potential for fibre wrinkling. Transverse fibre spreading can be avoided, when bending unidirectional laminates along an axis normal to the plane of the fibres, by providing side constraints. The loads needed to shape Plytron laminates are small and these are of minor importance when designing tools for manufacturing CFRT products. A model for predicting the behaviour of an idealised viscous beam has been developed, which provides an analytical expression for the forming load as a function of the forming speed, the die geometry and the sheet thickness. These factors all affect the stresses in CFRT materials as they are deformed. The model also establishes a useful basis for further theoretical work on kinematically constrained models, which take account of the highly non-linear visco-elastic nature of the matrix during forming. Molten uni-directional Plytron laminates exhibit a visco-elastic liquid behaviour when they are deformed. Future theoretical models should reflect this behaviour. The theoretical model provides a means to determine the apparent longitudinal shear and the transverse shear viscosity of the material as a function of the shear rate and the forming temperature. The apparent shear viscosity increases with decreasing temperature and rises dramatically when recrystallisation of the matrix commences. As the physical structure of the polymer changes, its rheological behaviour changes from that of a visco-elastic liquid to a visco-elastic solid. The rate dependence, or shear thinning behaviour, of Plytron laminates can be suitably modelled by a powerlaw relationship. Finally, it would appear that the shear thinning effect observed for both the longitudinal and transverse shear viscosities depends solely on the rheological behaviour of the matrix, as the ratio l z T / l Z L remains constant.
Acknowledgements The authors wish to thank the Foundation for Research, Science and Technology (New Zealand) for providing the funds to carry out this research. They are also thankful to Borealis (Norway) and Mitsui-Toatsu (Japan) for their support.
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References [1] Cogswell, F.N., "The Processing Science of Thermoplastic Structural Composites", Int. Polymer Processing, 4, pp. 157-165, 1987. [2] Wang, E.L., Gutowski, T.G., "Laps and Gaps in Thermoplastic Composites Processing", Composites Manufacturing, 2, pp. 69-78, 1991. [3] Cogswell, F.N., Leach, D.C., "Processing Science of Continuous Fibre Reinforced Thermoplastic Composites", SAMPE Journal, May, pp. 11-14, 1988. [4] Martin, T.A., Bhattacharyya, D., Pipes, R.B. "Deformation Characteristics and Formability of Fibre-Reinforced Thermoplastic Sheets", Composites Manufacturing, 3/3, pp. 165-172, 1992. [5] Groves, D.J., Bellamy, A.M., Stocks, D.M., "Anisotropic Rheology of Continuous Fibre Thermoplastic Composites", Composites, No. 2, pp. 75-80, 1992. [6] Scherer, R., Friedrich, K., "Inter- and Intra-Ply Slip Flow Processess during Thermoforming of CF/ PP-Laminates", Composites Manufacturing, 2/2, pp. 92-96, 1991. [7] Goshawk, J.A., Jones, R.S., "Structure Reorganization during the Rheological Characterization of Continuous Fibre-Reinforced Composites in Plane Shear", Composites, 27A, pp. 279-286, 1996. [8] Soil, W., Gutowski, T.G., "Forming Thermoplastic Composite Parts", SAMPE Journal, 24/3, pp. 15-19, May 1988. [9] Tam, A.S., Gutowski, T.G., "Ply-Slip during the Forming of Thermoplastic Composite Parts", Journal of Composite Materials, 23, pp. 587-605, June 1989. [10] Hou, M., Friedrich, K., "Stamp Forming of Continuous Carbon Fibre/Polypropylene Composites", Composites Manufacturing, 2/1, pp. 3-9, 1991. [11] Rogers, T.G., Bradford, I.D.R., England, A.H., "Finite Plane Deformations of Anisotropic ElasticPlastic Plates and Shells", Journal of Mech. Phys. Solids, 40/7, pp. 1595-1606, 1992. [12] Bradford, I.D.R., England, A.H., Rogers, T.G., "Finite Deformations of a Fibre-Reinforced Cantilever: Point-Force Solutions", Acta Mechanica, 91, pp. 77-95, 1992. [13] Evans, J.T., "A Simple Continuum Model of Creep in a Fibre Composite Beam", Journal of Applied Mechanics, Trans. ASME, 60, pp. 190-195, 1993. [14] Rogers, T.G., O'Neill, J.M., "Theoretical Analysis of Forming Flows of Fibre-Reinforced Composites", Composites Manufacturing, 2, 3/4, pp. 153-160, 1991. [15] Spencer, A.J.M., "Deformations of Fibre-Reinforced Materials", Oxford University Press, London, 1972. [16] Pipkin, A.C., Rogers, T.G., "Plane Deformations of Incompressible Fibre Reinforced Materials", Journal of Applied Mechanics, Transactions of the ASME, Sept., pp. 634--640, 1971. [17] Rogers, T.G., "Rheological Characterization of Anisotropic Materials", Composites, 20/1, pp. 21-27, 1989. [18] Dykes, R.J., Martin, T.A., Bhattacharyya, D., "Determination of Longitudinal and Transverse Shear Behaviour of Continuous Fibre-Reinforced Composites from Vee-Bending", Proc. 4th International Conference on Flow Processes in Composite Materials, The University of Wales, Aberystwyth, Wales, 1996. [19] Martin, T.A., Bhattacharyya, D., Collins, I.F., "Bending of Fibre-Reinforced Thermoplastic Sheets", Composites Manufacturing, 6, pp. 177-187, 1995. [20] Martin, T.A., "Forming Fibre Reinforced Thermoplastic Composite Sheets", Ph.D. Thesis, Dept. Mech. Engineering, The University of Auckland, New Zealand, 184 pp., July 1993. [21] Martin, T.A., Bhattacharyya, D., Pipes, R.B. "Computer-aided Grid Strain Analysis in Fibrereinforced Thermoplastic Sheet Forming", In: Computer Aided Design in Composite Material Technology III, ed. S.G. Advani et al., pp. 143-163, 1992. [22] Zahlan, N., O'Neill, J.M. "Design and Fabrication of Composite Components; the Spring Forward Phenomenon", Composites, 20/1, pp. 77-81, 1989. [23] Lodge, A.S., "Elastic Liquids", Academic Press, New York, pp. 101-122, 1964. [24] Bird, R.B., Armstrong, R.C., Hassager, O., "Dynamics of Polymeric Liquids", Vol. 1: "Fluid Mechanics", John Wiley and Sons, New York, 1987.
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[25] Rogers, T.G., Pipkin, A.C., "Small Deflections of Fibre-Reinforced Beams or Slabs", Journal of Applied Mechanics, Trans. ASME, pp. 1047-1048 Dec., 1971. [26] Spencer, A.J.M., "Plane Strain Bending of Laminated Fibre-Reinforced Plates", Quarterly Journal of Mechanics and Applied Mathematics, 25, Part 3, pp. 387-400, 1972. [27] Pipes, R.B., "Anisotropic Viscosities of an Orientated Fibre Composite with Power-Law Matrix", Journal of Composite Materials, 26, pp. 1536-1552, 1992. [28] Christensen, R.M., "Effective Viscous Flow Properties for Fibre Suspensions under Concentrated Conditions", Journal of Rheology, 37, pp. 103-121, 1993. [29] Binding, D.M., "Capillary and Contraction Flow of Long (Glass) Fibre Filled Polypropylene", Composites Manufacturing, 2, pp. 243-252, 1991.
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Composite Sheet Forming edited by D. Bhattacharyya 9 Elsevier Science B.V. All rights reserved.
Chapter 10
Thermoforming Processesfor Knitted-Fabric-Reinforced Thermoplastics." New Manufacturing Techniquesfor LoadBearing, Anisotropic Implants J. MAYER and E. W l N T E R M A N T E L Biocompatible Materials Science and Engineering, Department of Materials, Swiss Federal Institute of Technology, ETH Zurich, Wagistrasse 23, 8952 Schlieren, Switzerland
Contents Abstract 404 10.1. General aspects of anisotropic biomaterials for load-bearing implants 404 10.2. Knitted-carbon-fiber-reinforced composite materials 405 10.2.1. Introduction 405 10.2.2. Fiber architecture and fiber orientation distribution 407 10.2.3. Experimental details 409 10.2.4. Mechanical properties 412 10.2.4.1. Young's modulus and static strength 412 10.2.4.2. SEM study of tensile failure behavior 413 10.2.4.3. Discussion of structure-properties relations 415 10.3. Net-shape forming of knitted fabrics for load-transmitting implants shown for an ulnar osteosynthesis plate 419 10.3.1. Introduction 419 10.3.2. Experimental details 420 10.3.3. Structure and properties of the net-shape manufactured osteosynthesis plate 422 10.3.4. Comparison of homoelasticity in FEM calculations and strain gauge measurements 10.4. Deep drawing of knitted-fiber-reinforced organo-sheets 428 10.4.1. Introduction 428 10.4.2. Experimental details 428 10.4.3. Flow behavior 429 10.4.4. Correlation between plastic flow and fiber orientation distribution 431 10.5. Discussion 432 10.5.1. Structure-properties relationship 432 10.5.2. Thermoforming 433 10.5.3. Biocompatibility aspects and applications 434 10.6. Summary and conclusions 435 Acknowledgements 435 References 436
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Abstract In this chapter the application of specific thermoforming techniques for knitted carbon-fiber-reinforced thermoplastic composite materials to medical implants is described. In the first part of this chapter, the properties of the fiber architecture in knitted-fabric-reinforced composites and their influence on the mechanical properties are outlined in order to provide a basic understanding of the potential of knitted fabrics as reinforcement as well as their medical and engineering applications. Therefore, the influence of fiber orientation distribution, matrix and interphase properties on the mechanical behavior is discussed in detail. In a second part, a new net-shape bulk forming technique is described in which the coherence of a knitted fabric was used to manufacture a typical load-bearing implant, in this case an osteosynthesis plate, using a single-step technique and, thereby, reinforcing the countersunk holes of the osteosynthesis plate which are the mechanically critical load induction and joining areas. The effect of a knittedfiber architecture on the mechanical properties and the homoelasticity of the plate, is compared to the behavior of a laminated and a stainless steel plate. In a third part, the forming behavior of knitted-fabric-reinforced organo-sheets is described for the application in diaphragm deep drawing. The influence of multiaxial drawability and coherence of knitted fabrics as well as of flow conditions on the structure-properties relations in the deep drawn part are discussed. In general, this chapter should be seen as an introduction in thermoforming techniques for knittedfabric-reinforced thermoplastics with regard to structure-properties relations and failure characteristics. 10.1. General aspects of anisotropic biomaterials for load-bearing implants Anisotropic biomaterials are developed in order to functionally mimic the structure of biological tissue, i.e. bone. In general, the biocompatibility of an implant has to be defined with regard to its recipient tissue. Therefore, biocompatibility is surface and structural compatibility of an implant [1-3]. Structural compatibility includes optimal load transmission at an implant/materials interface. It is, therefore, suggested that anisotropic materials would offer a greater potential of biocompatibility than metals do, as mechanical properties can be adjusted closer to bone. Consequently, a load-bearing implant is defined to behave in a homoelastic manner when it approaches the stiffness of bone with the intention to minimize the strain mismatch between bone and implant. To achieve surface compatibility, the polymer matrix should expose a surface with appropriate surface energy and micro-structure. The implant surface should be completely covered with a continuous matrix layer in order to prevent a potential release of fiber particle debris during implantation. Many matrices, such as polysulfone [4,5], nylon [6] or even epoxy resins [7-9], carbon [10] and others [11-14], have been discussed in the literature. Currently highlighted are thermoplastic matrices with special focus on polyetheretherketone, because of its long-term chemical stability and the existence of appropriate processing techniques. Polyetheretherketone is
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well characterized as bulk polymer and as a matrix for carbon fibers [15] with regard to structure and morphology [16-18], processing [19-24], mechanical properties [25-29], chemical stability [30] and biocompatibility [31-35]. Other properties of anisotropic, non-metallic, materials must also be considered. These include the absence of metal ions such as nickel or chromium in order to prevent allergic reactions, adjustability of X-ray transparency by adding contrast medium to the polymeric matrix, full compatibility with NMR and CT diagnostic procedures.
10.2. Knitted-carbon-fiber-reinforced composite materials 10.2.1. Introduction
Weft and warp knitted fiber architectures have been intensively studied during the last decade because of their superb drapability and free forming capability. In warp knitting, the multiaxial warp knits made by either the K. Mayer or the Liba techniques are the most promising textile precursors [36-39]. They combine the controlled lay-up of uni-directional reinforcing fibers with the drapability of warp knits. However, textile deformability for draping is restricted to inter-laminar shear, whereas tensile deformation is hindered by the uni-directional orientation of the continuous fibers. Furthermore, the fiber orientation distribution in these composites is usually two-dimensional, which implies their well known sensitivity towards interlaminar shear loading. Weft knitting has been explored mainly for glass-fiber-reinforced duromeric resins [36, 39-43]. The knitted textiles investigated are characterized by their high area weight and by their small loop dimensions. The composites were therefore made from only a few, typically 2-3, knit layers. From comparative investigations of different two- and three-dimensional fiber architectures, Drechsler et al. [39,45] concluded that it is fiber architecture that determines the mechanical properties of the composite. Thus knitted-fiber-reinforced composites showed that strength is reduced to a much larger extent than modulus with increasing curvature of the fibers in the loop. Energy absorbance during impact is also increased. The mechanical properties are enhanced if the area weight of the knits and the number of knit layers are increased. By drawing of the knit prior to consolidation, the mechanical properties in the drawing direction could be slightly increased. Planck [44] showed that 80% of the Young modulus of a woven-textile-reinforced composite could be achieved by a drawing ratio of 40% in the wale direction. Owen and Rudd [36,40] compared the properties of weft knitted glass fiber and random mat-reinforced polyesters. In the wale direction the mechanical properties of knitted-fiber-reinforced composites were 50% higher, whereas in course direction they were 30% lower than in the random mat-reinforced material. They correlated these findings successfully with the fiber orientation distribution in the knit loop by applying a single fiber approach for modulus calculations. Chou et al. [41] investigated the influence of the knit orientation in the stacking sequence. For knitted-fabric-reinforced composites, they found up to 80% of the strength and Young's modulus and 320% of the energy absorbance during impact of 1 • 1 woven fabric reinforced composites. Ramakrishna et al.
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[42,43] also pointed out that the mechanical properties and the failure behavior are strongly correlated to fiber architecture and the properties of the knitted fabric. The manufacturing methods employed by several investigators [36,41,45,46] are based on duromeric resins. These processes are readily compared to those of resinimpregnated swirl mats on the basis of their mechanical properties and material costs. For mass production, compression molding of sheet molding compounds, reaction injection molding and resin transfer molding were investigated [47]. Employing RTM techniques, Rudd et al. [36] found a superior drapability and less fiber washing for knitted fabrics than for swirl mats. Hickman et al. [46], Brandt [45] and Drechsler [39] proposed net-shape knitting to manufacture integral helmets [45] or three-dimensionally reinforced profiles [46]. However, it has to be considered that net-shape knitting is much more time-consuming than circular knitting and is mostly restricted to glass or aramide fibers. Hickman [46] proposed using the stretchability of weft knitted fabrics to selectively reinforce load induction zones such as holes. In the investigations [48-52] reported by the present authors a contrary approach has been proposed: 1. The composite is built up from a multitude of knit layers. 2. The reinforcing knitted fabric should possess loops that are as large as possible. This is supposed to have two main effects. First, the curvature of the fibers and the portion of Unbound fibers between the interlocks is maximized. Second, these free fibers can be easily oriented by drawing. The maximum uni-directional drawing ratio achievable correlates to the loop size an the titer of the yarn. For a knitted 3K carbon fiber yarn which had a loop height of about 6 mm in the undrawn state, a drawing ratio of 150% could be reached. Thus, a drawinginduced strengthening and stiffening effect could achieved to the extent that the mechanical properties of woven fabric reinforced composites can be approached or even overcome [51]. 3. The area weight should be the lowest possible in order to achieve an overlapping effect. Due to the overlap of interlocks with aligned fibers from the neighboring 5 to 10 knit layers, the mechanically weak interlock is bridged over. Furthermore, fiber volume contents of more than 50% can be achieved. 4. The mechanical properties are entirely determined by the fiber orientation distribution (FOD) in the composite. This FOD is three-dimensional because of the interpenetration of the knit layers during consolidation. As a result, the anisotropy is reduced compared to that in composites reinforced with straight fibers [53]. These composites show no delamination [52]. 5. Thermoplastic resins are preferred because of their toughness [52], thermoforming potential [54] and biocompatibility, key properties for their application as medical materials in load-bearing implants, i.e. in bone plates [51, 55]. 6. The drapability of knitted fabrics is improved when larger loops and low area weight fabrics are used. 7. To manufacture volume parts with load induction areas and complex outer shapes, net-shape forming of circular knitted fabrics should be preferable to net-shape knitting.
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TABLE 10.1 Properties of thermoplastic matrix systems
Tensile modulus [GPa] Ultimate strength [MPa] Elongation at break [%] Melting point [~ Glass transition temp. [~ Density [gr/cm 3] Crystallinity [%] Water uptake at 20~ [%]
PA 12 Atochem
PEEK ICI
PEMA R6hm
1.4-1.6 52 240 178 40-45 1.10 30 1.5%
3.6 92 50 334 143 1.28 35 0.5%
1.1-1.3 40--45 3-4 45-55 1.13 amorphous 1.8 (37~
8. Coherence and drawability of weft-knitted fabrics allow almost unhampered thermal shaping of pre-consolidated thermoplastic sheets. The flowing behavior is almost isotropic and therefore allows metal-like shaping techniques such as stamping or deep drawing. These investigations were focused on circular weft-knitted carbon fibers as reinforcement for thermoplastic matrix systems, i.e. polyetheretherketone (PEEK), polyamide 12 (PAl2) and polyethylmethacrylate (PEMA). Different impregnation techniques have been investigated. These include powder impregnation (for PEMA), intermingling (for PEEK) or the use of a F.I.T. material (for PAl2) where the carbon yarn is impregnated with the matrix powder and coated with a matrix tube. Table 10.1 shows typical properties of the matrices, i.e. polyetheretherketone PEEK, polyamide 12 PAl2 and polyethylmethacrylate PEMA, used by the authors for biocompatible composites. Knitting of carbon fibers in large loops (up to 10 mm), was feasible due only to the development of a new knitting technique, the so-called "contrary technique" [56,57] which reduces the stressing of fibers during the loop build-up phase to prevent fiber breakage entirely. In the following section, the properties of the fiber architecture and its influence on the mechanical properties mentioned above are outlined in order to illustrate the potential of knitted fabrics as reinforcement as well as their medical and engineering applications. 10.2.2. Fiber architecture and fiber orientation distribution
The appearance of a single layer of the circular weft knit made by "contrary knitting" is shown in fig. 10.1. It is characterized by large loops and low area density. The basic structure is periodic and has a high symmetry with perpendicular mirror planes and screw axes [58]. Composites made from uni-directional lay-ups of circular knitted tubes are balanced and, therefore, show no distortions because of residual thermal stresses. This is due to the third mirror plane which is given by the in-plane symmetry of the knit tube. The fiber orientation distribution in the composite is orthotropic (Laue symmetry: mmm) and, therefore, mechanical structure modeling
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Fig. 10.1. Appearance of a single-knit layer of weft-knitted 3K carbon fiber yarn (left) with low area weight and large loops. Definition of knit directions, loop height and width as well as the unit cell are indicated. Model (right) for the stacking sequence of knit layers: interlocks of the layer "a" are bridged over with aligned parts of the loop in layer "b". Local strain field is determined by the higher stiffness of aligned parts in loops. Stress level in interlock areas are small according to their stiffness and, thus much smaller than in the aligned fibers between the interlocks [3,50].
based on the unit cell properties is possible with six independent elastic constants being considered for the calculation. To visualize the knit structure in the composite, a 100-~tm copper filament was co-knitted and X-rayed after consolidation. X-ray investigations of consolidated composites confirmed the shifted stacking sequence as indicated in fig. 10.1 (left image). The stacking sequence of individual knit layers is characterized by the fact that interlocks of one layer are bridged over with aligned parts of the loop in a next layer. The local strain field is then determined by the greater stiffness of the aligned part. During consolidation, the knit layers interpenetrate and a three-dimensional fiber structure is built up, as shown in the cross-section in fig. 10.2. Complete consolidation with fiber volume contents of more than 55% has been achieved. The coherence of the knit structure guarantees the fiber orientation even at high drawing ratios. Drawing induces a uniform deformation of the loops. In addition, it increases the number of fibers aligned to the drawing direction while also straightening them in the same direction. These effects lead to a strain stiffening and strengthening effect, as shown in fig. 10.6. However, drawing also enhances the anisotropy, thus reinforcing the drawing direction. Because of coherence, the fiber portion perpendicular to the drawing direction is considerably reduced. Using image processing, the loops in the X-ray knit image are cut into single curved fiber segments and thereby approach the shape of the knit loop with straight short fibers [1,59]. This results in a two-dimensional fiber orientation distribution which neglects thickness effects in the composite. X-rays in fig. 10.3 illustrate the influence of drawing in wale and course direction on the two-dimensional fiber orientation distribution (fig. 10.4). Direction and ratio of textile deformation prior to consolidation determines the fiber distribution and thus the anisotropy as well as
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Fig. 10.2. Cross-section of knitted-fiber-reinforced composites. Overview of a composite made from 16 knit layers (left, as seen at original magnification of 50x), indicating the macroscopic through-thickness orientation of the fibers and detailed view (right, as seen at original magnification of 200x) of the threedimensionality of the fiber orientation distribution [3,48]. the mechanical properties of the composite. However, projection effects cause an overestimation of the drawing rates so that a more precise estimation of the mechanical properties requires that the three-dimensional fiber orientation distribution be known. Accordingly, the orientation and ellipticity of the fiber cross-sections in metallographic cross-sections were analyzed using an image-analyzing system [59-62]. In fig. 10.5, the projection of the three-dimensional fiber orientation distributions in the knit plane and the perpendicular plane are illustrated for a composite drawn in wale or course direction respectively. The azimuth is the in-plane angle and the elevation is the out-of-plane angle of the analyzed fiber. As indicated, the interpenetrating effect during consolidation induces a mean out-of-plane orientation of about 15 ~ which is not significantly influenced by the drawing ratio. Because of the fiber curvature in the interlocks, small fiber portions can also be observed perpendicular to the knit plane for both cases (compare with the cross-section images in fig. 10.2).
10.2.3. Experimental details Two different fiber-matrix systems were mainly investigated: HT-fibers combined with polyethylmethacrylate (T300/PEMA, powder bath impregnated) and HT-fibers/
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Fig. 10.3. X-ray images of a co-knitted copper filament to be analyzed with regard to their plain fiber orientation distribution. Drawing ratios: (a) undrawn, (b) 68% in the course direction, (c) 40% in the wale direction, (d) 20% biaxially [48]. polyetheretherketone (AS4/PEEK, commingled yarn). The advantage of PEMA for surgical applications is its low glass transition point of approximately 65~ which allows the adaptation of the manufactured part to a special individual geometry by heating it up in hot water [55,58]. On the other hand, PEEK is known as a matrix material with excellent performance regarding adhesion, toughness and strength. Furthermore, the properties of AS4 fiber-reinforced laminates (APC2, ICI) have been abundantly documented in the literature. Knitted T300 (3K, Toray) fibers were powder-impregnated with a polyethylmethacrylate (PEMA) powder (mean grain size 20 ~tm). Before impregnation, the knit was washed in THF and subsequently oxidized in air for 15 min at 500~ [50,58,63] in order to improve adhesion between fiber and matrix. Twelve knitted tubes were stacked and consolidated in a hot press at 190~ and 1.8 MPa for 30 min resulting in sheets of about 3-ram thickness. A commingled yarn of PEEK (BASF, Germany) and AS4 (Hercules, UK) was used as precursor material, twelve knitted tubes were also stacked and consolidated
Thermoformh~g processes for knitted-fabric-reinforced thermoplastics fiber fraction [%]
411
fiber fraction [%]
20
20
16
16
12
12
8
8
4
4
0
0
4
4
8
8
wale direction
12 16
12 16
20
20
.......
undrawn
.......
40% drawn in wale direction
20% biaxially drawn in wale and course direction
-
68% drawn in course direction
Fig. 10.4. Effect of the mode of drawing on loop-shape and the correlating fiber orientation distribution, through image analysis. The corresponding X-rays are shown in fig. 10.3. Drawing induces a higher anisotropy in the fiber orientation distribution [1].
Y
fiber fraction [%] 2O
fiber fraction [%] 20
15
15
10
10
5
5
0
0
5
5
10
10
wale "ection
15 20
wale direction
15 20
40% drawn in wale direction
.....
68% drown in course direction
Fig. 10.5. Projection of the three-dimensional fiber orientation distributions after uniaxial drawing in the wale and in the course direction respectively. The out-of-plane orientation (15 ~ is not affected by the both drawing directions [48].
in a hot press at 390~ and 2.1 MPa for 45 minutes resulting in a sheet of about 3 mm thickness. The knit was drawn 13% in the wale direction before consolidation. Tests were performed in course and wale directions to selected directions to study the influences of knit structure, knit drawing, fiber matrix adhesion and matrix properties on the mechanical properties. Stress/strain behavior was analyzed during
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tensile and four-point bending loading according to DIN 29971 [58]. Measurements were performed in a universal testing machine (Zwick 1474) with a testing speed of 0.5 mm/min.
10.2.4. Mechanical properties The mechanical properties of knitted-fiber-reinforced composites are determined by the knit parameters, i.e. type, size and deformation of the loop. To improve mechanical properties, it is important to orient the knit layer and/or the direction of loop stretching according to the main load direction. This aligns the number of locally straightened and therefore load-bearing fibers in the force direction (compare with figs. 10.3 and 10.4) and improves strength and stiffness.
10.2.4.1. Young's modulus and static strength Depending on plastic deformation during hot forming, stiffness and strength of knits reach or even exceed the properties of 1 x 1 plain weaves according to the direction of the main deformation (fig. 10.6). These properties are anisotropic and their anisotropy can be controlled by drawing during textile preforming or plastic hot deformation. To allow a comparative estimation, all values have been recalculated for 40 vol. % fibers. 0 ~ corresponds to the UD fiber direction and the weft direction of the knit. 90 ~ or warp is the direction perpendicular to that. Before testing, the knits were deformed in either the wale or the course direction with subsequent testing in the direction of deformation as well as perpendicular to it. The anisotropy was calculated from plane fiber orientation distribution using a single fiber approach (see fig. 10.7). The anisotropy of the knitted-fabric-reinforced
Fig. 10.6. Comparison of the mechanical properties of 0 ~ UD, 1 x 1 plain weave and weft knit. The comparison is based on experimental data for HT-carbon-fiber-reinforced PEEK, but are recalculated for a fiber volume content of 40% [1].
Thermoformhzg processes for knitted-fabric-reinforced thermoplastics 30
25
" _o.
J
.
.
.
.
.
.
.
i
.
~1 56% drawn -t ................- - ~ , - - ~ - - - i n w a l e d i r e c t i o n
.
.
.
.
'
413
'
I .....................~................................... ..-.....t ........
15
.....................
10
...................................................
.....in c
tlonj,
~~
.: ~ ........
i _
0
"
0
~
|
J
|
'
20
!
|
!
|
~
40
i
60
|
"T
i
|
80
angle to wale direction
Fig. 10.7. Influence of knit deformation on the anisotropy of the materials stiffness that has been calculated from the plain fiber orientation distribution resulting from the short fiber analysis of the Xray images [1].
composites may be increased by drawing. With the application of biaxial deformations, the anisotropy will approach the behavior of woven-fabric-reinforced materials. The comparison of the anisotropy between uni-directional, 1 x 1 woven and undrawn weft knit reinforced composites (fig. 10.8) indicates a smooth anisotropy for knitted structures, because the fibers in the curved fiber structure are homogeneously distributed (compare fig. 10.4). Recent investigations [64] have shown that the angle-dependency of the mechanical properties can be correlated to a cosine function so that the amplitude is the difference between minimum and maximum properties corresponding to the wale and course direction. The shift of the function is given from the minimum value. (See fig. 10.9.)
10.2.4.2. SEM study of tensile failure behavior Strength and microscopic fracture behavior revealed significant differences between the failure behavior of reinforced polyethylmethacrylate and reinforced polyetheretherketone. On the basis of SEM investigations, three conditions which determine the failure behavior were found: (1) stress transfer between knit loops bridging over each other, (2) adhesion between fiber and matrix and (3) toughness of the matrix. Under optimal conditions, the mechanical behavior is entirely determined by the fiber orientation distribution using the fiber properties. In the case of brittle matrix or reduced adhesion, fracture mechanisms were dominated more by matrix failure, debonding and the textile structure, which all lead to a considerable reduction in strength.
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J. Mayer and E. Wintermantel
x l weave
~-, 0.8
LU
0.6 undrawn knit
_= "o 0 E
Iii~,!!
0.4
0.2
~ ! "1"'
9
o 0
I
I
10
20
30
j I
40
50
unidirectional 0 ~
I 60
I "I 70
80
, 90
Angle ~ between applied force and textile structure
Fig. 10.8. Comparison of the anisotropy between uni-directional, 1 x 1 woven and undrawn weft-knitreinforced composites. The curve of the knitted fabric was calculated from the fiber orientation distribution of a undrawn loop using the single fiber approach (E is proportional to cos 4 4>) [1].
5ol
I
I
I
I
I
-i 400 1
350
4O
(/)
m= :3 "O O
--
Young's modulus
n
30
E o~ 20 C
/
%
-
-
250
-
-
200
-
15o
E ~ j::: % r
tensile strength
O >-
300
U3
_=_
~:
11)0
~i 0
50
I
I
I
I
I
15
30
45
60
75
o 90
angle to wale direction [o]
Fig. 10.9. Angle-dependency of a knitted-carbon-fiber-reinforced polyamide 12 which has been slightly drawn in the wale direction prior to consolidation. The angle-dependency was fitted with a powered cosine function [3].
PEMA-T300: The macroscopic failure behavior of knitted-fiber-reinforced PEMA is illustrated in fig. 10.10. For both test directions the stress strain curves indicate deviation from the linear elastic behavior before the maximum stress is reached. Specimens tested in the course direction show this effect at lower strains. The
416
J. Mayer and E. Wintermantel
Fig. 10.11. SEM of undrawn PEMA-T300: after tensile tests in the wale direction (left), primary failure occurs by fiber failure of the aligned fibers in the loop (--+ 1). Fiber pull-out and debonding in interlocking area can be recognized ( ~ 2). After tensile test in course direction (right), primary failure occurs by transverse cracking in the fiber-matrix interphase at the surface of the roving (--+ 1) and by subsequent debonding along the loop. The applied force direction is indicated [1,3,50]. TABLE 10.2 Comparison of tensile strength measured according to D I N 29971 and calculated from the plain fiber orientation distribution. The calculation of strength is based on the amount of fibers which are oriented between an angle of 4-7.5 ~ to the force direction. Fiber tensile failure is considered to be the first failure criterion
PEMA-T300 undrawn PEEK-AS4 13% drawn in the wale direction
Test direction
Strength [MPa] calculated
Strength [MPa] measured
wale direction course direction
190 150
149.1 -I- 17.8 79.3 4- 11.5
wale direction course direction
325 75
316 4- 33 105 + 7.0
moduli with those calculated from the two- and three-dimensional fiber orientation distributions based on X-ray data and cross-section analysis of drawn PEMA-T300. A single-fiber approach is used for both calculations [36,40,48]. The computation based on the three-dimensional distribution allows a conservative estimation of the moduli, whereas the X-ray analysis gives an overestimation. This is mainly due to projection effects which neglect those fiber portions that are out-of-plane.
Thermoforming processes for knitted-fabric-reinforced thermoplastics
417
Fig. 10.12. SEM-micrograph of PEEK-AS4 13% drawn in the wale direction. After tensile test in the wale direction (left), primary failure by brittle tensile failure of fibers that are aligned to the applied force direction ( ~ 1). Debonding and transverse cracking cannot be observed. Very short pull-out fiber lengths indicate good adhesion between fiber and matrix. After tensile test in the course direction (right) failure characteristics are comparable to those in the wale direction. Brittle tensile failure of aligned fibers is considered as primary failure (---~ 1). Debonding was not observed. Transverse cracking along the loops occurs as secondary failure.The applied force direction is indicated [50].
Fig. 10.13. Calculation of the plane Young's modulus distribution using a single fiber approach: calculation based on the three-dimensional fiber orientation distribution (right, see fig. 10.5) and based on the two-dimensional fiber orientation distribution (left, see fig. 10.4) [48].
418
J. Mayer and E. Wintermantel
The failure behavior of knitted-carbon-fiber-reinforced composites can be understood from the models proposed in fig. 10.1 and fig. 10.14. The stacking sequence of individual knit layers (fig. 10.1) is characterized by the fact that interlocks of one layer are bridged over with aligned parts of the loop in a next layer. The local strain field is then determined by the greater stiffness of the aligned part. Previous investigations indicated that the plain distribution of Young's modulus is linearly correlated to the FOD [49-51,58]. In a first failure criterion, the FOD was used to calculate the stresses for first fiber tensile failure. The results are summarized in table 10.2 showing the first failure criterion to be fiber tensile failure. They indicate that this first failure criterion works only for the PEEK-AS4 system in which the matrix failure strain clearly exceeds the local strain concentrations in transversally loaded areas. SEM investigations (figs. 10.11 and 10.12) demonstrate the influence of fibermatrix adhesion and matrix toughness on the failure behavior. As a result, a failure model has to be proposed (see fig. 10.14) that considers a strain concentration criterion: the fiber strength can only be used if strain concentration in transversally loaded areas does not exceed matrix failure strain. Otherwise, first failure occurs by transverse cracking in this area and continues by subsequently debonding into the
Fig. 10.14. Failure model for knitted, multilayer composites. The first failure criterion is based on a strain concentration criterion: the local strain level exceeds the failure strain of the matrix (left) and the properties of the fiber cannot be entirely used (example: PEMA-T300 system). The matrix failure strain is considerably higher than local strains (right) and the properties of the fiber are entirely used, then the failure behavior is brittle and fiber-dominated (example: PEEK-AS4 system) [50].
Thermoform&g processes for knitted-fabric-reinforced thermoplastics
419
interlock area. This causes ongoing disintegration of the composite. However, the proposed model is not able to distinguish whether matrix or interphase has to be considered as the weak link. With regard to the influence of fiber-matrix adhesion, recent degradation experiments on knitted-carbon-fiber-reinforced polyamide 12 [49] indicated that selective aging of the interphase causes changes in the failure behavior comparable to that of the effect of matrix toughness.
10.3. Net-shape forming of knitted fabrics for load-transmitting implants shown for an ulnar osteosynthesis plate 10.3.1. Introduction
The most common osteosynthesis techniques focus on rigid fixation of the fracture sites using stiff plates made from stainless steel [65] or titanium alloys. To achieve primary bone healing without callus formation, considerable high axial pressures are needed [66]. Homoelastic osteosynthesis is an alternative approach which integrates fast mechanisms of bone healing such as callus formation [67]. Several concepts have been discussed, such as the use of resorbable materials [68,69] or reduction of plate stiffness by the change of material or of design [70]. It has been found that the amount of external callus depends on plate stiffness [71]. The formation of callus results in higher fracture strength of the healed bone, compared to primary bone healing [72,73]. In addition, several authors have observed less bone resorption underneath homoelastic plates [67,71,72], thus correlating loss of bone density with the degree of stress protection. The premise that local bone necrosis due to high contact pressure of the plate also leads to bone resorption, is controversial [74-76]. It has been reported [77-81], that the residual strength of healed bone shows a maximum before resorption by stress shielding or necrosis takes place. Therefore, an early removal of the plates would be advantageous, the precondition being the observation of bone structure by X-ray. The achievement of a stable homoelastic fixation requires an axial stiffness comparable to that of bone [70,82]. In contrast, bending and torsional stiffness should be high to guarantee proper fixation of the fracture. In order to fulfill these conditions, numerous carbon-fiber-reinforced, non-resorbable composites have been evaluated & vitro a n d / n vivo since 1975 [4,7,9,11,14]. Multistep manufacturing methods, i.e. hot pressing and machining techniques, were used for continuous fiber-reinforced laminates. Because of the fair competitiveness of these techniques for medium scale series, injection molding of short fiber-reinforced composites has been investigated [83]. The low fiber content, problems with the control of fiber distribution and the fatigue behavior restricted the application of injection molded plates. The possibility of hot shaping the plate because of the discontinuity of the fibers was considered to be an advantage for clinical application. Thermal shaping of laminated plates [55] is restricted to intra- and inter-laminar shearing and intra-laminar transverse flow because of the rigidity of the reinforcing carbon fibers.
420
J. Mayer and E. Wintermantel
Based on the previous discussion, the requirements for a homoelastic osteosynthesis system made from anisotropic biomaterials would be as follows: 1. Three-dimensional fiber orientation because of the accumulated complex loads in an osteosynthesis. 2. Axial stiffness low enough to stimulate external callus formation because of axial micromovements in the fracture gap. 3. X-ray transparency to allow determination of the earliest possible time of removal. 4. Shaping of the plate during operation, but with minimal impact to the recipient tissue. 5. Costs comparable to metal plate manufacturing. 6. Use of screws made from composites to prevent corrosion. In the following section the results of the development of such an osteosynthesis plate is discussed. The development of cortical bone screws from continuous carbonfiber-reinforced PEEK has been described elsewhere [1]. Cortical bone screws were manufactured from carbon-fiber-reinforced PEEK in a net shape melt extrusion process. In this process, uni-directionally reinforced planks were heated above the melting point of the matrix material prior to forming and then transferred to the screw-forming cavity. The bone screw had a core-diameter of 3 mm, a fiber content of 62 vol. % and a matrix-coated surface. The fiber orientation in the screw was defined by the flow conditions during injection. In the screw head and in the upper part of the threaded bolt, fibers were generally aligned along the screw axis. Towards the tip of the screw, fibers in the core zone became more circularity oriented. In a skin zone with an approximate thickness of 0.7 mm fibers still followed the longitudinal profile of the thread. An average tensile strength of 460 N/mm 2 was determined. Young's modulus in the axial direction of a melt-extruded bone screw decreased from approximately 40 GPa at the screw head to approximately 5 GPa at the tip.
10.3.2. Experimental details The net-shape manufacturing technique for a six-hole ulnar osteosynthesis plate (fig. 10.15) uses two characteristic properties of knitted fiber structures: drapability and coherence, which allows the molding-in holes by widening single stitches. The plate is made from weft-knitted carbon-fiber-reinforced PEEK in a single step netshape pressing technique [48,54]. Net-shape pressing is defined as the thermo-induced forming of a raw material in one production step without the need for further processing. The commingled knitted fabric was rolled and pushed over the spikes of the die (fig. 10.16). After all four side walls were inserted, a stamp was lowered onto the knitting. This procedure obviated the need for cutting the fibers, as the loops were distorted into a circular fiber alignment around the spikes. This had a self-reinforcing effect with complete coating of the polymer surfaces.
Thermoforming processes for knitted-fabric-reinforced thermoplastics
421
Fig. 10.15. Six-hole ulnar osteosynthesis plate, made from knitted-carbon-fiber-reinforced PEEK in a single step net-shape pressing technique [1].
Fig. 10.16. Net-shape pressing die for the osteosynthesis plate. The knit was rolled up and pushed over the countersunk hole forming pins by widening the single loops [1].
The pressing cycle for the weft-knitted intermingled PEEK/AS4 yarn (BASF/ Hercules) was as follows: heat-up to 390~ at a rate of 18~ dwell period at 390~ for 30 min, pressure 17.5 MPa, cooling rate 10~ Bending strength and modulus were determined by a 4-point bending test (DIN 29971, 2 mm/min) at 25~ The support span was 97 mm and the pressure span was 41.2 mm. A model for the calculation of the bending modulus and the strength was derived according to the geometry of the plates, in which the plates were considered to be beams with two different cross-sections, a massive section and a reduced section, representing the holes [1,48]. The fiber volume content was measured by gravimetric and volumetric
422
J. Mayer and E. Wintermantel
methods [54]. The failure mechanisms were observed by scanning electron microscopy (SEM). To compare net-shape pressing with common lamination technologies, a laminated osteosynthesis plate of identical geometry was made from carbon-fiberreinforced PEEK laminates (APC2-AS4 from ICI). The plates had stiff outer shells (0/0/45/-45/-45/45/0/0) and weak cores ((45/-45/-45/45/0/45/-45/0/-45/45)s) to achieve a high bending (Eb = 107 GPa) to axial (Ea = 60 GPa) modulus ratio. The properties of these plates have been discussed in previous publications [1,55]. A stainless steel plate (six-hole ulnar, Aesculap, Germany ) was used as reference.
10.3.3. Structure and properties of the net-shape manufactured osteosynthesis plate In the critical cross-section of the countersunk hole, a forced fiber orientation along the plate axis as well as a slightly improved fiber content due to the formed holes can be observed (fig. 10.17). This effect results in an improvement of the mechanical properties compared to drilled plates machined from pressed knittedfiber-reinforced plates [48]. Figure 10.17 illustrates the findings for bending modulus of osteosynthesis plates made from stainless steel which had been UD-laminated, machined, and net-shape knitted. The Young modulus of cortical bone is added to fig. 10.17 in order to demonstrate a possible mechanical approach to homoelasticity with knit reinforced thermoplastics. The spatial fiber orientation around the hole, as indicated in fig. 10.18, induces a failure behavior characteristic of the net-shape plate. Compared to the laminated plate, an improvement of damage tolerance is observed in the stress-strain curves in which pseudoplastic failure is considerably increased (fig. 10.19). The correlated stainless steel
plate
(UD 01+4.5 ~
. . . . . .
I
L
I~l
i
cortical bone ~ - - ~ 1
t
t
I
I
0
50
100
150
200
~1 250
Bending modulus [G Pa] Fig. 10.17. Bending moduli of osteosynthesis plates, made from stainless steel and AS4/PEEK composites (UD-laminated and machined, knitted net-shaped and knitted machined). The Young modulus of cortical bone is added as a reference [1].
Thermoforming processesfor knitted-fabric-reinforced thermoplastics
423
Fig. 10.18. Cross-section of a plate as indicated. The hole-forming process induces a fiber alignment along the plate axis (left) and a trough thickness orientation (right). stainless steel plate
~, 1200
+,,_A-..
,,- 800
laminated plate
,=I
~, 750
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~, 600
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0
2.5 5.0 7.5 10.0 elongation [%]
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~ 400 brittle failure
250 0
net-shape plate
0
i
200 t=
2.5 5.0 7.5 10.0 elongation [%]
0
0
2.5 5.0 7.5 10.0 elongation [%]
Fig. 10.19. Comparison of the stress-strain curves for the stainless steel plate (left), the laminated plate (middle) and the net-shape pressed plate (right) [1].
stress-strain curves illustrate the different failure behavior of these materials. While steel plates became plastically deformed, laminated plates showed a well localized brittle failure. Knitted-fabric-reinforced plates have enhanced failure strain due to pseudoplastic damage accumulation as demonstrated in the following SEMs. These findings correlate with an increase of the area of damage as illustrated in the SEM images in fig. 10.20, and with the dislocation of the failure area beside the smallest cross-section, as illustrated in fig. 10.21. Primary failure occurs at the compression site: in the laminated plate, local fiber buckling of the outer 0 ~ plies (1) with subsequent delamination (2) and compression failure of the inner 0 ~ plies (3) is observed, whereas in the net-shape plate (PEEK-AS4), failure occurs beside the smallest crosssection (compare fig. 10.21) The crack path is guided by the fiber orientation in the loop. Primary failure is a compression failure of fibers which have been well aligned to the plate axis and are underneath the plate surface.
424
J. Mayer and E. Wintermantel
Fig. 10.20. Comparison of the primary failure behavior of the osteosynthesis plates at their compression site. Laminated plate (left) with well localized compression ( ~ 1) and delamination failure ( ~ 2) and netshape plate (right) where the crack path is guided by the fiber orientation along the bundles ( ~ ) [48].
Fig. 10.21. Tensile failure side of the net-shape pressed osteosynthesis plate beyond the smallest crosssection in the countersunk hole ( ~ ) [1].
An additional effect of the net-shape processing is the complete coating of the plate surface with the matrix polymer by wetting the mold surface so that release of carbon fiber particles is prevented. This is illustrated in fig. 10.22, where the surface qualities of a drilled and a molded-in hole are compared on the bases of SEMs.
Thermoforming processesfor knitted-fabric-reinforced thermoplastics
425
Fig. 10.22. Surface quality of the net-shape pressed plate (left) being completely matrix coated and of the machined plate (right) showing carbon-fiber debris (--+) [1,48].
10.3.4. Compar&on of the homoelasticity & FEM calculations and stra& gauge measurements Finite element modeling (FEM) and finite element analysis (FEA) was used to evaluate the properties of osteosynthesis plates made of anisotropic carbon-fiberreinforced thermoplastics. The calculations were performed in CAEDS(I-DEAS (version 4.1) and its Integrated Finite Element Solver (IFES). The material properties were isotropic (steel plate E = 210 GPa, laminated plate E = 107 GPa, knitted plate 33 GPa, cortical steel screw 210 GPa (Aesculap, Germany), cortical composite screw 40 GPa [1], bone 18 GPa). The basic set-up of the F E M / F E A procedure is shown in fig. 10.23 [1,48]. A 3-D model of bone, plate and screws generated from linear solid brick and wedge elements (fig. 10.23) formed the basis for plate/bone system deformation analysis and for evaluation of the stress shielding effect. The bone is modeled as a thick-walled tube. Load transfer from bone to plate is modeled with gap elements between plate and screws. The constraints of the gap elements allowed the transmission of compression forces only on the countersunk holes. Friction forces were not considered as the calculations were restricted to relaxed osteosynthesis of reconsolidated bone. Between bone and screws, shared knots are used to fuse the neighboring elements. The calculations focused on global effects such as stress shielding and strain distribution. The calculated strain distributions in the plate/screw/bone model were verified in a 4-point bending test of a relaxed and reconsolidated ostesynthesis. In fig. 10.24 the set-up of the test and the location of the strain gauges is indicated. The
426
J. Mayer and E. Wintermantel
original screw-plate contact
model of screw-plate contact
plate
~
$
screw .
gap elements
bone
gap elements
Fig. 10.23. 3-D plate/screws/bone FE model (quarter model) of the osteosynthesis system. Brick elements are used for plate (upper), bone and screws (lower). Screws are connected by gap elements to the plate (lower fight) and shrunken elements to the bone [48].
Fig. 10.24. Four-point bending test of osteosynthesis. The location of the strain gauges is indicated (P1-P5, K1-K6). Load applied was between 100 N and 1,000 N [1].
Thermoforming processesfor knitted-fabric-reinforced thermoplastics
427
strain gauges on the plate were used to measure the strain distribution along the plate in order to show the reinforcing effect of the molded in holes. The strain gauges on the bone gave the dislocation of the neutral bending axis. A woven-glass-fiber reinforced tube (E = 18 GPa) was used as a model bone and plates were fixed on the tube with cortical steel screws. FE calculation as well as strain gauge measurements indicate the most intensive stress protection for the steel plate (fig. 10.25). The reinforcing effect of the net-shape process is seen only in the strain gauge measurements. The strains around the holes are even smaller for the net-shape plate although the Young modulus of the laminate is more than 30% higher than those of the knitted plates. It was not possible to take the influence of local stiffening of the net-shape pressed plate into account for the isotropic FEM-calculations. The stress protection effect of an osteosynthesis plate is indicated by the dislocation of the neutral bending axis of the plated bone. In a homoelastic osteosynthesis, the bending axis of the bone should undergo a minimal shift from its neutral position. Table 10.3 shows the shift of the bending axis for the three osteosynthesis systems. These were calculated from the FEM model and measured by strain gauges. The osteosynthesis plate, which was made by net-shape forming of the knitted fabric, reveals the smallest shift and thus is expected to have the best homoelasticity. However, the difference between laminated and knitted plate is smaller in the strain gauge measurements than in the calculated ones. This is due to the local reinforcement of hole areas. Furthermore, the variation of the Young modulus along the plate axis of the knitted plate seems to compensate for the variations in the bearing cross-section. Therefore, the strain distribution in the bone during osteosynthesis becomes more homogeneous when using a net-shape plate as compared to a
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Fig. 10.25. Comparison of the strains in the plate calculated in the FE model (left) and measured by the strain gauges (right) [1].
428
J. Mayer and E. Wintermantel
TABLE 10.3 Shift of the neutral bending axis in a plated osteosynthesis. Comparison of calculations using FEM to strain gauges measurements Shift of the neutral bending axis
Finite element calculation
Strain gauge measurements
Steel plate Laminated plate Net-shape plate
83% 28 % 11%
92% 48 % 34%
laminated or even a steel plate. The advantageous ratio of bending and torsional stiffness to axial stiffness of the laminated plate cannot, however, be achieved in the net-shape forming process because of the homogeneous through thickness properties of the knitted-fiber-reinforced material.
10.4. Deep drawing of knitted-fiber-reinforced organo-sheets 10.4.1. Introduction Consolidated sheets of knitted-fabric-reinforced thermoplastics as semi-finished products for thermoforming techniques, i.e. deep drawing, are discussed below. The authors would like to stress on that knitted fabrics feature a unique deformation behavior in comparison to woven or braided fabrics. In contrast to those fabrics that are built up from nearly straight yarns, knitted fabrics allow tensile as well as shear deformations. In the case of uniaxial loading, the tensile deformability can be far more than 100% until the knit attains a maximal density and the interlocks are densely packed. Biaxial tensile drawing is possible until the curved fibers in the loop become straightened and the density of the knit reaches a minimum. At this point, further deformation can only be applied by shear in a manner analogous to woven textiles. The authors suggest that the deformation behavior of a knitted-fiber-reinforced sheet during thermoforming is dominated by the textile deformation characteristics of a knitted fabric. In order to substantiate this hypothesis, the deep drawing behavior of a knitted-fiber-reinforced sheet was investigated using a lab-scale diaphragm technique. Although diaphragm forming is not suited for mass production, it is expected to be adequate for knitted-fabric-reinforced thermoplastics. This technique was selected in order to compare the forming characteristics of knitted-fiberreinforced sheets to those of uni-directional and angle-plied fiber-reinforced sheets [84,85]. As the principles of diaphragm forming and related phenomena have been discussed in detail by many authors [86-90], the discussion concentrates on the specific features of knitted fabrics.
10.4.2. Experimental details The organo-sheets were hot pressed from a knitted carbon fiber (T300, 3K) reinforced polyethylmethacrylate. An unreinforced polyethylmethacrylate sheet which
Thermoforming processes for knitted-fabric-reinforced thermoplastics
0~
~ 6 [ [ [
7
.
x-raymarker
22.5 ~
5
a =~\~\7\
deep dra~ng
~
biaxial strain
~ 90 ~
tangential
429
~radia!~rain
~5
\ ) ~\
orientation of the knit
Fig. 10.26. Orientation of the knitted fabric according to the co-ordinates and location of the X-ray markers which were used to define a grid on the sheet [48,51].
had identical dimensions was used as reference material. The knitted fabrics had a loop height of 4.4 mm and a width of 6.1 mm. 12 knit layers were stacked to construct the sheet. The global deformation field was measured from a grid which was defined by silver dots (diameter: 1 mm) applied beforehand (fig. 10.26). To determine the local fiber orientation distribution, the deformation of the loops was monitored by a co-knitted copper filament (100 ~tm diameter) as a tracing element for X-rays. The fiber orientation distribution was determined by the image analyzing technique mentioned above [1,48]. The plain Young's modulus distribution was then calculated using the single fiber approach proposed by Krenchel and Rudd [36,58] The set-up of the diaphragm die and a deep-drawn cone are shown in fig. 10.27. The forming process was carried out in an autoclave at 150~ by applying a deep draw pressure of 1 MPa for 20 min against vacuum. The sheet was held between the two diaphragm foils by an internal vacuum. The rim of the sheet was permanently clamped during deformation. 10.4.3. Flow behavior
The cone geometry of the die was shaped entirely by the sheet without the occurrence of any wrinkles or other instabilities. The outer surface of the sheet, which was in direct contact with the metal die, was smooth. The inner surface showed a noticeable roughness that correlated with the loop geometry of the knit. A smooth inner surface cannot be obtained because the softness of the diaphragm foil is unable to withstand the internal stresses of the knit in the softened matrix. Consequently, changes in the thickness of the sheet over the cone could not be measured. The deformation behavior of the knitted-fabric-reinforced sheet is qualitatively visualized in fig. 10.27. The deformation of the color dots locate maximum strains at the tip of the cone whereas the clamped outer ring remained undeformed. The material contacts the die at its outer circumference first, where deformation is hindered by friction between diaphragm foils and die. The regularity of the dots in the circumference indicates the almost isotropic flow during deep drawing.
430
J. Mayer and E. Wintermantel
Fig. 10.27. Set-up of the diaphragm die (upper) with clamping ring (A), vacuum foils (B), deep-drawn sheet (C), vacuum ring (D) and the cone-forming die (E). The cone (lower) is shown with applied color dots to illustrate the isotropy of the deformation behavior [48].
The tangential and radial strains were calculated from the deformation of the grid on the basis of the relative dislocation of the silver dots. The values for an unreinforced matrix sheet and for the knitted-fabric-reinforced sheet are shown in fig. 10.28. The radial strains along the evolution line of the cone reveal material maximum strains at the tip (torus 1) and minimal strains at the outer border for both. The unreinforced polymer sheets showed an isotropic deformation behavior as illustrated in the circular deformation distribution in the fourth torus. According to the contact conditions to the die, the strain is concentrated at the tip of the cone. The knitted fabric also allowed an isotropic deformation with respect to the circumference, although at higher strain levels in the outer torus, as indicated in
Thermoforming processesfor knitted-fabric-reinforced thermoplastics 60
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torus
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unreinforced cone, deep drawing pressure: 300 kPa
knitted fabric reinforced cone, deep drawing pressure: 1000 kPa
Fig. 10.28. C o m p a r i s o n o f the d e f o r m a t i o n d i s t r i b u t i o n o f a k n i t t e d - f a b r i c - r e i n f o r c e d c o n e a n d o f a n u n r e i n f o r c e d c o n e a f t e r d i a p h r a g m f o r m i n g . Left: s t r a i n s a l o n g the e v o l u t i o n line o f the cone. R i g h t : c i r c u l a r s t r a i n d i s t r i b u t i o n in the f o u r t h torus. T h e o r i e n t a t i o n o f the k n i t t e d f a b r i c is i n d i c a t e d b y the i c o n [48,51].
fig. 10.28 (left). The deformation behavior along the evolution line of the cone is strongly influenced by the textile deformation characteristics of the weft-knitted fabric. The sheet can flow isotropically as long as the deformation limit of the knit is not reached. Weft knitted fabrics allow uniaxial strains of more than 150%, but these limits are heavily reduced when a biaxial strain component is added. Under equibiaxial deformation the maximum strains are limited to about 40%. As soon as this condition is fulfilled at the tip of the cone, biaxial deformation of the sheet is blocked in these areas. Due to the coherence of the knitted fabric, any further deformation that would be required to form the cone is transferred to the outer areas. This explains the observation (fig. 10.28, left), that the maximum strains of the knitted-fabric-reinforced cone is considerably lower than those of the unreinforced cone. However, due to coherence of the knit, strain is more evenly distributed over the entire surface and is therefore at a higher level.
10.4.4. Correlation between plastic flow and fiber orientation distribution The analysis of the fiber orientation distribution by means of co-knitted copper wire is illustrated in fig. 10.29, where the modulus distribution was calculated for different characteristic areas of the cone. It demonstrates the equibiaxial distribution at the tip of the cone. At the outer border, the knit is uniaxially drawn with a certain biaxial component. The area which was drawn mainly in the wale direction shows a shift of the minimal modulus from 90 ~ (course direction) to 70 ~ to wale. This illustrates the biaxial strain component which had been induced by the coherence of the knitted fabric and which guaranteed constant tangential tensile stresses in the sheet
J. Mayer and E. Wintermantel
432
6O
radial strain as main component: loop drawn in wale direction
n
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= 50
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E p
~ 40
C 0
undrawn material
e q u i - ~ , ~
N 30
biaxially drawn loops in wale and course direction
0
20
0
10
20
30
40
50
60
70
80
90
angle to wale direction [~
Fig. 10.29. Anisotropy of the knitted-fabric-reinforced cone after diaphragm deep drawing. The distribution of the Young modulus was calculated from the fiber orientation distribution using a singlefiber approach (fiber volume 40%). The tip of the cone reveals equibiaxial deformation. The border exhibits uniaxial deformation with a tangential component as shown for an area drawn in the wale direction [48,51].
during deformation. Those coherence-induced stresses are seen to prevent the formation of radial instabilities.
10.5. Discussion
10.5.1. Structure-properties relationship In accordance to Planck [44] it was shown that the mechanical properties of knitted-fabric-reinforced composites depend directly on the geometry of the loop and therefore can be controlled by the drawing of the textile. It was possible to correlate quantitatively the rate of drawing of the knitted textile with the observed strain strengthening and stiffening effects. In contrast to dense knitted fabrics used by other authors [36,39,40], the low area density of our fabrics allowed the formation of a three-dimensional fiber orientation at fiber volume contents of more than 50% by the interpenetration of a multitude of stacked knit layers. Thus, the mechanical behavior of dense- and loose-knit reinforced composites obeys different rules. In a composite built up from few, dense knit layers, the mechanical properties correlate with the properties of the knitted fabric and thus increase with the area density of the knit [39]. When the composite is constructed from a multitude of interacting lowdensity knits with large loops, the mechanical properties are directly determined by the three-dimensional fiber orientation distribution in the consolidated material. They are no longer influenced by the textile properties of the knitted fabric. However, the authors would like to point out, that while on one hand, a good adhesion between fiber and matrix is the precondition to achieve a failure strength
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that correlates to the fiber orientation distribution, on the other hand, the use of thermoplastic matrices may be of advantage. Several authors [39-41,44] propose net-shape knitting techniques using flat-bed machines to produce complex shaped parts such as helmets. However, it should be taken into account that net-shape knitting has a much lower productivity than circular knitting and that it could not be applied successfully to carbon yarns. In the second part of this chapter it was demonstrated that circular weft knits could be used in net-shape manufacturing processes by net-shape formation of the desired part, instead of net-shape knitting its preform. In order to improve the low mechanical properties of composites reinforced by dense knits, Rudd [40] and Drechsler [39] proposed the use of more stable knit types or reinforcement by weft and warp inserted yarns. However, the insertion of straight yarns would considerably reduce the drapability and would restrict the tensile deformation capability in weft and warp direction. The application of low density, large loop weft knits allows the achievement of mechanical properties which correspond to the fiber orientation distribution and which can be considerably enhanced by uniaxial drawing. Furthermore, the orthotrope symmetry of a weft knit makes it susceptible to the modeling strategies already established for composite materials. However, several aspects of the structure-properties relationship are still unknown. Fatigue and creep properties, damping, crash and impact properties have to be investigated. The development of suitable engineering and structure modeling algorithms which integrate the influence of loop deformation by draping or thermoforming into the prediction of the mechanical properties, is thought to be a prerequisite for the successful application of these materials.
10.5.2. Thermoforming Coherence of the weft-knitted fabric is considered to be one key factor in processing. The coherence defines the fiber orientation distribution as well as the homogeneity of fiber distribution during thermoforming processes even when large strains are applied. During thermoforming, the strain field is homogenized in the material flow because the coherence of a locally fully deformed knit will tend to distribute strain to neighboring areas. The drawing-induced orientation of the loops may allow the achievement of selective strengthening and stiffening (compare fig. 10.29). Net-shape forming was introduced as a single-step thermoforming technique that enabled the production of complex shaped parts including molded-in holes or, more generally, load induction areas. Net-shape forming can be applied for thin parts such as sheets or pipes as well as for voluminous parts that cannot normally be realized from composite materials. The number of manufacturing steps might also be reduced considerably by employing this technique, so that it is expected to have a positive impact on production costs. However, the internal stresses in the loops hinder the dry consolidation of the fabric. During consolidation a volume reduction from the textile to the composite of more than 200% may occur and should be considered in the set-up of the processing technique. The directional formation of holes, load induction or notched areas in knitted-fiber-reinforced composite induces
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an orientation of fibers and will therefore selectively reinforce those critical areas. The reinforcing effect of molded-in holes on fatigue properties and notch factors for woven-fabric-reinforced composites has been shown by several authors [91-95]. In organo-sheet forming, the deep drawing behavior of knitted-fabric-reinforced sheets is characterized by their coherence, their drawability and their drapability. While these characteristics denote the main advantages, they also present inherent problems for the control of the manufacturing process. The drapability enables unhampered shaping of free forms, but the control of the fiber orientation becomes difficult when no well-defined forces or constraints, i.e. by clamping, act on the textile. Based on the experiences with co-knitting of copper wires for X-ray tracing, co-knitting of a stiff metal wire could be applied to stabilize the textile deformation of the knit during draping due to the plastic deformation of the wire. However, galvanic corrosion should be considered combining carbon fibers with metal wires. The coherence of the fiber architecture prevents wash-out of fibers even during extensive viscous flow. The transverse contraction behavior of the knitted fabric induces internal biaxial strains which hinder formation of faults, but it can limit the extent of applicable deformation. As mentioned above, it can help to distribute the deformation field in order to prevent local thinning due to strain concentrations. The internal stresses in the fibers which are introduced by the curvature in the loop influence the consolidation behavior. For these to be overcome, a pressure of about 1 MPa has to be applied. The achievement of smooth surfaces requires the use of rigid male and female dies. Consolidated preforms lose their consistence and their stiffness as soon as the matrix melts. Therefore, handling of preheated preforms needs special tools as well as the adaptation of the processing conditions which guaranties the presence of a defined force.
10.5.3. Biocompatibility aspects and applications It has been shown that the reinforcement of implants with knitted-fiber architectures enhances the structural compatibility in terms of homoelasticity and smoothness of the anisotropy compared to laminated materials or metals. Molding-in and selective reinforcement of the countersunk holes as well as the process-induced threedimensional fiber orientation improves failure security and reduces the sensitivity of the implant to variability in load cases between individuals. Process-integrated sealing of the implant surfaces with the polymer matrix prevents the possible release of fibrous particles and contributes to the reduction of manufacturing costs. Preliminary investigations proved the inter-operative, thermal adaptability of the bone plates [55]; however, an amorphous matrix has to be used to enable plastic deformation at the lowest possible temperature. The viscosity of the matrix has to be as high as possible to prevent deconsolidaton due to the eigenstresses in the compressed-fiber architecture. The concept of net-shape forming of implants can be transposed to other loadbearing implants or to surgical instruments whenever homoelasticity, X-ray transparency, and MRI-compatibility have to be combined with high multiaxial loads and complex implant geometries. The net-shape pressing technique could be preferentially
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applied to complex-shaped bone plates for spine surgery or to stem material for artificial joints. However, for contact with bone, the material has to be coated by bioactive hydroxyapatite coatings, to be applied by plasma spraying [95,96] or biomimetic coating [97]. Using light or thermally curing matrix resins, knittedfiber-reinforced materials can be used for individually adapted dental implants, i.e. dental bridges or for the reconstruction of bone substitutes in reconstructive surgery.
10.6. Summary and conclusions It was the aim of the authors to propose knitted fabrics as reinforcement structures for high-performance thermoplastic composite materials. These materials are considered to be suited for applications of composite materials whenever net-shape forming or thermoplastic sheet forming is of special importance for the overall performance of the product. It has been shown that knitted-fabric-reinforced organo-sheets may allow drawing ratios greater than 100% isotropically. The drapability of the knitted fabric should pose no major drawbacks to the realization of complex-shaped parts. The mechanical properties will approach those of woven-fabric-reinforced materials whenever the knit is uniaxially drawn during thermoforming. The net-shape forming technique will display its main advantages in the manufacturing of bulk parts which have been hardly feasible so far by conventional composite processing techniques. This technique is characterized by the simplicity with which parts possessing threedimensional fiber orientations and selectively reinforced areas, i.e. holes or notches, can be built up. The drapability of knitted fabrics permits the realization of almost any shape including formed-in holes or integration of inserts. Critical areas may be strengthened selectively by local drawing. The low density of these fabrics enables the shaping of the outer contours of a part as an alternative to cutting and may contribute to a reduction of manufacturing-related waste. Such preforms can be thermally fixed by the co-knitting of a low melting yarn or by a curing binder. The preforms can then be impregnated by GMT, RTM or SRIM technologies. The productivity of the "contrary knitting technique" may allow its integration in an on-line production processes such as to produce consolidated semi-finished parts using a double-belt press. The manufacturing costs for knitted textiles made by circular weft knitting are comparable to or even lower than those for woven cloth. A combination of these factors make knitted-fabric-reinforced sheet materials appear to be suited to mass production technologies.
Acknowledgements The authors would like to acknowledge the contributions of all students and colleagues at the Chair of Biocompatible Materials Science and Engineering which were involved in the research on knitted-carbon-fiber-reinforced composites during the past 6 years. The work was supported by A. Buck, Technische Strickerei Produkte, Germany, and Aesculap AG, Germany.
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[88] Niedermeier, M., Ziegmann, G. & Flemming, M., An experimental analysis of the forming behavior of continuous fiber reinforced thermoplastics with the diaphragm stretch forming test, 39th Int. SAMPE Symp. '94, Anaheim, USA, 1994, pp. 1821-31. [89] Groves, D.J., Bellamy, A.M., Stocks, D.M., Anisotropic rheology of continuous fiber reinforced thermoplastic composites. Composites, 23 (1992) 75-80. [90] Tam, A.S. & Gutowski, T.G., Ply-slip during the forming of thermoplastic composite parts. J. Comp. Mater., 23 (1989) 587-605. [91] Ghasemi Nejhad, M.N. & Chou, T.W., Compression behaviour of woven carbon fiber reinforced epoxy composites with molded-in and drilled holes. Composites, 21 (1990) 33-40. [92] Zimmerman, M., Parsons, J.R. & Alexander, H., The design and analysis of a laminated partially degradable composite bone plate for fracture fixation. J. Biomed. Mater. Res., 21-A3 (1987) 345-61. [93] Chang, L.W., Yau, S.S. & Chou, T.W., Notched strength of woven fabric composites with mouldedin holes. Composites, 18 (1987) 233--41. [94] Gunderson, S.L. & Whitney, N.M., Insect cuticle microstructure and its applications to advanced composites. Biomimetics, 1 (1992) 177-97. [95] Schepers, E. J.G. & Pinruethai, P., A comparative study of bioactive glass and porous hydroxyapatite particles in periodontal bone lesions. Bioceramics Vol. 6, eds. Ducheyne, P. & Christiansen, D., Butterworth-Heinemann, Oxford, UK, 1993, pp. 113-16. [96] Ha, S.-W., Mayer, J. & Wintermantel E., Micro-mechanical testing of hydroxylapatite coatings on carbon-fiber-reinforced thermoplastics. Bioceramics Vol. 6, eds. Ducheyne, P. & Christiansen, D., Butterworth-Heinemann, Oxford, UK, 1993, pp. 489-93. [97] Ha, S.-W., Reber, R., Eckert K.-L., Mayer, J. & Wintermantel, E., Precipitation of hydroxylapatite coatings in simulated body fluids: a novel technology for coating carbon fiber reinforced thermoplastic composites. J. Art. Org. 17 (1994) 430.
Composite Sheet Forming edited by D. Bhattacharyya 9 Elsevier Science B.V. All rights reserved.
Chapter 11
The Forming of Thermoset Composites H a o r o n g LI and T i m o t h y GUTOWSKI Laboratory for Manufacturing and Productivity, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Contents Abstract 441 11.1. Introduction to thermoset forming 442 11.2. Kinematics 446 11.2.1. Introduction and differential geometry results 446 11.2.2. Ideal shears to form a hemisphere 449 11.2.2.1. In-plane shear 449 11.2.2.2. Inter-ply shear 450 11.2.3. Ideal shears to form a curved C-channel 451 11.2.3.1. In-plane shear 451 11.2.3.2. Inter-ply shear 454 11.3. Thermoset forming experiments and forming limit analysis 455 11.3.1. Laminate wrinkling and its modeling 455 11.3.2. Material rheology and scaling laws for compressive forces 458 11.3.3. Laminate buckling resistance 460 11.3.4. Diaphragm stiffness 461 11.3.5. Order of magnitude analysis 463 11.3.6. Thermoset forming experiments and forming limit diagrams 464 11.4. Concluding remarks 468 11.4.1. Reinforced diaphragm forming 469 11.4.2. Inflated tool diaphragm forming 470 References 471
Abstract F o r m i n g has the potential to replace the time-consuming and labor-intensive h a n d lay-up processes as a cost-effective alternate for the m a n u f a c t u r i n g of a variety of thermoset composite products. In this chapter, d i a p h r a g m forming of thermoset composites is reviewed, with the emphasis on kinematics and forming limit analysis. The unique properties of the composites lead to kinematic constraints so that the conformance of laminates to complex geometries would ideally be achieved by viscous shearing mechanisms, a m o n g which the two most i m p o r t a n t such modes are in-plane shear, where adjacent fibers slide past one another, and inter-ply 441
442
H. Li and T. Gutowski
shear where plies slide relative to each other. General methods to calculate the ideal shear strains required to form a given part are obtained by applying the theory of differential geometry. A three-point bending test is employed to understand the constitutive laws governing the deformation. The nonlinear elastic behavior of the diaphragm materials is modeled by using the established biaxial stress theory of rubbers. The balance between the mechanisms that cause and prevent wrinkling leads to the preliminary forming limit diagrams, which would allow us to predict the occurrence of undesirable modes such as laminate wrinkling. The chapter concludes with some innovative development of the diaphragm forming process at MIT.
11.1. Introduction to thermoset forming
Although many of the initial developments of advanced composites sheet forming used thermoplastic materials, it is possible to form thermoset systems by essentially the same techniques [1-4]. In fact, the largest production forming applications of advanced composites may be for the Boeing 777 empennage which are thermoset composites (see fig. 11.1). This chapter will focus on diaphragm forming of thermoset materials. This process is similar to that used for thermoplastics; however, lower temperatures and pressures are used for thermosets. This leads to simpler equipment and tooling, and reusable diaphragms. After forming, thermosets must be cured,
Fig. 11.1. Forming press for 777 advanced composite trailing-edge beams (courtesy of the Boeing Commercial Airplane Group).
The forming of thermoset composites
443
either on the forming tool or on a separate curing tool. Thermoplastics, on the other hand, solidify by cooling. Figure 11.2 shows a schematic of the diaphragm forming process. Layers of prepreg are laid-up in various directions on a platform according to design requirements and trimmed into a preform shape. This stage can be performed by hand or by various automated means. Then, the preform is placed between two high elongation rubber diaphragms, which are the supporting materials. Effective contact of the diaphragms with the preform is realized by drawing vacuum between them. Finally, by applying vacuum from beneath the bottom diaphragm and/or positive pressure on the top, the preform is deformed over the tool. In most cases, vacuum alone is sufficient to form the thermoset composite part. Figure 11.3 shows some parts of complex shape formed by double diaphragm vacuum forming of epoxy resin/graphite fiber composites. Several variations of this process exist which are particularly suited to thermoset composites, including (1) using only the top diaphragm (called "drape" forming) [5], and (2) inflating the bottom diaphragm from beneath the tool ( called "inflated tool" forming) [6]. The major differences between forming of thermoset and thermoplastic composites root from the different chemical and mechanical properties of the matrix materials. The high temperatures for forming the thermoplastic composites (e.g. 370~ for ICI APC-2 PEEK) require high temperature diaphragms [7]. The commonly used materials of superplastic aluminum or high temperature polymers (e.g. polyimides) provide excellent support during the forming process, and then are usually discarded after the part cools. Because of their high stiffness, relatively high pressures are employed (of the order of 1 MPa). The combined effect of high diaphragm stiffness and high pressure has a positive effect of suppressing laminate wrinkling, but it can also lead to part thickness variation. Thermoset composites, on the other hand can
Fig. 11.2. Schematic representation of the diaphragm forming process of thermosets.
444
H. Li and T. Gutowski
Fig. 11.3. Thermoset matrix parts made by the diaphragm forming process: (a) chassis for a radiocontrolled model car, (b) scale model automotive body, and (c) roller blade.
The forming of thermoset composites
445
be softened for forming at relatively modest temperatures (around 40~ This allows the use (and reuse) of high elongation rubbers such as silicones and, in some cases, latexes. These softer diaphragms can be formed using only vacuum (0.1 MPa). The lower pressures and softer diaphragms used with thermosets are less effective at suppressing laminate wrinkling, but have the positive effect of causing only minor thickness variation. For example, it has been observed that diaphragm formed thermoplastic composites can suffer thickness changes of the order of 17% to 200% [7], while the variation in thermoset parts is usually less than 7% [4]. Such variation is typical of parts produced on one-sided tooling. To improve the stiffness of the softer rubber diaphragms, selective reinforces such as steel rods, woven rods, and screens, etc., have been added [8]. This method effectively turns the diaphragm itself into a composite, and can greatly expand the formability range for the process (see fig. 11.4). Despite the differences discussed above, the forming mechanisms for thermoset and thermoplastic composites have much in common. For example, the deformation mechanisms are similar and the kinematics, especially the ideal kinematics are identical. Furthermore, the lower heating and pressurization requirements for thermosets make it easier to perform multiple forming experiments. Hence, while all of the discussion in this chapter is developed for thermoset composites, much of it is of a fundamental nature and may apply (under the right circumstances) to thermoplastics as well. A major barrier to the broad application of advanced composite forming processes in industry is the occurrence of a variety of failure modes such as in-plane fiber buckling, fiber misalignment, and laminate wrinkling [2,3]. Formability prediction tools are needed for a robust design and efficient manufacturing process planning. Unfortunately, the current collective knowledge of the deformation mechanisms in forming of advanced composites is quite incomplete. The technical complexities in the modeling of forming processes are related to the complex nature of the materials to be formed, as well as the large-scale, partially unsupported nature of the deformation. This chapter will review the research in this area.
Fig. 11.4. Reinforced diaphragm forming process: (a) 122 cm long part formed using reinforced diaphragms; (b) close-up of reinforcement.
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H. Li and T. Gutowski
Section 11.2 introduces the deformation modes of laminates in the forming process and identifies the desirable modes. Quantification of the deformation modes leads to the recognition of important shearing mechanisms. With the theory of differential geometry, general calculation methods for ideal mappings of various parts are obtained, and ideal shares are calculated for hemisphere and curved Cchannels. Section 11.3 combines the data from thermoset forming experiments with the constitutive behavior of the composites and the rubber diaphragm. This leads to preliminary forming limit diagrams. Finally section 11.4 presents concluding remarks and introduces some innovations to the original diaphragm forming process of thermoset composites. 11.2. Kinematics
11.2.1. Introduction and differential geometry results The unique kinematic characteristics of aligned continuous fiber composites are directly related to their structure. Under forming conditions, the fibers can be regarded as inextensible in their longitudinal direction and the composite incompressible. If, in addition to these two assumptions, one assumes that even fiber spacing is maintained (this is the "ideal composite assumption", see, for example, [9,10]) the required strains to obtain a given deformation can, in principle, be directly calculated [3,9-12]. When these results are then used with appropriate constitutive equations, statements can be made concerning the build-up of stress in the formed part. This same approach could be used with intermediate shapes during the entire forming sequence. Failure occurs when there is an alternative, lower energy deformation mode available which still satisfies the constraints of the forming process. In this section we will outline the basic "ideal" deformation modes for the composite. We start by defining two local co-ordinate systems as shown in fig. 11.5. In the material co-ordinate system, axis 1 lies along the fiber direction; axis 2 is
Fig. 11.5. Illustration of the material co-ordinate system, 1, 2, 3, and part co-ordinate system,/z, v, (.
The forming of thermoset composites
447
perpendicular to the fibers and within the plane of a lamina; and axis 3 is vertical to the ply. (Hereinafter the terms "ply" and "lamina" are used interchangeably.) In the part co-ordinate system, axis ( is normal to the laminate; axes /z and v are two orthogonal directions within the laminate. Under the kinematic constraints noted above, the deformation modes of the composite laminate during forming are quite limited; essentially three shear modes within a lamina (illustrated in fig. 11.6) and one between plies (illustrated in fig. 11.7). For a part to be formed with only single curvature such as in simple bending, the necessary modes are: longitudinal through-the-thickness shear 1-'13 (fig. l l.6a), transverse through-the-thickness shear F23 (fig. l l.6b), and inter-ply shear (fig. 11.7) in the same order of magnitude as F13 and/or F23. The inter-ply shear, where plies slide relative to each other, can be denoted by 1-'v3 if v is defined as the direction of interply relative displacement. Parts with doubly curved geometries, on the other hand, are much more difficult to form and hence will be of major interest in the following discussion. The conformance of aligned fiber composites to these complex geometries requires an additional important shear mode: longitudinal in-plane shear F12 (fig. 11.6c), where adjacent fibers slide past one another within the lamina plane. The corresponding 3
3
2
2
(a)
)
(c) Fig. 11.6. Illustration of shear modes (a) 1-'13, (b) 1-'23 , and (c) I"12.
.... .
oo oo
'
'
.
.
. II
~
| i
....
b~6Vdc L
i
.....
i
|
1
v
Fig. 11.7. Illustration of inter-ply shear mode F3v.
448
H. Li and T. Gutowski
&ter-ply shear 1-'u3 (fig. 11.7) is of considerably larger magnitude than in singly curved parts because of the different F12 between adjacent plies with different fiber orientations. If these shearing mechanisms are prohibited, then failure modes such as undesirable thickness variation, in-plane fiber buckling and laminate wrinkling may occur. The goal of forming can be simply put as enabling the desirable deformation modes while suppressing the undesirable modes. Since ideal inter-ply shear is directly determined by the ideal in-plane shear mapping of the corresponding plies, any calculation of inter-ply shear must be preceded by a detailed study of ideal in-plane fiber mapping. Here, we will outline the calculation of F12 and 1-'u3, for two important part shapes: hemispheres and curved Cchannels. In general, the required strains for ideal composites can be calculated directly only for relatively simple shapes [12]. More complex shapes would require a properly constructed CAD drawing [10-11]. To determine the in-plane shear 1-'12, consider the shear along a fiber element (fig. 11.8). If a fiber slips a total distance, 8, relative to its neighbor with inter-fiber spacing, h, then the total shear for the fiber can be written as F12 = ~
(11.1)
This can be related to the geodesic curvature, Xg, of the fiber by [10,11] L AI"12 -- / Kg(S)
ds
(11.2)
0
where L is the length of the fiber, and s is measured along the fiber. Furthermore, the above integral can be related to the Gaussian (or double) curvature K for the part surface over some region R, enclosed by M smooth curves Ci with exterior angles Oi (one of them representing the fiber of interest) by the Gauss-Bonnett theorem (fig. 11.9) [10,13]
d,+j'j/< dA-2 c
R
M
- oi i=1
• B
nr
Illlll L
F
Fiber element before deformation
Fiber element after deformation
Fig. 11.8. Quantitative definition of in-plane shear.
(II.3)
The forming of thermoset composites
449
01
Fig. 11.9. Illustration of Gauss-Bonnet theorem.
Hence by using this result, the required shear can be determined from the part shape (K) and the fiber orientation (Oi). This procedure can be used to calculate the required ideal shears for a variety of complex shapes [10].
11.2.2. Ideal shears to form a hemisphere 11.2.2.1. In-plane shear [10] We wish to determine the in-plane shear for the fiber identified by the ideal path C 3 in fig. 11.10. Here the arcs of the closed path C are as follows: C1 is a semicircle on a great circle; C3 is a semicircle representing the fiber path of interest; and C2 and Ca are the connecting arcs that lie on a great circle. The geodesic curvature of arcs C~, C2 and Ca are all zero. The line integral in the Gauss-Bonnet theorem thus reduces to that of the fiber path C3. The exterior angles sum to 2zr, and since the Gaussian or total curvature of the hemisphere is 1
K -- R2
(11.4)
C4
C2
- - q b I---Fig. 11.10. Illustration of fiber paths
on a hemisphere.
450
H. Li and T. Gutowski
the Gauss-Bonnet theorem thus reduces to -
l Xg ds -
1
K A - - ~ (~rRb) - Jr sin 4~
(11.5)
c3
or IF121 - zrsin 4~
(11.6)
11.2.2.2. Inter-ply shear [3]
Knowledge of the in-plane shear pattern for a given part geometry allows the preform shape to be calculated. Figure 11.1 l a shows the preform for a hemisphere with an orthogonal grid. Referring to the co-ordinate system with origin at the center of the preform, and considering a [0o/90 ~ laminate one can identify a series of points on both the 0 ~ and 90 ~ plies that have the same (x, y) co-ordinates. These points will move relative to one another after forming. Figure 11.11 b shows the location of the points A and B, after forming. The positions of these points after forming are determined assuming that the individual plies achieve ideal in-plane shear patterns. The relative movement of the points is given by, (11.7)
M - fiR
where R is the radius of the hemisphere, and/3 is the azimuthal angle between the points (X1, YI, Z1) and (X2, Y2, Z2) after forming. From the dot product of the vectors,
[1
fl -- COS-1 ~-~(XlX2 -[- Y1 Y2 + 2122)
]
(11.8)
or ) + sin 02 cos 01sin ( 02 ) + C 02 COS01
sinOlcosO2sin(Oo2
fl -- COS-1
cos01 cos02COS cos01
(Oo:)
cos c
(11.9)
02
Z
//
VI
1
So,
i/JI
.
Y
A&B x (a)
(b)
Fig. 11.11. (a) Preform shape for a [0090~ hemisphere, and (b) quadrant of formed part showing the relative motion of points A and B.
The forming of thermoset composites
451
Figure 11.12 shows the relative inter-ply movement for a quadrant of a hemisphere. The maximum occurs when x = y ~ 0.934R. At this location/~ ~ 0.297. Therefore the maximum relative inter-ply movement M m a x is given by M m a x ~ 0.297R
(11.10)
This represents a rather large displacement. For example for a 7.5 cm radius hemisphere, the maximum inter-ply movement necessary to form an ideal [0~ ~] part is of the order of 2.25 cm. In practice this movement should occur in the resinrich layer that is found on the surface of most prepreg plies. The inter-ply shear required is calculated by dividing this relative movement by the thickness of the inter-ply layer, which for most materials is of the order of a few fiber diameters. This calculation yields a value for shear that is on the order of 1,000 or roughly two or three orders of magnitude larger than the in-plane shear.
11.2.3. Ideal shears to form a curved C-channel [3] 11.2.3.1. In-plane shear Figure 11.13 shows the kind of C-channel we are concerned with in this work; the two contours are arcs of concentric circles with radii R1 and R 2. Figure 11.14 shows the inner flange of radius R1. Note that the geodesic curvature of the fiber paths is not changed by unrolling the flange onto the flat. We consider the 0 ~ plies where the fibers run along the length of the C-channel, i.e. in the x-direction. The appropriate ideal fiber mapping is the one where shear is not required on the top face of the Cchannel; all shearing occurs on the flanges. The tangent angle c~ on the top is duplicated on the flange [11]. Therefore the shear at point P is simply the angle enclosed by the arc, which is c~. We can obtain a more general expression for the shear at any point on the flange by considering the geometry shown in fig. 11.15. The distance between fibers fo and fp is, A -- R I ( 1 - c o s or)
Fig. 11.12. Plot of inter-ply displacement on one quadrant of a hemisphere.
(11.11)
452
H. Li and T. Gutowski
R1
Fig. 11.13. Dimension of curved C-channel.
~
0
P
P (x=otR1) X
Inner Flange-Top View
Unrolled Flange
Fig. 11.14. Illustration of a fiber path that passes along both the top and the inner flange of a curved Cchannel.
/oN
(a
_O t~
/~
Ptx=o~R1)__ -x
S
Fig. 11.15. Parameters used to determine the shear along a fiber segment, A - A o , on the inner flange of a curved C-channel.
The forming of thermoset composites
453
The distance lateral to the fibers is given by Sn--OAo
(11.12)
PA = Sn - A = Sn - RI(1 - c o s a )
(11.13)
Then,
and the co-ordinates of the point A are given by Xa = Xp + PA sinc~ = Rlc~ + [Sn - RI(1 - cosot)] sinc~
(11.14)
YA = PA cos c~ = [Sn - R1 (1 - cos a)] cos ot
(11.15)
The length of the fiber segment A o - A is given by
If
k, dot ] + \ dot J
dot
(11. 16)
O
Therefore the shear at any location along the fiber on the inner flange can be related to the fiber length by the expression, If = 2R1 sin F12 + ( S n -
R1)F12
(11.17)
A similar analysis yields the following expression for the outer flange: If = 2R2 sin F12 - (Sn + R2)Fle
(11.18)
When R1 (or R2) and Sn are known, for any given fiber length we can calculate the shear using Newton's iterative method. A useful result, however, is that the maximum shear required to form a C-channel is simply the angle of the enclosing arc a. Figure 11.16 shows an ideal fiber mapping for a 90 ~ ply where in-plane shear is allowed on the top face. This is consistent with some experimental observations for large parts. F r o m kinematics again the maximum in-plane shear required is simply or. To determine the inter-ply shear, however we need to know the positions of all fibers
l
02
O,
Fig. 11.16. Top view of a C-channel 90~ mapping.
454
H. Li and T. Gutowski
after forming. Referring to the co-ordinate system given, the parametric equations describing the curve P - B are, x = R1 cos01 - Rl(Ctl - 01) sin01 y = R1 sin01 + Rl(Ctl - 01)cos01
(11.19)
w h e r e 0 ~<01 ~
(11.20)
v = Rz(sin or2 - sin 02) sin 02 where 0~ ~<02 ~<~2, in which 0~ can be d e t e r m i n e d by n
u] I
-- R 2 sin or2
0~=0~
11.2.3.2. I n t e r - p l y shear
T h e fiber m a p p i n g for a [0~ ~ lay-up on the inner flange o f a c u r v e d C - c h a n n e l is s h o w n in fig. 11.18. T h e relative inter-ply m o v e m e n t s between the plies at p o i n t A in the 0 ~ a n d 90 ~ fiber directions are d e n o t e d 8o a n d 89o. It can be s h o w n t h a t u .,q
C
C'
,
02
Fig. 11.17. C-channel 90~ mapping: outer flange. 01
Sn
P (x=o~R1)
Io I i
Fig. 11.18. Illustration of inter-ply displacement at point A on the inner flange of C-channel.
Theforming of thermoset composites
80 -- If -- (Riot + 1 6 -1R 6
PA
455
sin ot)
,~-(R 1 -+-Sn)ot3
(11.21)
1ot3
and
890----R1 ~- ot+
PA sin ) ~ R1 +PA cos ot - Sn
~,~-1S n ~R1 + Snot2
(11.22)
2 R1 1 -- Snot2 2 Hence the relative displacement between 0 ~ and 90 ~ plies at point A is
(Sint)A ~,~V/82 -+-820 "~ ~ (1 R1 ot3) 2-~-(~ Snot2)
(11.23)
= -1otZv/R2ot2 + 9S 2 6 For the outer flange, for a point where in-plane shear of a 0 ~ fiber is ot, a similar analysis leads to the following result:
8int ~,~ 1 ot2v/R2ot2 + 9Sn2
(11.24)
For the example of a curved C-channel that is 61 cm long and has a 244 cm inner radius, the in-plane shear is approximately 0.125, while the maximum relative movement between the plies is of the order of 0.11 cm. Dividing the latter by a typical inter-ply spacing yields an inter-ply shear value of the order of 50.
11.3. Thermoset forming experiments and forming limit analysis 11.3.1. Laminate wrinkling and its modeling
Forming experiments show that the quality of a specific part is strongly influenced by the geometry of the part [2,3,10]. Even for parts with single curvature, under inappropriate conditions, simple bending may induce failure modes such as in-plane or out-of-plane fiber buckling and tow-splitting in some layers of a laminate. On the other hand, laminate wrinkling, the out-of-plane failure mode that involves all the layers, primarily occurs in the forming of doubly curved parts and usually dominates among the failure modes. Since the majority of practical composite parts feature double curvature, laminate wrinkling will be the topic of this section. Figure 11.19 shows comparisons between parts with and without laminate wrinkling.
456
H. Li and T. Gutowski
Fig. 11.19. (a) Illustration of laminate wrinkling on a 16-ply [0~ ~ hemisphere, (b) the same hemisphere formed without wrinkles, and (c) C-channels with and without laminate wrinkling.
A reasonable model for laminate wrinkling requires, among other things: (1) welldefined constitutive equations for the materials involved in the deformation, including the laminate and the diaphragms, and (2) a proper instability failure criterion. Up to now, most models have treated the composites as anisotropic viscous fluids (see reference [1] for a good review) and the corresponding stability analyses are viscous [14,15]. When studying the pure composite shear modes these models are invaluable. However, for more complicated modes of deformation the composite is viscoelastic. For example, even in simple bending one sees a viscoelastic transient. Hence, in our analysis presented here, we will make the simplifying assumptions that the shear behavior of the composite is viscous (with an initial yield stress), but the
Theformingof thermosetcomposites
457
buckling behavior is (initially) elastic [3,4]. We make the simplification that the stored elastic stress, primarily in the fibers, leads to the laminate wrinkling phenomena. This assumption is based upon some of our observations, including in some cases, spring-back in the formed laminates. As a first approximation, we will assume the shear strain mapping for an ideal composite, then, with the appropriate constitutive equations for the material deformation behavior, it is possible to estimate the magnitude of the stresses induced in the composite during forming. This information, in combination with the stiffness properties of the diaphragm and the composite's inherent resistance to buckling, then forms the basis of the wrinkling scaling laws which are presented in this section. The critical condition that leads to elastic laminate wrinkling can be developed by using an energy analysis [16]. Figure 11.20a shows a free body diagram for a simplified composite column with length Lb and constraining diaphragm during forming, and fig. l l.20b shows the geometry of a buckled composite. The work of deformation AT is given by AT -- P . A
(11.25)
The strain energy A U of the system is
(11.26)
A U - - A U d 'l- A U c
where the superscripts d and c represent the diaphragm and composite respectively. The strain energy associated with stretching of the diaphragm is given by A U d ~ 2FD. A
(11.27)
and that required for bending of the composite is Zb
l M2
kbE(t)I
. 2 E ( t ) I d x - ~ L~
A Uc -
A
(11.28)
0
FD.gl.--I
Lb
(a) I
I
A
(b)
Fig. 11.20. (a) Element of composite and diaphragm material before forming, and (b) the composite element after deformation.
458
H. Li and T. Gutowski
where M is the bending moment, I is the moment of inertia, E(t) is the time-dependent stiffness, and kb is a constant determined by boundary conditions. The critical load, Pent, for wrinkling occurs when AT = A U, which leads to Pcrit -- 2FD +
kbE(t)I
L-------Tb
(11.29)
11.3.2. Material rheology and scaling laws for compressive forces The shearing behavior of the aligned graphite fiber/epoxy prepregs was characterized by a three-point bending test which produced a through-the-thickness shear. By assuming the material is transversely isotropic, and by adding multiple plies to add inter-ply effects, one can measure the 1-'13 and I'12 material response directly, and infer the 1-'v3response [17]. The test configuration and typical results for AS4/3501-6 prepreg tested at different rates of deformation are shown in fig. 11.21. The tests were carried out on an Instron 1125 mechanical test machine. All results show a transient rise in stress followed by a steady state. The rise time was of the order of 1 second for the slower cross-head speeds with a sample length of 5 cm. (Note that in general the rise time is a viscoelastic phenomenon, and depends upon the square of the sample length; see [18]). The average strain rate dependence can be observed, for example, by plotting the maximum force at steady state versus the maximum shear rate, as shown in fig. 11.22. These results show a power-law rate dependence at low temperatures and high shear rates, and a significant reduction in rate and temperature effects at high temperatures and/or low shear rates. Apparently, the material behavior is dominated by the polymer in the former regime, and the fiber network in the later. For example, time shifting the data shown in fig. 11.22 shows a distinct deviation from the WLF equation at high temperatures and low shear rates. From these results an effective power law viscosity can be estimated for high rate and low temperature regime, and an effective yield stress can be estimated in the high temperature and/or low rate regime. The addition of multiple plies showed that the effective in-plane viscosity and the inter-ply viscosity are comparable at low shear
Fig. 11.21. Drape test data for AS4/3501-6 graphite/epoxy material tested at different deflection rates.
The forming of thermoset composites
459
1000
oD 100
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ell
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ii
. . . . . . . .
0
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9
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*
46"C [
[]
.u ,
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o[]
9
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n
I
|
.0001
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9
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9' ' " ' I
.001
'
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.01
. . . . . .
I
9
. . . . . . .
.1
1
Max. Shear Rate [1/sec]
Fig. 11.22. Strain rate dependence of maximum load for drape testing of AS4/3501-6 graphite/epoxy material at different temperatures. rates (within 10%). The apparent shear viscosity for the Hercules AS4/3501-6 for various temperatures and crosshead speeds is shown in fig. 11.23. Assume that the width of a laminate is w and the thickness of each ply of prepreg is H, then the two principal contributors to the in-plane compressive force in a composite with Np plies can be scaled as (11.30)
F12 ~ NpHw(ro + mFT2)
10 7
B [] O
.0025/sec @ .0125/sec [] .025/sec
.125/sec .25/sec
10 6
i
[]
10 s
10 4 I
0.00306
0.00324 1/T [1/Kelvin]
Fig. 11.23. A p p a r e n t viscosity o f A S 4 / 3 5 0 1 - 6 material.
0.00; 342
460
H. Li and T. Gutowski
and "n Fv3 ~ UpLw(ro + mFv3)
(11.31)
and are illustrated in fig. 11.24. The labeling system corresponds to the material and part co-ordinate systems shown in fig. 11.5.
11.3.3. Laminate buckling res&tance The forces that oppose wrinkling deformations originate from the inherent resistance of the material itself and from the restraining force supplied by the diaphragm as it is stretched. The trends in the wrinkling data consistently suggest that even without the support of a diaphragm, the composite material can resist wrinkling to some extent. In an effort to evaluate this inherent wrinkling resistance, a number of buckling tests were carried out on cross-plied samples. The test geometry was based on an Euler buckling test column with fixed ends. Samples were either 8 or 16 plies thick and the test direction was chosen to be that which was weakest, i.e. in the 45 ~ direction on a [0~ ~ laminate and at 22.5 ~ on a [0~176 ~ lay-up. Samples were loaded instantaneously and were deemed to be wrinkled when a lateral deflection of 6 mm was reached. Results of room-temperal~ure (22~ buckling tests are shown in fig. 11.25. The critical Euler buckling load for this test geometry is given by
4zrZE(t)I Pcrit --
L~
4zrZE(t)wNpH 3 --
lZL 2
(11.32)
where I = moment of inertia, L b = length of column, E(t)= time-dependent stiffness of the composite material, and 1 ~
w
H
F12
L.
L
v3 Fv 3
Fig. 11.24. Relevant dimensions used to develop scaling laws for in-plane and inter-ply shear forces.
The forming of thermoset composites
461
4000
]
3500
r~ ' 8 p l i e s [ 9 16 plies]
3000 -mpm r
2500
o 2000 ,..1 1500
9 []
9
500
O0
0o
0
50
100
[]
150
200
9
0
250
[]
300
350
400
450
500
Time(sec) Fig. 11.25. Buckling d a t a for AS4/3501-6 prepreg material.
11.3.4. Diaphragm stiffness To estimate the buckling resistance supplied by the diaphragm requires estimation of the diaphragm tension transverse to the potential buckle as given in eq. (11.29). This in turn can be estimated from a knowledge of the diaphragm strain during forming, and its constitutive behavior. To determine the critical diaphragm strains a grid pattern was printed on the diaphragm and the deformations measured after forming. For hemispheres the critical diaphragm strain is in the circumferential direction, while for curved C-channels, it is along the length of the flange. For the purposes of calculating diaphragm tension forces we designate the critical stress (in the direction transverse to the potential wrinkle) as Crll. The generalized expression for the stress in the 1-direction of biaxial tension of a rubber membrane is given by [19], O'll - -
(
2 X2 - ~~ ' l k l
+ X2
(11.33)
where U is the strain energy function for an elastic solid and l lX2 are the extension ratios in the 1- and 2-directions. The invariant terms I1 and/2 are given by, I1 - X12 + X 2 + X 2 1
1
(11.34)
1
/2 -- 7X + X--~q- X--~
(11.35)
From conservation of volume, ~.1~.2~, 3 --
1
(11.36)
462
H. Li and T. Gutowski
Hence, 1
Ao
~'1 = ).2~.3
(11.37)
A
where Ao and A correspond to the original and current cross-sectional areas of the diaphragm. We can therefore write an expression for the nominal stress in the 1-direction as, a]~l - 2(~.1 _ ~ 1 )(0~1 + ~2 0~2) ~.l,k2
(11.38)
A similar expression can be derived for the nominal stress in the 2-direction: a~.2 - 2()~2 - ~ 1 )(0~1 + ~.12 OU) ~,1~,2 -~2
(11 39)
In biaxial testing ~.1 and )~2 can be varied appropriately to fix the invariant terms and the relevant partial derivatives of the strain energy function, U, can be evaluated. For convenience we can rewrite eq. (11.38) as O'~1--
)'1 - ~--~5~2 ),1),2
1 +--
ao
(11.40)
where ao=2~ 1
OU
(11.41)
o/1
OU/OI2 (11.42)
-k = OU/011
Rivlin and Saunders [20] determined the appropriate quantities for vulcanized rubber. They found that OU/OI1is approximately constant and independent of 11 and I2. OU/OI2 is independent of/1 but is a linear function of I2. 1/k is thus a weak linear function of I2 only. Thus, from Rivlin and Saunders data, 1
= 0.152 - 0.0368 x 12
(11.43)
For uniaxial tension )'2 -- )'3 -- ~
1
(11.44)
and eq. (11.40) can be rewritten,
O'~1--()'1- ~12)( 1 + ~~l)aO
(11.45)
We can now fit this equation to uniaxial test data for the different thicknesses of silicone rubber diaphragms to determine the values of the constant a0. These values
The forming of thermoset composites
463
are shown in table 11.1. Agreement between the tests and the model is shown in fig. 11.26. Table 11.2 shows measured values for the extension ratios for formed hemispheres of different sizes. It also shows the constants and invariants needed to calculate a~l from eq. (11.40). Note that )~1 is circumferential extension close to the edge of the part and )~2 is measured in the orthogonal direction at the same location. The diaphragm tension force is then simply:
FD -- DWa~l
(11.46)
where D is the undeformed thickness of the diaphragm, and w is the width.
11.3.5. Order o f magnitude analys& In this section, we will employ an order-of-magnitude analysis to estimate the relative importance of F12 versus Fv3, and 2FD versus the buckling resistance of the composite. Consider a laminate subject to wrinkling under certain forming conditions. Assume that L is the length of the laminate in the critical direction which is perpendicular to the wrinkle, and w is the width of the laminate (parallel to the wrinkle). Both of these dimensions scale with the size of the part. The n u m b e r of plies is Np, and the thickness of each ply is H. Then, the ratio of inter-ply and in-plane viscous shear forces can be estimated by
Fv3 ,,~ NpLwm[" n3 ,~ L[" n3 F12
NpHwm[" ~2
HI" 72
L (Fv3) n
(11.47)
H \F12/
TABLE 11.1 a0 for different thickness diaphragms, from uniaxial test data Diaphragm thickness (mm) a0 (kN/m2)
0.397 574
0.794 745
1.588 896
3.175 952
TABLE 11.2 Extension ratios and material constants for the deformation of silicone rubber diaphragm material over parts of different sizes ~'1
~'2
~'3
12
1/k
Hemisphere R = 5.1 cm 6.4 cm 8.9 cm 11.4 cm
1.03 1.04 1.17 1.29
2.25 2.50 2.85 3.10
0.43 0.38 0.31 0.25
6.48 7.87 11.54 16.60
0.13 0.12 0.11 0.09
C-channel L = 30.5 cm 61.0 cm 121.9 cm
1.22 1.02 1.09
2.25 1.42 1.68
0.36 0.69 0.55
8.40 3.55 4.55
0.12 0.14 0.14
464
H. Li and T. Gutowski
3.5
0.040 cm Test 9 0.079 cm Test i 0.159 cm Test 9 0.318 cm Test 0.040 cm Calculated 0.079 cm Calculated . . . . 0.159 cm Calculated " " " 0.318 cm Calculated
3.0 t 2.5 2.0 r~ r~
1.5
om
1.0
9
0.5
0.0 -b 0
1
2 Extension
3
Ratio
Fig. 11.26. Simulation of uniaxial tensile response of diaphragm rubber.
We know that: H ~ 10 - 2 cm, 1"v3/I"12 ~ 10 2 to 3, and n ,~ 0 to 0.45. For a part of dimension L --, 101 cm, this leads to
Fv3
L (I'v3)n
F12 ~ n
\V12]
lO1 10 - 2 .
( 1 0 2 to 3) 0 to 0.45., 103
to
104
(l 1.48)
Hence, according to this analysis, the inter-ply viscous shear stress is the main source of compressive force within a laminate. The relative magnitude of the two resistance forces is approximated by
2FD
~ ~ Felastic
2. 12Dwcr]"1L~
4rcZE(t)wN~H3
6Da~'l L~ -
7rZE(t)N~H3
(11.49)
Since D ,~ 10 -1 cm, cry'1 ~ 106 Pa, L b ~ 100 cm, E(t)~ 107 Pa (room temperature), Np ~ 100 to 101, a ~ 1.6, and H ,-~ 10 -2 cm 3, we get 2Fo ~ 102 to 104
(11.50)
Felastic
Hence, diaphragm tension provides the main support against laminate wrinkling.
11.3.6. Thermoset forming experiments and forming limit diagrams To explore the general validity of these scaling laws a series of experiments were conducted, and then forming limit diagrams were constructed by plotting each result in terms of the diaphragm tension FD and the relative compressive force Fv3. This is shown in fig. 11.27 for a series of [0o/90 ~ hemispheres. The range of forming and part parameters is given in table 11.3 [3]. As can be seen, a relatively clear demarcation between "good" and "wrinkled" parts exists. The figure graphically illustrates
The forming of thermoset composites
465
40
O "~ 3O
O O
O(ID
D
@ @
O
@
Z o
o~
20 o
[] o x o O O 0 9
O @
,1~ 111 x .!=1
O 0
9
0.0
00 I
.
0.2
I
9
I
0.4 Relative
R=0.97cm Good R=1.35cm G o o d R=2.61cm G o o d R=5.16 cm G o o d R=7.70 cm G o o d R=8.97 cm Good R=11.5 cm Good R=8.97 cm Wrinkled R=11.5 cm Wrinkled 9
0.6
Compressive
I
0.8
'
1.0
Force
Fig. 11.27. Forming limit diagram for [00/90~ hemispheres (see text). T A B L E 11.3 P a r a m e t e r ranges for experiments on hemispheres
Variable
Lower limit
Upper limit
Temperature F o r m i n g time D i a p h r a g m thickness D i a p h r a g m extension ratio N u m b e r of plies Radius
20~ 1.5 min 0.04 cm 1.03 2 0.97 cm
95~ 4 h 0.159 cm 1.3 32 11.5 cm
that increased diaphragm tension suppresses wrinkling, while increased inter-ply shear resistance leads to wrinkling. In a separate set of experiments, a series of curved C-channels with the range of parameters listed in table 11.4 were formed [3]. In this case, the preform was constructed with [0o/90o/+45 ~] ply orientations. This arrangement is substantially more difficult to form than the [0o/90 ~] or [+45 ~] arrangements with the same number of plies. The reason for this is related to the additional constraints imposed by the [0~ 90o/+45 ~ lay-up. That is, the [00/90 ~ and [+45 ~ arrangements can deform by a trellising type of deformation which substantially reduces the required inter-ply shear (by the mechanism of thickness variation). However, for the [00/90o/+45 ~] lay-up this ability to avoid large inter-ply displacements is severely restricted. As a consequence the in-plane compressive force for the [0o/90o/+45 ~] parts appears to increase by about an order of magnitude over the [00/90 ~] and [+45 ~] parts. Evidence of inter-ply shear for a [0~176 ~ C-channel and the lack of it for a [00/90 ~ one is shown in fig. 11.28.
466
H. Li and T. Gutowski
TABLE 11.4 Parameter ranges for experiments on C-channels. Variable
Lower limit
Upper limit
Temperature Forming time Diaphragm thickness Diaphragm extension ratio Number of plies Length Web width Flange depth
20~ 1.5 min 0.04 cm 1.05 2 30.5 cm 5.1 cm 5.1 cm
95~ 1h 0.159 cm 1.8 32 121.9 cm 20.3 cm 20.3 cm
Fig. 11.28. Comparison of the inter-ply shear occurring at the edge of (a) a [0o/900/+45~ C-channel and (b) a [0o/90~ C-channel with no apparent inter-ply shear. The data is shown in fig. 11.29. Again a clear organization between good and wrinkled parts is apparent. In this case, however, all of the parts are of the same size. When larger C-channels (scaled up by a factor of 2 in every dimension, but with the same enclosed angle 2ct) were formed they did not superimpose on fig. 11.29. In short, the larger C-channels were harder to make than the small ones by a factor larger than that suggested by our previous scaling laws. This difference appears to be due to a shift in the shear patterns for the small and large parts as shown in fig. 11.30. For small parts significant shear in the web is allowed, whereas for the large parts all of the shear occurs in the flanges. The pattern for the large C-channel is similar to the ideal shear shown in the figure, but at a reduced level. The large part data can be superimposed if eq. (11.31) is multiplied by an additional empirical length scale as given in eq. (11.51). A new plot using the empirical length scale for both large and small C-channels is shown in fig. 11.31.
The forming of thermoset composites
467
80 8 plies Good 16 plies Good 20 plies Good mE] 24 plies Good 8 plies Wrinkled 9 16 plies wrinkled o
nC]o
60
z o r~
b~
4O
o Do
O
[]
~ 20 9i,,,0
[] [] 0
9
i
0.0
9 9
l
0.2
9
i
0.4
'
0.6
i
0.8
1.0
Relative C o m p r e s s i v e Force Fig. 11.29. Forming limit diagram for [00/900/+45 ~ 30.5 cm long C-channels.
Ideal Shear C-channel 61 cm long C-channel 31.5 cm long
, 9
[]
0.3
0.2
[]
or3
9
mmmm
mn
[]
t~
0.1
~ 0.0
D D
rm 9
.
, .
[]
[]
i .
,
.
D
,
I
m':'
[] []
o
J
[]
[]
n
Imll--ll u
-
|
~
o |
l
o !-
i
9
u
9
i
9
-1.0 -0.8 -0.6 -0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0 N o r m a l i z e d Transverse Distance, S n / S n , m a x Fig. 11.30. Comparison of ideal fiber and actual fiber shears for C-channels.
Fv3 ~ NpZw(To -k- mlqn3)
(L)2 Zcha r
(11.51)
Note that a similar size scaling effect for the actual shear was also observed for hemispheres. The total fiber shears for a series of hemispheres of different sizes is shown in fig. 11.32. The basic trend is that the fibers follow the ideal shear up to about half-way down the side of the part. Toward the edge, the fibers deviate
H. Li and T. Gutowski
468 70 []
6O
[] 30.5cm Good 30.5 cm Wrinkled 61 cm Good R 61 cm Wrinkled
•12
so ,......,
o 4o
,1".4
3O 0
~o ~ 20
O
n
[]
[]
.i.=,
[]
m
KI
[]
o
9
I
o.o
0.2
9
9 I
'
I
"
0.4 0.6 Relative Compressive Force
I
0.8
1.0
Fig. 11.31. Forming limit diagram for [00/900/4-45 ~ C-channels of two different sizes, plotted using the empirical length scale correction given in eq. (11.51).
Ideal 9 R=1.35 cm
o R-2.61 cm O R=8.97 cm
O R=ll.5 cm
3.0
~9) oO
2.0 ~r3
0
9
~., 1.0
0
o
oO .od9
O 0.0
0.0
0.2
0.4
0.6
0.8
1.0
Normalized Transverse Distance, Sn/Sn, max Fig. 11.32. Comparison of ideal fiber and actual fiber shears for [00/90 ~ hemispheres.
substantially from the ideal. As with the C-channels, larger parts obtain shear values closer to ideal. 11.4. Concluding remarks
In this chapter we have explored the kinematic approach to understanding the forming behavior of advanced composites. This method seems to have been first introduced by Pipkin and Rogers [9] for flat composites. Here, we have outlined
The forming of thermoset composites
469
several important results from extending this technique to the forming of complex shaped parts [3,10,11]. Clearly, the most appealing feature of this approach is that the deformation strains can be obtained directly from part shape geometry. Following this line of reasoning one can show the enormous magnitude of the required inter-ply strains required to make ideal complex shaped parts, such as hemispheres and curved C-channels. We then applied the kinematic approach with an elastic buckling analysis to explore the laminate wrinkling phenomenon. In this approach, the viscous behavior of the composite shows up as the mechanism by which the composite can slide into its ideal configuration. The assumption that the laminate behaves elastically in the buckling phenomenon is based in part on our observation that the laminate can sustain a finite compressive load for a time much longer that the time scale for the wrinkling phenomenon. In general, however, we would expect that the laminate will display both elastic and viscous behavior and this simplification will not be able to capture this entire range. Nevertheless, the resulting "forming limit" diagrams seem to give partial confirmation of the utility of this approach, clearly showing the influence of the various forming parameters on laminate wrinkling. Our work here has, however, clearly shown the limits of the kinematic approach, both from the direct measurement of part strains and from our inability to collapse wrinkling data for curved C-channels of different sizes. These two observations may, in fact, be related as we suggest in the text of this chapter. Nevertheless, we have clearly observed the deviation from ideal kinematics, due to the availability of other lower energy modes of deformation. Future work on the forming of composites will have to address this issue. Unfortunately, the deviations from ideality which we have observed have been in the "wrong" direction. That is, it appears that large parts are harder to make than one would predict by this approach (or smaller ones are easier). However, the net effect is that in order to extend the range of the forming process some new variations on the process will need to be developed. To close this chapter we will outline here two variations on the diaphragm forming process which we have developed at MIT. The intention of this work is to help extend the range of forming of advanced composites.
11.4.1. Reinforced diaphragmforming [8] In this process a stiff, directional reinforcement is placed with one or both diaphragms to suppress the out-of-plane wrinkling. A part with woven reinforcement made in our laboratory is shown in fig. 11.4. This technique has the potential to greatly expand the range of parts that can be made. This is illustrated by the results of experiments on the Toray T800H/3900-2 carbon/epoxy system. The thickest 61 cm long C-channel that could be made from this material system by the standard diaphragm forming technique had only 2 plies. Using the reinforced diaphragm forming process a quasi-isotropic lay-up that was 32 plies thick was successfully formed. This was achieved using a non-optimized reinforcement, and without employing any special procedures for controlling the deformation. If this technology
470
H. Li and T. Gutowski
is combined with sensing techniques and control strategies, the range of part sizes could be further increased. 11.4.2. Inflated tool diaphragm forming [6]
This process provides a means by which thicker parts can be made sequentially, a number of plies at a time. It also allows the parts to be formed directly on to the tool. Thus the part can be taken directly to autoclave cure without the need for removing it from the forming tool. An inflated "tool" is used to support the lower surface of the composite laminate. The tool is placed above the lower of two diaphragms. The lower diaphragm is inflated around the tool to the extent that it forms a surface on which the preform may be supported (fig. 11.33a). The stiffness and/or thickness of this diaphragm may be tailored in such a way that when it is fully inflated it forms a suitable surface on which the preform will rest. The preform is located accurately relative to the tool and the top diaphragm is placed above it. The diaphragm is sealed around its perimeter to a plate which ideally should be adjustable in the vertical direction to allow it to be brought in close proximity to the tool. The forming process is initiated by inflating the lower diaphragm until it makes contact with the sides of the tool and the preform (fig. 11.33b). The forming process is advanced by the application of pressurized air to the top of the upper diaphragm, and the simultaneous bleeding off of air from beneath the bottom diaphragm (fig. 11.33c). In this way the rate and sequence of deformation can be controlled along with the throughthe-thickness pressure that is applied to the laminate. When the process is complete the pressure beneath the lower diaphragm is released so that the lower diaphragm
Fig. 11.33. Schematic of inflated tool forming process.
The forming of thermoset composites
Fig. 11.34. Inflated tool drape forming: (a) early stage of the forming sequence; (b) 8-ply [0~176 successfully formed by the inflated tool forming process.
471
~ part
retracts fully to allow the composite to make contact with the tool (fig. 11.33d). The above steps are then repeated until the desired part thickness is reached. Preliminary experiment set-up is shown in fig. 11.34. By this technique we have been able to sequentially form very thick parts several plies at a time. References [1] C.L. Tucker III, "Forming of Advanced Composites", Chapter 9, Advanced Composites Manufacturing (ed. T.G. Gutowski), John Wiley, New York (1997). [2] S. Chey, "Laminate Wrinkling during the Forming of Composites", M.S. Thesis, Department of Mechanical Engineering, MIT (1993). [3] H. Li, "Preliminary Forming Limit Analysis for Advanced Composites", M.S. Thesis, Department of Mechanical Engineering, MIT (1994). [4] T.G. Gutowski, G. Dillon, S. Chey and H. Li, "Laminate Wrinkling Scaling Laws for Ideal Composites", Composites Manufacturing, 6 (1995) 123. [5] A.S. Tam, "A Deformation Model for the Forming of Aligned Fiber Composites", Ph.D. Thesis, Department of Mechanical Engineering, MIT (1990). [6] T.G. Gutowski, G. Dillon, S. Chey and H. Li, "Apparatus and Method for Diaphragm Forming", United States Patent Application Serial No. 08/433/125 (1995). [7] M.R. Monaghan, P.J. Mallon, C.M. OBr/tdaigh and R.B. Pipes, Journal of Thermoplastic Composites, 3 (1990) 202. [8] T.G. Gutowski, G. Dillon, H. Li and S. Chey, "Method and System for Forming a Composite Product from a Thermoformable Material", United States Patent No. 5,578,158 (1996). [9] A.C. Pipkin and T.G. Rogers, "Plane Deformation of Incompressible Ideal Fiber-Reinforced Composites", Journal of Applied Mechanics, 38 (1971) 634. [10] A.S. Tam and T.G. Gutowski, "The Kinematics for Forming Ideal Aligned Fibre Composites into Complex Shapes", Composites Manufacturing, 1 (1990) 219. [11] T. Gutowski, D. Hoult, G. Dillon and J. Gonzalez-Zugasti, "Differential Geometry and the Forming of Aligned Fibre Composites", Composites Manufacturing, 2 (1991) 147. [12] K. Golden, T.G. Rogers and A.J.M. Spencer, "Forming Kinematics of Continuous Fibre Reinforced Laminates", Composites Manufacturing, 2 (1991) 267. [13] D.J. Struik, "Lectures on Classical Differential Geometry", Dover, New York, 2nd edition (1961). [14] B.D. Hull, T.G. Rogers and A.J.M. Spencer, "Theory of Fibre Buckling and Wrinkling in Shear Flows of Fibre-Reinforced Composites", Composites Manufacturing, 2 (1991) 185.
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[15] B.D. Hull, T.G. Rogers and A.J.M. Spencer, "Theoretical Analysis of Forming Flows of ContinuousFibre-Resin Systems", Flow and Rheology in Polymer Composites Manufacturing (ed. S.G. Advani), Elsevier, Amsterdam (1994). [16] S.P. Timoshenko and J.M. Gere, "Theory of Elastic Stability", McGraw-Hill, New York, 2nd edition (1961). [17] E.T. Neoh, "Drape Properties of Thermosetting Prepregs", M.S. Thesis, Department of Mechanical Engineering, MIT (1992). [18] A.S. Tam and T.G. Gutowski, "Ply-Slip during the Forming of Thermoplastic Composite Parts", Journal of Composite Materials 23 (1989) 587. [19] I.M. Ward, "Mechanical Properties of Solid Polymers", John Wiley, New York, 2nd edition (1971). [20] R.S. Rivlin and D.W. Saunders, Phil. Trans. of the Royal Society A, 243 (1951) 251.
Composite Sheet Forming edited by D. Bhattacharyya 01997 Elsevier Science B.V. All rights reserved.
Chapter 12
Roll Forming of Sheet Materials S.J. MANDER,* S.M. PANTON, R.J. DYKES and D. B H A T T A C H A R Y Y A Composites Research Group, Department of Mechanical Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand
Contents Abstract 474 12.1. Introduction 474 12.2. Roll forming equipment and tooling 476 12.2.1. Feeding device 477 12.2.2. Roll forming mill 477 12.2.2.1. Form tooling 477 12.2.2.2. The rolling mill 479 12.2.3. Auxiliary equipment 480 12.2.4. Operating conditions 481 12.2.4.1. Roll lubrication 481 12.2.4.2. Roll quality and variation in material properties 481 12.2.4.3. Roll setting 482 12.2.4.4. Common roll forming defects 482 12.3. Conventional form roll design 483 12.3.1. Finished section design 483 12.3.2. Strip width 484 12.3.3. Roll schedule and flower pattern design 484 12.3.3.1. Section orientation 485 12.3.3.2. Sequencing of bends 486 12.3.3.3. Bending at each stage 487 12.3.3.4. Means of forming bends 489 12.4. Computer-aided design in roll forming 489 12.5. Deformation analysis of roll forming 491 12.5.1. The deformed surface in roll forming 491 12.5.2. Modelling the strain distribution during roll forming 495 12.5.3. Finite element analysis of roll forming 498 12.6. Roll forming of thermoplastic material 498 12.6.1. Materials 499 12.6.2. Roll forming equipment and procedures 500 12.6.3. Quality assessment of formed parts 501 12.6.3.1. Prediction of spring-back/forward in FRTPs 504 12.6.4. Temperature control 506 12.6.4.1. Development of cooling model 507 *Currently at McKinsey & Company, Sydney, NSW, Australia. 473
474
S.J. Mander et al.
12.6.5. Deformation length and strain development 12.6.5.1. Strain measurement 510 12.6.5.2. Deformation length 511 12.7. Concluding remarks 512 Acknowledgements 513 References 513
510
Abstract
Roll forming is one of the most versatile forming processes, capable of producing a wide range cross-sections from sheet materials by passing a strip of material through successive pairs of rolls. The scope of this secondary deformation process can even be further advanced with the addition of suitable auxiliary operations. The first part of this chapter gives a brief introduction to the process itself in the context of metallic sheets and explains the basic concepts of roll forming. The latter part shows the possibility of roll forming continuous fibre-reinforced thermoplastic (FRTP) sheets and identifies the important parameters influencing the success of the process. The inlet temperature of the strip, the cooling rate and the fibre architecture appear to have the most significant effects on the product quality in relation to fibre buckling, product curvature and material spring-back or spring-forward. The in-situ measurement of membrane strains in the product, while roll forming FRTP sheets, shows a distribution similar to that obtained in metallic sheets. The deformation zones under various roll stations appear to be generally larger in the case of composite sheets; however, the effects of different geometric parameters again seem to follow the trend obtained with metallic sheets. 12.1. Introduction
In roll forming a sheet of material is formed into some desired shape by feeding it through successive pairs of rolls, arranged normally in tandem (fig. 12.1) [1]. For many years this process has been used in various sheet metal industries to produce different cross-sections and it is interesting to note that a sketch of a small, crude roll forming device was found in one of Leonardo da Vinci's notebooks written early in the fifteenth century. However, cold roll forming in its present form is a relatively young forming process and its use did not become widespread until the demand for better and faster methods of producing sheet metal parts was recognised. It is a very versatile secondary deformation process capable of producing a wide range of cross-
.~-~
Fig. 12.1. A typical roll forming process (from reference [1]).
Roll forming of sheet materials
475
sectional configuratiorfs and finds application in automobile, aircraft, construction and general manufacturing industries. Even the space industry has recognised its advantages [2]. It is a high productivity process capable of achieving close quality control and economy in labour and materials, and has the ability to produce long, continuous lengths and profiles of a complexity beyond the capacity of simple press bending process. Auxiliary operations such as fly-cutting, seam welding, etc. often widen the scope of the roll forming process and make it part of a bigger system, resulting in a very high volume of material being processed. Roll forming has a number of advantages over rival techniques such as extrusion, press braking or hot rolling. The process is more flexible than the other processes with the possibility of forming a large range of profiles using clever roll designs. For metallic materials the process can be almost continuous, the only restriction on lengths coming from the ease of handling or transporting. In the future, roll forming could even play a vital role in fabricating space structures. An automated beam builder has already been developed, around three seven-station roll forming lines, which could be used in space to produce the beams that would form the building blocks of space stations orbiting the Earth [3]. Roll forming equipment, tooling and power requirement being relatively expensive, the viability of the process is assessed by considering the equipment costs, production rates and its volume [1]. Like many other material forming processes, roll forming relies heavily on an experience base that has evolved over the years as its practitioners became more skilled. This has also been reflected in the published literature, much of which has been devoted to outlining empirical rules which have found favour within the industry. Many of these rules are very useful and cannot yet be replaced by a more scientific approach. A comprehensive discussion of various aspects of form roll design can be found in the collection of articles edited by Halmos [4]. However, due to the inherent complexity of the process that involves three-dimensional bending, stretching and some non-unique boundary conditions, roll forming has remained under-studied compared to many other forming processes. Moreover, the overall success of the process is also dependent on the material input and the operating conditions at the time of rolling, as described in fig. 12.2. The number of parameters describing the roll forming process of even a simple section cannot be underestimated and certain steps are to be followed right from the design stage of a particular machine to make it more versatile and economical in the future. In the last thirty years or so, the research scenario has changed somewhat with the appearance in the literature of a number of scientific papers, both experimental and analytical, along with many papers of generally descriptive nature. As expected, there has also been a growing trend towards the use of computer-aided design and manufacturing of form rolls. These packages have considerably reduced the tedium of calculation and drawing present in the traditional form roll design but few, if any, have attempted to substantially improve the understanding of the basic mechanics of the roll forming process. The possibility of using expert systems to tap the existing body of roll forming expertise is yet to be fully explored. Due to the relatively recent availability of fibre-reinforced thermoplastic sheets significant commercial interest has been generated in the development of low-cycletime, mass manufacturing techniques. Because thermoplastics can be "reshaped" by
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Fig. 12.2. Factors influencingthe roll forming process (from reference[18]). the application of temperature and relatively low forces, they are suitable for the adaptation of established sheet forming techniques, as has been discussed in earlier chapters. This chapter will be primarily devoted to the introduction of roll forming as a sheet forming process and its possible application in the manufacture of fibrereinforced thermoplastic sheet components. The discussion has been divided into three main sections: a brief introduction to equipment and tooling design; the application of computer-aided design and manufacturing, and roll forming of thermoplastic composite sheets. In "Concluding remarks" comments will be made on the commercial viability of roll forming thermoplastic sheets and some of the important factors that are to be considered in future by the reinforced sheet manufacturers and their users.
12.2. Roll forming equipment and tooling In this section attention will be given to the equipment and tooling used in roll forming of metallic sheets to familiarise readers with various terminologies and the research carried out to improve the general understanding of the process. Later, while the roll forming of thermoplastic sheets is being discussed, references will be made to these practices and findings. A sheet metal roll forming line primarily consists of a feeding mechanism (uncoiler), the actual forming unit containing a number of forming stations, and a power unit. The output side which constitutes the third main unit, normally contains auxiliary forming (optional) and straightening equipment and a shearing mechanism.
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Furthermore, coil joiners and handling equipment for decoilers and rolls may also be necessary to complete a truly semi-continuous line. 12.2.1. Feeding device
The main function of this unit is to feed the sheet material properly into the roll forming unit with minimal damage to the sheet and minimising the demand on the entry guides. There are various types of decoilers used in industry, depending on the total cost of the roll forming mill. The hydraulically operated mandrel-type (to fit the coil bore) decoilers are common for reasonably priced machines, though many small operators use simple manual decoilers and depend very much on the guide rolls for the accuracy of alignment and feed. 12.2.2. Roll forming mill 12.2.2.1. Form tooling The actual forming unit contains form rolls mounted on suitable stands and a power unit that drives the forming rolls. The drive input can be given to top or bottom rolls and it is often desirable to keep one roll free. The roll stands are of basically two types: overhung or cantilever type, where the roll spindles are supported at one end (fig. 12.3), and the outboard support type where the spindles are supported at both ends (fig. 12.4). The cantilever-type roll stands are used for thinner sections, whereas the outboard-type stands are more rigid and can accommodate heavier sections. Either type may be designed with fixed or adjustable lower spindles. The form rolls actually render the shape and are normally paired with a top and a bottom roll (fig. 12.5a), but sometimes side rolls are also used, as shown in fig. 12.5b. Often the form rolls are manufactured in segments to accommodate the varying product dimensions with minimal roll cost, a simple example for a channel section
Fig. 12.3. Overhung roll stand.
Fig. 12.4. Spindle type roll stand.
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L
!
(a~
I
.... i....i i.... iiii-I ---
...
I
(b) Fig. 12.5. (a) Top and bottom rolls. (b) Top, bottom and side rolls.
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is illustrated in fig. 12.6. In addition, the split rolls allow easy machining and replacement of the worn-out segment where the wear is unevenly distributed. 12.2.2.2. The rolling mill Having considered in the previous section the tooling which imparts form to coldrolled sections, it will be enlightening to briefly describe the machine on which the rolls are mounted the mill or rolling mill. Furthermore, it will be seen later that the rolling mill must be kept in consideration when form rolls are designed. Whilst a detailed analysis of rolling mill design would be inappropriate for this chapter, those seeking a more exhaustive description of the mechanical design of general rolling mills should consult Tsclikov and Smirnov [5]. The primary purpose of the rolling mill is obviously to provide the framework on which the form rolls can be mounted, driven and controlled. Mills can be designed for producing one specific section or, more commonly, for producing a variety of sections, within the constraints on power, size of section and the number of passes imposed by the tool design. The elements of rolling mill can be described in three main sections: (i) structure, (ii) power and transmission, and (iii) executive functions. Structure. The machine structure consists of two main elements, the base and the roll stands. The base should provide sufficient rigidity to restrict the roll stand deflections to a minimum. In addition, common sense dictates a design which eases maintenance whilst restricting access to moving parts. As discussed earlier, the roll stands support the spindles on which the form rolls are mounted and can be mainly of spindle and overhung type. They also contain the mechanisms by which roll positions can be adjusted (or set) relative to one another. The size and complexity of sections that can be produced on any mill is constrained by the inter-stand distance and the number of available passes. While the
TOP ROLL
SPACER
(t.EV'T)
flOP)
BOTIOM ROLL
(LEFT)
SPACER
fnOaqOM)
TOP ROLL
fRIGHT)
BO'I'IOM ROLL
fRIGHT)
Fig. 12.6. Example of the use of split rolls for the production of channel section with varying base widths.
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inter-stand distance (pass or pitch length) normally remains fixed on a mill, the number of passes available can be made more flexible by designing the mill's structure in a modular fashion, allowing additional passes to be "bolted on" to a mill. Power and transmission. The form rolls are driven by an electric motor through some form of transmission system. Power capacity can provide a constraint on the sections which can be rolled on a machine, particularly with thicker materials. Individual passes are driven from the motor by some form of linkage, commonly in the form of chaining. The rolling speed is important both technologically and economically and is dependent on the type of material and section. Thus a generalpurpose mill requires some form of gearing and/or speed controller to allow variation in rolling speed. Executive functions. The previous sections have described how rolls are mounted and driven. The final element in a rolling mill is clearly to incorporate control (or executive) functions, for initiating and terminating rolling operation and for synchronising the process with auxiliary equipment.
12.2.3. Auxiliary equipment All rolling mills are combined with auxiliary equipment of some description, it is necessary either to ensure as continuous a flow of the strip as possible through the mill or to alter the nature of the finished product. Whilst the equipment mentioned below cannot be considered a comprehensive listing, it includes the more widely used equipment and is representative of the diversity of equipment available. The strip is commonly transported and stored in coils which are first mounted onto a strip decoiler that rotates and unwinds the coils prior to passing through the rolls. Depending on the material used and the section required, it may be necessary to pass the strip through a straightening set-up. This consists of a series of fiat rolls, which are offset and which progressively increase the longitudinal radius of curvature until any residual curvature in the section is effectively eliminated. The major impediment to continuous flow of strip occurs when a coil ends. A method of overcoming this is to create a stockpile of stored metal between the decoiler and the mill in a loop accumulator. This allows sufficient time for a new coil to be positioned and the ends of the coils to be butt-welded together, thus allowing virtual continuous production. Roll forming is a very flexible process - - virtually any profile of constant thickness can be produced. Auxiliary equipment can also be used to alter the nature of the section, hence giving the process more flexibility. To ease transportation, finished sections are either cut to length or formed into a coil. The cutting operation is performed using a shear. This can be done by stopping the mill, or by using a "flying shear", where the shear cuts whilst moving at the same velocity as the strip, thus allowing continuous production. Joining equipment can also be incorporated into the mill, an example being seam welding (for instance, in tube rolling processes). Also sections may be notched, pierced or embossed before or after rolling, thus allowing a diverse range of products to be produced without the need for finishing operations.
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12.2.4. Operating conditions The previous sections have introduced the process of cold roll forming and the tooling and equipment required in cold roll forming. It remains to describe the conditions under which roll forming is carried out. The operating conditions will be described under three headings: 1. roll lubrication 2. roll quality and variation in material properties 3. roll setting
12.2.4.1. Roll lubrication The decision whether to use a lubricant and, if so, about the type of lubricant and the method of application, must be given careful consideration on the basis of the type of material being rolled, the rolling speed, the tooling material and the likelihood of damage to the tooling and material surfaces. Careful lubricant selection can reduce sectional distortion due to heat; it can give a better surface finish by reducing scuffing and by "flushing" away debris from between the rolls; additionally it can reduce wear and prolong tool life. However, whilst the application of a lubricant can have these numerous advantages, unnecessary or incorrect application of a lubricant can similarly result in significant disadvantages such as sectional distortion due to differential friction and cooling, surface defects, staining, blushing, blistering and peeling, and the need for auxiliary degreasing operations. So many sections are rolled without lubricant. Care must also be taken while forming a precoated material with special surface finish as the chemical effects of the lubricant may damage the surface. There are numerous types of lubricant for cold roll forming and several methods of application. It would be inappropriate to detail these, so readers requiring further detailed information should consult the papers written by Ivaska, tabulating the type of lubricant suitable for specific metals [6] and investigating and analysing the most common lubricant problems [7].
12.2.4.2. Roll quality and variation in material properties Two factors, which contribute to the sectional quality but are difficult to accommodate within the tooling design, are the roll quality and the variation in material properties. Roll form tooling is necessarily designed on assumptions regarding the material being formed. In practice the material properties will vary from these nominal values, so the finished section for the same tooling will alter. This is a particular problem where the material's tendency to return to its original shape (spring-back) is altered, or the material thickness alters. Roll quality can affect the finished section in two main ways. First, as the roll surface finish deteriorates, the finished section acquires marks and surface defects. Secondly, if the rolls have been inaccurately machined or the rolls have worn, then the finished section profile will clearly differ from the required finished section.
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12.2.4.3. Roll setting The setter, the person who sets the rolls on the mill, provides the link between the design and development of tooling and the finished section production. The importance of roll setting should not be u n d e r e s t i m a t e d - a good setter can salvage a poor design and can suggest those minor modifications to the design that will improve the final section quality. Similarly, the quality of roll design can become irrelevant if the rolls have been set carelessly. Therefore, a setter must combine a considerable level of practical skill with a thorough knowledge of the forming process. In practice, rolls are normally positioned on the mill one pass at a time. As each pass is installed, strip is fed through the rolls and bent towards the required shape. The setter attempts to ensure that the section exiting the pass is as free from sectional defects as possible. Thus the roll stages are built up successively with minor modifications being made to the roll design as required There has been little scientific study of the factors involved and the methods employed by the setter in achieving the best section from any given roll design. Thus roll setting has remained a mystery in a similar way to roll design. So, whilst there are more subtle factors involved, the main methods employed by the setter seem to be moving the passes out of line, varying roll pressures, varying the pass length (if possible) and using inter-pass forming devices. 12.2.4.4. Common roll forming defects There are three main types of produce defect. These are: 9 surface markings 9 longitudinal curvature 9 edge buckling These problems are all caused by excessive strain occurring in the sheet material during forming. If these defects are severe enough the product may be rejected. Surface markings. Surface markings detract from the aesthetic qualities of the product. They generally do not affect the product in any other way. In an article published by Jimma and Ona [12] four types of common markings are identified. These are shown in fig. 12.7. The markings shown in figs. 12.7a, b and d are caused by excessive deformation occurring in the final stages of forming. Figure 12.7c shows a type of marking known as the "orange peel" effect and this appears during the initial or middle stages of forming. Highly polished materials such as bright steel, stainless steel and aluminium are particularly susceptible to this type of flaw and they tend to lose their lustre and become whitish in colour. Longitudinal curvature. This occurs when the strain induced in the top and bottom of the channel section are not equal. The formed product curves in the longitudinal direction, as shown in fig. 12.8. Curvature must be minimised as the product formed should ideally be straight. Long items such as roofing iron should only contain a small amount of curvature while short products, such as wall fittings, may allow some curvature. Edge buckling. Edge buckling generally occurs in products containing large flanges. The section buckles resulting in a corrugated edge. This is obviously unacceptable as the final product must have a straight edge. Another buckling
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<)fl(lf] I
(a) Deep narrow scoring near the flange edge.
(b) Long shallow scores near the flange centre
.
(c) Fine indentation marks at the bend site
(d) Narrow scoring near the flange edge
Fig. 12.7. C o m m o n surface defects that appear in roll formed products.
(a) Positive curvature.
(b) Negative curvature.
Fig. 12.8. Longitudinal curvature.
(a) Edge buckling.
(b) Mid web buckling.
Fig. 12.9. Buckling defects.
problem is the so-called "oil can" effect. This occurs in sections containing large webs and consists of a series of small buckles in the web centre. This also is not acceptable in the final product. These two types of buckling are shown in fig. 12.9.
12.3. Conventional form roll design 12.3.1. Finished section design Clearly the initial input to the form roll production process must be (paradoxically) some finished shape to work towards. It is perhaps misleading to
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refer to this as the "finished section design" since very often the section has already been designed to meet the requirements of its eventual usage and form roll designers take this definition as their starting point. However, it is often desirable for the finished section designer to work closely with the form roll designer since it is important to design with manufacturing aspects in mind. Design for manufacture is particularly important when designing sections suitable for roll forming since an intelligent section design will be easier (hence cheaper) to roll form and also will produce a better-quality section. Whilst the designer is ultimately constrained by the required service properties of the section, it would be advantageous to consider the following points when finished section is being designed. One important factor in roll forming which can often be altered with little effect on the sectional properties is the inside radius. As a general rule, the minimum ratio of radius to thickness for different materials is guided by the material characteristics and reference tables can be found in various handbooks. Large radii should be avoided wherever possible, since spring-back increases with the ratio of radius to thickness. Blind bends, which cannot be reached by top and bottom rolls, should also be avoided wherever possible because such bends provide considerable production difficulties. Legs which are short with respect to material thickness should be similarly avoided, wherever possible. Deep narrow slots are also not desirable as the rolls required to form such profiles are prone to breakage. 12.3.2. Strip width
Approximate strip widths for roll formed sections can be calculated from section details and they provide sufficient accuracy for section flower development and roll design. The great majority of section designs consists of two types of element, namely linear and circular elements. Although more complex sections can be rolled, clearly the design and manufacture of form rolls will become correspondingly more complex. One of the initial routine exercises performed by the form roll designer is to calculate the strip width since this determines the dimensions of the rolling mill on which the section will be produced and allows the procurement of the material. The calculation is done by breaking down the entire profile into individual circular and linear elements and determining each element's stock length separately. If the bend radius is reasonably large (generally ~>20, the material thickness (t) may be assumed to remain constant and the strip arc length is calculated at the mid-thickness plane. However, if the bend radius is less than 2t, thinning should not be ignored and tables, based on observed results and common experience, are consulted to calculate the bend allowances [1,7]. 12.3.3. Roll schedule and flower pattern design
It must be appreciated that a final section can be produced by following various deformation routes and the determination of this route or the roll-pass schedule is
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Fig. 12.10. Roll flower pattern for an equal leg channel section.
(a)
(b)
Fig. 12.11. Roll flower patterns for top hat section: (a) formed from channel section, (b) formed from flat strip. the most important aspect of designing a roll forming mill. An erratic design involves the loss of material and time, and may end up producing unsaleable products. Even a simple C-section or a channel section may be formed through various roll schedules and, due to the appearance of the flange position at different stages, the cumulative pictorial representation of the sections is commonly referred to as flower diagram (fig. 12.10). The example of a slightly more complex "top-hat" section is shown in fig. 12.11, which illustrates the forming of the section from a channel or a flat strip following different roll schedules. The membrane and shear strain distributions in the sheet for producing the same section vary with the selected roll schedule and their implications will be discussed later. Flower pattern design may conveniently be considered by splitting it into four main areas: (i) section orientation, (ii) sequencing of bends, (iii) bending at each stage, and (iv) means of forming bends.
12.3.3.1. Section orientation Any section's orientation is determined by a number of (often conflicting) considerations careless orientation may result in tooling, production and quality problems. As with many other aspects of roll forming, section orientation is often a compromise, some of the factors influencing the designer's final decision are discussed in the following paragraphs. Whenever a bend is not formed by both male and female rolls, it is termed a blind bend or fresh-air bend (fig. 12.12). Such bends often result in sectional inaccuracy and are thus to be avoided if at all possible. Careful orientation may reduce fresh-air bending. Spring-back (the metal's tendency to return to its original shape) is a common problem in roll forming and there are several methods for overcoming this. Where spring-back is likely to be a problem, careful section orientation will allow the use of side rolls to "overbend" a leg. The section's orientation will be influenced by the preferred vertical centre line. The vertical centre line is a theoretical line whose position relative to the centre of the
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/
: ,
~/" BENDING
Fig. 12.12. Example of fresh air bend.
machine does not change. Where mention is made of the left-hand side or right-hand side of a section, this refers to the position relative to the section's centre line. The vertical centre line is itself often a compromise, the main criteria for choosing it being balancing horizontal forces each side of the guide line, allowing metal movement by forming rather than drawing, and selecting a vertical centre line that passes through the deepest part of the section. The surface finish of pre-finished material can be damaged by careless orientation. The pre-finished material should be orientated so as to minimise "rubbing velocity" (the relative roll velocity between opposite surfaces) and also if possible, to aid operator inspection. Staining or differential cooling may occur on sections if coolant trapping is possible this may be minimised by careful orientation of a section. Auxiliary operations such as notching, piercing, embossing and welding can obviously dictate a section's orientation. Cut-off tooling should also be considered, since with proper orientation it may be possible to eliminate additional deburring operations by altering the position of the burr. The factors mentioned above cannot be considered exhaustive, merely representative, since any section may produce unique problems and may require a unique solution demanded by equipment, tooling personnel and setting constraints.
12.3.3.2. Sequencing of bends The order in which the designer chooses to form the bends is termed the sequencing of the bends. In very simple sections such as channels and angles, there is no choice in the sequencing (since there is only one bend), but in complex sections there can be a very large number of possible permutations. Ideally, designers work from the centre line outwards in sequencing, which would mean that a bend once formed would never be subject to further deformation. However, when sequencing bends, a designer must consider a large number of
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often conflicting factors. Hence it is often not possible to work from the centre line outwards since there are often good reasons to form by other sequences, for instance to (a) avoid fresh air bending, (b) reduce metal movement, (c) avoid excessively large bending moments, or (d) improve the smoothness of the flow of the material.
12.3.3.3. Bending at each stage After having decided how a section will be orientated and the sequence by which bends will be formed, it is necessary to decide how much bending should take place at each stage. While there are a number of techniques to aid the designer in deciding how much forming to perform, there is no one, generally accepted, method. The determination of roll schedule is complicated but, as mentioned earlier, is extremely important for the overall success of the design. As a consequence much of the research, both experimental and theoretical, has been dedicated to study the influence of the roll schedules on product quality [8-18]. However, the research has essentially been restricted to the study of "basic" profiles such as channel, top-hat and circular sections. Though more complex profiles can be degenerated into these simpler sections to understand the fundamental deformation mechanics, the experience of the tool designer plays a vital role in the final determination of the roll schedule and only a few guide rules are available to the designers [7,19]. Some salient features of roll design will be highlighted in the following sections so that the roll forming of fibre-reinforced sheets can be discussed in a proper context. Number of forming stages and roll angles. For reasons of economy, it is important to select a roll schedule with the minimum possible number of form rolls giving an acceptable range of product quality. As the terminology and the basic principle change little, the discussion will be based on a simple channel section. The most common practice is for the designers to use the forming angle method which assumes a smooth process of deformation from the beginning to the end of the roll forming mill (fig. 12.13), and the strip edge follows a straight line. The so-called forming angle
Rolt sta•
splndte c
e
H__
~
1 N
~_~ L = H~cotr N = L/d
L where H r d L N
= = = = =
Final section height. the Forming angte. • rol[ s t a t i o n pl• [ength. the r e q u l r e d ?ormlng length. • number o? rolt s t a t i o n s required.
Fig. 12.13. Diagram of forming angle method.
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is a statement of the material's formability, and for steel it is generally assumed to be 1~ Now presetting the pass length of the mill, the number of roll stations can be calculated by dividing the forming length (determined from the known flange height, H, of the final section) by the pass length. One additional roll station is included for the shape conformance of the final profile. However, this method has been criticised by researchers [11,16] for ignoring the intermittent nature of the deformation. Furthermore, to minimise the distortion of the product it is necessary to keep the deformation at the last forming stage to a minimum and the first roll station which is often used essentially to ensure proper feeding of the strip material into the roll forming machine, is usually not to have a large fold angle. These constraints clearly suggest that the hypotenuse of the forming angle diagram should not be a straight line; however, the actual shape of this flow pattern varies with the style of a designer. Ona et al. [13] suggested a cubic polynomial (fig. 12.14), to describe the deformation height y = A x 3 -Jr-B x 2 Jr- C x -Jr-D
(12.1)
which satisfies the following boundary conditions at x = 0 and x = N, d y / d x = 0 (12.2)
at x = O, y = O, and at x = N, y = H(1 - c o s 0o) The angle at any particular station is given as cos Ot = 1 + (1 - cos 00) [i2(2i - 3 N ) / N 3]
(12.3)
N is determined by an empirical method from a large database, grouped under different section types. Apparently this method works well for reported sections
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.
.
.
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.
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.
.
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I
]-~
RI,RP__P ,. nI AlzA2,...,An !
RI - R2 = R3 -- ... - R . = constant Al - A2 = A3 = . . . - A~ = constant Fig. 12.15. Constant
inside radius
method
of forming
bends.
F
R1 A ~ R2 I
AIR1 = A2R~ = AsR3 = ... = A . R . = c o n s t a n t Fig. 12.16. Constant
element
length method
of forming
bends.
with some minor modifications from the designer but suffers from the major drawback of depending on the limited design style and data of a particular company.
12.3.3.4. Means of forming bends After making a decision on the sequencing and magnitude of the bends, it is necessary to decide on the means of forming bends. In general, the only parameter within the designer's control is the inside radius of the bend (although it is possible to control the shape of the bends too). The most common methods of forming bends are the constant inside radius method (fig. 12.15), and the constant element length corner method (fig. 12.16). Each method has its advantages: the constant inside radius method reduces springback and distributes deformation more evenly between passes, whereas the constant element length method reduces wear on the rolls and the likelihood of "trapping" of material. The designer should obviously decide which method is more suitable for each job; in practice, however, individual designers seem to have preferences and become skilled in minimising the disadvantages of a particular method.
12.4. Computer-aided design in roll forming There can be little doubt that the development of CAD/CAM techniques has revolutionised many aspects of roll forming. Almost all of the advantages of the computer are relevant to form roll design. In particular, the computer can carry out large numbers of complex calculations effortlessly, store large amounts of information and retrieve it quickly, produce high-quality graphical and text documentation and allow simple editing to an existing design. Amongst the benefits that have resulted from CAD/CAM systems for roll forming, have been reduced lead times, fewer calculation errors, improved design, the development of form roll databases and more efficient manufacture.
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Originally, many roll forming organisations chose to develop their own in-house systems. This perhaps reflects the idiosyncratic nature of roll forming, in which opinions vary from company to company as to what facilities a CAD/CAM system should offer. To a large extent, then, such systems effectively computerised existing procedures and standards within particular roll forming organisations. CAD/CAM systems have not only automated many of the relatively routine, nondesign tasks, but have also resulted in improved designs. Firstly, the removal of these peripheral tasks has allowed the designer to concentrate on pure design. Secondly, since the data input to these systems has been reduced, it is very simple to alter the design, thus making it possible to examine several designs and choose the most suitable one. In terms of design analysis, there is no doubt that the CAD systems have resulted in greatly enhanced visualisation through 3-D wire frame constructions. It may fairly be said, though, that deformation analysis is still very much in its infancy, and that form roll design is still largely based on the individual designer's experience of material properties, mill properties and the roll forming process. The basic components of roll forming CAD/CAM systems tend to be fairly similar, and can be summarised by fig. 12.17. The finished section, flower pattern, roll design and roll machining software form a definite route for data to flow. More advanced components, such as enhanced visualisation, deformation analysis, and data management, provide supporting functionality which may be present depending on the nature of the system. The purpose of the finished section software is to fully define the finished section and to perform the strip length calculations. The vast majority of roll-formed sections consist of linear sections separated by circular arcs, and finished section definition consists of these plus the metal thickness. Flower pattern design is central to roll design, and consists of nominating the elements to be bent at each stage, the amount of bending to take place at each stage, and manner in which the element is to be deformed. Typically flower patterns are visualised as in fig. 12.18, although various plan, side and three-dimensional views are also often employed.
Finished Section
~
~/
-oo s
Flower Pattern Software Additional Features
Roll Machining | ~ Software ..... !
.....
Roll Design
Deformation Analysis Enhanced Visualisation Data management etc.
Basic Software
Fig. 12.17. Typical components of a roll forming C A D / C A M system.
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Fig. 12.18. Typical flower pattern generated by the COPRA | CAD system. (Courtesy of DataM GmbH.)
The purpose of roll design software is to define and draft the form rolls and to quantify the roll profile into a data file. In addition to the section and flower pattern data, the pass height and roll centre-to-centre distance, the drive and clearance surfaces and the side roll contour (if required) must be defined. The machining of the rolls is generally performed using a numerically controlled lathe. Future development may be anticipated in the area of computer-aided deformation modelling. The current restrictions on this are the intense computer resources demanded to achieve realistic results. There is also some uncertainty about the interpretation of strain results to assess forming defects and about the effects of residual stresses on forming limits.
12.5. Deformation analysis of roll forming Deformation analysis in roll forming has traditionally considered two areas: modelling the deformed surface during roll forming and modelling the strain distribution in the sheet during roll forming. Often, but not always, the former has been seen as a stepping stone to the latter. This has perhaps been a little misdirected: roll formers have for decades been skilled in judging the severity of deformation from the deformed shape of the sheet, and could learn much from visualising the deformed surface at the design stage. The more recent application of finite element methods, has the advantage of determining the deformed surface and the strain distribution simultaneously, and undoubtedly far more work can be anticipated in this direction.
12.5.1. The deformed surface in roll forming In this section it is shown that the dominant effect on the shape of the deformed surface is the shape of the rolls. Since this effect is purely geometric, the deformed surface can be approximated relatively accurately with very simple material models. It will be seen later that this is not the case when determining the strains in the sheet. For this reason, an accurate surface model is no guarantee of accurate strain prediction. As a simple model, and neglecting spring-back, the roll forming process can be considered to consist of three regions. For instance, if we consider a single pass in the roll forming of a channel section, and remove all rolls but the lower forming roll of the pass (fig. 12.19), we can identify three regions: region I where the channel does
492
S.J. Mander et al.
-\~ \\\\i I
Fig. 12.19.Typicalregionsin the roll formingprocess. not deform, region II where the channel does deform but the sheet is not in contact with the rolls, and region III where the channel deforms and is in contact with the rolls. The combined length of regions II and III is conventionally known as the deformation length of the pass. It should be noted that the deformation length is not related to the pass length (the combined length of regions I, II and III). The influence of the roll size on the deformation length is indicated by a sometimes quoted "rule of thumb" that the deformation length is equal to the centre-to-centre distance between the lower and upper rolls. By considering the work done in forming a channel to consist of transverse bending and longitudinal stretching, and minimising to find the deformation length, Bhattacharyya et al. [16] were able to derive a very useful expression for the deformation length of a channel section:
L-
V 3t
(12.4)
where L is the deformation length, a is the flange length of the channel section, t is the sheet thickness and A0 is the bend angle increment during the pass. It will be noted that since a rigid-perfectly plastic material was assumed, all of the properties in the expression are geometric. Despite this, the expression proves to be surprisingly accurate for a variety of different materials, as demonstrated by fig. 12.20. It may be noted that eq. (12.4) is a special case for a channel section, and that using the work
Roll forming of sheet materials
70
60
THEORETICAL STEEL ALb')zINItrg
X
9
493
X
z.r 9~
50
N
40
~4
,(~ X
10
20
30
40
50
DEFORMATION LENGTH (mm)
Fig. 12.20. Deformation length, theoretical and experimental results.
expressions derived by Panton et al. [20], the general case for any section with a single active bend is given by
L -- V/8J?O
(12.5)
where J is the polar second moment of the section about the fold line. Although eq. (12.4) proves to be an accurate estimate of the forming length for channel sections it does not accurately model the actual shape of the deformed surface. This is unsurprising since it neglects the geometric effects of the rolls in region III. To model the deformed surface we introduce a concept known as the "bend angle curve" [21]. Returning to fig. 12.19, if we make the assumption that the channel flange remains straight as it moves through the pass, the deformed surface is entirely described by the variation of the bend angle 0, along the pass length, the bend angle curve. The effect of the roll geometry can be mapped onto the bend angle curve at any section by the construction shown in fig. 12.21, and this defines the shape of the bend angle curve in region III. To determine the entire bend angle curve, it is necessary to determine the shape of the bend angle curve in region II, and the transition points between regions I/II and II/III. The shape of the bend angle curve in region II may be determined using the minimum energy method described in reference [16]. The transition points may be determined by assuming continuity of slope along the bend angle curve. The resulting bend angle curve is derived entirely geometrically; despite this it gives accurate results, as shown in fig. 12.22. Note that the geometric effect of the rolls is clearly evident in fig. 12.22. It should also be noted that the maximum slope of the bend angle curve occurs at the point of tangency between regions II and III, i.e. at the section where the sheet first contacts the rolls. We will see later that this indicates that the longitudinal strain will be a maximum at this point. Alternative methods of modelling the deformed surface use empirical or semiempirical methods. Kiuchi [22], for example, uses a semi-empirical approach. The
494
S.J. Mander et al.
:r
BASE OF CHANNEL
A'
YI
O'
I 1 !
02
~
1:3 Z U.I 113
....
ZA
0
(FORMING DIRECTION)
F i g . 12.21. Mapping the effects of the roll onto the bend angle curve.
90
-
, .
,
,
,,
::
:
-
80
A
60
"
"
A,
A ....
&
a
': 0''0 --i " ~
0
0
25
50
75
100
1~5
150
Distance from previous roll centre (mm) F i g . 12.22. Experimental results and theoretical predictions for the bend angle curve.
forming path of any point on the cross-section (fig. 12.23), is defined by a sinusoidal shape function - - presumably defined from empirical data. A family of surfaces may be generated by varying the value n, and the actual deformed surface is determined by a minimum energy method. It may be noted that although the geometric effects of the rolls are not directly included, the choice of the sinusoidal shape function models them indirectly.
Roll forming of sheet mater&&
495
[ ((I+1)th Stand) x
( x(z) = Xl + (X2-X,) s(z) P~ (X ~,~'~
y(z) = Y1 + (Y2-YI) s(z)
~
nd)
s(z) = sin ((rd2) (z/L)")
Fig. 12.23. Kiuchi model of the deformed surface in roll forming.
The well-known COPRA | CAD system appears to use an entirely empirical method for modelling the deformed surface. The forming path of a point is defined by a polynomial, the coefficients of which are determined from empirical data.
12.5.2. Modelling the strain distribution during roll forming With the exception of the finite element methods described in the following section, all methods of determining roll forming strains share the characteristic of further processing a model of the deformed surface. An accurate model of the deformed surface does not guarantee an accurate strain distribution. It may be shown [23], for instance, that for any roll formed surface, there is a family of potential strain distributions, and this makes it possible for the deformed surface to be modelled quite accurately but for the strain prediction to be highly inaccurate (this is demonstrated, for instance, by Zhu [24]). The simplest method for determining the strain distribution from a deformed surface is to assume that transverse sections of the strip remain plane orthogonal and a constant distance apart during roll forming. This situation has been analysed in detail by Panton et al. [20], for a general case, but in this chapter, the discussion is restricted to the basic concepts by considering the forming of a channel section (fig. 12.24a). Consider the forming path of a point on the tip of the channel, it lies in a cylindrical surface whose development can then easily be obtained (fig. 12.24b). It is clear that the length of the forming path on the tip edge is longer than that of a point in the base of the section, and that longitudinal stretching must take place. If it is assumed that the bend angle changes by an angle dO over a distance dz, then the longitudinal strain can easily be determined by the triangle in the plane development: 1 r2(d0)2
et = ~
dzz
(12.6)
496
S.J. Mander et al.
D
///
//,,
/
/
/ /
/
/
<-
i/""~'~"\\\ !/i ,
x__ [
/r'--~nzl
"~
-~./-
~-. . . . Web
0
~
//
/,"
y
l~I~.nge
. . . .
~A//
~-v//--)"
iI ./ ,',\ l //-" ~,, '~/B
(b)
\
p-.o,
_.
~
<
Ai
<< <
'
. <
-
-
/
f
/
~
~ /
/ /
/
/-4
,
/
N
/-----7 ~/
// / /
//
/
,_
A
,
f
/
_~
/~'~~ / .~ ~.o~
~or~ B
Fig. 12.24. Forming of a channel section.
where r is the distance from the fold line. The value dO/dz is the slope of the bend angle curve and, as was noted previously, the bend angle curve is steepest at the section where the sheet first contacts the roll; this is where the longitudinal strain is at its maximum. This is in good agreement with experimental results which show that the longitudinal strain generally reaches a maximum somewhere just before the centre line of the roll pass. Equation (12.6) and the bend angle curve can be used in conjunction to show the effect of various roll forming parameters on the longitudinal strain. This produces some interesting results [25], for instance, it can be shown that as the flange length of the channel increases, the longitudinal strain will, after a certain level, start to decrease (fig. 12.25a). This is because, as the flange length increases, the deformation length will also increase and hence the value dO/dzwill decrease. Although the results are counter-intuitive, they are in agreement with experimental results (fig. 12.25), and in line with the knowledge of experienced roll designers. The shear strain in roll forming a channel section may also be determined. For instance consider the plane which is normal to ON at the point N. Since this plane is the tangent plane of the cylindrical surface at N, the forming path of the tip edge 20" n
O - 20" 4.0, -- 3.4,
' ~
3.1,
a=10
b=20mm
a=15
t=0.6mm
_j
4
-
3
-
2
--?
m
lange ]. .9
-o.5
-
o
a=20
1.6
o.61~
FREE
0
i.....
"
web
] ---1
Fig. 12.25. Experimental results for the longitudinal strain in the roll forming of channel sections with varying flange lengths.
Rollforming of sheet materials
497
must be perpendicular to ON. Hence the shear strain must be zero. The same is not true for other sections. Consider, for instance, the roll forming of a top hat section (fig. 12.26a). The outer flange of the section lies on a cylindrical surface which may be developed (fig. 12.26b). In this case we note that there is a shear strain, and considering the triangle in the development, we note that it is equal to the product of r and dO/dz. In general [20], we find that the shear strain at a point is given by dO Y =Pdzz
(12.7)
where p is the perpendicular distance between the tangent to the section at a point and the fold line. It may be noted that, as with the longitudinal strain, the shear strain is related to the slope of the bend angle curve. However, whereas longitudinal strain is related to (dO/dz) 2 [20], the shear strain is related to dO/dz. Since the value of dO/dz is normally small, this means that the shear strain is potentially far larger than the longitudinal strain. Correspondingly, the work done in shear of metals may be significantly larger than the work done in longitudinal stretching. Kiuchi [22], and later Duggal et al. [27], used a somewhat better version of the plane sections remain plane assumption, in which the transverse planes were not required to remain a constant distance apart. In this situation, the distance between planes is determined so that the overall longitudinal force in the sheet remains constant throughout each individual pass. In practice, the overall longitudinal force is normally only known for one pass (the first, where it is zero). In practice, there is no reason to suppose that plane sections remains plane during roll forming. By examining the family of strain distributions which exist for any roll formed surface [23], some interesting results emerge. It can be shown, for instance, that for any roll formed surface there will be a case where there is no longitudinal strain (in which case there will be shear strain), and there will be a case where there is /-- ... //t I
," /
,
/
,
/ /
:
/
/ ,," /
/
L
,'
A2
I /,/ _ I/b :+d
l
u__./__:
7'/-
/ /n~:n-g-~/ //
ln.n~__ /TI
. . . .
./
Fig. 12.26. Forming of a top hat section.
/
/
">
/
t
1.'~
/"/
498
S.J. Mander et al.
no shear strain (in which case there will be longitudinal strain). With metals, experimental results indicate that the real situation normally lies somewhere between these two cases. It remains to be seen whether the extreme directional properties of fibre composite materials push the strain distribution to one of the extreme cases, in which case quite accurate strain predictions may be possible geometrically using approaches like the Chebyshev net technique.
12.5.3. Finite element analysis of roll forming Although the simplified models described previously will continue to provide insight into the roll forming process, and may one day lead to a scientific basis for form roll design, the future of roll forming deformation modelling is likely to be increasingly provided by finite element analysis. The results of McClure and Li [28], and Heislitz et al. [29], show the possibilities of the method, e.g. fig. 12.27. At present, it must be said that few roll forming companies have the technical and computational resources necessary for such studies.
12.6. Roll forming of thermoplastic material Due to the relatively recent availability of continuous fibre-reinforced thermoplastic (FRTP) sheets, a large amount of commercial interest has been generated in the j
,:t+ i l I,
't
~t,
I1--T
20
dig
[ ,~-a, M
i "
:"
~"i
.o,.,
!
1o
............. '
. . . . . . . .
"'"
i'
t't__ '~ 0
-
t/,.Ii' M. JZXN l/
.0,.,
1
.
~h
i~-
Strain gauge position (mm)
I [
Distance from leading edge (mm)
J
Fig. 12.27. Experimental (left) and finite element (right) longitudinal strain results for the single pass roll forming of trapezoidal sections with differing fold angles (from reference [28]). Dotted line indicates the roll central plane.
Roll forming of sheet materials
499
development of low cycle time manufacturing techniques. It is generally accepted that thermoplastic composites lend themselves to mass production methods, due to their ability to be "reshaped" simply by the application of heat and relatively low forces. The other benefits, such as long shelf life, ease of handling and the potential for recycling, are also well recognised. In spite of this, it is felt that the benefits of FRTP composites are unlikely to be fully realised until manufacturing techniques have become more conducive to mass production. Recent efforts in forming FRTP sheets have centred on variations of pressure forming and diaphragm forming techniques [30-32]. Other methods have transposed standard sheet metal forming operations, such as matched die stamping [33] and incremental forming [34,35], to these materials. Despite the large number of processing methods now available, there still remains a well recognised demand for the ability to produce long narrow profiles, such as top hat and channel sections, from FRTP sheets. An obvious manufacturing method offering high production rates for such sections is the semi-continuous roll forming widely used in the sheet metal industry. Whilst the practice of roll forming is well established, as mentioned earlier in this chapter, the deformation mechanism that involves three-dimensional bending and stretching, is not yet fully understood. It is felt that this process, with some modifications, lends itself to be used in the production of narrow and wide profile FRTP composite sections and goes some way towards realising their full benefits. Preliminary trials of roll forming of FRTP composites have been reported by Cattanach [36], but the degree of success and the detailed methodology of these trials remains unpublished. Recently Mander et al. [37] have reported a systematic study on the roll forming of FRTP composite sections using a modified metal roll forming equipment, and have shown that useful structural sections can be easily produced at 10 m/min with a clear possibility of achieving higher line speeds. This section of the chapter outlines the findings of continuing research carried out at Auckland University. The effects of various roll schedules and other forming parameters on process reliability and product quality are closely examined. Particular attention is given to the determination of the deformation length and the development of strain in the strip. A novel method of obtaining real time strain measurements during the deformation process is introduced, which offers researchers and designers a unique opportunity to understand the deformation process of FRTP sheets. 12.6.1. Materials
Fibre-reinforced thermoplastic composite sheet materials have become increasingly popular in recent times and are now available in a variety of different product forms, ranging from stiff board-like preimpregnated sheets, to drapable commingled woven fabrics. The materials presently available can be broadly classified into two groups: those which require impregnation as part of the manufacturing process, and those that are supplied in preimpregnated (prepreg) form. Interestingly a number of manufacturers, such as Vetrotex, supply their material in either form depending on the requirements of the particular user.
500
S.J. Mander et al.
The results of a series of unpublished tests performed by the authors have shown that the roll forming process can be successfully applied to a wide range of commercially available composite sheet products. However, the experimental investigation presented here uses Plytron | (a continuous uni-directional glass fibre (v/v 35%)/ polypropylene (PP) prepreg) laminates consolidated from four plies of prepreg sheet, giving a nominal strip thickness of 2 mm. The strips were consolidated at 180~ under a 6 kPa vacuum for 20 minutes. Roll forming test specimens were cut to 1000 • 90 mm so that the control axis was aligned with the rolling direction. 12.6.2. Roll forming equipment and procedures To facilitate the roll forming investigation of F R T P sheets, a commercially available five-stand horizontal beam raft sheet metal roll forming machine was employed. The roll former was fitted with a variable speed drive to the interconnected bottom rolls. After preliminary trials a friction drive for the top roll of each stand was installed to ensure the matching of its angular velocity with that of the bottom roll. The salient features of the roll forming equipment are provided in table 12.1. A top hat section, as shown in fig. 12.29a, was selected for this study and initial trials were conducted using an industry standard flower pattern (flower A, fig. 12.28) used for producing this section in aluminium alloy or mild steel. A second flower pattern (flower B, fig. 12.28) was subsequently investigated, both roll configurations having a constant driving diameter for top and bottom rolls. To enable the F R T P sheet to be formed, a pre-heating oven was positioned on wheels close to the first stage and in line with the centre line of the roll former. The F R T P sample was placed on a Mylar T M strip in a drawer mechanism which was constructed in a way that it also served as a feed guide for the roll former. To enable the temperature of the F R T P strip to be measured during the preheating stage and subsequent roll forming operation, a fine 0.75-mm K-type thermocouple wire was consolidated in the centre core between the second and third layers along the laminate centre line with the hot junction at a depth of 200 mm from the leading edge. An optical sensor was placed prior to the first stage of the roll former TABLE 12.1 Roll former specifications Type: Number of stands: Shaft diameter: Stand width Drive: Speed control: Roll stand pitch: Vertical adjustment: Roll material: Feed stock table:
Horizontal beam raft (light-medium gauge) 5 40 mm 350 mm Bottom- chain via 3-phase AC 410 volt T o p - friction via bottom roll Infinitely variable speed, 0.1-10.0 m/min 275 mm Manual screw thread- 75 mm 1040 Steel, DelrinT M plastic 400 mm • 2000 mm
Roll forming of sheet materials
I
~
......
Flower A
Roll Station
ao
1
25
2
47
3
67
48
4
81
76
5
90
90
,.
po 0 ,, 2I
501
Flower B
47
13~ ,..0 21
.,. 67
48
(x ~
30
90 .... 90
90 90
Fig. 12.28. Roll station flower diagrams.
which enabled the position and speed of the strip to be measured throughout the forming operation. The optical sensor was triggered by strips of aluminium foil attached to the upper surface of the test sample at measured intervals. A traditional experimental approach was adopted in which each of the various roll forming parameters was examined individually while the other variables were kept constant. An optimum inlet temperature was determined which was used for all subsequent trials. Further trials investigated the influence of laminate architecture. The effect of roll forming line speed and a second roll schedule were also examined. A simple cooling ring, which consisted of a coiled perforated copper tube through which the cooling agent was passed under pressure, was installed at the exit side of stage 5. This cooling ring was then positioned between stages 4 and 5 for use with the second roll schedule (flower B). For all tests the same preheating sequence was used, namely the sample was preheated to and held for 5 min at 175~ then cooled in a controlled manner to the desired entry temperature and fed into the roll forming machine. All the roll formed samples were then compared for conformity to the desired geometrical shape (fig. 12.29a), in accordance with the normal procedures for roll formed sections. The deviation from intended formed angle 8 is defined in figs. 12.29b and c. A negative value for 3 is universally referred to in sheet forming as spring-back and hence a positive value of 3 can be termed as negative spring-back or spring-forward. The extent of curvature was measured as the amount of offset over the span length. With the condition of the fibres being an important consideration in FRTP composite structures, a visual inspection was made of the laminate to establish the degree of out-of-plane fibre buckling and in-plane fibre wrinkling.
12.6.3. Quality assessment of formed parts As with any forming operation it is important to be able to produce parts which meet high tolerances. The quality of each successfully rolled sample was assessed L--
, (3)
50mm
.d
I
Web
b)
J
c)
Fig. 12.29. Dimensions of top hat section used in this study, with definition of deviation from intended geometry 8.
502
s.J. Mander et al.
with regard to two important criteria: (i) deviation from that of its intended geometry, and (ii) severity of fibre buckling and wrinkling. All samples that were successfully roll formed were checked for conformance to the desired geometry, the results of this are shown in table 12.2. A visual rating out of 10 was given for the extent of in-plane fibre wrinkling of each sample with a high score indicating a significant presence of wrinkling. From these results it is evident that a larger scatter of formed geometry exists for those samples formed at temperatures other than at an inlet temperature of 140~ Based on this it was decided that the preferred roll forming inlet temperature for this study was 140~ Table 12.2 shows that the degree of in-plane wrinkling is greater at the higher inlet temperatures which is believed to be symptomatic of a lower than desirable matrix viscosity. The relatively high rating for wrinkling at 130~ is given due to the presence of out-of plane buckling, particularly in the flange 1 (F1)/flange 2 (F2) bend regions indicating delamination. The presence of buckling in this region may be a result of lower sample temperature due to its proximity to the strip edge. However, from temperature readings given by thermocouples situated in the middle of each flange, there appears to be no discernible temperature gradient measured across the width of the strip. It was observed during the trials that there was a considerable amount of relaxation after each stage and an appreciable amount of spring-back between F1 and F2 after exiting the roll former. This spring-back is contrary to the normal experience of spring-forward in F R T P bends [38]. A possible cause for this may be a large thermal gradient through the thickness of the sample caused by the insulating effect of the Mylar film attached to the bottom (or outer) surface. From the temperature readings it was found that, for all inlet temperatures studied, the exit temperature of the sample was such that the matrix was still in a partially molten phase, i.e. recrystallisation had not begun or was incomplete. Because of this high exit temperature the formed sections remained in a deformable or compliant state. To alleviate this problem an external cooling facility was positioned at the exit of the roll former. Some of the results of these trials, using roll schedule B, are given in table 12.3, where the relevant details for a roll formed aluminium strip are also given. Due to this introduction of cooling, some improvement in the formed geometry has been achieved; but still a large amount of variation in formed part geometry is present, particularly in the F1/F2 region. TABLE 12.2 Conformance measurements at various inlet temperatures. Line speed 5 m/min, flower A, laminate [4-15~ Temperature (~
8OFt/web(o)
8~ F2/F1 (~
Offset (mm/m)
Fibre wrinkling
170 160 150 140 130
+3 to-3 +2 to -2 +2 to -2 +1 to -1 +3 to -3
-5 -5 -10 -10 -10
2.0 0.8 0.6 0.5 -6.0
3 2 1 1 4
to-15 to -10 to -15 to -15 to -15
Roll forming of sheet materials
503
TABLE 12.3 Conformance measurements with various cooling mediums. Line speed 5 m/min (*10 m/min), flower B, laminate [+ 15~ inlet temperature 140~ Cooling medium
8~ F1/web (~
8~ F2/F1 (o)
Offset (mm/m)
Wrinkling score
None Pressurised air Water spray Pressurised air*
+2 +2 0 0
-6 -6 -2 -2
1 1.5 0 0
1 1 1 1
to to to to
-2 +5 -2 -2
to to to to
+12 -9 +5 +5
From these trials it may be concluded that the exit temperature of the strip is a significant factor in producing a component with minimal deviation from its intended geometry. It is also evident that samples formed with the water spray cooling have improved conformance to the intended geometry with a lower range of deviation. These results are expected if the sample temperatures are considered (fig. 12.30), where it may be seen that for the water cooling trials, the exit temperature is 100~ and recrystallisation has been completed. For the sample formed at 10 m/min the results are still acceptable in spite of the higher exit temperature. This establishes that higher line speeds are possible provided the temperature is correctly managed. It must also be noted that in using flower B roll schedule, the deformation process can be completed in four stages with a greater degree of deformation at stage 1, which is higher than what is practical in roll forming metallic sheets. It is generally recognised that when thermoforming FRTP sheets, fibre orientation can have a large influence on the quality of the formed part [39]. Having determined an appropriate roll schedule and temperature regime for the roll forming of Plytron, various laminate architectures were examined with the results shown in table 12.4. Of particular interest are the ratings for in-plane fibre wrinkling where a definite trend is
Fig. 12.30. Sample temperature profile during roll forming operation with cooling facility between stages 4 and 5, flower B, line speed 5m/min, laminate [+15~ .
504
S.J. Mander et al.
TABLE 12.4 Conformance measurements for various laminates. Line speed 5 m/min, flower B, inlet temperature 140~ pressurised air cooling Laminate
8~ F1/web (~
8~ F2/F1 (~
Offset ( m m / m )
Wrinkling score
[+15~
+2 +2 +0 +3 +2
-6 -4 -6 -4 -7
1.5 2.4 1 1.8 2.2
1 4 6 0.5 8
[+30~ [+45~
[0~176 s [90~176 s
to to to to to
+5 +4 +2 +5 +5
to to to to to
-9 -9 -11 -7 -12
seen, namely an increase in the presence of wrinkling with an increase in the fibre angle of the outer ply. As shown in fig. 12.31, the wrinkling is typically located on the inner radius of a bend, where compressive strains are to be expected, and most notably on the inner surface of the web. For the [45~ and [0~176 s out-of-plane buckling is evident on the inner radius of F 1 / F 2 , suggesting possible delamination.
12.6.3.1. Prediction of spring-back/forward in FRTPs The geometric conformance of a component is an important consideration in any forming operation. One well documented phenomenon in the processing of FRTP materials already observed in the roll forming trials, is the spring-forward effect which is primarily due to the anisotropic thermal properties of this class of materials notably the often large difference between longitudinal and transverse coefficients of thermal expansion. For uni-directional laminates, the spring-forward has been simply expressed mathematically by Zahlan and O'Neill [38] in the following manner: A0 -- (C~T-- C~L)0AT
(12.8)
Fig. 12.31. The influence of fibre architecture on formed part quality. Note that as the surface fibre orientation becomes more obtuse to the rolling direction the in-plane wrinkling effect becomes more severe.
Roll forming of sheet materials
505
where aT and aL are the transverse and longitudinal coefficients of thermal expansion respectively, 0 is the forming or mould angle and AT is the difference between the forming temperature and final temperature. Published experimental results generally support the above expression. However, Martin [39] and Hou et al. [40] give conflicting evidence for the influence of laminate stacking sequence on the magnitude of spring-forward. They also show that the forming speed has some influence on the magnitude of spring-forward but attribute this to fibre migration due to a lower than desirable matrix viscosity. A useful tool for predicting the distortion of bend angle in continuous fibrereinforced composite components is the finite element (FE) method. Zahlan and co-workers have shown how the technique may be used to successfully model the spring-forward, in thin carbon fibre/PEEK top hat sections. A similar analysis to that presented by Zahlan and co-workers has been undertaken by the authors in an attempt to model the distortion observed in the roll formed top hat sections presented earlier. In the interests of brevity the full details of the analysis are not given here. In keeping with the approach of Zahlan and co-workers, the model has been constructed in a (R, 0) co-ordinate system, as shown in fig. 12.32, such that only a single bend region is described. The numerical analysis may be simplified somewhat by ignoring the effects of any thermal gradients occurring within the component, thereby assuming a uniformly changing temperature field. The validity of this assumption is questionable, considering the thickness of the sample and the cold forming rolls utilised to form the bends but would seem reasonable for mouldings of "thin" cross-section. One further point that may be made regarding the modelling constraints is that the bend region is free to undergo unrestrained deformation, thereby rendering the solution independent of any specified elastic properties. However, the same could not be said if any type of elastic constraint is imposed, either internally or externally. Such would be the case in the presence of any thermal gradients, any localised crystallisation or any geometric constraints imposed by tooling.
130~
Fig. 12.32. Thermallyinduced distortion of uni-directional Plytron laminate formed to 90~
506
S.J. Mander et al.
Using this method the numerically derived spring-forward for the 90 ~ bend region has been calculated as approximately 1.5 ~, which is significantly lower than any of the results observed earlier in the roll forming trials. The most likely reason for this discrepancy may be the uniformly changing temperature field assumed to be acting over the entire section. As a comparison, one might consider the effect of a thermal gradient resulting in non-uniform property variation through the thickness of the region. In effect the deformation of this "more realistic" type of problem would become significantly dependent on the specified elastic properties, unlike the unconstrained deformation of the model described above. Clearly the effects of recrystallisation, or solidification from the melt, have a significant effect on the degree of spring-forward experienced in a forming operation and must be considered if components which meet high tolerances are to be successfully produced.
12.6.4. Temperature control In any forming operation involving FRTP composites, the forming temperature is invariably an important factor and traditionally the forming operation has been carried out at temperatures above the Tm of the matrix material (~160~ for PP). It has been shown [37,41] that polypropylene-based Plytron is capable of being formed at temperatures well below Tm providing the sample has been heated above Tm and then formed before the polymer matrix is able to reach its maximum degree of crystallinity. This is also supported by other researchers [42] who have applied differential scanning calorimetry (DSC) techniques, the results of which are shown in fig. 12.33. The large peaks and troughs shown in the graph are indicative of phase changes in the material. Upon cooling from the melt (cooling curve in fig. 12.33) recrystallisation begins at T1 (approximately 125~ and is completed at T2 (approximately 95~ During this stage the polypropylene matrix is in a supercooled liquid phase. This recrystallisation from the melt phenomenon allows a relatively large processing window, of nearly 60~ below Tm for polypropylene. The roll forming operation is particularly well suited to exploiting this material property, 25 r Cooling
g2o
T~
lO
r
Forming window
25
,
__,
20 .~
"~ 15
o
, ,
Tm
15
long
Heating ::~
5
5
o 0
25
50
75
100
125
150
175
200
225
*~
o 250
Temperature (~ Fig. 12.33. DSC heating and cooling curves for Plytron 35% v/v glass/polypropylene after reference [39].
Roll forming of sheet materials
507
being a non-isothermal deformation process performed in multiple stages over a relatively long forming time. While interpreting the results of the roll forming trials at various inlet temperatures it became apparent that the reliability of the process should be a major consideration. For samples formed at an inlet temperature of 160~ and above, there is approximately a 60% failure to complete the pass through all stages of the roll former. This is considered to be due to the relatively low stiffness of the composite caused by a lower viscosity of the molten matrix above Tm, resulting in the composite being unstable during the process. Conversely those strips formed at an inlet temperature below 135~ also demonstrated high failure rates (approximately 50%), but this can be largely attributed to the practical difficulties in supplying test pieces at a temperature close to the recrystallisation temperature of the matrix to the roll former. However, for samples formed at inlet temperatures of 150~ and 140~ a much lower failure rate was encountered (typically < 5%). It is important to note that most FRTP forming processes involve three sequential stages in their operation: heating, deformation, and cooling. To avoid the problems associated with material property variation, the deformation stage is typically carried out under conditions which are close to isothermal. In roll forming however, the actual forming process proceeds under highly non-isothermal conditions, as the deformation and cooling stages occur simultaneously. A number of authors have investigated the problems associated with non-isothermal forming of simple shapes [43,44]. Martin [43] has performed a series of simple vee-bending experiments that have highlighted the need to couple both deformation and cooling rates. It has been shown that if cooling advances at a rate which greatly exceeds that of the deformation then localised freezing is likely to occur, resulting in a component that has not fully realised the desired shape.
12.6.4.1. Development of cooling model The reliability of the roll forming process has been demonstrated to be highly dependent on the strip temperature being within the forming window indicated in the DSC graph shown in fig. 12.33. Hence it is desirable to be able to predict the temperature variation in the strip so that inlet temperatures, line speeds, and other important operating parameters can be incorporated into the development and design of such a roll forming process. With this in mind, a simple numerical cooling model is proposed that enables such a prediction to be made. Consider a strip of material as it passes through a roll forming machine at a constant line speed, v. In this analysis a local Cartesian co-ordinate system is adopted where the y- and z-directions are chosen to coincide with the strip's width and longitudinal axis respectively. The x-direction is aligned to coincide with the thickness which is assumed to remain constant throughout the deformation. In the present analysis it is assumed that the temperature variation in any given crosssection (in the x-y plane) will depend on its thermal history in addition to its current boundary conditions. Consider the section, indicated by the shaded region in fig. 12.34, moving through time and space and possessing an arbitrary temperature
508
S.J. Mander et al.
~Z
~
W Prevlous TemperQture
RolLing Dlrectlon
~ ~ -~J
J
F
i
tlme
e
t
d
~
Increment
Current Temperature Field
~
V
f Fig. 12.34. The current temperature field is assumed to depend on the current surface boundary conditions and the previous temperature field.
distribution. After some time increment, At, the same section will have moved through a distance Az and will have assumed a new temperature field. Initially, such a cross-section will possess a uniform temperature equal to that of the applied inlet temperature. As the section moves forward, the temperature field changes as the outer surfaces of the strip are exposed to various cooling mediums. If we invoke the assumptions made above, and also assume that the through-thickness temperature variation remains constant, then the equation governing the flow of heat can be simply expressed by 02T Ox2
=
10T
(12.9)
ot Pt
where ct is the thermal diffusivity, given by a kx/,OC p. The first- and second-order partial derivatives in eq (12.9) can be simply approximated by applying standard numerical finite difference methods. The transient temperature field for a section can then be evaluated at any time and simply related back to its displacement in the zdirection through the line speed. The model has also been constructed to incorporate a temperature-dependent thermal diffusivity. For instance, the value of a used in the evaluation of the temperature of the ith node after j + 1 time increments, is simply based on the temperature of the same node at the previous time step, Ti.j =
Oli,j+ 1 = ol( Ti,j)
The relationship between the through thickness thermal diffusivity, ct, and temperature in Plytron has been investigated at ICI Ltd. [42]. Their experimental data have been roughly approximated using a series of linear interpolation functions. These have been plotted in fig. 12.35, which serves to illustrate the relationship used in the analysis.
Roll forming of sheet materials
509
0.18 0.16 ~
0.14 el 0.12
".~ 0.08 C~ 0.06
~ 0.04
~
0.02 0 0
50
100
150
200
Temperature (*C) Fig. 12.35. Thermal diffusivity plotted as a function of temperature.
Boundary conditions. As the hot composite strip passes through the process it is subjected to two forms of cooling at its surface. The first of these is natural convective air cooling which occurs between the roll stands and involves the dissipation of heat to the working environment. The second form of cooling involves direct contact between the strip surface and the cold forming rolls, resulting in heat being conducted away from the strip. In both cases some information regarding the heat transfer coefficient, h, is required in order to accurately describe the surface boundary condition. However, in the absence of any such data a number of simplifying assumptions can be made. Between the roll stands, the surface of the strip is assumed to be insulated from any thermal effects arising from the surrounding environment. This can be somewhat justified by considering the thermal properties of the material in question. Both the glass fibres and polypropylene matrix are relatively poor conductors of heat and can thus be generally classified as good insulators. Moreover, an insulated boundary condition is relatively simple to implement in a numerical finite difference model such as the one described above. The conduction of heat away from the strip due to contact with the forming rolls is rather more complicated, the actual time depending largely on the line speed and deformation length of the strip. Rather than adopting a complicated boundary condition that would vary with position and time, an effective roll contact time is assumed, whereby the entire width of the strip is subjected to a temperature equal to that of Tro.. Therefore the boundary conditions may be summarised as: (i) Air cooling (between roll stands) 0T
m 0
OX x=O,h
(ii) Roll cooling over an effective contact time
TIx=o,h =
Troll
The input parameters used in the model are detailed in table 12.5.
510
S.J. Mander et al.
T A B L E 12.5. Roll forming variables used in the cooling model Inlet temperatures: Line speed: Roll temperature: Effective roll contact time: Strip thickness: Pitch distance:
130, 140, 150, 160~ 5 m/min 20~ 0.3 s 2 mm 275 mm
The results of the model are presented in fig. 12.36. Good correlation can be seen between the experimental temperature variation at the core and that predicted by the model. It appears as if the assumptions used in the analysis are valid for temperature measurements taken at, or around, the mid-plane of the strip. However, if more accurate predictions of temperature variation around the strip surface are required then it would be necessary to alter the applied boundary conditions. As already stated, the application of a more realistic surface condition relies upon some knowledge of a heat transfer coefficient. The model does, however, provide the framework within which to interpret the influence of various important processing parameters such as line speed, the number of roll stations, and inlet temperature. 12.6.5. Deformation length and strain development 12.6.5.1. Strain measurement
In order to quantify the amount of strain during the roll forming process, realtime strain measurement experiments have been conducted. This has been achieved by using a novel technique which involves bonding electrical resistant strain gauges 180 170 160
v
f~ 150
~
Model 130oC
P
14o
130
.'~.-~ 9
~
A
A 140oC
~
13 150~ X 160~
~,
120 110 100
2
3
4
5
Roll Stand # Fig. 12.36. A comparison between the numerical cooling model and experimentally obtained results.
Roll forming of sheet materials
511
onto fibre bundles arranged in a ribbon form. These ribbons are then consolidated, with the appropriate electrical connections, into the laminate with the fibre ribbon aligned with the fibres in the outer lamina. The samples can then be roll formed at the predetermined conditions enabling the recording of real-time strain measurements. An example of a typical strain profile is shown for a [0~176 laminate in fig. 12.37. This strain profile is in reasonable agreement with Zhu's [45] measurements for sheet metals at similar deformations. In the interest of brevity the full details of this technique are not given here. This method of real-time strain measurement in molten FRTP composites is of invaluable use to the manufacturing personnel, as it enables for the first time in-situ strain measurement during the deformation of FRTP sheets.
12.6.5.2. Deformation length In published sheet metal forming trials, the deformation length is often measured in a static condition after the sample has been partially deformed, either in situ, or once it has been removed from the forming rolls. However, due to the requirement that FRTP sheet be formed in a molten state, this approach would not seem to be viable. This is due to the large amount of relaxation which occurs in the deformation zone prior to the thermoplastic matrix recrystallising or solidifying. To this end it is necessary to measure the deformation length instantaneously with the aid of photography. This has been achieved by positioning two 35-mm SLR cameras at the inlet side of stage 1. While this method of measurement is inherently less accurate than those afforded by static measurement techniques, it is considered that it provides satisfactory confidence levels of • In general the results of the deformation length trials followed the same trends as those commonly observed for sheet metals. Not surprisingly, as the deformation, or roll angle is increased from 20 ~ to 40 ~, the deformation length of all samples is observed to increase, as shown in fig. 12.38. As a comparison, the deformation length, as predicted for metallic sheet material using eq. (12.4), has also been plotted on the same graph. It is interesting to note the similarity in trends between the deformation lengths of the FRTP material and those predicted using the theory
1250
9
9
,
m
,
9
9
,
,
.
9
,
.
.
o
,
.
.
9 9
i
750
r~
-250 -750
: Roll Stand # 1
" 2
: 3
: 4
: 5
Fig. 12.37. Longitudinal membrane strain of [0~176 laminate as a function of position within the roll former measured at the interface of laminae 2/3 and 3/4. Inlet temperature 140~ line speed 5 m/min, no cooling.
512
S.J. Mander et al.
300
,.••
250 13 13
200
~, ..-" X o"" .41 ~176
....---
.-'II 9
o
[o~176
Q
[+15~
X x
[:1:45~ [9o~
O
~o 15o
Theory 35ram ....... Theory 4 0 m m
I00
,
:
I0
20
. . . . . . .
:
:
30
40
50
Deformation Angle (degrees) Fig. 12.38. Comparison of deformation lengths of various laminates and those lengths predicted using existing sheet metal theory.
derived for sheet metals, particularly when one considers that the theoretical model for metals is independent of any material properties. However, the results have to be interpreted with caution given the overwhelming differences in the constitutive behaviour of both materials [46,47]. 12.7. Concluding remarks
This chapter has briefly introduced the basic considerations of roll forming sheet materials and has clearly eastablished the possibility of applying the technique in the area of continuous fibre-reinforced thermoplastic sheets. With the recent advent of these relatively cheap sheet materials based on propylene and aramid 6, this development would be significant from the high-productivity, manufacturing point of view. Entry temperature and the cooling rate have been shown to have a significant influence on the quality of the manufactured product and its production viability. These parameters encompass many other importants issues in roll forming, such as the forming speed, feeding mechanism and shape conformance. Thus, unlike in metal sheets, the heating system becomes one of the prime design concerns in roll forming of composites and can prove to be the biggest barrier to achieving high production speeds. The ply lay-up is another complexity which is absent in sheet metals but for composite materials, it influences the design of feeding mechanism and subsequently affects the shape conformance of formed products. However, at the same time composite sheets appear to accommodate large deformations at various roll stands and also lend themselves to be roll formed with relatively small load requirement, resulting in the possibility of designing lighter and smaller rolling mills. The springback and spring-forward phenomena may also be wisely used having mutually cancelling effects to produce little distortions on the manufactured items. It is interesting to note that the development of membrane strains, while roll forming FRTP sheets,
Roll forming of sheet materials
513
follows a t r e n d similar to t h a t o b t a i n e d in metallic sheets. T h e d e f o r m a t i o n zones u n d e r v a r i o u s roll stations, h o w e v e r , a p p e a r to be generally larger in the case o f c o m p o s i t e sheets.
Acknowledgements T h e a u t h o r s wish to a c k n o w l e d g e the g r a n t s received f r o m the F o u n d a t i o n for R e s e a r c h Science a n d T e c h n o l o g y ( N e w Z e a l a n d ) a n d D u P o n t ( U S A ) . T h e y are also t h a n k f u l for the help received f r o m H o r t o n I n d u s t r i e s (NZ), Borealis ( N o r w a y ) a n d Mitsui-Toatsu (Japan).
References [1] Hobbs, R.M., Duncan, J.L., Roll Forming, Metals Engineering Institute, American Society of Metals, Ohio, USA (1979). [2] Gold, R., Roll forming is out of this world, Precision Metal, 40 (1983) pp. 11-15. [3] O'Leary, M., Space Industrialisation, CRC Press, Boca Raton, FL, USA (1982) pp. 102-105. [4] G.T. Halmos (ed.), High Production Roll Forming, Society of Manufacturing Engineers, Marketing Services Department, Dearborn, MI, USA (1983). [5] Tsclikov, A.I., Smirnov, V.V., Rolling Mills, Pergamon Press, Oxford, UK (1965). [6] Ivaska, J., Lubricants help expand roll forming capabilities, High Production Roll Forming, ed. G.T. Halmos, SME Marketing Services Department, Dearborn, MI, USA (1983) pp. 107-109. [7] Ivaska, J., Analysis of lubrication problems in roll forming, High Production Roll Forming, ed. G.T. Halmos, SME Marketing Services Department, Dearborn, MI, USA (1983) pp. 110-122. [8] Suzuki, H., Kiuchi, M., Nakajima, S., Experimental investigations on cold roll forming process, Report of the Inst. of Ind. Sci., Univ. of Tokyo, Tokyo, Japan, 22, (1972), pp. 84-172. [9] Kokado, J.I., Onada, Y., On longitudinal curvature and transition of strain of sheet metal in forming a groove with wide flanges by cold roll forming, Kyoto Univ. Faculty of Eng. Memoirs, Kyoto, Japan, 36, No. 4 (1974), pp. 443-457. [10] Suzuki, H., Kiuchi, M., Nakajima, S., Ichidayama, M., Takada, K., Experimental investigation of cold roll forming process II, Rep. of the Inst. of Ind Sci., Univ. of Tokyo, Tokyo, Japan, 26, No. 8, (1978), pp. 291-346. [11] Noble, C.F., Sarantidis, T.M., A study of cold roll forming, Proc. 1st Int. Conf. on Rotary Metal Working Processes, London, UK (1979), pp. 411-424. [12] Jimma, T., Ona, H., Optimum roll pass schedules on the cold roll forming process of symmetrical channels, Proc 21st Int. M.T.D.R. Conf., (1980), pp. 63-67. [13] Ona, H. Jimma, T., Fukaya, N., Experiments into the cold roll forming of straight symmetrical channels, J. Mech. Working Technology, 8, No. 4 (1983) pp. 273-292. [14] Bhattacharyya, D., Smith P.D., The development of longitudinal strain in cold roll forming and its influence on product straightness, Advanced Technology of Plasticity, JSTP, Tokyo, Japan, 1 (1984), pp. 422-427. [15] Kiuchi, M., Analytical study on cold roll forming process, Report of the Inst. of Ind. Sci., Univ. of Tokyo, 22, No. 2 (1972), pp. 1-43. [16] Bhattacharyya, D., Smith, P.D., Yee, C.H., Collins, I.F., The prediction of deformation length in cold roll forming, Journ. Mech. Work. Tech., 9 (1984), pp. 181-191. [17] Kiuchi, M., Koudabashi, T., Automated design system of optimal roll profiles for cold roll forming, Proc 3rd Int. Conf. on Rotary Metalworking Processes, Kyoto (1984), pp. 423-428. [18] Bhattacharyya, D., Panton, S.M., Research and computer-aided design in cold roll forming, Modernization of Steel Rolling, ed. Zhao Linchun et al., International Academic PublishersPergamon Press, Oxford, UK (1988) pp. 464-471.
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[19] Bhattacharyya, D., Smith, P.D., Cold roll f o r m i n g - engineers' dilemma and artisans' art, Trans. Inst. Prof. Engineers NZ, 12, No. 3 (1985), pp. 179-191. [20] Panton, S.M., Zhu, S.D., Duncan, J. L., Fundamental deformation types and sectional properties in roll forming, Int. J. Mech. Sci, 36 (1994), pp. 725-735. [21] Panton, S.M., Zhu, S.D., Duncan, J.L., Geometrical constraints on the roll forming of channel sections, Proc. Instn Mech. Eng., Part B: J. Engng Manufact., 206 (1992), pp. 113-118. [22] Kiuchi, M., Koudabashi, T., Development of simulation model of roll forming process: multipurpose CAD system for roll forming process, J. Japan Soc. Technol. Plasticity, 27, No. 306 (1987), pp. 874-881. [23] Panton, S.M., Duncan, J.L., Zhu, S.D., Longitudinal and shear strain development in cold roll forming, Journal of Materials Processing Technology, 60 (1996), pp. 219-224. [24] Zhu, S.D., Theoretical and experimental analysis of roll forming, Ph.D. Thesis, University of Auckland, Auckland, New Zealand (1993). [25] Zhu, S.D., Panton, S.M., Duncan, J.L., The effect of geometric variables in roll forming a channnel section, Proc. Instn Mech. Eng., Part B: J. Engng Manufact., 210 (1996), pp. 127-134. [26] Chiang, K.F., Cold roll forming, ME Thesis, University of Auckland, Auckland, New Zealand, 1984. [27] Duggal, N., Ahmetoglu, M.A., Kinzel, G.L., Altan, T., Computer aided simulation of cold roll forming - - a computer program for simple section profiles, Journal of Materials Processing Technology, 59, (1996), pp. 41-48. [28] McClure, C.K., Li, H., Roll forming simulation using finite element analysis, Manufacturing Review, 8 (1995), pp. 114-122. [29] Heislitz, F., Livatyali, H., Ahmetoglu, M.A., Kinzel, G.A., Altan, T., Simulation of roll forming process with the 3-D FEM code PAM-STAMP, Journal of Materials Processing Technology, 59 (1996), pp. 59-67. [30] Mallon, P.J., O Bradaigh, C.M., Pipes, R.B., Polymeric diaphragm forming of complex curvature thermoplastic composite parts, Composites, 20 (1989), pp. 48-56. [31] Smiley, A.J., Diaphragm forming of carbon fiber reinforced thermoplastic composite materials, Ph.D. Dissertation, University of Delaware (1988). [32] Okine, R.K., Analysis of forming parts from advanced thermoplastic composite sheet materials, J. Thermoplast. Compos. Mater., 2, (1989), pp. 50-56. [33] Friedrich, K., Hou, M., Stamp forming of continuous carbon fibre/polypropylene composites, Composites Manufacturing, 2, (1991), pp. 3-9. [34] Strong, A.B., Hauweller, P., Incremental forming of large fibre reinforced thermoplastic composites, 34th International SAMPE Symposium, (1989), pp. 43-54. [35] Miller, A.K., Gur, M., Peled, A., Payne, A., Menzel, E., Die-less forming of thermoplastic matrix continuous fibre composites, J. Compos. Mater., 24, (1980), pp. 346-381. [36] Cattanach, J.B., Cogswell, F.N., Processing with aromatic polymer composites, Developments in Reinforced Plastics, Vol. 5. ed. G. Pritchard, Elsevier Applied Science Publishers, (1986), pp. 1-38. [37] Mander, S.J., Bhattacharyya, D., Collins, I.F., Watson, P., Roll forming of fibre reinforced composite sheet, Proc. Tenth Int. Conf. on Composite Materials, Vol. III, ed. A. Poursartip and K. Street, Woodhead Publishing Ltd., Cambridge, UK (1995) pp. 413-420. [38] Zahlan, N., O'Neill, J.M., Design and fabrication of composite components: the spring-forward phenomenon. Composites 20, (1989), pp. 77-781. [39] Martin, T.A., Bhattacharyya, D., Deformation characteristics and formability of fibre reinforced thermoplastic sheets. Composites Manufacturing 3 (1992), pp. 165-172. [40] Hou, M., Friedrich, K., Stamp forming of continuous carbon fibre/polypropylene composites, Composites Manufacturing, 2, No. 1, (1991), pp. 3-9. [41] Krebs, J., Bhattacharyya, D., Friedrich, K., 3-D component manufacturing with fibre-reinforced composite sheets, to be published in Composites: Part A. [42] Cuff, G., Fibre Reinforced Industrial Thermoplastic Composites, Woodhead Publishing Ltd., Cambridge, UK (1995). [43] T.A. Martin, Forming fibre-reinforced thermoplastic sheets, Ph.D. thesis, University of Auckland (1993).
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[44] Soil, W., Gutowski, T.G., Forming thermoplastic composite parts, SAMPE Journal, 24, No. 3, May (1988), pp. 15-19. [45] Zhu, S.D., Theoretical and experimental analysis of roll forming. Ph.D. thesis, University of Auckland (1993). [46] Martin, T.A., Bhattacharyya, D., Collins, I.F., Bending of fibre-reinforced thermoplastic sheets, Composites Manufacturing, 6, (1995), pp. 177-187. [47] Martin, T.A., Mander, S.J., Dykes, R.J., Bhattacharyya, D., Bending of continuous fibre-reinforced thermoplastic sheets, this volume, pp. 371-401.
This . Page Intentionally Left Blank
A U T H O R INDEX
Numbers refer to pages on which the author (or his work) is mentioned. Numbers between brackets are the reference numbers. No distinction is made between first author and co-author(s).
Beaussart, A.J., 348 [44]; 348 [45] Becraft, M.L., 352 [52] Bellamy, A.M., 173 [10]; 373 [5]; 428 [89] Bellet, M., 86 [35] Berglund, L.A., 60 [30] Bergsma, O.K., 176 [13]; 254 [18] Bersee, H.E.N., 362 [71]; 363 [71] Bhattacharyya, D., 148 [52], [53]; 159 [59]; 226 [7]; 234, 236 [14]; 238, 239 [17]; 240 [14]; 241 [17]; 242 [14], [17]; 243 [14]; 372 [4]; 377 [18], [19]; 382 [21]; 383 [19], 388 [19]; 476 [18]; 487 [14], [16], [18], [19]; 488 [16]; 492 [16], 493 [16], 499 [37], 503 [39], 505 [39], 506 [37], 506, [39], [41]; 512 [46], [47] Billoet, J.L., 179 [17] Binding, D.M., 352 [50]; 352 [51]; 364 [50]; 364 [51] Bird, R., 390 [24] Bird, R.B., 329-331,335, 348 [8] Birley, A.W., 78 [3] Blanlot, R., 179 [17] Bowden, F.P., 200, 203, 205 [37] Braden, T.D., 419 [80] Bradford, I.D.R., 374 [11], [12] Bradley, J., 404 [13], [9]; 419 [9] Bradley, J.S., 404, 419 [7] Bradley, S., 231 [13] Brady, D.G., 50 [8] Brandis, H., 249 [3] Brandt, J, 405, 406 [45] Breuer, U., 147 [51]; 159 [62] Brinker, W.O., 419 [80]; 419 [81] Broutman, L.J., 140, 144 [48] Brown, S.A., 405 [33] Bruschke, M.V., 324, 358 [1]; 358 [59] Buchanan, J.S., 419 [76]
Acrivos, A., 342, 343 [30] Adkins, J.E., 83 [30]; 228 [11] Advani, S.G., 100 [22]; 234, 235 [15]; 254 [17]; 324 [1], [2]; 341 [28]; 345 [36]; 354 [54]; 358 [1], [57], [58], [59]; 364 [36] Ahmetoglu, M.A., 497 [27]; 498 [29] Akeson, W.H., 404, 419 [14], [70], [82] Alexander, H., 419 [68]; 434 [92] Allard, R., 80 [13] Allgoewer, M., 419 [65] Allred, R.E., 54 [19] Altan, T., 497 [27]; 498 [29] Alvarez, R.A., 419 [72] Amiel, D., 404, 419 [14] Andersson, P.A., 409 [61] Andro, R., 86 [35] Apalset, K., 419 [71] Armstrong, R.C., 329-331,335 [8]; 390 [24] ,~strfm, B.T., 358 [57] Bafna, S.S., 358 [60] Bahadur, S., 199 [34] Baird, D.G., 358 [60] Balasubramanyam, R., 359 [63] Barlow, C.Y., 405 [17]; 405 [23] Barnes, A.J., 95 [8]; 158 [58] Barnes, H.A., 339 [23] Barnes, J.A., 100 [23]; 115, 125 [39]; 170, 172 [6]; 359 [64] Barone, M.R., 198 [31]; 359 [62] Bartenev, G.M., 199 [33] Bartolomucci, J.P., 50 [11] Bartos, O., 80-83, 86 [18] Batchelor, G.K., 345, 347 [37] Batchelor, J., 78 [3] Bathe, K.J., 80 [10] Beaussart, A., 254 [32] 517
518
Author index
Buck, A., 407 [56]; 428 [84] Bunsell, A.R., 54 [18]; 55 [18] Butler, D., 419 [80] Caddell, R.M., 4, 6 [1]; 239 [18] Cai, Z., 358 [56]; 361 [66] Cakmak, M., 252 [15] Caldwell, D.L., 43 [4] Canavan, R.A., 258, 277, 307 [40] Cantwell, W., 405 [22] Cantwell, W.J., 405 [26] Carley, J.F., 83 [26], [27] Carlsson, L.A., 94 [6]; 109 [33] Carreau, P.J., 331,360 [13] Cattanach, J.B., 95 [8]; 158 [58]; 228 [9]; 250 [13]; 499 [36] Caulk, D.A., 198 [31]; 359 [62] Chang, I.Y., 325 [4]; 364 [73] Chang, L.W., 434 [93] Charrier, J.M., 80 [13] Chawla, J., 140, 144 [48] Chey, S., 254 [20]; 442 [2], [4]; 443 [6]; 445 [2], [4], [8]; 455 [2]; 457 [4]; 469 [8]; 470 [6] Chou, S., 405, 406, 433 [41] Chou, T.W., 123 [41]; 434 [91], [93] Christensen, R.M., 342, 343, 361,362 [31]; 397 [28] Christie, G.R., 221 [5]; 226 [7] Chu, E., 218, 219 [1] Chuang, Y., 410 [63] Claes, L., 404 [4], [10]; 419 [4] Coffin, D.W., 285, 303 [57]; 348 [42], [43]; 361 [42]; 362 [43] Cogswell, F.N., 36 [2]; 58, 59, 60, 66 [2]; 93, 95 [1]; 100 [18], [23]; 101 [1]; 115, 125 [39]; 165 [1]; 170 [5], [6]; 172 [6]; 173 [1]; 174 [5]; 181 [1]; 189 [5]; 228 [9]; 249 [2]; 250 [13]; 258, 259, 283, 307 [2], 348 [46], 351 [46], 359 [64], 362 [68], 362 [69], 372 [1], 372 [3], 382 [1]; 499 [36] Collins, I.F., 226 [7]; 377 [19]; 383 [19]; 388 [19]; 487, 488, 492, 493 [16]; 499, 506 [37]; 512 [46] Colton, J.S., 363 [72] Comtet, J.J., 404 [8] Corcoran, S.J., 419 [68] Cordey, J., 419 [75] Coutts, R.D., 404, 419 [14]; 419 [70] Creasy, T.S., 345, 364 [36]
Cuff, G., 97, 100 [12]; 228 [9]; 506, 508 [42] Cutolo, D., 94 [3] Daniel, I.M., 57 [27] Darcy, H., 357 [55] Davies, P., 405 [22]; 405 [26] Day, R., 419 [73] De Angelis, F., 428 [84] de Haan, J., 407, 410, 412 [58]; 413 [64]; 415, 418, 429 [58] Delaloye, S., 428 [86] de Lorenzi, H.G., 80 [15], [16], [17]; 81, 82 [16] De Luca, P., 250 [6] Demarmels, A., 85 [34] Desiderato, R., 404 [6] de Waele, A., 331 [12] Diefendorf, R.J., 54 [20] Dillon, G., 254 [19], [20]; 442 [4]; 443 [6]; 445 [4], [8]; 446, 448, 451 [11]; 457 [4]; 469 [11]; 469 [8]; 470 [6] Drechsler, K., 405 [39]; 405 [45]; 406 [39]; 406 [45]; 432, 433 [39] Duckett, R.A., 409 [62] Duggal, N., 497 [27] Dumas, P., 404 [8] Duncan, J.L., 19 [4]; 218, 219 [1]; 221 [3], [4]; 474, 475, 484 [1]; 493 [20], [21]; 495 [20], [23]; 496 [25]; 497 [20], [23] Dutta, A., 252 [15] Dykes, R.J., 377 [18]; 512 [47] Eckert, K.-L., 435 [97] Eckold, G., 57 [26] Einstein, A., 341 [29] El'kin, E.I., 199 [33] Eng, B., 419 [73] England, A.H., 374 [11]; 374 [12] Ericson, M.L., 60 [30] Eringen, A.C., 83 [31] Esteghamatian, M., 79 [8] Ethridge, E.C., 419 [69] Evans, J.T., 374 [13] Fife, B., 405 [17] Fines, R.E., 50 [11] Fitzer, E., 100 [20] Fitzpatrick, J.E., 50 [13] Flemming, M., 428 [84], [88]
Author index
Francis, D., 405 [32] Frankel, N.A., 342, 343 [30] Fredrickson, G.H., 347 [41] Freundlich, H., 339 [22] Friedrich, K., 97 [13], [13]; 100 [15], [16], [21], [24], [26], [27]; 101 [28]; 111 [38]; 159 [59], [63], [64]; 181 [18], [19]; 254, 307 [31]; 373 [6], [10]; 386 [10]; 405 [21]; 499 [33]; 505 [401; 506 [411 Fritz, U., 409 [60] Fukaya, N., 487, 488 [13] Gachon, H., 179 [17] Galanty, P.G., 50 [7] Gautier, E., 419 [75] Gere, J.M., 457 [16] Ghafur, M.O., 80, 82 [19]; 86 [36], [38] Ghasemi-Nejhad, M.N., 405 [24]; 434 [91] Ghosh, A., 80 [13] Giacomin, A.J., 353 [53] Giesekus, H., 334 [18] Giorgetta, S., 410 [63] Goddard, J.D., 345 [35]; 347 [39], [40] Gogos, C.G., 77 [2]; 260 [44]; 359 [61] Gold, R., 475 [2] Golden, K., 254 [28]; 446, 448 [12] Gomez, M.A., 419 [70]; 419 [82] Gonzalez-Zugasti, J., 254 [19]; 446, 448, 451,469 [11] Goshawk, J.A., 373 [7] Green, A.D., 83 [29], [30] Green, A.E., 228 [11] Griese, R.A., 405 [29] Grosch, K.A., 200 [36] Groves, D.J., 173 [10]; 181, 210 [21]; 258, 259 [37]; 351, [49]; 373 [5]; 428 [891 Gunderson, S.L., 434 [94] Gur, M., 499 [35] Gutowski, T.G., 100 [25];, 165 [2]; 181 [25]; 252 [14]; 254 [19]; 254 [20], 254, 307, 314 [21]; 361 [65]; 361 [66]; 372 [2]; 373 [8]; 373, 379 [9]; 428 [90]; 442 [4]; 443 [6]; 445 [4]; 445 [8]; 446 [101; 446 [11]; 448 [10], [11], 449 [10], 451 [11], 455 [10], 457 [4], 458 [18], 469 [8], [10], [11]; 470 [6]; 507 [44] Gysin, H.J., 249 [4]
519
Ha, S.-W., 404 [3]; 406 [51], [53]; 407 [58]; 408, 409 [3]; 410 [58]; 412 [58]; 414, 415 [3]; 415 [58]; 416 [3]; 418 [51], [58]; 429 [51], [58]; 431,432 [51]; 435 [96], [97] Haessly, W.P., 80, 82 [19] Hailer, K.D., 234 [16] Halmos, G.T., 475 [4] Hancox, N.L., 53-55 [16] Hang, E., 250 [6] Hassager, O., 329-331,335, 348 [8]; 390 [24] Hastings, G.W., 404 [7]; 404 [9]; 419 [7], [9], [11], 404 Hatebur, A., 406, 410, 419, 422, 434 [55] Hauweller, P., 499 [34] Haworth, B., 78 [3] Hayes, W.C., 419 [77] Hearle, J.W.S., 348 [44], [45] Hehne, H.J., 404 [6] Heisley, F.L., 234 [16] Heislitz, F., 498 [29] Hench, L.L., 419 [69] Heppenstall, R.B., 419 [77] Herrington, P.D., 200 [38] Hickman, G.T., 406 [46] Hill, R., 14 [2] Hobbs, R.M., 474, 475, 484 [1] H6ger, A., 104, 119 [29] Holmstrom, T., 419 [79] Horn, W.J., 405 [30] Hosford, W.F., 4, 6 [1]; 239 [18] Hou, M., 97 [13]; 100 [13],[24], [26], [27]; 101 [28]; 159 [60], [61], [63], [64]; 373, 386 [10]; 499 [33]; 505 [40] Hoult, D., 254 [19]; 446, 448, 451,469 [11] Hsiue, E.S., 141 [49] Huisman, J., 254 [18] Hull, B.D., 254, 311 [26]; 456 [14], [15] Hull, D., 405 [42], [43]; 406 [42], [43] Htittner, W., 404 [4], [5]; 419 [4] Hylton, D., 85 [32] Ichidayama, M., 487 [10] ICI Thermoplastic Composite Handbook 1992, 207 [42] Igl, S.A., 82 [24] Ishai, O., 57 [27] Ishida, H., 405 [15] Ivaska, J., 481 [6], [7]; 484, 487 [7]
520
Author index
J/iger, H., 100 [20] Jammet, J.C., 86 [35] Jang, B.Z., 42 [3] Jeffreys, H., 334 [17] Jenkins, R.B., 419 [80] Jimma, T., 482, 487 [12]; 487, 488 [13] Johnson, A.F., 176 [15] Johnson-Nurse, C., 404 [7]; 404 [9]; 419 [7],
[9] Jones, A.D., 339 [22] Jones, R.S., 181 [22]; 359 [63]; 373 [7] Kaliakin, V.N., 255 [33] Kalpakjian, S., 140, 144 [48] Kamal, M.R., 341 [26]; 351,352 [47] Kaprielian, P.V., 181 [23]; 258 [39]; 310 [54]; 362, 363 [70] Karaharju, E., 419 [79] Kardos, J.L., 165 [4] Kardos, L.J., 405 [35] Karger-Kocsis, J., 406 [52]; 410 [63] Kausch, H.H., 405 [22]; 405 [26] Kazomer, D.O., 80 [17] Keefe, M., 234, 235 [15]; 254 [17] Kempe, G., 405 [27] Ketzer, V., 159 [62] Keuscher, G., 404 [5] Kinzel, G.A., 498 [29] Kinzel, G.L., 497 [27] Kinzl, L., 404 [10] Kirch, M., 413 [64] Kiuchi, M., 487 [10]; 487 [8], [15], [17]; 493, 497 [22] Kizior, T.E., 345 [34] Klenner, J., 249 [3] Klinkmiiller, V., 100 [21] Ko, F.K., 123 [41]; 405 [37] Koch, B., 406 [51]; 406 [53]; 406, 410 [55]; 410 [63]; 418 [51]; 419, 422 [55]; 429, 431, 432 [51]; 434 [55] Koch, P., 140 [47] Koch, S.B., 95 [9] Kokado, J.I., 487 [9] Korger-Roth, G., 405 [27] Kouba, K., 80 [18], [19]; 81, 82 [18]; 82 [19]; 83 [18]; 86 [18], [36], [37], [38] Koudabashi, T., 487 [17]; 493, 497 [22] Krauss, H., 405 [27] Krebs, J., 159 [59]; 506 [41]
Kroschwitz, J.I., 109 [32] Krueger, W.H., 364 [73] Kuczynski, K., 14 [3] Kutz, J., 405 [37] Lam, R.C., 165 [4] Lammerding, J.J., 419 [81] Latta, L.L., 419 [72] Laun, H.M., 339 [25]; 340 [25]; 352 [25] Laursen, T.A., 313 [56] Leach, D.C., 100 [18]; 372 [3] Lee, D., 224 [6] Lennon, J.J., 201,209, 212 [40] Levenetz, B., 404, 419 [14] Li, H., 254 [20]; 442 [3], [4]; 443 [6]; 445 [3], [4], [8]; 446, 450 [3]; 451,455, 457 [3]; 457 [4]; 464, 465, 469 [3]; 469 [8]; 470 [6]; 498 [281 Li, H.L., 140 [47] Lind, D.J., 158 [57] Liskova-Klar, M., 419 [78] Little, R.W., 419 [80], [81] Livatyali, H., 498 [29] Lobe, V.M., 338 [20]; 343, 344 [20] Lodge, A.S., 388 [23] Lothringer, K.S., 419 [70] Ludema, K.C., 199 [34] Lfischer, P., 408, 409, 415 [59] Lustinger, A., 405 [18], [20], [25] Maher, E.J., 211 [43] Mai, Y.-W., 159 [60] Mallon, P.J., 95 [7]; 139 [45], [46]; 142 [46]; 144 [45]; 170 [7]; 179, 181 [16]; 181 [27], [28], [7]; 198 [30]; 201,209, 212 [40]; 249251 [5]; 259 [43]; 271 [48]; 277, 285 [5]; 286 [501; 287 [51, [481; 289 [51; 308, 311 [531; 314 [53]; 443, 445 [7]; 499 [30] Mander, S.J., 499 [37]; 506 [37]; 512 [47] Mgmson, J.A.E., 93 [2]; 405 [28] Marangou, M., 80 [13] Marciniak, Z., 14 [3]; 19 [4] Martin, T.A., 148 [52-54]; 150 [54]; 226 [8]; 234, 236 [14]; 238 [8], [17]; 239 [17]; 240 [14]; 241 [17]; 242 [14], [17]; 243 [14]; 372 [4]; 377 [18-20]; 382 [21]; 383 [19], [20]; 388 [19]; 503, 505, 506 [39]; 507 [43]; 512 [46], [47]
Author index
Matsuoka, S., 333 [16] Matthews, J.V., 404, 419 [14] Maxwell, J.C., 333 [14] Mayer, J., 404 [1]; 406 [48-55]; 407 [58]; 408 [1], [50], [59]; 409 [48], [59]; 410 [48], [50], [55], [58], [63]; 411 [1], [48]; 412 [1], [58]; 413 [1], [64]; 414 [1]; 415 [58], [59]; 416 [1], [50]; 417 [48], [50]; 418 [49], [50], [51], [58]; 419 [49], [55]; 420 [1], [48], [54]; 421 [1], [48]; 422 [1], [48]; 422 [54], [55]; 423 [1]; 424 [1], [48]; 425 [1], [48]; 426 [1], [48]; 427 [1]; 428 [84]; 429 [1], [48], [51], [58]; 430, 431 [48]; 431 [51]; 432 [48], [51]; 434 [55]; 435 [96], [97] Mayer, R.M., 53-55 [16] McClure, C.K., 498 [28] McCormack, B., 405 [31] McDonald, W., 419 [73] McEntee, S.P., 250, 255 [12] McGuinness, G.B., 250 [10], [11]; 255 [10], [11]; 258 [40], [41]; 267 [47]; 277 [40], [41], 307 [40], [41] McKibbin, B., 404 [9]; 419 [9], [67] McLaren, R., 419 [73] McNamara, A., 405 [34]; 419 [83] Medwin, S.J., 327, 364 [6] Melander, A., 218, 219 [2] Menzel, E., 499 [35] Merrit, K., 405 [33] Metzner, A.B., 333 [15]; 345, 347 [33]; 347 [38] Mewis, J., 345, 347 [33] Middleton, V., 405 [36], [40]; 406, 415 [36], [40]; 429, 432 [36]; 432, 433 [40] Miles, J.N., 362 [67] Miller, A.K., 254, 307 [22]; 314 [22]; 499 [35] Miller, D.M., 55 [22] Milliken, W.L., 341 [27] Mirza, F.A., 82 [22], [23] Mitsoulis, E., 78 [4] Moet, A., 405 [33] Molden, G.F., 362 [67] Monaghan, M.R., 139, 144 [45]; 170 [7]; 181 [27], [28], [7]; 198 [30]; 259 [43]; 271 [48]; 286 [50]; 287 [48]; 308, 311,314 [53]; 443, 445 [7] Monasse, B., 86 [35] Mooney, M., 81, 84 [20] Morgan, R.J., 54 [19]
521
Morigaki, T., 361 [66] Morris, S.R., 181, 184 [26] Moulin, C., 405 [22] Moy, P., 50, 53 [6] Moyen, B., 404 [8] Mulholland, A.J., 170, 181 [7] Muller, G.W., 419 [77] Mfiller, M.E., 419 [65] Mfiller, U., 409 [60] Mullis, D.L., 419 [72] Murtagh, A.M., 175, 177 [12]; 179 [16]; 181 [12], [16], [27], [28]; 188 [12]; 201 [40]; 206, 208 [12]; 209 [40]; 211 [12]; 212 [40]; 213 [12]; 283 [49]; 308 [49], [53]; 311 [49], [53]; 314 [49], [53]; 317 [49] Murty, N.K., 362 [67] Mutel, A.T., 341 [26]; 351,352 [47] Muzzi, J., 194 [29] Muzzy, J., 94 [4] Muzzy, J.D., 363 [72] Nakajima, N., 173, 174, 181, 188 [11]; 258 [38]; 405 [19] Nakajima, S., 487 [8], [10] Neitzel, M., 147 [51]; 159 [62] Neoh, E.T., 458 [17] Nestor, T.A., 258, 277, 307 [40] Neugebauer, R., 404 [10] Newatz, G.M., 405 [20] Newaz, G.M., 405 [25] Newman, S., 406 [47] Nguyen, H.X., 405 [15] Nied, H.F., 80 [15], [16]; 81, 82 [16] Niedermeier, M., 428 [84], [85], [88] Nield, E., 405 [17] Nietert, M., 404 [5] Noble, C.L., 487, 488 [11] Norpoth, L., 94 [4], [29] Noser, G.A., 419 [81] Nunamaker, D.M., 419 [77] Br~tdaigh, C.M., 95 [7]; 100 [17]; 139 [45], [46]; 142 [46]; 144 [45]; 248 [1]; 249 [5]; 250 [5], [8-12]; 251 [5]; 254 [1]; 255 [8-12]; 258 [40], [41]; 259 [42]; 260 [8]; 261, 262 [8], [9]; 263 [1], [46]; 264 [42]; 267 [47]; 268 [46]; 271 [48]; 277 [5], [40], [41]; 282 [46]; 285 [5]; 287 [5], [48]; 289 [5]; 307 [40], [41]; 443, 445 [7]; 499 [30]
522
Author index
Oakley, D., 181 [22] Oakley, J.G., 353 [53] O'Conell, P.A., 409 [62] Oden, J.T., 80 [9], 82 [9] Ogden, R.W., 81, 84 [21] Okine, R.K., 254 [16], [32]; 325 [3]; 345 [36]; 348 [42], [44], [45]; 361 [42]; 364 [36]; 499 [321 O'Leary, M., 475 [3] Ona, H., 482, 487 [12]; 487, 488 [13] Onada, Y., 487 [9] O'Neill, J.M., 119, 127 [40]; 128 [42]; 181 [23]; 254 [27]; 310 [54]; 362,363 [70]; 374 [14]; 386 [22]; 502, 504 [38] Osswald, T.A., 82 [24] Ostrom, R.B., 95 [9] Ostwald, W., 331 [11] Oswald, H.J., 140 [47] O'Toole, B.J., 327 [5] Otsubo, Y., 339 [21] Owen, M.J., 405 [36], [40]; 406, 415 [36]; 415 [40]; 429 [36]; 432 [36], [40]; 433 [40] Owens, G.A., 158 [57] Ozisik, M.N., 107 [30] Paavolainen, P., 419 [79] Padscheider, J., 410 [63] Panton, R.L., 329, 330 [9] Panton, S.M., 476, 487 [18]; 493 [20], [21]; 495 [20], [23]; 496 [25]; 497 [20], [23] Parnas, R.S., 324, 358 [1] Parsons, J.R., 419 [68]; 434 [92] Parvizi-Majidi, A., 405 [24] Payne, A., 499 [35] Peacock, J.A., 405 [17], [23] Peled, A., 499 [35] Perdikoulas, J., 78 [7] Perren, S.M., 419 [66], [75], [76] Peters, D.M., 108 [31] Petitmermet, M., 407, 410, 412, 415, 418, 429 [58] Petrie, C.J.S., 337 [19] Phelan, F.R., 358 [58] Pickett, A.K., 250 [6] Pigliacampi, J.J., 55 [21] Pinruethai, P., 434, 435 [95] Pipes, R.B., 95 [7]; 100 [17], [19], [22]; 139 [45], [46]; 142 [46]; 144 [45]; 148 [52], [53]; 234 [14], [15]; 235 [15]; 236 [14]; 238, 239
[17]; 240 [14]; 241 [17]; 242 [14], [17]; 243 [14]; 249 [5]; 250 [5], [8-101; 251 [5]; 254 [17], [32]; 255 [8-10], [33]; 259 [42]; 260, 261 [8], 262 [8], [9]; 264 [42]; 271 [48]; 277, 285 [5]; 287 [48], [5]; 289 [5]; 342 [32]; 348 [42]; 348 [43]; 348 [44], [45]; 352 [51]; 358 [57]; 361 [42]; 362 [43]; 364 [51]; 372 [4]; 382 [21]; 397 [27]; 428 [87]; 443 [7]; 445 [7]; 499 [30] Pipkin, A.C., 255 [35]; 270 [35]; 311 [55]; 375, 379 [16]; 394 [25]; 446, 468 [9] Planck, H., 405, 433 [44] Potter, K.D., 141 [50] Powell, R.L., 341 [27] Pratte, J.F., 325 [4]; 364 [73] Prevorsek, D.C., 140 [47] Price, C.R., 339 [24] Queckborner, T., 250 [6] Rahn, B.A., 419 [75] Ramakrishna, S., 405 [42], [43]; 406 [42], [43] Ranganathan, S., 358 [58] Rasmussen, M.L., 150 [56] Reber, R., 413 [64]; 435 [97] Reddy, J.N., 150 [56] Reinicke, R., 159 [62] Richardson, J.J., 50 [7] Rivlin, R.S., 228 [10]; 462 [20] Robertson, R.E., 141 [49] Robroek, L.M.J., 94 [5]; 362 [71]; 363 [71] Rodriguez, F., 331,333, 349 [10] Rogers, T.G., 128 [42]; 173 [9]; 254 [25-28]; 255 [25], [35]; 256 [25]; 258 [39]; 270 [35]; 311 [26], [55]; 374 [11], [12], [14]; 375 [16]; 377 [17]; 379 [16]; 394 [25]; 446 [12], [9]; 448 [12]; 456 [14], [15]; 468 [9] Rosen, B.W., 56 [24] Rudd, C.D., 405 [36], [40]; 406 [36]; 415 [36], [40]; 429, 432 [36], [40]; 433 [40] Ruffieux, K., 406 [51], [53-55]; 410 [55]; 418 [51]; 419 [55]; 420, 422 [54], [55]; 429, 431 [51]; 432 [51]; 434 [55] Rumelhart, C., 404 [8] Ryan, M.E., 80 [13] Sabbaghian, M., 200 [38]
Author index
Sampson, S., 419 [77] Samuelsson, A., 254 [30] Santini, R., 404 [8] Sapega, A., 419 [77] Sarantidis, T.M., 487, 488 [11] Sarmiento, A., 419 [72] Sastry, A.M., 348 [44] Saunders, D.W., 462 [20] Savadori, A., 94 [3] Scardino, F., 405 [38] Schedin, E., 218, 219 [2] Schein, S.S., 419 [77] Schepers, E.J.G., 434, 435 [95] Scherer, R., 100 [15], [27]; 111 [38]; 181 [18], [19]; 254 [31]; 307 [31]; 373 [6] Schmidt, L.R., 83 [26], [27] Schneider, E., 419 [75] Schneider, R., 419 [65] Scholten, H.J., 419 [74] Schulten, T., 406, 410, 419, 422, 434 [55] Schwab, P., 419 [76] Schwarz, P., 409 [60] Scobbo, J.J., 173, 174, 181, 188 [11]; 258 [38] Scobo, J.J.R., 405 [19] Seferis, J.C., 405 [16], [28] Seguchi, Y., 419 [82] Seyer, F.A., 345 [34] Shaikh, F.M., 405 [30] Shalaby, S.W., 50, 53 [6] Shapery, R.A., 56 [23] Shaqfeh, E.S.G., 347 [41] Shrivastava, S., 80 [13] Shuler, S.F., 348 [42]; 352 [51]; 354 [54]; 361 [42]; 364 [51] Sickafus, E.N., 141 [49] Siegling, H.F., 405, 406 [45] Silvermann, E.M., 405 [29] Silvi, N., 78 [6] Simacek, P., 255 [33], [34]; 348, 361 [42] Simo, J.C., 313 [56] Simon, B.R., 419 [70], [82] Skirving, A.P., 419 [73] Slatis, P., 419 [79] Smiley, A.J., 100 [19]; 428 [87]; 499 [31] Smirnov, V.V., 479 [5] Smith, P.D., 487 [14], [16], [19]; 488, 492, 493 [16] Soeganto, A., 405 [30]
523
Soil, W., 373 [8]; 507 [44] Soil, W.E., 181 [24]; 252 [14] Solt~sz, U., 404 [6] Song, W.N., 82 [22], [23] Sowerby, R., 218, 219 [1] Spencer, A.J.M., 128 [42]; 173, 176, 181 [8]; 228 [12]; 254 [24], [26], [28]; 255 [24]; 311 [26]; 374 [15]; 394 [26]; 397 [26]; 446, 448 [12]; 456 [14], [15] Spescha, G., 410 [63] Steffee, A.D., 405 [33] Stocks, D.M., 173 [10]; 258, 259 [37]; 351 [49]; 373 [5]; 428 [89] Strong, A.B., 499 [34] Struik, D.J., 448 [13] Stiirmer, K.M., 419 [74] Suemasu, H., 101 [28] Sun, C.T., 181, 184 [26] Suzuki, H., 487 [8], [10] Tabor, D., 199 [32]; 200, 203, 205 [37] Tadmor, Z., 77 [2]; 260 [44]; 359 [61] Takada, K., 487 [10] Talbot, M.F., 254, 307, 314 [22] Tam, A.S., 100, 181 [25]; 254, 307, 314 [21]; 373, 379 [9]; 428 [90]; 443 [5]; 446, 448, 449, 455 [10]; 458 [18]; 469 [10] Tanaka, K., 199, 205 [35] Tanner, R.I., 85 [33] Tarr, R.R., 419 [72] Taske II, L.E., 50, 51 [10] Taylor, C.A., 80 [15], [17] Taylor, D., 405 [31] Taylor, R., 56 [25] Taylor, R.L., 80 [11], [12]; 263 [45] Tayton, K., 404, 419 [9] Tayton, K.J.J., 404 [12], [13] Terjesen, T., 419 [71] Thomas, K.L., 108 [31] Thompson, E.G., 254 [30] Throne, J.L., 76, 83 [1]; 200 [39] Timoshenko, S.P., 457 [16] Tobolsky, A.V., 49 [5] Tognini, R., 406 [51], [53]; 406 [54]; 418 [51]; 420, 422 [54]; 429, 431,432 [51] Toll, S., 409 [61] Treloar, L.R.G., 83 [25] Tsclikov, A.I., 479 [5] Tucker, C.L., 254 [29]; 341 [28]
524
Author index
Tucker, C.L. III, 442, 456 [1] Turner, R.M., 405 [34]; 419 [83] Uhthoff, H.K., 419 [781 Uralil, F.S., 405 [25] Vantal, M.H., 86 [35] Van West, B.P., 100 [22]; 176 [14]; 234, 235 [15]; 254 [17] Varughese, B., 94 [4]; 194 [29] Vetter, S., 407 [57] Vlachopoulos, J., 78 [4-7]; 80 [18], [19]; 81 [18]; 82 [18], [19], [22], [23]; 83 [18]; 86 [18], [36-38] Vlcek, J., 78 [6], [7] Vogel, J.H., 224 [6] Wagner, M.H., 85 [34] Wang, E.L., 372 [2] Warby, M.K., 80 [14] Ward, I.M., 83 [28]; 461 [19] Watson, P., 499, 506 [37] Watt, D.F., 79 [8] Watterson, E.C., 50 [12] Weeton, J.W., 108 [31] Weinberger, C.B., 345 [35] Weiss, A.B., 419 [68] Weiss, R., 404, 419 [4] Wenz, L.M., 405 [33] Wheeler, A.B., 359 [63] Wheeler, A.J., 165 [3] White, J.L., 333 [15]; 338 [20]; 343, 344 [20] Whiteman, J.R., 80 [14] Whitney, N.M., 434 [94] Wild, U., 413 [64] Willenegger, H., 419 [65] Williams, D.F., 405 [34]; 419 [83] Williams, D.J., 406 [46] Williams, R., 405 [32]
Windle, A.H., 405 [23] Wintermantel, E., 404 [1-3]; 406 [49], [50], 406 [51-55]; 407 [58]; 408 [1], [3], [50], [59]; 409 [3], [59]; 410 [50], [55], [58], [63]; 411 [1]; 412 [1], [58]; 413 [1], [64]; 414 [1], [3]; 415 [3], [58], [59]; 416 [1], [3], [50]; 417 [50]; 418 [49-51], [58]; 419 [49], [55]; 420 [1], [54]; 421,422 [1], [54], [55]; 423-427 [1]; 428 [84]; 429 [1], [51], [58]; 431,432 [51]; 434 [55]; 435 [96], [97] Wirz-Safranek, D.L., 95 [9] Wittich, H., 100 [16] Woo, S.L.-Y., 404, 419 [14], [70]; 419 [82] Woo, Y.K., 419 [70] Wood, R.D., 254 [30] Wu, C.J., 405, 406, 433 [41] Wu, R., 80 [13] Wu, X., 363 [72] Xiang, Wu, 181 [20] Yamada, Y., 199, 205 [35] Yau, S.S., 434 [93] Ye, L., 100 [21]; 159 [60] Yee, C.H., 487, 488, 492, 493 [16] Yeh, G.S.Y., 141 [49] Yuan, Q., 406 [52] Zahlan, N., 100 [15]; 119, 127 [40]; 181 [19]; 254 [31]; 307 [31]; 386 [22]; 502, 504 [38] Zamani, N.G., 79 [8] Zema, W., 83 [29] Zhang, Z.T., 221 [3], [4] Zhu, S.D., 493 [20], [21]; 495 [20], [23], [24]; 496 [25]; 497 [20], [23]; 511 [45] Ziegrnann, G., 428 [86], [88] Zienkiewicz, O.C., 80 [11], [12]; 254 [30]; 263 [45] Zimmerman, M., 434 [92]
SUBJECT INDEX Cetex fabric, 179, 180, 182, 196, 197, 211-214 Clamping pressure, 141, 145 Coefficient of friction, 200, 205, 213 versus sliding velocity, 209 Coefficient of thermal conductivity (CTC), 52 Commingled fibres, 93 Commingled materials, 59 Commingled reinforcement, 47 Composites, 29, 55-60 applications, 37 mechanical properties, 56-60 thermal properties, 55-58 Compression moulding, 93 Compressive strain contour map, 242, 244 Computer-aided design in roll forming, 489-491 Computer simulation of thermoforming, 75-89 Consistency index, 390 Consolidation quality and lay-up, 172 Constitutive relationships, 11, 86, 256 Continuous fibre-reinforced composites, 164 Continuous fibre-reinforced pre-consolidated laminates, 97 Continuous fibre-reinforced thermoplastics (CFRT), 92, 100, 125, 147, 227 formability, 372 rheological behaviour of, 371-401 sheets, 91-162, 218, 233, 498-512 Continuous glass-fibre reinforced thermoplastic composite, 96 Continuum mechanics analysis, 254 Contour maps, 225 Contrary knitting technique, 435 Controlled strain rate forming, 364 Couette flow, 311,329 Coulomb friction, 198 Covalent bonds, 32, 34 Crosslinks, 34 Crystal conformation, 32 Crystal structure, 4 Crystalline melting point, 49 Crystallinity and stamping velocity, 131 Crystallinity degree, 33, 34 Cubic Hermite basis functions, 222
Adhesion theory, 199 Admissible stress fields, 379-380 Aerodynamic fairing, 230 Amorphous structures, 33, 34, 35, 78, 104 Angular displacement, 129-130 Anisotropic biomaterials, 404-405 Anisotropic flow rules, 6 Anisotropic materials, yielding, 3-4 Anisotropy, 412-4 13 APC-2, 172, 173, 181, 182, 184, 187, 192-195, 197, 200, 201,203,204, 206, 208, 210-214, 286 Arrow diagram, 225, 229, 233-237, 241,243 Aspect ratio, 339, 340 Axial intra-ply shear deformation, 361-362 Axial stress, 272, 278, 280, 282, 283, 291,296-298 Barrelling effect, 166 Bend angle and stamping time, 119 Bend forming, 489 Bending, 19, 80, 371-401,487-489 constraint conditions and kinematics, 374-376 idealised viscous model, 374-376 under tension, 22-23 without tension, 19-20 Bending angle, 123, 127-130 Bends, sequencing, 486 Bi-cubic Hermite element, 222 Biocompatibility, 434-435 Blister fairing, 230-234 Buckling, 112, 116, 146, 239, 252, 383, 460, 483 CAD/CAM systems, 489-491 CAEDS(I-DEAS), 425 Calendering, 78 Capillary rheometer, 350-352 Carbon-black-filled polystyrene, 343 Carbon fibre/PEEK composites, 57, 58, 60 Carbon fibre/PEEK laminates, 130, 132 Carbon fibre/PP laminates, 109, 110, 133, 134 Carbon fibres, 40, 43, 97, 164, 181,328 Carreau relation, 332 Cauchy-Green deformation tensor, 81, 85 Cauchy-Green strain tensor, 220 C-channels, 451-454, 466, 467, 468
Darcy's law, 358
525
526
Subject &dex
Deep drawing, 17-19, 68 knitted-fibre-reinforced organo-sheets, 428-432 Deformation, principal element, 4-5 Deformation analysis of roll forming, 491-498 Deformation behaviour and rheological properties, 329 Deformation gradient tensor, 225 Deformation length, 492, 511-512 Deformation modes in thermoset composite forming, 446-455 Deformation processes, 232, 373 Deformation theory, 199 Degree of crystallinity, 33, 34 Deviatoric stresses, 5 Diaphragm failure, 254 Diaphragm forming, 69, 94, 95, 146-159, 169, 249, 250, 252 assessment and characterisation of thermoformed parts, 148-152 experimental comparisons, 286-303 experimental details and procedures, 96-100, 147-148 friction, 197-213 inflated tool, 470-471 large strain analysis technique, 148-150 materials employed, 96-97 occurrence of defects and thickness variations, 150-152 pre-consolidated laminates, 98-99 recommendations, 157-159 reinforced, 469-470 thermoset composites, 443 variation of forming parameters and rating of part quality, 152-157 Diaphragm stiffness, 461-463 Diaphragm viscosity, 270-273, 275-279, 281 Differential scanning calorimetry (DSC), 106, 130, 132, 133, 506 Diffuse neck, 13 Digitised mesh diagram, 242 Double-belt press (DBP), 45, 63-64 Drape forming, 443 Draping theory of textile fabrics, 234-238 Effective strain, 6 Effective stress, 6 Elastic limit, 7 Elastic modulus, 83 Elastic-perfectly plastic model, 11 Elongational flows, 336-337, 343-348 Elongational rheometers, 355 Elongational velocity, 338 Elongational viscosity, 354-355
Explicit finite element method, 250 Extensional deformation, 364 Extra stress, 256 Extrusion, 78 Fabrics, 123, 328 draping theory, 234-238 inter-ply slip, 179, 189, 190 intra-ply shear, 174-175 transverse flow, 166 Failure criterion, 418 Failure model, 418 FEFORM, 263, 264, 291,310 Fibre architecture, 407-409 Fibre distribution, 114 Fibre-filled melts, 344 Fibre migration, 382 Fibre movement studies, 133-135 Fibre orientation, 186, 198, 210, 236 Fibre orientation distribution (FOD), 406, 407-409, 415, 431 Fibre-reinforced fluid constitutive model, 176 Fibre reinforcement, 42, 327, 328, 382 Fibre rotation, 255 Fibre straightening factor (FSF), 179 Fibre volume fraction, 116-118 Fibre wrinkling, 383, 384 Filled polymers, 351-352 Filled systems, 341,343 Filled viscous fluids, 338-348 Filler aspect ratio, 339 Film stacked prepregs, 93 Finger strain tensor, 85 Finite element analysis osteosynthesis plates, 425-428 roll forming, 498 Finite element equations, 83 Finite element formulations, 80-83 Finite element methods, 75, 79 explicit, 250 implicit, 247-322 osteosynthesis plates, 425-428 simulation code, 254 strain analysis, 224 Finite element software package, 86 Finite element solution technique plane deformation, 308-309 plane stress problems, 261-263 Flow behaviour, 210, 429 Flow mechanism, 100, 311 Flow rules, 5 anisotropic, 6 Flow stress, 24 Flower diagram, 485
Subject index Folding, 70 Forming characteristics of sheet metals, 6-12 Forming limit diagram, 14, 15, 464-468 Forming limits for sheet metal, 12-15 Forming pressure, 156 Forming ratio, 154-156 Forming temperature, 110-115, 153-154 Forming velocity, 156 Friction, 197-213 diaphragm forming, 197-213 effects of surface temperature, 204 influence of normal load, 205 mechanisms, 199 theories, 199 Friction power-law model parameters, 214 Friction sled, press forming, 201-202 Friction tests, 200-213 twin platen arrangement, 202 Frictional behaviour, 163-216 Frictional forces in press forming, 169-170 Gauss-Bonnet theorem, 450 Giesekus model, 335 Glass fibre fabric/PEI laminates, 97, 125, 126, 141-143 Glass fibre/PP laminates, 109, 143-144, 146-159 Glass fibres, 39, 43, 96, 97, 328 Glass-mat-reinforced thermoplastics (GMT), 48, 71-72, 95 Glass transition temperature, 49, 83, 86 Glass/PA composites, 58, 59 Glass/PP composite, 58 Green-Lagrange strain tensor, 81 Green-Lagrange strains, 84 Grid strain analysis, 217-245 application, 219 diagnostic applications, 238-241 technique, 241 Heat conduction model, 108 Heat transfer, 52 Heating temperature profile, 109 Heating time in relationship to laminate thickness, 110 Hemispherical dome, 235, 238 Hooke's laws, 11 Hot stamping, 326 Hydrocarbon polymers, 30 Hydrodynamic friction, 198-199 Hydroforming, 68 Hydrostatic stress, 5 Ideal fibre-reinforced fluid (IFRF), 255-258, 308, 309, 311,313
527
Image processing, 408 Implant materials, 404-405, 419-428 Implicit finite element modelling, 247-322 Incompressibility constraint, 257 Inextensibility condition, 257 Inflated tool forming, 443, 470-471 Injection moulding, 326 In-plane fibre movement, 106 In-plane shear, 449, 451-454 Integrated Finite Element Solver (IFES), 425 Interface temperature, 198 Inter-laminar rotation, 101 Intermittent matched-die consolidation, 65 Intermolecular structure, 32 Inter-ply forming mechanisms, 251 Inter-ply shear, 373, 386, 447, 448, 450, 454-455 deformation, 362-364 mechanism, 307 Inter-ply slip, 100, 113, 114, 167-168, 177-197, 305, 306, 314, 317 effect of lay-up variation, 187 effect of normal pressure, 186 effect of processing temperature, 190 effect of temperature, 184 fabrics, 179, 189, 190 laminates, 191 mechanism, 201 power-law model parameters, 197 test set-up, 182 velocity increments, 183 Intramer structure, 31 Intramolecular structure, 31 Intra-ply shear, 101, 166, 173-177 Isotropic materials, yielding, 2-3 Jeffreys fluid, 335, 336 Joining, 93 K-BKZ model, 86-88 Kevlar, 41,328 Knitted-carbon-fibre-reinforced composite materials, 405-419 Knitted-fabric-reinforced thermoplastics, 403-440 Knitted fabrics, net-shape forming, 419-428 Knitted-fibre-reinforced composites, mechanical properties, 412-4 19 Knitted-fibre-reinforced organo-sheets, deep drawing, 428-432 Lagrangian strain tensor, 220 Laminate dimensions, 141-142 Laminate stacking sequence, 115, 143
528
Subject index
Laminate thickness and lay-up, 157 and stamper radius, 126 and stamping pressure, 118-119 Laminate wrinkling, 455-458, 464--465 Large strain analysis, 218-224 Lay-up and consolidation quality, 172 inter-ply slip, 187 and laminate thickness, 157 Least squares fitting, 224-226 Linear variable differential transformer (LVDT), 170, 191,286, 287 Long discontinuous fibres (LDF), 46, 240 Long fibre-reinforced composites, rheological properties, 323-369 Manufacturing techniques, 60-72 Matched-die consolidation, 62-63 Matched-die forming, 250, 252, 259, 364-365 Matched-die moulding, 66-67, 71-72 Material behaviour in thermoforming, 83-86 Material functions, 329 Mathematical modelling, 79 Matrices, 29-38, 48-53, 327 mechanical properties, 52-53 migration, 114 thermal and rheological properties, 49-52 thinning, 239 Maxwell equation, 333 Melt impregnation, 45-46 Membrane formulations, 80-82 Memory effect, 119 Mesh sensitivity, 291-293 Mooney-Rivlin model, 84 Mould geometry and stamping time, 120 Mould surface finish, 200 Mould surface/release agent, 198 Moulding compounds, 48 Multi-axial stress conditions, 11 Multi-directional sheet analysis, 282-283 Multi-ply laminates, 311 Multi-valued function, 221 Navier-Stokes equation, 359 Necking, 13, 14 Net-shape forming, 433, 434 Newton-Raphson method, 82 Newtonian fluid, 330, 332, 338, 359 Newtonian viscosity, 330 Non-linear elastic models, 83 Normal loading effects, 212 Normal pressure, 198 Nylons, 36
Ogden model, 84, 86-88 Optical microscopy, 105 Order-of-magnitude analysis, 463-464 Organic fibres, 41-43, 328 Organo-sheets, 428-432, 434 Oscillatory shear experiments, 373 Osteosynthesis plates finite element analysis (FEA), 425-428 finite element modelling (FEM), 425-428 Oven bags, 99 Parallel-plate "squeeze flow" apparatus, 170 Penalizing, 313-314 Petroleum pitch, 40 Piola-Kirchhoff stress tensor, 84 Plane deformation finite element solution technique, 308-309 modelling, 307-308 numerical solution, 305-315 Plane stress problems, 258-263 analysis application, 303-305 finite element solution technique, 261-263 problem formulation and solution scheme, 260-261 Plastic flow, 2-6, 431 Plastic strain, 4 Plastic work, 6 Plastic yielding, 2-3 Platen angle, 389, 396 Ploughing of asperities, 199 Plug-assisted forming, 76 Ply contact formulation, 311-315 Plytron, 95, 96, 99, 153, 226-230, 238, 239, 241, 242, 380, 387, 389-391,395, 396, 398 Poisson's ratio, 11, 57 Polyacrylonitrile (PAN), 40 Polyamide (PA), 36, 41,328, 339, 340, 407, 414 Poly(amide imide) (PAl), 38 Poly(aryl sulfone) (PAS), 38 Poly(butylene terephthalate) (PBT), 36 Polyesters, 36 Poly(ether ether ketone) (PEEK), 33, 38, 50, 51, 59, 70, 95, 205, 212, 328, 407, 410, 415, 417, 418, 421 Poly(ether imide) (PEI), 38, 114, 123, 328 Poly(ether ketone) (PEK), 38 Poly(ether ketone ketone) (PEKK), 38, 95, 328, 343 Poly(ether sulfone) (PES), 38, 328 Polyethylene (PE), 30, 33, 36, 42 Poly(ethylene terephthalate) (PET), 36 Polyethylmethacrylate (PEMA), 407, 410, 414--416 Polyimide (PI), 38
Subject index
Polyisoprene, 31 Polymer matrix. See Matrices Polymer morphology, 30 Polymer structures, influencing properties, 31-34 Poly(phenylene sulfide) (PPS), 38, 328 Polypropylene (PP), 30-33, 36, 96, 328 Polystyrene (PS), 49, 338, 343, 344 Polysulfone (PSU), 38 Powder impregnation, 46-47, 94 Power hardening law, 11 Power-law strain rates, 332 Pre-consolidated laminates, 97-100 diaphragm forming, 98-99 Preheating time, 107-109 Preimpregnated reinforcement. See Prepregs Prepreg lay-up, 61 Prepregs, 43-44 comparison of types, 47 consolidation, 61 film stacked, 93 Press forming, 14, 203-204, 249, 250 friction sled, 201-202 frictional forces in, 169-170 Pressure loading, 275-282 Principal element, 2, 6 deformation, 4-5 Principal strain, 5 Principal stresses, 2, 3, 5 Processing temperatures, 50 Pull-out tests, 189, 373 Pultruded bands, 94 Pultruded continuous carbon fibre (CF) polypropylene (PP)-tape, 97 Pultruded continuous glass fibre (GF) polypropylene (PP)-tape, 97 Pultrusion, 93, 326 Punch deformation experiments, 266 Punch experiments with circular unidirectional sheets, 264-265 Quasi-isotropic laminates, 284, 285, 289, 290, 299, 300 Quasi-isotropic preforms, 289 Quasi-isotropic sheet, 301 r-value, 8 Radial pressure, 273-275 Radial velocity, 265-273 Rayon, 40 Reaction injection moulding (RIM), 324 Reaction stress, 256 Reinforced diaphragm forming, 469-470 Reinforcement-matrix interaction, 42-43 Reinforcements, 38-43
529
impregnation, 35 mechanical properties, 54 thermal properties, 54 Relaxation, 332, 339-343 Relaxation modulus, 49 Relaxation parameter, 334 Relaxation phenomena, 329-336 Relaxation time, 333 Release agent, 198, 201, 210-211 Residual stresses, 22 Resin interlayer, 208 Resin percolation, 100, 165, 357-358 Resin thickness measurements, 207 Resin transfer moulding (RTM), 324, 406 Rheological measurement of composite shear viscosities, 258 Rheological parameters in thermoforming, 364-366 Rheological properties application in sheet forming, 356-366 continuous fibre-reinforced thermoplastic, 371-401 and deformation behaviour, 329 long fibre-reinforced composites, 323-369 measurement techniques, 348-355 Roll angles, 487 Roll forming, 70, 473-515 computer-aided design, 489-491 defects, 482-483 deformation analysis, 491-498 equipment and procedures, 500--501 equipment and tooling, 476--483 finite element analysis, 498 form roll design, 483-489 operating conditions, 481 quality assessment, 501-506 section orientation, 485-486 strain distribution during, 495-498 temperature control, 506-510 thermoplastic material, 498-512 Roll lubrication, 481 Roll quality, 481 Roll schedule, 484-485 Roll setting, 482 Rolling mills, 479-480 auxiliary equipment, 480 Rotational rheometer, 351 Rotational viscometers, 348-350 Rubber-die moulding, 67 Scaling laws, 458-460, 464 Semicrystalline polymers, 33-36 Semicrystalline thermoplastics, 49 Shear buckling, 249, 251,253, 263-285
530
Subject index
Shear deformation, 311 Shear flows, 329-336, 339-343 Shear magnification effect, 346 Shear modes, 342, 447 Shear rate, 342, 389, 390 modified constant shear rate tests, 392-399 Shear rheometers, 353 Shear stress, 270, 275,276, 279, 281,282, 332, 339 Shear thinning, 206, 330-331,339, 347 Shear viscosity, 51,353, 389, 391 Shearing characterisation, 163-216 Shearing velocity/shear stress relationship, 186 Sheet forming, 65-72 implicit finite element modelling, 247-322 mechanisms, 250 philosophy, 325-327 processing methods, 327 Sheet metal forming characteristics of, 6-12 limits for, 12-15 mechanics of, 1-25 Sheet stamping, 325 Single-curvature forming, 305-307 Sliding velocity versus coefficient of friction, 209 Solvent impregnation, 44-45 Specialty fibres, 42 Specific heat, 108 Spherical dome, 226-230 Spring-back, 19, 119, 193, 504-506 Spring-forward, 504-506 Squeeze flow viscometers, 353-354 Stability considerations, 285 Stamp forming, 94, 95, 97, 100-146 characterisation methods, 105-107 description of stamping process, 104-105 experimental details and procedures, 96-100, 102-104, 139-140 experimental set-up, 138-139 force analysis, 141 forming temperature, 110-115 materials employed, 96-97 mechanisms, 140-141 optimised processing window, 135-137 processing parameters, 103 recommendations, 145-146 Stamper radius and laminate thickness, 126 and stamping time, 122 Stamping pressure and laminate thickness, 118-119 and stamping time, 120, 121 and stamping velocity, 115, 123-124 and weaving of tracer wires, 135 Stamping time
and bend angle, 119 and mould geometry, 120 and stamper radius, 122 and stamping pressure, 120, 121 Stamping velocity, 112 and crystallinity, 131 and stamping pressure, 115, 123-124 Stokes flow, 262 Strain analysis technique, 148-150 Strain distribution, 236 during roll forming, 495-498 Strain energy function, 84 Strain-hardening, 13 Strain measurement, 51 0-511 Strain rate, 83, 336 sensitivity index, 24 Strain space diagram, 225, 226, 229, 230, 233,237 Strain tensor, 335 Stress equilibrium, 377 Stress growth, 334 Stress-strain characteristics, 7 Stress-strain curve, 7-13 Stress-strain equations, 129 Stress-strain properties, 11 Stress-strain relationship, 19 Stress tensor, 376 Stretch forming, 15-17 Structure-properties relationship, 415-4 19, 432-433 Superplasticity, 23-24 Surface coatings, 200 Surface fibre orientation, 200, 209 Surface release agent, 201 Surface resin layer, 208 Surface roughness measurements, 201 Tangential stress, 280, 283-285, 300-302 Tape laying, 64-65, 326 Tape winding, 93 Tear drop region, 232 Tensile failure behaviour, 413-415 Tensile strain contour map, 238 Tensile test, 6, 11, 12 Tension mass matrix, 263 Tepex, 95 Textile fabrics. See Fabrics Thermal analysis, 106-107, 130-133 Thermal conductivity, 108 Thermal expansion coefficients, 128 Thermal Mechanical Analysis (TMA), 107, 128, 130 Thermoforming, 93, 326 rheological parameters in, 364-366 Thermoplastics, 34
Subject index Thermoplastics (continued) composite sheet forming, 27-73 constituents, 29-48 polymer matrices, 34-38 sheet production, 77-78 Thermosets, 34 composite forming, 441-472 background, 442-446 deformation modes, 446-455 kinematics, 446-455 forming experiments, 464-468 Thick sheet formulations, 82-83 Time to surface degradation, 51 Tolerance to degradation, 51 Total Lagrangian (TL) formulation, 80 Tow straightening effect, 189 Transition temperatures, 49, 50 Transverse flow, 165, 170-173, 306 fabrics, 166 mechanism, 125 process, 100 Transverse shear behaviour, 392 vee-bending test, 393-394 Transverse shear viscosity tests, 395-399 Transverse squeeze flow, 358-361 Transverse stress, 274, 277 Trellis angle, 174, 175 Trellis deformation, 232 Trellis effect, 141, 167, 228 Trellis structure, 228, 230 Tresca yield criterion, 3 Trouton relation, 338 Ulnar osteosynthesis plate, 419-428 Ultimate tensile strength (UTS), 8 Unfilled polymer melts, 329 Uniaxial elongation, 337
531
Updated Lagrangian (UL) formulation, 81 Vacuum bag, 61-62, 148 Vacuum forming, 76, 326 van der Waals forces, 199 Vee-bending, 374 experimental procedures, 380-382 kinematic model, 378-379 modified test, 392 tests, 373 transverse shear behaviour, 393-394 Viscoelastic behaviour, 388 Viscoelastic fluids, 333 Viscoelastic models, 85 Viscoelastic response, 334 Viscoelasto-plastic constitutive relation, 86 Viscosity, 50, 329-336, 339-348 in elongational flows, 336-337 Viscosity ratio, 275, 279, 280, 398 von Mises yield criterion, 3 Wall thickness distribution, 142-143, 144 White-Metzner equation, 333 Williams-Landel-Ferry (WLF) transform, 200, 458 Work-hardening, 8 Woven fabric, 122 Yield locus, 18 Yield point, 7, 8 Yield strength, 8 Yield stress, 194-195 Yielding, anisotropic materials, 3-4 Young's modulus, 11, 412, 417, 418 Zero shear rate viscosity, 331
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