Composites with Micro- and Nano-Structure: Computational Modeling and Experiments (Computational Methods in Applied Sciences)

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Composites with Micro- and Nano-Structure

Computational Methods in Applied Sciences Volume 9

Series Editor E. Oñate International Center for Numerical Methods in Engineering (CIMNE) Technical University of Catalunya (UPC) Ediﬁcio C-1, Campus Norte UPC Gran Capitán, s/n 08034 Barcelona, Spain [email protected] www.cimne.com

Vladimir Kompisˇ

Composites with Microand Nano-Structures Computational Modeling and Experiments

123

Vladimir Kompisˇ Academy of the Armed Forces Liptovsky´ Mikula´ sˇ Slovakia

ISBN 978-1-4020-6974-1

e-ISBN 978-1-4020-6975-8

Library of Congress Control Number: 2008928508 All Rights Reserved c 2008 Springer Science + Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microﬁlming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1

Torsional Buckling of Single-Walled Carbon Nanotubes . . . . . . . . . . . A.Y.T. Leung, X. Guo, and X.Q. He

1

2

Equilibrium and Kinetic Properties of Self-Assembled Cu Nanoparticles: Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . Roberto Moreno-Atanasio, S.J. Antony, and R.A. Williams

9

3

Method of Continuous Source Functions for Modelling of Matrix Reinforced by Finite Fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ˇ Vladim´ır Kompiˇs, M´ario Stiavnick´ y, Mari´an Kompiˇs, Zuzana Murˇcinkov´a, and Qing-Hua Qin

4

Effective Dynamic Material Properties for Materials with Non-Convex Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Martin Schanz, Georgios E. Stavroulakis, and Steffen Alvermann

5

Modelling of Diffusive and Massive Phase Transformation . . . . . . . . . 67 Jiˇr´ı Vala

6

The Use of Finite Elements for Approximation of Field Variables on Local Sub-Domains in a Mesh-Free Way . . . . . . . . . . . . . . . . . . . . . 87 V. Sladek, J. Sladek, and Ch. Zhang

7

Modelling of the Process of Formation and Use of Powder Nanocomposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Alexandre Vakhrouchev

8

Modelling of Fatigue Behaviour of Hard Multilayer Nanocoating System in Nanoimpact Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Magdalena Kopernik, Lechosław Tre¸bacz, and Maciej Pietrzyk

v

vi

Contents

9

A Continuum Micromechanics Approach to Elasticity of Strand-Based Engineered Wood Products: Model Development and Experimental Validation . . . . . . . . . . . . . . . . 161 Reinhard St¨urzenbecher, Karin Hofstetter, Thomas Bogensperger, Gerhard Schickhofer, and Josef Eberhardsteiner

10 Nanoindentation of Cement Pastes and Its Numerical Modeling . . . . 181 Jiˇr´ı Nˇemeˇcek, Petr Kabele, and Zdenˇek Bittnar 11 Ductile Crack Growth Modelling Using Cohesive Zone Approach . . . 191 Vladislav Koz´ak 12 Composite (FGM’s) Beam Finite Elements . . . . . . . . . . . . . . . . . . . . . . 209 ˇ s Just´ın Mur´ın, Vladim´ır Kutiˇs, Michal Masn´y, and Rastislav Duriˇ 13 Computational Modal and Solution Procedure for Inhomogeneous Materials with Eigen-Strain Formulation of Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Hang Ma, Qing-Hua Qin, and Vladimir Kompiˇs 14 Studies on Damage and Rupture of Porous Ceramics . . . . . . . . . . . . . 257 Ioannis Doltsinis 15 Computation of Effective Cement Paste Diffusivities from Microtomographic Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 K. Krabbenhoft, M. Hain, and P. Wriggers Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

Introduction

This book contains selected, extended papers presented (with three exceptions) in the thematic ECCOMAS conference on Composites with Micro- and NanoStructure (CMNS) – Computational Modelling and Experiments held in Liptovsk´y Mikul´asˇ, Slovakia, in May 28–31, 2007 and sponsored by the Slovak Ministry of Education. Composite materials play important role in all mechanical, civil as well as in electrical engineering structures especially in the last two decades in connection with the nano-materials and nano-technologies. Of course the chemical engineering is a connecting element for all these applications. Computational methods and experiments are a basis for the Simulation-Based Engineering Science (SBES). SBES can play a remarkable role in promoting the developments vital to health, security and technological competitiveness of nations. Recent experimental and computational results demonstrate that materials reinforced with stiff particles and ﬁbres can obtain substantial improvements in stiffness, thermal conductivity and electro-magnetic properties. Materials reinforced with ﬁbres can have very different properties in different directions. The material can have very good conductivity in one direction and can be isolator in other directions. Most important reinforcing materials discovered less than 20 years ago, which are in the centre of interest in many universities and research institutions are the carbon nanotubes. With these new materials also importance of computational simulations and experimental veriﬁcations increases. The book contains 16 papers: The ﬁrst paper, by Leung et al., employs an atomic-scale ﬁnite element method to study torsion buckling of single-walled carbon nano-tubes (SWNTs). The dependence of critical torsion angle on the length is discussed and compared with conventional shell theory. Strain energy and morphologies of the SWNT are discussed. In the second paper, the authors study the structure and mechanical strength properties of Cu nano particulate aggregates. The inter-particle interaction models have

vii

viii

Introduction

been considered to account the long-range forces: electrostatic, van der Waals and coupled Johnson, Kendall and Roberts (JKR) and Brownian force models. The assemblies considered have poly-size distribution of particles. In the third paper the interaction of matrix with ﬁbres and ﬁbre with other ﬁbres in the ﬁbre-reinforced composites (FRC) is numerically simulated using 1D continuous source functions along the ﬁbre axis and 2D source functions inside the ends of ﬁbres. Numerical examples show that the interaction of the end parts of ﬁbres is crucial for evaluation of the mutual interaction of ﬁbres in the composite. Correct simulation of all parts is important for evaluation of stiffness and strength of the FRC. In the fourth paper the effective dynamic material properties for materials with microstructure are assumed. Micro-scale inertia are taken into account and numerical homogenization is performed. The frequency dependent macroscopic material parameters are found for frequency range from 0 up to 1 MHz. The ﬁfth paper describes modelling of diffusive and massive phase transformation of a multi-component system based on the principle of maximum dissipation rate by Onsager; the ﬁnite thickness of the interface between both phases can be respected. The mathematical analysis results in an initial-value problem for a system of partial differential equations of evolution with certain non-local integral term; the unknowns are the mole fractions of particular components. Paper number six deals with numerical implementation of local integral equation formulation of 2D linear elastic media with continuous variable Young’s modulus. Two meshless presentations of the formulation are described and accuracy, convergence and numerical stability are investigated. The seventh paper presents methods for the modelling of the processes that accompany obtaining and use of powder nano-composites. For this purpose, a number of physical-mathematical models, including the models of obtaining of nano-sized powders at “up down” processes, the models of the main steps of powder nanocomposites compaction and the models of deformation of powder nano-composites under the ambient action were developed. The eighth paper presents the basis of the nano-impact test and the idea of prediction of fracture occurrence, especially fatigue behaviour of nano-impacted materials. Finite element (FE) model is applied to identiﬁcation of the material model of hard nano-coating in the multilayer system. In the second part of the chapter a new approach to analysis of fracture phenomena is introduced as the fatigue criteria, which are used in simulation of nano-impact test. Results of simulations with the fatigue criteria implemented into the FE code for the analysed problem are discussed. The ninth paper contains a description of a continuum micromechanics model development and experimental validation of strand-based engineering wood products aimed to improve the mechanical properties of wood products. The model allows considering of the relevant (micro-) characteristics of the wood on the mechanical properties of panels. The tenth paper focuses on experimental investigations and numerical modelling of micromechanical behaviour of cement paste taken as a fundamental representative of building materials with heterogeneous microstructure. The experimental

Introduction

ix

nano-indention and its implications to evaluation of material properties are focused. Better descriptions of indentation based on analytical visco-elastic solution and ﬁnite element model with general visco-elasto-plastic constitutive relation are proposed. These models are used for simulation of indentation and for estimation of material parameters at micrometer scale. Paper number eleven studies the prediction of the crack growth of the ductile fracture of forgets steel 42CrMo4. Crack extension is simulated in sense of element extinction algorithm based on the damage model Gurson-Tvergard-Needleman and on the cohesive zone model. Determination of micro-mechanical parameters is based on the combination of static tests, microscopic observation and numerical calibration procedures by FEM. In the twelfth paper a longitudinal polynomial continuous variation of the stiffness properties is considered in the stiffness matrix of the beam element and presented for the analysis of the electric, thermal and structural ﬁeld. The transversal and longitudinal variation of the material properties is considered. In the thirteenth paper a novel computational modal and solution procedure are proposed for inhomogeneous materials with the eigenstrain formulation of the boundary integral equations. Each inhomogeneity embedded in the matrix with various shapes and material properties described via the Eshelby tensors can be obtained through either analytical or numerical means. As the unknowns appear only on the boundary of the solution domain, the solution scale of the inhomogeneity problem with the present model is greatly reduced. To enhance further the computational efﬁciency, the overall elastic properties using the present model with eigenstrain formulation are solved using the newly developed boundary point method for particle reinforced inhomogeneous materials over a representative volume element. The fourteenth paper deals with reliability of porous ceramics based on experimentation and analysis. Damage prior to rupture and aging because of corrosive activity in porous materials is discussed. Fracturing processes in the material that result to rupture under pore pressure are studied on the micro-scale by the numerical simulation model. Finally, in the last paper a computational framework for extracting effective diffusivities from micro-tomographic images is presented. Effective diffusivity of a cement paste whose microstructure has been digitized is derived, consistent homogenization and statistical testing and interpretation of results are highlighted. ˇ I would like to thank my colleague Dr. Stiavnick´ y for his help in preparation of the book. Vlado Kompiˇs

Chapter 1

Torsional Buckling of Single-Walled Carbon Nanotubes A.Y.T. Leung, X. Guo, and X.Q. He

Abstract This paper employs an atomic-scale ﬁnite element method to study torsional buckling of single-walled carbon nanotubes (SWNTs). As the torsional angle increases, a spiral will appear and the morphology of SWNT will change abruptly. For SWNTs with ﬁxed diameter and those with ﬁxed aspect ratio, the dependence of critical torsional angle on the length is discussed and compared with conventional shell theory, respectively. Strain energy grows initially as a quadratic function of the torsional angle, and then changes approximately linearly. The morphologies of the SWNT are illustrated in detail.

1.1 Introduction The excellent mechanical properties of carbon nanotubes (CNTs) have attracted enormous research interest for both science and technology applications [1–3]. The buckling behavior of CNTs is an important topic for both experimental investigation and theoretical research [4–8]. Compared with extensive experimental research on the buckling of CNTs under compression and bending, experimental studies on their torsional buckling are limited. For example, Yu et al. [9] used atomic force microscopy to observe a fully collapsed CNT section and a twisted and fully collapsed CNT. The limited research on torsional buckling concentrates mainly on theoretical research via molecular dynamics (MD) and continuum mechanics. Yakobson et al. [6] performed MD simulation of torsion of a SWNT and found that it ﬂattened into a straight spiral beyond the critical torsional angle. Shibutani and Ogata [10] presented an MD study on the torsional buckling of a cantilevered SWNT. The initial torsional deformation gives a linear relationship between torsional moment and torsional angle. After buckling, an anti-plane distorted structure is generated. Wang A.Y.T. Leung, X. Guo, and X.Q. He Department of Building and Construction, City University of Hong Kong, Hong Kong [email protected] V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

1

2

A.Y.T. Leung et al.

et al. [11] simulated the torsion of SWNTs via MD. After buckling the whole section collapses like a dumbbell and the atoms move with off-plane displacements. Sears and Batra [12] performed molecular mechanics simulation of torsional buckling of a very long SWNT and found that the critical shear strain agreed well with prediction of Donnell’s shell theory. Even for the axial compression and tension of CNTs, MD simulation is so computationally expensive that it is difﬁcult to scale up [13, 14]. Besides the great computational effort, MD simulation of torsional buckling of CNTs involves other difﬁculties such as realization of torque on an atomistic system and large deformation around the spiral. There is some investigation based on continuum mechanics to study torsional buckling of CNTs. For example, Li and Guo [15] employed conventional shell theory and eigenvalue buckling methodology to investigate large deformation behavior and post-buckling modes of SWNTs under torsion; Han and Lu [16] used Donnell’s shell theory to study torsional buckling of a DWNT embedded in an elastic medium. In the above continuum models, interatomic potential was not employed directly and multi-body interactions could not be considered accurately, so the dependence of critical torsional angle and concrete conﬁguration of buckled CNTs could hardly be achieved. Liu et al. [17, 18] proposed an atomic-scale ﬁnite element method. Using interatomic potential to consider the multi-body interactions, it is as accurate as molecular mechanics simulation and much faster than the conjugate gradient method. The present authors [19–21] employed it to study axial buckling and bending buckling of CNTs, and characterized the dependence of critical strain on the length, strain energy curve and the conﬁguration in post-buckling stages. Further, this paper employs it to study torsional buckling of SWNTs. Dependence of critical torsional angle on the length of SWNTs with a ﬁxed diameter is found to be consistent with conventional shell theory. Also, for armchair and zigzag SWNTs with ﬁxed aspect ratios, the critical torsional angles are achieved and compared with conventional shell theory. A (13, 0) SWNT is taken as an example of post-buckling. When the torsional angle is large enough, a spiral is initiated, its morphology changes abruptly, and strain energy has a slight drop. After that the strain energy increases approximately linearly with respect to the torsional angle, and the spiral continues developing. The morphologies of twisted SWNT are presented in detail.

1.2 Simulation Procedure For a straight SWNT in equilibrium, the center of left cross-section is taken as the origin, its longitudinal direction as the z axis, and its cross-section as the x − y plane. In step i, for the left boundary atoms, a rotation increment ∆i around the centre of the cross-section is applied; for the right boundary atoms, the reverse rotation increment ∆i is applied. Then, the x(i) and y(i) of the boundary atoms are kept ﬁxed, the z(i) of the left boundary atoms are kept ﬁxed, while the z(i) of the right boundary atoms are kept free, and the atomic-scale ﬁnite element simulation for the whole atomistic system is performed to achieve new equilibrium conﬁguration.

1 Torsional Buckling of Single-Walled Carbon Nanotubes

3

The covalent bonds among the atoms are described by Brenner et al.’s “second generation” empirical potential [22]. Each atomic-scale ﬁnite element consists of 10 atoms including the central atom, the three nearest-neighboring atoms and the six second nearest-neighboring atoms. A schematic diagram of the atomic-scale ﬁnite element, the associated element stiffness matrix and the non-equilibrium force vector are therein [17, 18]. The element captures the interactions among the central atom and the other atoms. All calculation is performed by ABAQUS via its UEL subroutine [23]. With the torsional angle increasing, a spiral will appear and morphology of the SWNT will change abruptly at a certain torsional angle, which is the critical angle for torsional buckling.

1.3 Numerical Results and Discussion First, dependence of the critical torsional angle on the length of SWNTs with a ﬁxed diameter is studied. Figure 1.1 shows various critical torsional angles of (13, 0) SWNTs at different lengths l, while their aspect ratios change from about 5 to larger than 20. From Fig. 1.1, it is observed that as the length increases, the critical torsional angle also increases. The proportional function is employed to ﬁt the critical torsional angle. The ﬁtting result is

ψ = 0.07144l

(1.1)

with a correlation coefﬁcient 99.1%. The ﬁtting shows that except the shortest SWNT, there is a good proportional relationship between the critical torsional angle and the length of SWNTs. The above result is compared with the shell theory

1.8

Critical torsional angle

1.6

y = 0.07144l

1.4

R2 = 99.1%

1.2 1.0 0.8 0.6 0.4 0.2 2

4

6

8

10 12 14 16 18 20 22 24 26

Length of SWNTs (nm) Fig. 1.1 Critical torsional angle changes with the length of (13, 0) SWNTs

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A.Y.T. Leung et al.

results. For torsional buckling of a shell, when overall beam buckling takes place, the critical torsional angle is [24]

ψcr = 2(1 + ν )π

(1.2)

and when cylinder-helix ﬂattening occurs, the critical torsional angle is [24]

ψcr = β l d −2.5

(1.3)

where β is a dimensional parameter that is dependent on parameters other than l and d. The latter occurs ﬁrst for l < 136d 5/2 nm, which is true for all the tubes here. From Eq. (1.3), for the SWNTs with a ﬁxed diameter, the critical torsional angle is proportional to the length, i.e. the critical torsional angle per unit length is independent of the length. Here, when the length is greater than 10 nm (aspect ratio is greater than 10), the critical torsional angle is approximately proportional to the length, which is consistent with the prediction of conventional shell theory. Secondly, dependence of critical torsional angle on length of SWNTs with a ﬁxed aspect ratio is investigated. For armchair SWNTs with a ﬁxed aspect ratio ∼7.6, the critical torsional angle is related to the length in Fig. 1.2. From Fig. 1.2, it is observed that as the length increases, the critical torsional angle decreases rapidly. The power function is used to ﬁt the critical torsional angle. The ﬁtting result is

ψcr = 12.96965l −1.47366

(1.4)

with a correlation coefﬁcient larger than 0.99. This result is also compared with the shell theory results. For the cylinder-helix ﬂattening of torsional buckling of a shell, the critical torsional angle is expressed as Eq. (1.3). For the current case, the aspect

1.8

Critical torsional angle

1.6

ycr = 12.96965l −1.47366

1.4

R2 = 0.99779

1.2 1.0 0.8 0.6 0.4 0.2 0.0

2

4

6

8

10

12

14

16

Length of SWNTs (nm) Fig. 1.2 Critical torsional angle varies with the length of armchair SWNTs with a ﬁxed aspect ratio ∼7.6

1 Torsional Buckling of Single-Walled Carbon Nanotubes

5

2.75

Critical torsional angle

2.50

ycr = 25.66043l −1.85374 R2 = 0.99284

2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25

3

4

5

6

7

8

9

Length of SWNTs (nm)

Fig. 1.3 Critical torsional angle varies with the length of zigzag SWNTs with a ﬁxed aspect ratio ∼7.66

ratio is a constant; after replacing the diameter d in Eq. (1.3) with c l (where c is inverse of the aspect ratio), one has

ψcr = β c−2.5 l −1.5

(1.5)

The ﬁtting on the present critical torsional angle, Eq. (1.4), shows that the critical torsional angle is proportional to l −1.47 , which is in good agreement with l −1.5 , predicted by Eq. (1.5). A similar study is performed for zigzag SWNTs with a ﬁxed aspect ratio ∼7.66, and the critical torsional angle varies with the length as shown in Fig. 1.3. It is observed that as the length increases, the critical torsional angle also decreases rapidly. The power function is employed to ﬁt the critical torsional angle. The ﬁtting result is ψcr = 25.66043l −1.85374 (1.6) with a correlation coefﬁcient larger than 0.99. The critical torsional angle is proportional to l −1.85 , so it is also in reasonable agreement with the prediction by Eq. (1.5). In the above two examples, the armchair SWNTs behave more like the conventional shell than the zigzag ones. The main reason might be that the diameter and length of the zigzag SWNTs here are too small compared with those of the armchair SWNTs. Next, the post-buckling behavior of SWNT is studied. A (13, 0) SWNT with a length of 23.23 nm is taken as an example. The average strain energy is calculated as the difference of the average energy per atom in the torsional and un-torsional system as a function of torsional angle as shown in Fig. 1.4. Its morphologies are presented in Fig. 1.5 [25]. At small torsional angle, the SWNT deforms linearly, consistent with Yakobson et al. [6] and Shibutani and Ogata [10], and the strain energy grows quadratically with respect to the torsional angle. At the critical torsional angle of 1.64, a spiral appears, as shown in inset of Fig. 1.4. When the SWNT buckles, there is a very small energy release, which is similar with the research of Yakobson et al. [6].

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A.Y.T. Leung et al.

Average strain energy (eV)

0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Torsional angle Fig. 1.4 The curve of average strain energy for the (13, 0) SWNT changes with the torsional angle

(a) torsional angle of 2.00

(b) torsional angle of 2.40

(c) torsional angle of 2.80

(d) torsional angle of 3.20

(e) torsional angle of 3.60

(f ) torsional angle of 4.00

Fig. 1.5 The morphologies of the (13, 0) SWNT in torsion

1 Torsional Buckling of Single-Walled Carbon Nanotubes

7

When the torsional angle increases, the spiral continues developing, as seen in Figs. 1.5a–f, and the morphologies are similar with those in Yakobson et al. [6], Wang et al. [11] and Shibutani and Ogata [10]. The strain energy increases approximately linearly with the torsional angle, in agreement with Yakobson et al. The above is quite similar with our former study on axial buckling and bending buckling of CNTs [19–21], where in each post-buckling stage the strain energy changes approximately linearly with the strain and the bending angle, respectively. When the torisonal angle reaches 4.0, the whole tube shortens by about 1.0% of the original length. For the same SWNT, in the MD simulation of Yakobson et al. [6], the critical torsional angle is larger than 2.5, and as they stated, it is obviously larger than the prediction of Eq. (1.3) (1.28 radian). In the present simulation, it is 1.64 and better than 2.5. An important reason for this improvement is the potential functional that is employed. The Brenner empirical potential [26] was employed in MD simulation of Yakobson et al. while the Brenner et al. “second generation” empirical potential [22] is employed in the present simulation. When Brenner empirical potential is employed in the present atomic-scale ﬁnite element simulation, the critical torsional angle is found to be 2.06. The difference of the present prediction and the prediction of the Eq. (1.3) comes from the treatment of the boundary, which is same with Yakobson et al. The circular ends deter the through ﬂattening necessary for the helix to form.

1.4 Conclusion In conclusion, torsional buckling of SWNTs is studied by the atomic-scale ﬁnite element method. At the critical torsional angle, a spiral will appear and torsional buckling as cylinder-helix ﬂattening will take place. For slender SWNTs with ﬁxed diameter, the critical torsional angle is approximately proportional to the length, consistent with the conventional shell theory. For armchair and zigzag SWNTs with the ﬁxed aspect ratio, dependence of the critical torsional angle on the length is in good agreement with the conventional shell theory, respectively. After buckling, the strain energy changes approximately linearly with the torsional angle, and the morphologies are illustrated in detail. Acknowledgements A.Y.T. Leung acknowledges the support from Research Grant Council of Hong Kong grant #1161/05E.

References 1. Qian D, Wagner GJ, Liu WK, Yu MF, and Ruoff RS (2002) Mechanics of carbon nanotubes, Appl. Mech. Rev. 55: 495–533. 2. Srivastava D, Wei CY, and Cho K (2003) Nanomechanics of carbon nanotubes and composites, Appl. Mech. Rev. 56: 215–230.

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3. Huang Y and Wang ZL (2003) Mechanics of carbon nanotubes, in Comprehensive Structural Integrity Handbook, B. Karihaloo, R. Ritchie, and I. Milne (eds), Elsevier, Amsterdam, Vol. 8, p. 551. 4. Falvo MR, Clary GJ, Taylor II RM, Chi V, Brooks Jr FP, Washburn S, and Superﬁne R (1997) Bending and buckling of carbon nanotubes under large strain, Nature (London), 389: 582–584. 5. Lourie O, Cox DM, and Wagner HD (1998) Buckling and collapse of embedded carbon nanotubes, Phys. Rev. Lett. 81: 1638–1641. 6. Yakobson BI, Brabec CJ, and Bernholc J (1996) Nanomechanics of carbon tubes: instabilities beyond linear response, Phys. Rev. Lett. 76: 2511–2514. 7. Srivastava D, Menon M, and Cho K (1999) Nanoplasticity of single-wall carbon nanotubes under uniaxial compression, Phys. Rev. Lett. 83: 2973–2976. 8. Ru CQ (2000) Effect of van der Waals forces on axial buckling of a double-walled carbon nanotubes, J. Appl. Phys. 87: 7227–7231. 9. Yu MF, Kowalewski T, and Ruoff RS (2001) Structural analysis of collapsed, and twisted and collapsed, multiwalled carbon nanotubes by atomic force microscopy, Phys. Rev. Lett. 86: 87–90. 10. Shibutani Y and Ogata S (2004) Mechanical integrity of carbon nanotubes for bending and torsion, Model. Simul. Mater. Sci. Eng. 12: 599–610. 11. Wang Y, Wang XX, and Ni XG (2004) Atomistic simulation of the torsion deformation of carbon nanotubes, Model. Simul. Mater. Sci. Eng. 12: 1099–1107. 12. Sears A and Batra RC (2004) Macroscopic properties of carbon nanotubes from molecularmechanics simulations, Phys. Rev. B. 69: 235406. 13. Liew KM, Wong CH, He XQ, Tan MJ, and Meguid SA (2004) Nanomechanics of single and multiwalled carbon nanotubes, Phys. Rev. B. 69: 115429. 14. Liew KM, He XQ, and Wong CH (2004) On the study of elastic and plastic properties of multi-walled carbon nanotubes under axial tension using molecular dynamics simulation, Acta. Mater. 52: 2521–2527. 15. Li C and Guo WL (2003) Continuum mechanics simulation of post-buckling of single-walled nanotubes, Int. J. Nonlinear Sci. Numer. Simul. 4: 387–393. 16. Han Q and Lu G (2003) Torsional buckling of a double-walled carbon nanotubes embedded in an elastic medium, Eur. J. Mech. A. 22: 875–883. 17. Liu B, Huang Y, Jiang H, Qu S, and Hwang KC (2004) The atomic-scale ﬁnite element method, Comput. Methods Appl. Mech. Eng. 193: 1849–1864. 18. Liu B, Jiang H, Huang Y, Qu S, Yu MF, and Hwang KC (2005) Atomic-scale ﬁnite element method in multiscale computation with applications to carbon nanotubes, Phys. Rev. B. 72: 035435. 19. Leung AYT, Guo X, He XQ, Jiang H, and Huang Y (2005) Post-buckling of carbon nanotubes by atomic-scale ﬁnite element, J. Appl. Phys. 99: 124308. 20. Guo X, Leung AYT, Jiang H, He XQ, and Huang Y (2007) Critical strain of carbon nanotubes: an atomic-scale ﬁnite element study, J. Appl. Mech. 74: 347–351. 21. Guo X, Leung AYT, He XQ, Jiang H, and Huang Y (2008) Bending buckling of singlewalled carbon nanotubes by atomic-scale ﬁnite element, Composites Part B: Engineering, 39: 202–208. 22. Brenner DW, Shenderova OA, Harrison JA, Stuart SJ, Ni B, and Sinnott SB (2002) A secondgeneration reactive empirical bond order (rebo) potential energy expression for hydrocarbons, J. Phys.: Condens Matter. 14: 783–802. 23. ABAQUS, ABAQUS Theory Manual and Users Manual, 2005, version 6.5, Hibbit, Karlsson and Sorensen, Pawtucket, RI, USA. 24. Timoshenko SP and Gere JM (1961), Theory of Elastic Stability, 2nd ed., McGraw-Hill, New York. 25. Humphrey W, Dalke A, and Schulten K (1996) VMD – visual molecular dynamics, J. Mol. Graphics. 14: 33–38. 26. Brenner DW (1990) Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond ﬁlms, Phys. Rev. B. 42: 9458–9471.

Chapter 2

Equilibrium and Kinetic Properties of Self-Assembled Cu Nanoparticles: Computer Simulations Roberto Moreno-Atanasio, S.J. Antony, and R.A. Williams

Abstract We present a study of the inﬂuence of interparticle interactions on the kinetics of self-assembly and mechanical strength properties of Cu nanoparticulate aggregates. Three types of commonly used inter-particle interaction forces have been considered to account for the attraction between particles, namely electrostatic forces, van der Waals forces and the JKR cohesion model. These models help to account for the forces generated due to surface treatment of particles, a process commonly used in fabricating composite particles. The assembly formed using the electrostatic interaction force model has 50% of the particles positively charged and the remaining particles are negatively charged. All the assemblies considered here have a polydisperse size distribution of particles. To be able to compare the bulk properties predicted between these models, the maximum force required to break the interparticle contacts (pull-off force) is kept identical in all the systems considered here. Three assemblies were generated. The assemblies were allowed to self-assemble based on the three interaction force models as mentioned above. We have studied some of the key properties of self-assembled Cu aggregates obtained by using the above mentioned models. The study shows that, although the pull-off force between particles is identical, variations in the long-range forces between particles signiﬁcantly affect the structural properties and mechanical strength of the self-assembled nanoaggregates. The approach adopted here forms a basis on which to further probe the bulk behaviour of self-assembled particulates in terms of their single-particle properties.

2.1 Introduction The study of nanocomposites is a major and growing topic spanning many scientiﬁc disciplines including materials science and chemistry. The reason for an increasing Roberto Moreno-Atanasio, S.J. Antony, and R.A. Williams School of Environmental, Process and Materials Engineering, University of Leeds, Leeds, LS2 9JT, UK [email protected], [email protected], [email protected] V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

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interest in the development of composite materials at the nanosize scale is found in the unusual properties that these materials may exhibit due to their small size. On one hand a decrease in particle size produces an increase in the ratio of surface area to volume and thus an increase in the strength of the surface interactions when compared with volume interactions. On the other hand, when a nanocomposite material is formed the properties of the new materials might be due to the fact that one or each individual component grants its properties to the new material or in other cases the properties of the new materials are radically different from those properties shown by the individual components. Therefore, many of the properties of the nanocomposites and especially the interparticle forces present between nanoparticles depend on the processing routes by which the nanocomposites are fabricated, with special attention paid to the surface treatment. A typical nanocomposite material may be a combination of inorganic and organic components, the former usually being a mineral and the latter a polymer [16, 27]. Some organic materials, when free in solution are able to self-assemble, due to the interaction between some of the radicals of the molecules. They can form complex reversible structures which can be controlled by means of an external trigger such as pH [1]. The use of some of these organic molecules to fabricate nanocomposites allows an easy control of the interparticle interactions and therefore they show a great potential for developing new materials with high mechanical resistance or speciﬁc functionalities. Some of the organic compounds that cap the inorganic components can form hydrogen bonds in solution which allows self-aggregation of the nanoparticles. This is the case with organic acids or bases which are joined to minerals such as copper or gold through a covalent interaction [4, 10]. As shown in the literature [30] the adhesion between particles through hydrogen bonds can be described by using the well known JKR model [15]. In other cases, the use of more than one type of organic molecule allows the combination of deprotonised acids with protonised amine groups which produces a net attraction between the different organic coatings. Alternatively some coatings could induce hydrophobic interactions between particles, which can also be described by the JKR model [5]. However, although many of these aggregates are formed in solution, their structural integrities remain after they are taken out from solution. The above mentioned studies clearly show the potential of engineering particulate systems by controlling the interparticle interactions through appropriate processing routes. Computer simulation work of the bulk properties of micro- or nano-composite materials can be performed using a discrete approach. In the discrete approach, systems are modelled at a particle level using either Molecular Dynamics or Distinct Element Method [3, 19, 20]. In addition, a mesoscopic approach [27] or the combination of macroscopic and microscopic approaches [22] can be used for the analysis of the mechanical behaviour of nanocomposite materials. Here we present a comparative computational analysis of the inﬂuence of the type of interaction on the self-assembly and mechanical strength of nanocomposites made of copper nanoparticles coated with an organic compound.

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There are several models available to deﬁne the inter-particle cohesiveforces. Some of these models account for a long-range contribution such as van der Waals and electrical forces and others ignore it as in the case of the JKR model. The presence of long-range forces could signiﬁcantly inﬂuence the kinetics of self-assembly and the mechanical strength of nanoparticulate systems, an aspect which is not yet well studied. We address this problem in the present study using three-dimensional Distinct Element Method (DEM) simulations. We simulate the self-assembly of nanoparticulate systems by considering some of the most commonly used interparticle force-separation relationships. The strength of the interparticle forces are chosen in such a way that the maximum pull-off force acting between the particles is the same, while the nature of the forces acting between the particles depends on the particular type of model used (i.e. electrostatic, van der Waals and JKR). Organically coated copper nanoparticles are chosen as a model material in this study.

2.2 Computer Simulations 2.2.1 Distinct Element Method Distinct Element Method (DEM) is a computer simulation technique (Cundall and Strack 1979) that treats particles individually by assigning them speciﬁc physical (e.g. density and size) and material properties (e.g. elastic modulus, friction, Poisson’s ratio) and describing their motion by using Newton’s law. DEM was created by Cundall and Strack [7] to describe rock mechanics problems. Since then this technique has become increasingly popular [24] and has been used in many different disciplines such as food science [6] and particle technology [3, 21], the latter one being the most popular of them. DEM works cyclically and therefore the state of the system is updated each time step. During a time step particle forces and therefore accelerations are considered to remain constant. Particles positions are calculated according to Newton’s Law and using an explicit ﬁnite difference scheme. The time step chosen should be smaller than the time needed by Rayleigh waves to propagate from a given point in a particle to the diametrical opposite point of the particle [14]. The equation of motion used in the simulation is an extension of the Langevin equation including terms for contact forces (elasticity, damping and friction) and long range forces (van der Waals, electrostatic). Therefore, the motion equation for a particle of mass, m, can be written in the form: m

d 2x = FContact + FLong range + FBrownian + FDrag dt 2

(2.1)

where x is the particle position vector which is calculated by the double integration of the net forces acting on the particle.

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In our simulations the attractive interactions between particles are governed by theories of contact mechanics (JKR model for cohesive particles) and long range interaction forces (electrostatic and van der Waals forces) depending on the speciﬁc case.

2.2.2 Interparticle Forces 2.2.2.1 Contact Forces If within any given time step the position of two particles is such that the particles overlap, the overlap distance is considered as an effective particle deformation. The particle deformation is estimated as the difference of the distance between particle centres and the sum of the radii of the particles. From the value of deformation obtained from the relative particle positions the applied force can be estimated by using Hertz Law as [14]: 4 PN = E ∗1/2 R∗1/2 δ 3/2 (2.2) 3 where R∗ and E ∗ are the reduced particle radius and elastic modulus which are deﬁned as a function of the individual particle radii, R1 and R2 , and elastic moduli, E1 and E2 , of the two particles in contact: (1 − ν12 ) (1 − ν22 ) 1 = + E∗ E1 ∗ E2 ∗

(2.3)

1 1 1 = ∗+ ∗ ∗ R R1 R2

(2.4)

If the model of Johnson, Kendall and Roberts (JKR) is used to model interparticle cohesion, the normal applied force, PN , in the contact is calculated through the stiffness of the contact according to the expression provided by Thornton and Yin [29]: 1/2 a3 2E ∗ a 3 − 3 aC3 ∆PN = (2.5) kN = 1/2 ∆δ 3 aC 3 − a3 with aC being

3PADH R∗ (2.6) 4E ∗ where PADH is the adhesion force and is equal to the negative of the pull-off force, POFF or force required to break a contact. According to the JKR model [15] the pull-off force or force required to break the contact between two particles is given by: aC3 =

2 Equilibrium and Kinetic Properties of Self-Assembled Cu Nanoparticles

POFF = −PADH = −3πγ R∗

13

(2.7)

where γ is the surface energy or energy per particle and unit area required to separate two particles by an inﬁnite distance [13]. If a tangential traction is acting on the particle and the particle cohesion is deﬁned by the JKR model [15], then the model of Savkoor and Briggs [26] is used to calculate the contact area radius between the particles (see [28] for more details about the tangential deformation model).

2.2.2.2 Long Range Forces Two types of long range forces have been selected to simulate the attractive interaction between particles: electrostatic forces in air and van der Waals forces. These forces decay with distance in different ways. Electrostatic forces decay with the square of the distance between particle centres. However, Van der Waals forces decay with the square of the distance between particle surfaces and therefore they decay faster with distance than the electrostatic forces. Figure 2.1 shows a schematic diagram of the way in which the interparticle forces between two particles, each of 100 nm in diameter, decay with distance. The value of the dispersion of the Brownian forces used in the simulations of the JKR model (see below) and the particle weight has been added for comparison. The van der Waals force equation is given by Hamaker [9] and vastly used in the literature [2]. The exact van der Waals equation given by Hamaker [9] has been implemented in the simulation using in this work. However, when the separation distance between particles, d, is much smaller than the particle size, the van der Waals force FV dW reduces to:

Fig. 2.1 Schematic diagram of the decay of electrostatic and van der Waals forces with the distance between the centres of two particles of 100 nm in diameter. The average particle weight and the value of the standard deviation of the Brownian forces used in the simulations have been added for comparison

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FV dW = −

HR∗ 6d 2

(2.8)

Since Eq. (2.8) tends to inﬁnity when the separation distance between the surfaces of the two particles tends to zero an experimentally determined minimum cut-off distance dmin is assigned equal to four Angstroms [17, 18]. In order to compute the force between two particles, the volume around each particle is scanned at a maximum distance of 400 nm. At this distance the value of the van der Waals forces has already decayed around ﬁve orders of magnitude for the properties of the particles used in this work. The electrostatic forces have been simulated using Coulomb’s expression and assuming that the particles have a constant absolute value of surface charge density. Therefore the force between two particles of radii R1 and R2 whose centres are separated a distance, r, is given by: FElec =

1 4πσ 2 R1 R2 ε0 r2

(2.9)

where σ is the surface charge density, ε0 is the permittivity of the vacuum and the minus sign corresponds to the case in which the two particles have opposite charges. In order to compute the electrostatic force between two particles the space around each particle is scanned at a maximum distance of 1 µm (10 particle diameters). At this distance the electrical force between two particles has decayed to a value of around 1% of the value when the two particles are in contact (Fig. 2.1).

2.2.2.3 Brownian Forces The Brownian forces are commonly used to describe the random motion of particles suspended in ﬂuids [8, 11, 23]. However, their use in the present work is not to simulate directly a ﬂuid environment but to induce collision between nanoparticles in the system in which the JKR model (cohesive contact force model) was used. The average value of the Brownian forces is zero and their dispersion is given by Eq. (2.10) for the simulations presented here. < F(t)2 > =

1.55x10−22R ∆t

(2.10)

2.2.3 Simulation Details Three different systems have been considered to study the effect of the interparticle interaction on the kinetics of self-assembly and mechanical strength. Each of them is made of 5,858 spherical particles whose sizes follow a Gaussian distribution with a mean value of 100 nm (Table 2.1). The material properties are given in Table 2.2 and they correspond to copper.

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Table 2.1 Particle size distribution Particle size (µm)

Frequency

80 85 90 95 97 100 103 105 110 115 120

44 172 458 823 932 1,000 932 823 458 172 44

Table 2.2 Material properties Material property

Value

Density (kg/m3 ) Elastic modulus (GPa) Poisson’s ratio () Friction () Surface energy (J/m2 ) Hamaker constant (J) Surface charge density (C/m2 )

8,930 120 0.35 0.3 0.044 40 × 10−20 0.0034

At the beginning of the simulations the particles are randomly positioned within a prescribed working cubical space and elastic boundaries are set up. The initial particles velocities are set to zero. Once the particles have been randomly positioned within the working space, the motion of the particles is initiated by the action of the long range interparticle forces as described in electrostatic, van der Waals or coupled JKR and Brownian force models. In each system the aggregation has been produced by the inﬂuence of a different kind of interaction. In the ﬁrst case, the particles aggregate only when they come into contact and follow the JKR model [15]. In the second case, the self-assembly is produced by van der Waals forces with the value of Hamaker constant for Cu in air. In the third case particles are considered to have a constant surface charge density where half of the particles are positively charged and the other half negatively charged. In order to be able to compare the three cases, the value of pull-off force between particles is kept identical. Thus, by equating Eqs. (2.7) and (2.8) an expression for the surface energy can be obtained in the form:

γ=

H 18π d 2

(2.11)

Therefore, the value of γ obtained using Eq. (2.11) is 0.044 J/m2 (Table 2.2).

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For the system in which electrical forces are being used the equivalent charge density can be obtained by equating Eqs. (2.8) and (2.9).

σ2 = ±

ε0 H 24π d 2R∗

(2.12)

Therefore, an equivalent value of σ equal to 0.0034 C/m2 is obtained by using Eq. (2.12) and considering a mean particle diameter of 100 nm.

2.3 Simulation Results 2.3.1 Self-Assembly of Copper Nanoparticles Figure 2.2 shows the evolution of the average coordination number as a function of time. It can be seen that the electrostatic forces produce the fastest increase in coordination number until reaching a plateau value which is very similar to the case in which van der Waals forces are acting. However, the self-assembly in the presence of the Brownian forces is the slowest of all cases. The origin for these differences is attributed to the way in which these forces decay with distance. Electrostatic forces decay slower than van der Waals forces. Finally, Brownian forces are random in direction and magnitude and therefore it can be considered that their interaction range is virtually null. The slower the forces decay with distance the faster the selfassembly process is. Figure 2.3 shows the time evolution of the size of the largest fragment in the system as deﬁned by the ratio of the number of particles contained in the fragment, N(t), to the total number of particles in the system, N0 . During the ﬁrst stages of the

Fig. 2.2 Evolution of the coordination number with time for the three assemblies

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Fig. 2.3 Evolution of the size of the largest fragment in the system for the three systems studied

simulation the size of the largest fragment varies very slowly with time, however a rapid increases in the size of this fragment is observed until a steady state value is achieved. In addition, particles aggregate faster in the system in which electrical forces are acting between particles. However, the slowest self-assembly process corresponds to the case in which the JKR model coupled with Brownian forces has been used. These results corroborate the correlation between the kinetics of selfassembly and the way in which the interparticle forces decay with distance as has already been shown for Fig. 2.2. It has also been observed that in the three systems studied here the size of the second largest fragment never contained more than 10% of the total number of particles in the system. This indicates that the self-assembly process occurs by addition of small clusters to the largest cluster rather than by collision of clusters of similar sizes. It is worth mentioning that in the assembly in which electrical forces are operating, particles of opposite charges come together to form the aggregate as could be expected. This phenomenon can be seen when the pair correlation function is determined as given in Eq. (2.13). g(r) =

V n(r) 4π r2 N∆r

(2.13)

Where n(r) is the mean number of particles within a shell of width ∆ r at a distance r from the origin, V is the volume of the system and N is the number of particles in the system. Figure 2.4 shows the g(r), or pair correlation function, for the assembly with electrical forces and for the cases of particles with the same charges, particles with opposite charges and for all the particles in the system. The g(r) for particles with opposite charges show its ﬁrst maximum at around 100 nm and the g(r) for particles with the same charges has its ﬁrst maximum at around 150 nm. The

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Fig. 2.4 Pair correlation function, g(r), of the particles in the system in which electrical forces are acting. The plot contains the g(r) function for particles with opposite sign of the charges, for particles with the same sign of the charges and for all the particles in the system Table 2.3 Comparison of the ﬁnal states of the self-assemblies Assembly

Pack. fract.

Coord. no.

Mean stress (kPa)

0.20 0.24 0.24

2.13 4.32 4.27

4.0 350 146

JKR model Van der Waals force model Electrical force model

g(r) that considers all particles in the system shows the maxima of the curves corresponding to the same and the opposite charges as would be expected. Since the average particle diameter is 100 nm it is clear that each particle tries to surround itself by particles with the opposite charge. This characteristic suggests that despite the fact that the particles have been randomly positioned, and therefore each particle should have the same number of neighbours with similar and opposite charges, during the self-assembly process a segregation of the material occurs. This segregation makes each particle to be only surrounded by particles with opposite charges. Table 2.3 shows a comparison of packing fraction, coordination number and value of the mean stress (σ11 + σ22 + σ33 )/3, in the assemblies. The stresses have been determined using the expression as given by Satake [25]:

σi j =

2 V

M

2

M

∑ RNni n j + V ∑ RT nit j 1

(2.14)

1

where Rni is the i component of the radius vector of the centre of the particle to the contact, Nni and T t j are the i and j components of the normal and tangential contact forces, respectively and M is the number of interparticle contacts. The sum of the forces in each contact of the assembly is divided by the volume of the assembly, V , and therefore the information is given on the whole state of the stress of the system.

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The assemblies in which long range forces are acting are better packed and their coordination number and the value of the mean stress are larger than the assembly in which the interparticle interaction is given by the JKR model (Table 2.3).

2.3.2 Analysis of the Mechanical Strength of Self-Assemblies In order to analyse the mechanical strength, the unconﬁned yield stress (UYS) of the assemblies has been determined. Each assembly was placed in between elastic platens at the top and bottom surfaces as schematically shown in Fig. 2.5. The side boundaries were removed and the top platen was moved downwards. The number of interparticle contacts and the force on the platen were monitored during the compression process. The sample was considered to have failed when a sharp peak on the platen force accompanied by a signiﬁcant drop in the number of contacts was observed. The force at this point divided by the platen area was taken as the UYS. The UYS was determined for different velocities of the platen. Figure 2.6 shows the UYS for values of compression velocities between 1 and 20 m/s. The UYS seems to be larger in the system in which the particles interact using the JKR model and smaller in the system in which electrostatic forces were used. However, these differences in the values of UYS are smaller for large values of compression velocities. As the compression velocity is reduced the differences in UYS between the three systems increases. Amongst the possible reasons for the low value of UYS in the assembly with electrical forces could be found in the attractive/repulsive nature of these forces. This dual nature can make neighbouring particles decrease the net strength of the system with respect to the other assemblies in which the interparticle forces are always attractive. For values of compression velocity smaller than 1 m/s no peak was found during uniaxial compression or in the best of cases only a very small peak accompanied with a very small drop in the number of interparticle contacts occurred. In contrast, the number of interparticle contacts increased with time during compression indicating a continuous deformation of the material. Therefore, no data are shown in Fig. 2.6 for values of compression velocities lower than 1 m/s.

Fig. 2.5 Schematic diagram of the compression of the assemblies

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Fig. 2.6 Evolution of the unconﬁned yield stress for the three assemblies

Fig. 2.7 Evolution of the contact ratio and force on the top platen as a function of the strain during uniaxial compression at the value of velocity of 5 m/s for the assemblies in which (a) the JKR model has been used, (b) van der Waals forces are acting (c) electrical forces are acting

As an example we show the force and contact ratio versus strain curves at the value of compression velocity of 5 m/s for the three assemblies (Fig. 2.7). It can be seen that despite the fact that the values of UYS are not drastically different, the failure occurs at very different values of strain. In addition, the drop in the number of interparticle contacts is similar for the case of the assemblies in which the JKR model has been used and the one with van der Waals interactions (around 20%

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Fig. 2.8 Evolution of the work carried out during uniaxial compression at the value of velocity of 1 and 5 m/s for the assemblies in which (a) the JKR model has been used, (b) the van der Waals forces are acting (c) the electrical forces are acting

drop in the number of interparticle contacts). However, the drop at peak strength is slightly larger for the assemblies in which electrical forces have been considered. Figure 2.8 presents the work done for producing the failure of the assemblies for a value of compression velocity of 1 and 5 m/s. The work has been obtained by integrating the curve of force versus displacement of the platen. The work required to produce the failure of the aggregates increases with the compression speed. The work obtained for the assembly in which electrostatic forces are acting between particles is more than one order of magnitude higher than the work obtained in the assembly in which the interaction is deﬁned by the JKR model. Therefore, it is clear that the longer the range of the interaction, the larger the work required to produce the failure of the assembly.

2.4 Discussion We have presented a comparative analysis of the inﬂuence of the types of interactions on the self-assembly and mechanical strength of particulate solids. Three cases have been considered. In the ﬁrst case a contact interaction force as deﬁned by the JKR model coupled with Brownian force has been used. In the other two cases, long

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range forces have been considered based on van der Waals and electrostatic force models. However, van der Waals forces have a shorter range of interaction than the electrostatic forces since they decay faster with the distance between particles. Therefore, the effective range of the forces decreases in the following order: electrostatic, van der Waals and JKR-Brownian model. These interactions can be controlled in experiments by using different coatings for the inorganic molecules and therefore creating nanocomposites made of organic and inorganic components [4]. We presented results on the inﬂuence of these interactions on the bulk mechanical strength of Cu nanoparticulate aggregates. First of all, we should consider that the JKR model and the van der Waals forces are two different ways of accounting for the cohesion between particles. However, the use of these two models for simulating self-assembly and mechanical strength produces drastic differences. The self-assembly process is very slow and the ﬁnal coordination number and the value of the mean stress are much lower in the case in which the JKR-Brownian model has been used (Fig. 2.2). It is obvious that the self–assembly behaviour is signiﬁcantly inﬂuenced by the long-range force acting between nanoparticles. In addition, the way in which elasticity and cohesion are described in the system in which the JKR model has been used and the system in which van der Waals forces were used is different. In the former the deformation between particles is based on an energy balance of elastic, mechanical and cohesion energy as described by the JKR model. In the latter case elasticity and cohesion are considered as the arithmetic sum of the elastic force (as deﬁned by the Hertz model) and the attractive van der Waals force. Although there are some ways of combining the JKR model and the van der Waals forces, a deeper analysis of the inﬂuence of the use of these models on the self-assembly and mechanical strength of nanocomposites is needed. A comparison with experimental results is also required. When we compare the self-assembly of nanoparticles in the system modelled using electrical forces with the system in which van der Waals forces were used, we observe that the self-assembly is slightly faster than in the former case than in the latter case although the ﬁnal coordination number and value of contact stresses are similar. The self-assembly properties for system in which cohesion is represented by the JKR model coupled with Brownian forces drastically differ from other models used here. In the case of system with electrical force, the self-assembly process is about three orders of magnitude faster than in the case of JKR-Brownian model based simulations. It is worth mentioning on the lack of a clear quasi-static regime for low values of compression velocity – only continuous deformation of the aggregates was observed at very low compression velocities. This could be due to two factors: a low level of consolidation and the nature of the interparticle interaction. If the compression rate is small and the level of consolidation is also small the particles can produce permanent deformation of the material, forming a new bond or by simply sliding over another particle. With a higher level of consolidation or a high strain rate the particles would not have so much space or time, respectively, to ﬁnd a new position and the material would accumulate stresses until the bulk failure is produced. Finally, the most important difference that has been shown for the mechanical strength of

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23

the assemblies is the difference in the value of work required to produce the failure of the assemblies. Therefore, the work required to break the assembly with electrical forces is around 40 times larger than the work required for producing the failure of the assembly in which the cohesion is based on the JKR theory. The larger the range of the interparticle forces, the larger the distance that the particles need to be separated for the interaction with other particles to cease and therefore the larger the work needed to produce the failure of the assembly. However, a more detailed analysis of the inﬂuence of the range of the interparticle forces on the crashing stress of the particles is needed.

2.5 Conclusions We have presented a comparative analysis of the inﬂuence of van der Waals forces, electrostatic forces and interparticle adhesion as described by the JKR model, on the self-assembly and mechanical properties of Cu nanoparticulate aggregates by using computational modelling. The fastest rate of particle self-assembly as well as the highest values of number of contacts per particle are seen in the assemblies in which long range interparticle forces are acting (electrostatic and van der Waals forces). In contrast, the assembly in which the JKR model has been used produces a structure with a very low coordination number as a consequence of the short range of the cohesive interaction. In addition, the work required to produce the failure of the assemblies also depends on the type of interparticle interaction. The slower the interparticle forces decay with distance, the larger the work required to produce the mechanical failure of the self-assemblies. The generic approach adopted here links the interparticle forces to the bulk strength of the self-assembled aggregates. However, a more detailed study of the relationship between the work to produce the failure of the system and the range of interparticle interactions at the quasi-static conditions is needed in the future. The methodology presented here aids the control of interparticle interactions which is needed to create new nanocomposite materials with a speciﬁc level of strength. Acknowledgements We gratefully acknowledge EPSRC (EP/D027411/1) and MiMeMip, UK for their support to this work.

References 1. Aggeli A, Bell M, Carrick LM, Fishwick CWG, Harding R, Mawer PJ, Radford SE, Strong AE, Boden N (2003) pH as a trigger of peptide β-sheet self-assembly and reversible switching between nematic and isotropic phases. J Am Chem Soc 125: 9619–9628 2. Anderson MT, Lu N (2001) Role of microscopic physicochemical forces in large volumetric strains for clay sediments. J Eng Mech 127: 710–719

24

R. Moreno-Atanasio et al.

3. Antony SJ, Sultan M (2007) Role of inter-particle forces and inter-particle friction on the bulk friction in charged granular media subjected to shearing. Phys Rev E 75: 031307 4. Boal A, Gray M, Illhan F, Clavier G, Kapitzsky L, Rotello V (2002) Bricks and mortar selfassembly of nanoparticles. Tetrahedron 58: 765–770 5. Bose K, Moreno-Atanasio R, Antony SJ, Ding Y, Biggs SR, Ghadiri M (2005) Direct Measurement of the effect of adhesion on powder ﬂow behaviour: experimental and DEM investigations. In: Garc´ıa-Rojo R, Hermann HJ, McNamara S (eds) Powder and Grains, vol 1. Taylor & Francis, London, pp. 555–558 6. Cordelair J, Greil P (2004) Discrete Element modeling of solid formation during electrophoretic deposition. J Mater Sci 39: 1017–1021 7. Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Geotechnique, 29: 47–65 8. Foss DR, Brady JF (2000), Brownian dynamics simulation of hard-sphere colloidal dispersions. J Rheol 44(3): 629–651 9. Hamaker HC (1937), The London-Van der Waals attraction between spherical particles. Physica IV 10: 1058–1072 10. Han L, Luo J, Kariuki NN, Maye MM, Jones VW, Zhong CJ (2003) Novel Interparticle Spatial Properties of Hydrogen-Bonding mediated nanoparticle assembly. Chem Mater 15: 29–37 11. Heyes DM, Branka AC (1994) Brownian Dynamics simulations of self-diffusion and shear viscosity of near-hard sphere colloids. Phys Rev E 50: 2377–2380 12. Hong CW (1998) From long range interaction to solid-body contact between colloidal surfaces during forming. J Eur Ceram Soc 18: 2159–2167 13. Israelachvili J (1985) Intermolecular and Surface Forces. Academic, London 14. Johnson KL (1985) Contact Mechanics. Cambridge University Press, Cambridge 15. Johnson KL, Kendall K, Roberts AD (1971) Surface energy and the contact of elastic solids. P Roy Soc Lond A 324: 301–313 16. Kotsilkova R (2005) Processing-structure-properties relationships of mechanically and thermally enhanced smectite/epoxy nanocomposites. J Appl Polym Sci 97: 2499–2510 17. Kuhle A, Sorensen AH, Oddershede LB, Busch H, Theil Hansen L, Bohr J (1997) Scaling in patterns produced by cluster deposition. Z Phys D 40: 523–525 18. Lee JM, Curran C, Watkins KG (2001) Laser removal particles from silicon wafers using UV, visible and IR radiation. Appl Phys A 73: 219–224 19. Martin CL, Bouvard D (2004) Isostatic compaction of bimodal powder mixtures and composites. Int J Mech Sci 46: 907–927 20. Martin CL, Bouvard D (2006) Discrete Element Simulations of the Compaction of Aggregated ceramic Powders. J Am Chem Soc 89: 3379–3387 21. Moreno-Atanasio R, Antony SJ, Ghadiri M (2005) Analysis of the ﬂowability of cohesive powders using Distinct Element Method. Powder Technol 158: 51–57 22. Munjiza A, Latham JP (2004) Comparison of experimental and FEM/DEM results for gravitational deposition of identical cubes. Eng. Computation 21: 249–264 23. Pugnaloni LA, Ettelaie R, Dickinson E (2005) Brownian dynamics simulation of adsorbed layers of interacting particles subjected to large extensional deformations. J Colloid Interf Sci 287: 401–414 24. Richards K, Bithell M, Dove M, Hodge R (2004) Discrete-element Modelling: methods and applications in environmental sciences. Phil Trans Roy Soc A 362: 1797–1816 25. Satake M (1982) Fabric Tensor in granular materials. In: IUTAM Conference in Deformation and Failure of granular Materials, Delft, The Netherlands, pp 63–68 26. Savkoor AR, Briggs GAD (1977) The effect of the tangential force on the contact of elastic solids in adhesion. Proc Roy Soc Lond A 356: 103–114 27. Scocchi G, Posocco P, Fermeglia M, Pricl S (2007) Polymer-Clay Nanocomposites: A Multiscale Molecular Modeling Approach. J Phys Chem B 111: 2143–2151 28. Thornton C, Randall CW (1988) Applications of theoretical contact mechanics to solid particle system simulation. In: Stake M, Jenkins JT (eds) Micromechanics of Granular Materials. Elsevier, Amsterdam, pp 133–142

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29. Thornton C, Yin KK (1991), Impact of elastic spheres with and without adhesion. Powder Technol 65: 153–166 30. Vezenov DV, Noy A, Rozsnyai LF, Lieber CM (1997) Force titration and ionization State sensitive imaging of functional groups in aqueous solutions by Chemical Force Microscopy. J Am Chem Soc 119: 2006–2015

Chapter 3

Method of Continuous Source Functions for Modelling of Matrix Reinforced by Finite Fibres ˇ Vladim´ır Kompiˇs, M´ario Stiavnick´ y, Mari´an Kompiˇs, Zuzana Murˇcinkov´a, and Qing-Hua Qin

Abstract Fibres are the most effective reinforcing material. Simulation of the interaction of matrix with ﬁbres and ﬁbre with other ﬁbres is a most important problem for understanding the behaviour of ﬁbre-reinforced composites (FRC). Large gradients in all displacement, stress and strain ﬁelds and their correct simulation for near and far ﬁeld action are essential for effective computational modelling. Because of the large aspect ratios in ﬁbre type reinforcing particles, methods using volume discretization are not efﬁcient. Source functions (forces, dipoles, dislocations) describe correctly both near and far ﬁeld activities and thus help to simulate all interactions very precisely. The method of continuous source functions allows us to satisfy the continuity of ﬁelds between very stiff ﬁbres and much more ﬂexible matrix by 1D continuous functions along the ﬁbre axis and local 2D functions in the end parts of a ﬁbre with only few parameters. Two types of examples with rows of non-overlapping sheets of ﬁbre and with overlapping ﬁbres show that the interaction of the end parts of ﬁbres is crucial for evaluation of the mutual interaction of ﬁbres in the composite. Correct simulation of all parts is important for evaluation of stiffness and strength of the FRC. ˇ Vladim´ır Kompiˇs and M´ario Stiavnick´ y ˇ anik, Liptovsk´y Mikul´asˇ, Department of Academy of the Armed Forces of General M.R. Stef´ Mechanical Engineering, Slovak Republic [email protected] Mari´an Kompiˇs ˇ Siemens Program and System Engineering s.r.o., 01001 Zilina, Slovakia [email protected] Zuzana Murˇcinkov´a Department of Technical Devices Design, Faculty of Manufacturing Technologies of Technical University Koˇsice with seat in Preˇsov, Slovakia [email protected] Qing-Hua Qin Department of Engineering, Australian National University, Canberra ACT 0200, Australia [email protected] V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

27

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3.1 Introduction Composites of the future will offer many advances over those in use today. Composite materials reinforced by ﬁbres are important materials possessing excellent mechanical and also thermal and electro-magnetic properties. The outstanding properties of mechanical strength of carbon nano-tubes (CNT) are well known. This allows them to be used as possible reinforcing materials [1]. Reinforcement with nano-tubes facilitates the production of very strong and light materials. These properties of CNTs have attracted the attention of scientists all over the world. Understanding the behaviour of such composite materials is essential for structural design. Computational simulations play an important role in this process [25]. In computational simulations, boundary-type solution methodologies are now well established as alternatives to prevailing domain-type methods such as FEM [3, 33], because of the computational advantages they offer by way of reduction of dimensionality, good accuracy for the whole domain, and simplicity of data preparation for the model. The BEM [2, 5] is the most popular and efﬁcient boundary solution procedure, formulated in terms of boundary integral equations (BIEs). In BEM the integral identities are applied over elements discretizing the boundary of the domain. However, the use of elements in the BEM, with evaluation of weakly singular, strongly singular, hyper-singular and quasi-singular integrals, is a cumbersome and non-trivial task. The integration of elements containing singularity requires special integration models. If a singularity is close to the element (i.e. the element with quasi-singularity), then the integrals with large gradients in points closest to the singularity must be computed by high order quadratures, or by another special technique in order to obtain good accuracy. The boundary contour method (BCM) [20, 22] represents an effort to improve efﬁciency by transferring the surface integrals into line integrals by application of Stokes’ theorem for 3D problems. The boundary node method (BNM) [21, 32] is a combination of the moving least squares (MLS) approximation scheme and the standard BIE method. This method divorces the traditional coupling between spatial discretization (meshing) and interpolation as commonly practised in the FEM or in the BEM. Instead, a “diffuse” approximation, based on MLS approximants, is used to represent the unknown functions and surface cells, with a very ﬂexible structure used for integration. Thus, the BNM belongs with boundary meshless methods. Using the virtual boundary method and radial basis functions (RBF), the boundary point collocation method has been proposed to construct a boundary meshless formulation [29, 31], in which the boundary conditions and body forces are enforced and coupled with the analogue equation method to construct a boundary-type meshless method for analysing nonlinear problems [30]. Hybrid-Trefftz methods [8, 12, 15, 27] are also boundary-type methods. They use a set of trial functions, singular or non-singular, which a priori satisfy the corresponding linear part of the governing differential equation inside the (sub)domain

3 Method of Continuous Source Functions

29

(the large element). Other independent functions maintain the continuity between the subdomains (in a weak and strong sense). The method of fundamental solutions (MFS) [9, 14] is a boundary meshless method which does not need any mesh. In linear problems, only nodes (collocation points) on the domain boundaries and a set of source functions (fundamental solutions) in points outside the domain are necessary to satisfy the boundary conditions. MFS has certain advantages over the BEM, as it completely avoids the need for any integral evaluation and it leads to very simple formulations in some problems. However, large numbers of both collocation points and source functions are necessary if the shape of the domain is complex and moreover, the resulting system of equations is bad conditioned in some problems. The source functions serve as the trial functions and must be placed outside the domain. The location of the source functions is vital to both the accuracy and the numerical stability of the solution. The MFS can be also included among Trefftz-type methods. A novel boundary-type meshless method – the boundary point method (BPM) was developed in [18]. The BPM is based on the direct formulation of conventional and hypersingular BIEs employing favourable features of both the MFS and BEM. It is well known that for the integration of kernel functions over boundary elements, the shorter the distance between the source and ﬁeld points, the more difﬁcult it is to evaluate them accurately because of the properties of the fundamental solutions. In the formulation, “moving elements” are introduced by organizing relevant adjacent nodes in order to describe the local features of a boundary such as position, curvature and direction, over which the treatment of singularity and integration can be carried out, a beneﬁt not only for the evaluation of integrals in the case of coincidence points, but also for the versatility afforded by using unequally spaced nodes along the boundary. In special problems like composite materials reinforced with short micro-/ nanoﬁbres, all the methods mentioned above require very many elements or boundary points to obtain a sufﬁciently accurate solution. In such problems the materials of the matrix and ﬁbres have very different electro-magneto-thermo-mechanical properties and very large gradients are present in all ﬁelds in matrix and in ﬁbres as well. Domain formulations also require billions of equations after numerical discretization to simulate the decaying effects with increasing distance from the ﬁbre and the gradients along the boundaries. Boundary formulations can simulate the decaying effect well, but they also require a large number of equations to simulate the large gradients on the inter-domain boundaries. Both near and far ﬁeld effects are important. Near ﬁeld effects are important for evaluation of the strength of the composite and far ﬁelds are important for correct evaluation of the stiffening effect. The fast multi-pole method (FMM) [11] was developed to increase the efﬁciency of numerical models. A FMM based on the Taylor expansion of kernel functions (the fast multipole boundary integral equation method (FMBIEM)) [7, 10, 19, 23, 24, 26] was developed to solve the problem of composite materials reinforced by many small particles, considerably accelerating BEM solutions. However, near ﬁeld integrals still have to be solved by classical BEM and the boundaries are also discretized by elements.

30

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In this paper, a novel method of continuous source functions (MCSF) for the modelling of problems such as composites reinforced with short ﬁbres is presented. The source functions (forces and dipoles [4]) are continuously distributed along the ﬁbre axis (i.e. outside of the domain) and their intensity is modelled by 1D quadratic elements along the axis. Moreover, 2D distribution of the source functions are used to satisfy the continuity in the end part of the ﬁbre. The method can be included among BPMs, as inter-domain continuity is satisﬁed in the discrete points of the inter-domain boundary in the least squares (LS) sense. The model presents a signiﬁcant reduction (even by several orders) in the resulting system of equations compared to FEM, BEM, and other known mesh reducing methods. The MCFS requires integration over 1D and 2D elements. Analytic integration using symbolic manipulation is used for evaluation of the quasi-singular integrals occurring in the models, and it is a very efﬁcient tool for the evaluation of such integrals. Numerical integration is used for 2D elements because of complex form of the integrands. Recently a procedure of improved efﬁciency for such problems was also developed [18]. If the source domain and the ﬁeld point (e.g. the collocation point) are far from each other, the source values can be replaced by their resulting value and the computation is again considerably reduced. The effect is similar to that used in the FMBIEM but the algebraic manipulation is simpler, based on the principles of mechanics in this case instead of the Taylor expansion used in FMBIEM. The model is applied to simulation of the interaction of matrix-ﬁbre-ﬁbre for regularly distributed straight ﬁbres in a patch inside the matrix. Two different problems are studied: (1) ﬁbres distributed in rows without overlap of the ﬁbres and (2) with overlap of the ﬁbres.

3.2 Modelling of Composite Material Reinforced with Short Fibers Let us consider a linear elastic material reinforced by regularly distributed “short” straight ﬁbres. Let the cross sectional dimensions of a ﬁbre be much smaller than its length, the tensional (axial) stiffness of the ﬁbre is much higher than the stiffness of the matrix and ideal cohesion between the matrix and the ﬁbres is assumed in the present model. Then a continuity condition between the matrix and a ﬁbre can be introduced by zero strains (rigid ﬁbre) in the longitudinal direction along the ﬁbre boundary and by zero difference of the displacements in each pair of points on opposite ﬁbre boundaries in the perpendicular direction to the ﬁbre axis. If the ﬁbres are straight then an alternative continuity condition can be expressed by the difference between displacements of a point on the ﬁbre surface and a point in the middle cross-section along the ﬁbre axis. All displacement, strain and stress ﬁelds are split into a homogeneous part corresponding to constant stress and strain acting in the matrix without the ﬁbres and local ﬁelds corresponding to the stiffening effect. For simplicity, isotropic material properties are assumed in this paper.

3 Method of Continuous Source Functions

31 fibre

distributed forces and dipoles

x3

Fig. 3.1 Fibre-matrix interaction simulated by source functions placed inside the ﬁbre (the numbers indicate 1D elements)

The interaction of the matrix and a ﬁbre is simulated by source functions placed inside the ﬁbre along its axis. The source functions are continuous forces and dipoles (Fig. 3.1) acting outside the domain (matrix). The ﬁeld of displacements in an elastic continuum caused by a unit force acting in the direction of the axis x p is given by the Kelvin solution as it is known from BEM [2, 5, 6] 1 1 (F) U pi = [(3 − 4ν ) δip + r,i r,p ] (3.1) 16π G (1 − ν ) r where i denotes the xi coordinate of the displacement, G and ν are shear modulus and Poisson’s ratio of the material of the matrix. δij is the Kronecker’s delta and r is the distance between the source point s, where the force is acting and a ﬁeld point t, where the displacement is expressed, i.e. r=

√

ri ri , ri = xi (t) − xi (s)

with the summation convention over repeated indices and r,i = ∂ r ∂ xi (t) = ri r

(3.2)

(3.3)

is its directional derivative. The gradients of the displacement ﬁelds are the corresponding derivatives of the ﬁeld (3.1) at the point t (F)

U pi, j = −

1 1 [(3 − 4ν ) δ pi r, j − δ p j r,i − δi j r,p + 3r,i r, j r,p ] 16π G (1 − ν ) r2

The second derivative of nth power of the radius vector is deﬁned as n n n−1 n n r,k , j = r,k r,k δ jk − r, j r,k j = r

(3.4)

(3.5)

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V. Kompiˇs et al.

The strains are (F)

E pi j =

1 1 1 (F) (F) U pi, j + U p j,i = − 2 16π G (1 − ν ) r2

[(1 − 2ν ) (δ pi r, j + δ p j r,i ) − δi j r,p + 3r,i r, j r,p ]

(3.6)

and the stress components ij of this ﬁeld are (F)

(F)

S pi j = 2GE pi j +

2Gν 1 1 (F) δi j E pkk = 1 − 2ν 8π (1 − ν ) r2

[(1 − 2ν ) (δi j r,p − δ j p r,i − δipr, j ) − 3r,i r, j r,p ]

(3.7)

The displacement ﬁeld of a dipole can be obtained from the displacement ﬁeld of a force by differentiating it in the direction of the acting force, i.e. (D)

(F)

U pi = U pi,p = −

1 1

2 − r,i + 2 (1 − ν ) r,p δip 3r,i r,p 16π G (1 − ν ) r2

(3.8)

The summation convention does not apply over the repeated indices p here or in the following relations. Recall that the derivatives in the direction perpendicular (see, Eq. (3.4)) to the force deﬁne a force couple [4, 13], which can be also used in some problems. These derivatives have the physical meaning of corresponding couples of forces acting at a point. The gradients of a dipole displacement ﬁeld (3.8) are (D)

Upi, j = −

1 1

2 + 3r,i r, j −15r,i r, j r,p 16π G (1 − ν ) r3

2 +2 (1 − 2ν ) δip (δ j p − 3r, j r,p ) + 6r,i r,p δ j p + δi j 3r,p −1

(3.9)

and corresponding strain and stress ﬁelds are 1 (D) 1 1 (D) (D) [−15r,i r, j r,p + 3r,i r, j U pi, j + U p j,i = − E pi j = 2 16π G (1 − ν ) r3 + 2 (1 − 2ν ) δip δ j p + 6ν (δip r, j r,p + δ j p r,i r,p ) 2 + δi j 3r,p −1 ] (3.10) (D)

2Gν 1 1

(D) 2 δi j − δi j (1 − 2ν ) 2δip δ jp + 3r,p δi j E pkk = − 1 − 2ν 8π (1 − ν ) r 3 2 (3.11) + 6ν r,p (r,i δ jp + r, j δip ) + 3 1 − 5r,p r,i r, j ] (D)

S pi j = 2GE pi j +

If unit forces acting at source points (i.e. the fundamental solution satisfying the homogeneous equilibrium equations in the whole domain with the exception of the source point alone) are located in discrete points outside the solution domain for computational models, and also the collocation points (i.e. the points at which the boundary conditions have to be satisﬁed) are chosen at some discrete points

3 Method of Continuous Source Functions

33

of the domain boundary, then the method of solution is known as the method of fundamental solutions (MFS) [9, 14]. This method is very simple. It does not need any elements or any integration and thus is a fully meshless method. These functions are Trefftz functions and they serve as interpolators in the whole domain. Note that any other Trefftz functions can be used for this purpose [15]. On the other hand, dipoles are very effective tools for the modelling of composites reinforced by spherical or ellipsoidal particles [16, 28], and if the density of particles is small a single triple dipole can very effectively simulate a particle. The efﬁciency of the model is higher than that using the FMBIEM [7, 10, 19, 23, 24, 26] as integration is not necessary. Note that a dipole located inside a particle, i.e. outside of the domain represented by the matrix, gives both zero resulting force and moment along the particle boundary and thus the global equilibrium is not destroyed by local errors, as it can be when using MFS [15]. However, the location of the source points is vital for the best simulation of continuity and equilibrium along interdomain boundaries. However, if the ﬁbres are thin then satisfaction of continuity of displacements, strains and tractions on the surface between the matrix and ﬁbres and corresponding displacements and strains along the ﬁbre would require a very large number of source points to simulate the interaction. Moreover, in the end parts of a ﬁbre the ﬁelds have very large gradients [17], which increases the difﬁculties with accuracy and numerical stability of the solution. In our models, continuous distribution of source points is used for simulation of the interaction. This method is here called the method of continuous source functions (MCSF). It is possible to use both distributed forces and distributed dipoles along the ﬁbre axis (1D distribution) and oriented in the axis direction in the model. Their role is mainly to satisfy continuity in the ﬁbre axis direction. Continuity in directions perpendicular to the ﬁbre axis is served mainly by the continuous dipoles along the ﬁbre axis, but directed perpendicularly to the ﬁbre axis. Recall that continuously distributed dipoles are derivatives of continuously distributed forces. The ends of a ﬁbre can be in the form of half spheres or cylinders. It is important to satisfy the boundary conditions (b. c.) in these parts also. Without taking these b. c. into account, the source functions located along the ﬁbre axis can give incorrect results in evaluation of the stiffening effect. Special b. c. have to be speciﬁed for tube-type particles such as carbon nanotubes. Instead of displacements, zero tractions must be prescribed in such cases. The distribution is approximated by piecewise quadratic functions with C0 continuity between the elements. The following integrals need to be evaluated b a

p xns xs − x f m +r dxs = f x f 2 y2 + xs − x2f

(3.12)

where x is the coordinate along the ﬁber axis, the subscripts s and f denote the source and ﬁeld point and exponents n, m, p and r are integer numbers. y is the distance of the ﬁeld point from the axis. The integral (3.12) is transformed for better manipulation to

34

V. Kompiˇs et al. b+x f

a+x f

x + xf

n

(y2 + x2 )

xp

m +r 2

dx = f x f

(3.13)

The integrands are quasi-singular with very large local gradients and they are evaluated analytically by symbolic manipulation in MATLAB. Quadratic elements were chosen as the best way of approximation by polynomial functions. It was found that they can better approximate the b. c., as the end parts of ﬁbres transmit the largest forces from the matrix and it was preferable to use large gradients in these parts of ﬁbres when small elements were used for the parts with large gradients than to use larger elements of high order polynomials. Although the integrals give large gradients at the ends of ﬁbres, i.e. if x f → a, or x f → b, the C0 continuity of elements permits a smooth solution to be obtained.

3.3 Numerical Results and Discussion Two different problems were simulated in order to study the interaction of ﬁbres with matrix and also the interaction of ﬁbres: (1) a patch of non-overlapping rows of ﬁbres as shown in Fig. 3.2, and (2) a patch of overlapping rows of ﬁbres according to the Fig. 3.3. In the examples the modulus of elasticity of the matrix was E = 1,000 and the Poisson ratio ν = 0.3. The matrix was reinforced by a patch of straight rigid cylindrical ﬁbres. The length of ﬁbres was L = 100 and L = 1,000 and the radius x3

fibre of interest x1

D3 L

Fig. 3.2 Patch of non-overlapping rows of ﬁbres

D1

BA

B

fibre of interest

BA

Fig. 3.3 Patch of overlapping rows of ﬁbres

B

3 Method of Continuous Source Functions

35

R = 1. The distance between ﬁbres was ∆1 = ∆2 = ∆3 = 16 and for longer ﬁbres also ∆3 = 200 in the ﬁbre direction. The ﬁbres in the patch contained approximately 1% of the volume of the composite material. The patches of ﬁbres consisted of 5 × 5 × 7 ﬁbres in the presented examples and the “ﬁbre of interest” (FOI) was chosen in the centre to study the interaction of the ﬁbre with matrix and with the other ﬁbres as well. The domain is assumed to be loaded by far ﬁeld stress σ33∞ = 10 in the direction (x3 ), which is also parallel to ﬁbres’ axes. The model of the ﬁbre used in these examples contained fewer than 100 unknown parameters (intensities of the source functions) and about 200 collocation points. The problem is solved by the least squares (LS) method. In order to reduce the number of unknown parameters it was assumed that the intensities of source functions are identical in all ﬁbres. That is of course not correct, as the ﬁbres at the patch boundaries will transmit larger loads than those in the middle. An iterative procedure can be used to correct the simulation. But for the purpose of quantitative evaluation of the inﬂuence of ﬁbre reinforcement the models give sufﬁcient information. Some results are presented in the next ﬁgures. All displacement, strain and stress ﬁelds given in the ﬁgures are the local components of corresponding ﬁelds. Recall that the far ﬁelds have to be added in to obtain the total quantities. Figures 3.4–3.10 show the local ﬁelds in the vicinity of the ﬁbre of interest (the coordinates’ origin is in the middle of the ﬁbre) for L = 1,000 R with overlay and the distance ∆3 = 200 R. Displacement differences of the points on the ﬁbre boundary were linear along the ﬁbre (Fig. 3.4). As the LS method was used in the procedure, the errors were examined as shown in Figs. 3.4 and 3.5. The circles denote nodal

6

4

2

0

−2

−4

−6 −500 −400 −300 −200 −100

0

100

Fig. 3.4 Local displacements along a ﬁbre (L = 1,000 R)

200

300

400

500

36

V. Kompiˇs et al. 0.25 0.2 0.15 0.1 0.05 0 − 0.05 − 0.1 − 0.15 − 0.2 − 0.25 −500 −400 −300 −200 −100

0

100

200

300

400

500

Fig. 3.5 Errors in local displacements along ﬁbre (L = 1,000 R) disp along / parallel to fiber

6

4

2

0

−2

−4 −6 0

500

1000

1500

2000

2500

Fig. 3.6 Displacement ﬁeld parallel to ﬁbre axis (L = 1,000 R)

points in distributed source functions (ﬁctive forces of the Kelvin functions and dipoles) along the ﬁbre axis. Two different models were used: one with discontinuities (A – red) near the ends of neighbouring ﬁbres where the ﬁelds have large gradients, and one with continuous distribution of source functions (B – blue) along whole ﬁbre axis.

3 Method of Continuous Source Functions

37

shear stress along / parallel to fiber

60

40

20

0

−20 −40 −60

0

500

1000

1500

2000

2500

2000

2500

Fig. 3.7 Shear stress parallel to ﬁbre axis (L = 1,000 R) stress along / parallel to fiber

10 0 −10 −20 −30 −40 −50 −60 −70

0

500

1000

1500

Fig. 3.8 Stress in ﬁbre direction parallel to ﬁbre axis (L = 1,000 R)

Displacements, shear stresses and stresses in the ﬁbre direction along the ﬁbre (dashed line) and in the middle between the ﬁbre of interest and the neighbouring ﬁbre are shown in Figs. 3.6–3.8, respectively. Figures 3.9 and 3.10 contain the distribution of intensities of ﬁctive forces along the ﬁbre and the forces in the ﬁbre cross section given by the integral of forces. The discontinuous model contains large local

38

V. Kompiˇs et al. 3000

loads along fiber

2000 1000 0 −1000 −2000 −3000 −4000 −5000 −500 −400 −300 −200 −100

0

100

200

300

400

500

400

500

Fig. 3.9 Intensity of ﬁctive forces along ﬁbre (L = 1,000 R)

10

x 104

forces along fiber

8

6

4

2

0

−2 −500 −400 −300 −200 −100

0

100

200

300

Fig. 3.10 Forces in ﬁbre cross section (L = 1,000 R)

forces in the vicinity of the discontinuities. The maximal forces in the middle of the ﬁbre are important for evaluation of the strength of the ﬁbre. Both continuous and discontinuous models give similar results and the difference in maximal values is less than 3%.

3 Method of Continuous Source Functions

39

Figure 3.11 shows the inﬂuence of both the gap between the ﬁbres and the overlap for longer ﬁbres. As could be expected, the conﬁguration with overlap gives a much larger reinforcing effect. The ﬁctive forces are concentrated at the end parts of ﬁbres only in the case without overlap of ﬁbres (Fig. 3.12). A conﬁguration without overlap can occur when long ﬁbres are broken by large forces.

9

x 104

forces along fiber

8 7 6 5 4 3 2

c 1

d 0 −500 −400 −300 −200 −100

0

100

200

300

400

500

Fig. 3.11 Forces in ﬁbre cross section with overlap ∆3 = 200 R (red), ∆3 = 16 R (black); without overlap ∆3 = 200 R (cyan), ∆3 = 16 R (blue) by (L = 1,000 R) loads along fiber 8000 6000 4000 2000 0 −2000 −4000 −6000 −8000 −10000 −500 −400 −300 −200 −100

0

100

200

300

400

Fig. 3.12 Intensity of ﬁctive forces along ﬁbre in without overlap (L = 1,000 R)

500

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Different transmission of load between matrix and ﬁbres in the cases with and without overlap can be observed in shorter ﬁbres (L = 100 R), as can be seen from Figs. 3.12–3.19. The forces in the ﬁbres with overlap are greater than without overlap, but the difference is not as great as occurs with longer ﬁbres. Two aspects are important for accurate simulation in computational models: (1) how accurately the compatibility conditions in the inter-domain (matrix-ﬁbre)

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −50

−40

−30

−20

−10

0

10

20

30

40

50

20

30

40

50

Fig. 3.13 Local displacements along a ﬁbre (L = 100 R) 0.05 0.04 0.03 0.02 0.01 0 −0.01 −0.02 −0.03 −0.04 −0.05 −50

−40

−30

−20

−10

0

10

Fig. 3.14 Errors in local displacements along ﬁbre (L = 100 R)

3 Method of Continuous Source Functions

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disp along / parallel to fiber

0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 0

50

100

150

200

250

Fig. 3.15 Displacement ﬁeld parallel to ﬁbre axis (L = 100 R) with and without overlap shear stress along / parallel to fiber

15

10

5

0

−5

−10

−15

0

50

100

150

200

250

Fig. 3.16 Shear stress parallel to ﬁbre axis (L = 100 R) with and without overlap

boundary are satisﬁed, and (2) the numerical stability of the ﬁctive source functions. Instability of the source functions can be observed in the end parts of ﬁbres and between the discontinuous parts of source functions (Fig. 3.9). The instability in the end parts can result in inaccurate estimation of forces in ﬁbre cross-section. The instability shown in the Fig. 3.9 does not inﬂuence the results greatly. It was observed

42

V. Kompiˇs et al. strain along / parallel to fiber

0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1 −0.12

0

50

100

150

200

250

Fig. 3.17 Strain in ﬁbre direction parallel to ﬁbre axis (L = 100 R) with and without overlap loads along fiber

500 400 300 200 100 0 −100 −200 −300 −400 −50

−40

−30

−20

−10

0

10

20

30

40

50

Fig. 3.18 Intensity of ﬁctive forces along ﬁbre (L = 100 R) with and without overlap

that too ﬁne 1D elements in the parts of ﬁbres with large gradients can lead to instability of the source functions whereas too course elements decrease the accuracy of the primary variables. Both these features can destroy the accuracy of estimation of secondary ﬁelds (stresses, strains, forces in ﬁbre cross section). No general rule has yet been found for choosing the nodal points for 1D source distribution.

3 Method of Continuous Source Functions

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forces along fiber

1800 1600 1400 1200 1000 800 600 400 200 0 −200 −50

−40

−30

−20

−10

0

10

20

30

40

50

Fig. 3.19 Forces in ﬁbre cross section (L = 100 R) with and without overlap

Extreme shear forces between the ﬁbre and the matrix can lead to de-bonding of the ﬁbre or to de-cohesion and re-cohesion at the ends and also in the middle of a ﬁbre close to another ﬁbre in materials reinforced with nanotubes, which are typical and very efﬁcient novel reinforcing materials. This middle part of the ﬁbre will carry the largest forces, which can lead to fracture of the ﬁbre. The forces in the ﬁbre can exceed the largest stresses in the matrix and the bonding stresses on the matrix-ﬁbre interface by several orders.

3.4 Conclusions The MCSF enables us to simulate the interactions both of matrix with stiff reinforcing ﬁbres and of ﬁbre with other ﬁbres very effectively. Computational experiments have shown that very large gradients in all ﬁelds occur not only in the end parts of ﬁbres, but also in points close to the ends of neighbouring ﬁbres, more precisely in points on the line perpendicular to the axis of the neighbouring ﬁbre. The large gradients in the end parts of ﬁbres also inﬂuence numerical models. If polynomial interpolation of source functions is chosen in the models then ﬁner division of the continuous function needs to be deﬁned in these parts. Numerical models used for simulation of all matrix-ﬁbre and ﬁbre-ﬁbre interactions must maintain the gradients in order to correctly estimate the interactions. Because methods using volume discretization such as FEM and ﬁnite volume method (FVM) can smooth out the gradients, very ﬁne meshes would be necessary for numerical models.

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The extreme shear forces between ﬁbre and matrix can lead to de-bonding or to de-cohesion in the end parts and also in the middle of a ﬁbre close to another ﬁbre in materials reinforced with ﬁbres or nanotubes, which are typical and very efﬁcient novel reinforcing materials. This middle part of the ﬁbre will be subjected to the largest forces, which can lead to its fracture. The forces in the ﬁbre can result in stresses which can be larger by several orders than the largest stresses in the matrix, or in the bonding on the matrix-ﬁbre interface. The optimal aspect ratio of ﬁbres can be found for speciﬁc working stress/strain conditions in each part of a structure according to the strength of all ﬁbre, matrix and bond between ﬁbre and matrix. This is another attractive property of this kind of composite material. Two types of problems simulated in our experiments show, as expected, that overlapping ﬁbres reinforce the matrix more effectively. Fibres distributed without overlap can suffer breakage when subjected to high tension. In the present model we have considered straight ﬁbres, rigid in the axial direction. It is not complicated to consider some more general cases, using this model as the ﬁrst step in the iteration process. However, more effort will be necessary to include features like boundary conditions for more complex shaped regions, curved ﬁbres, nonlinear effects in matrix, and bonding properties. Acknowledgements This research was partially supported by grands APVT-20-035404 and NATO RTA 001-AVT-SVK. The ﬁrst two authors thank for this support.

References 1. Atieh MA et al. (2005) Multi-wall carbon nanotubes/Natural rubber nanocomposites. AzoNano – Online Journal of Nanotechnology, 1, pp. 1–11 2. Balaˇs J, Sl´adek J and Sl´adek V (1989) Stress Analysis by Boundary Element Methods. Elsevier, Amsterdam 3. Belytschko T, Liu WK and Moran B (2000) Nonlinear Finite Elements for Continua and Structures. Wiley, Chichester 4. Blokh VI (1964) Theory of Elasticity. University Press, Kharkov 5. Brebbia CA, Telles JCF and Wrobel LC (1984) Boundary Element Techniques – Theory and Applications in Engineering, Springer, Berlin 6. Cheng AHD and Cheng DT (2005) Heritage and early history of the boundary element method. Journal of Engineering Analysis with Boundary Elements, 29, pp. 268–302 7. Fu Y, Klimkowski KJ and Rodin GJ (1998) A fast solution method for three-dimensional many-particle problems in linear elasticity. International Journal for Numerical Methods in Engineering, 42, pp. 1215–1229 8. Gaul L and Moser F (2002) A Hybrid Boundary Element Approach Without Singular Boundary Integrals. In: Selected Topics in Boundary Integral Formulations for Solids and Fluids, Kompiˇs V. (ed.), Springer, Wien, pp. 107–116 9. Golberg MA and Chen CS (1998) The Method of Fundamental Solutions for Potential, Helmholtz and Diffusion Problems. In: Boundary Integral Methods – Numerical and Mathematical Aspect, Golberg MA. (ed.), Computational Mechanics, Southampton, pp. 103–176 10. Gomez JE and Power H (1997) A multipole direct and indirect BEM for 2D cavity ﬂow at low Reynolds number. Journal of Engineering Analysis with Boundary Elements, 19, pp. 17–31

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11. Greengard FL and Rokhlin V (1987) A fast algorithm for particle simulations. Journal of Computational Physics, 73, pp. 325–348 12. Jirousek J and Wroblewski A (1997) T-elements: State of the art and future trends. Archives of Applied Mechanics, 3, pp. 323–434. 13. Kachanov M, Shaﬁro B and Tsukrov I (2003) Handbook of Elasticity Solutions. Kluwer, Dordrecht 14. Karageorghis A and Fairweather G (1989) The method of fundamental solutions for the solution of nonlinear plane potential problems. IMA Journal Numerical Analysis, 9, pp. 231–242 ˇ 15. Kompiˇs V and Stiavnick´ y M (2006) Trefftz functions in FEM, BEM and meshless methods. Computer Assisted Mechanics and Engineering Sciences, 13, pp. 417–426 16. Kompiˇs V, Kompiˇs M, Kaukiˇc M and Hui D (2006) Singular Trefftz Functions for Modelling of Material Reinforced by Hard Particles. Proceedings of the Fifth International Conference on Engineering Computational Technology, Topping, B.H.V., Montero, G. and Montenegro, R. (eds.), CD-ROM, Civil-Comp Press, Dun Eaglais 17. Kompiˇs V, Kompiˇs M and Kaukiˇc M (2007) Method of continuous dipoles for modeling of materials reinforced by short micro-ﬁbers. Journal of Engineering Analysis with Boundary Elements, 31, pp. 416–424 18. Ma H and Qin QH (2007) Boundary point method for linear elasticity based on direct formulations of conventional and hypersingular boundary integral equations, Computers and Mathematics with Applications, submitted 19. Mammoli AA and Ingber MS (1999) Stokes ﬂow around cylinders in a bounded twodimensional domain using multipole-accelerated boundary element method. International Journal of Numerical Methods in Engineering, 44, pp. 897–917 20. Mukherjee S (2002) The Boundary Contour Method. In: Selected Topics in Boundary Integral Formulations for Solids and Fluids, Kompiˇs V (ed.), Springer, Wien, pp. 117–150 21. Mukherjee S (2002) The Boundary Node Method. In: Selected Topics in Boundary Integral Formulations for Solids and Fluids, Kompiˇs V (ed.), Springer, Wien, pp. 151–180 22. Mukherjee S and Mukherjee YX (1998) The hypersingular boundary contour method for three dimensional elasticity, ASME Journal of Applied Mechanics, 65, pp. 300–309 23. Nishimura N (2002) Fast multipole accelerated boundary integral equations. Applied. Mechanics Review, 55, pp. 299–324 24. Nishimura N, Yoshida K and Kobayashi S (1999) A fast multipole boundary integral equation method for crack problems in 3D. Journal of Engineering Analysis with Boundary Elements, 23, pp. 97–105 25. Oden JT et al. (2006) Simulation – Based Engineering Science: Revolutionazing Engineering Science Through Simulation. Report NSF Blue Ribbon Panel 26. Peirce AP and Napier JAL (1995) A spectral multipole method for efﬁcient solution of largescale boundary element models in elastostatics. International Journal for Numerical Methods in Engineering, 38, pp. 4009–4034 27. Qin QH (2002) The Trefftz Finite and Boundary Element Method. WIT, Southampton ˇ 28. Stiavnick´ y M, Kompiˇs V, Kaukiˇc M (2007) Global dipole model for composite reinforced by micro/nano particles. International Conference on Computational Modeling and Experiments of the Composites Materials with Micro-and Nano-Structure, May 28–31 2007, CD-ROM 29. Wang H, Qin QH and Kang YL (2005) A new meshless method for steady-state heat conduction in anisotropic problems and inhomogeneous media. Archive of Applied Mechanics, 74, pp. 563–579 30. Wang H and Qin QH (2006) A meshless method for generalized linear and nonlinear Poissontype problems. Engineering Analysis wit Boundary Elements, 30, pp. 515–521 31. Wang H, Qin QH and Kang YL (2006) A meshless model for transient heat conduction in functionally graded materials. Computational Mechanics, 38, pp. 51–60 32. Zhang JM, Tanaka M and Matsumoto T (2004) Meshless analysis of potential problems in three dimensions with the hybrid boundary node method. International Journal for Numerical Methods in Engineering, 59, pp. 1147–1160 33. Zienkiewicz OC and Taylor RL (2000)The Finite Element Method, 5th edn. ButterworthHeinemann, Oxford

Chapter 4

Effective Dynamic Material Properties for Materials with Non-Convex Microstructures Martin Schanz, Georgios E. Stavroulakis, and Steffen Alvermann

Abstract The determination of macroscopic, effective properties of microstructured materials is referred to as homogenization. Usually in homogenization, it is assumed that on the microscale inertia effects can be neglected. Here, contrary to these approaches, inertia effects are taken into account, leading to a frequency dependent microscopic behavior. Additionally to this effect, non-convex microstructures are considered. It is assumed that the microstructure can be modeled as a beam framework in frequency domain which is exactly solved by a boundary integral formulation. Further, it is assumed that the structures to be treated are made up of identical unit cells. However, due to the inertia effects a mean value of the microscopic response calculated for several unit cells is used. Under these micromechanic assumptions on the macroscopic scale a frequency dependent, i.e. viscoelastic and auxetic behavior is expected. Hence, an analytical homogenization is presumably not possible. Therefore, the homogenization is performed numerically formulated as an optimization process. The classical technique SQP and soft computing methods, in particular a Genetic Algorithm, are used. The frequency dependent macroscopic material parameters are found for a frequency range from 0 up to 103 kHz for a seven parameter model using as well fractional derivatives. The system responses on micro- and macroscale show a good agreement for the considered frequency range. Both optimization strategies are able to ﬁnd adequate material parameters on the macroscale but the SQP needs, as expected, reliable starting values. On the contrary the Genetic Algorithm is more robust but much slower. Martin Schanz and Steffen Alvermann Institute of Applied Mechanics, Graz University of Technology, Technikerstr. 4, 8010 Graz, Austria [email protected] Georgios E. Stavroulakis Department of Production Engineering and Management, Technical University of Crete, 73100 Chania, Greece [email protected] V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

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4.1 Introduction Nearly all common materials present a certain heterogeneous microstructure. The determination of macroscopic, effective properties of such microstructured materials is referred to as homogenization. The objective of the homogenization process is to ﬁnd macroscopic properties of a homogeneous material which represents the behavior of the non-homogeneous microstructure. The methodologies to evaluate the effective mechanical properties of heterogeneous materials have attracted attentions from many researchers. A comprehensive overview of different approaches is given in [12, 20] or from a computational point of view in [27]. Many materials exhibit a cellular or foam-like structure in order to provide a certain amount of stiffness with minimal weight. Cork, for example, is a common cellular material having a closed-celled hexagonal (honeycomb) architecture. In addition to natural materials, man-made cellular structures are used in aerospace structures, composite plates, or other lightweight applications. Due to the increasing importance of these materials, several studies have been performed to evaluate their properties. The effective elastic properties of regular isotropic triangular grid structures as well as square cells have been analyzed by Gibson and Ashby [11] and Torquato et al. [26]. Analysis of more general grid structures were provided by Hohe and Becker [14] who uses an energetic concept for the homogenization process. A comprehensive state of the art publication on homogenization methodologies is given by Hohe and Becker [15]. For microstructures composed of beams, it has been postulated that non-convex shapes (with re-entrant corners) are responsible for a negative Poisson’s ratio effect. Several authors have published papers concerning this effect which is also referred to as auxetic material behavior [25]. Materials with a negative Poisson’s ratio are expected to have interesting mechanical properties such as high energy absorption [5] and fracture resistance which may be useful in applications such as packing material, knee and elbow pads, robust shock absorbing material, ﬁlters or sponge mops. Nevertheless, until now, no industrial applications have been realized. From a continuum mechanics point of view, there is no restriction for Poisson’s ratio to be positive. This is known for a long time, but nobody made an effort to investigate this behavior. In fact, the earliest example for a material with a negative Poisson’s ratio was published in Science in 1987, ‘Foam structures with a negative Poisson’s ratio’ by Lakes [16]. He converted a synthetic foam from its conventional, positive Poisson’s ratio state to one having a negative Poisson’s ratio by a relatively simple process [4]. In fact, the fabrication process is published in the Internet in form of a ‘cooking recipe’ [17]. Since then several new negative Poisson’s ratio materials have been developed and fabricated (see, e.g. [1, 6, 21]). Most publications on homogenization assume on the microscopic level that the inﬂuence of inertia can be neglected [20] which is mostly motivated by the scale separation which is assumed to be large as stated by Miehe [19]. Only a few approaches to study the effect of micromechanical inertia are published, e.g. [18]. In this paper, a dynamical homogenization approach is presented, where dynamical means that, contrary to other approaches, inertia effects on the microscale are

4 Effective Properties of Non-Convex Microstructured Materials

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not neglected. Hence, the micromechanic calculations are performed in frequency domain with harmonic excitations. Therefore, a frequency-dependent behavior is expected also on the macroscale. Accounting for this frequency-dependence, a viscoelastic constitutive law is applied on the macroscale. Since the viscoelastic constitutive law requires a number of parameters to be found, the homogenization is formulated as an optimization problem, i.e. ‘ﬁnd macroscopic material parameters that describe the micromechanical behavior as good as possible’. The paper is arranged in the following way. First, the calculation of the considered microstructure is elaborated. The constitutive equation to be used for the macroscopic scale is then presented in Section 4.3. Section 4.4 deals with the used methodologies for the optimization, a classical SQP and a Genetic algorithm. Finally, some examples are presented and discussed.

4.2 Calculation of the Unit Cell Within this work, cellular solids are considered which consist of simple beams. Such structures are found quite often in the literature, because they can represent sandwich cores or simpliﬁed models of foam-like materials. It is assumed that the material is periodic, i.e. it consists of equally shaped cells. For sandwich cores, this assumption is well justiﬁed, real materials, of course, have an imperfect microstructure, where neighboring cells have a slightly different geometry. But, in order to calculate effective, homogeneous material properties, this model should be sufﬁciently accurate to represent the microstructure. If the assumption of periodicity is made, it is very easy to ﬁnd a representative volume element (RVE), because if the microstructure consists of identical units, the smallest unit contains all information and is therefore representative. If on the microscale the dynamic effects are neglected, i.e. it is modeled static, effective properties can be calculated from this single unit cell. However, in a dynamic calculation in frequency domain one unit cell does not sufﬁce to obtain representative results. The reason is that different modes of deformation (structural modes) may occur when using a different number of unit cells. This can be explained by considering the example given in Fig. 4.1. A unit cell is loaded with a force on the left and right

Fig. 4.1 Example calculation of 1 and 3 × 3 unit cells

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hand side. The cell is calculated in frequency domain, i.e. the load is harmonic and thus, the resulting strain ∆ (4.1) ε= is also harmonic. Another calculation is done using 3 × 3 cells. Both strains are plotted over a frequency range of 0–106 Hz in logarithmic scale in Fig. 4.2. On the left hand side, the whole frequency range is plotted, whereas on the right hand side, only a small range is plotted. It can be seen that the results from both calculations are not identical. At approximately 3.4 · 105 Hz, an eigenfrequency is missing in the single unit cell calculation and at 2.3 · 105 Hz, the eigenfrequencies of both calculations are different. Therefore, it is necessary to calculate multiple unit cells to obtain characteristic microscale results. A simple and pragmatic approach is to calculate a speciﬁc number of unit cells and then average the results. The frequency dependent response of the unit cell above as well as in the following is based on a frame model using the Timoshenko theory for the beams. This reﬁned theory is used to take into account the shear deformation and the rotatory inertia which can not be neglected for higher frequencies [8]. Further, for the numerical model the Boundary Element Method for beams [2] is used because it solves the governing equations exactly independent of the load in contrast to a Finite Element model.

4.3 Macroscopic Constitutive Equation Since inertia effects are taken into account on the microscale, the response of the unit cell is frequency-dependent. Thus, the effective properties on the macroscale must also feature a frequency-dependence. One of the easiest constitutive laws exhibiting a frequency-dependence is the linear viscoelasticity. The stress-strain equation of a linearly viscoelastic material can be expressed in the form of a differential equation (see, e.g. the book by Christensen [7]) N

∑

k=0

pk

M d αk d αk σ = q i j k ∑ dt k εi j . dt k k=0

(4.2)

In (4.2), M and N can have different values and must be adapted to the application, so their choice is somewhat arbitrary. Also, fractional derivatives as denoted by d αk may be used to better ﬁt a larger frequency range [9]. Details on the deﬁnition of fractional derivatives can, e.g. be found by Podlubny [22] and their application in viscoelastic constitutive equations, e.g. in [3]. In frequency domain, viscoelastic effects are described by complex moduli and, especially, fractional derivatives become simple. They are presented by a fractional power [22]. Applying a Fourier-transformation on (4.2), the time dependence is replaced by a frequency dependence. In the present application, the constitutive equation

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0.001 1 unit cell 3x3 unit cells

strain

1e-05

1e-07

1e-09 0

500000

1e+06

frequency (a) Large frequency range 1e-05 1 unit cell 3x3 unit cells

strain

1e-06

1e-07

1e-08 200000

250000

300000

frequency (b) Small frequency range Fig. 4.2 Strain of 1 and 3 × 3 unit cells

350000

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σi j (ω) =

E(ω) ν (ω)E(ω) εi j (ω) + δi j εkk (ω) (1 + ν (ω)) (1 + ν (ω)) (1 − 2ν (ω))

(4.3)

is chosen, i.e. a isotropic material is assumed. For the Young’s modulus, in the following, either the viscoelastic model E(ω) =

q1 (iω) + q2 (iω)2 + q3(iω)3 + q4 (iω)4 + q5 (iω)5 + q6(iω)6 + q7 (iω)7 p1 (iω) + p2 (iω)2 + p3 (iω)3 + p4 (iω)4 + p5(iω)5 + p6 (iω)6 + p7(iω)7 (4.4)

with 14 parameters or the model 1 + q1(iω)α1 + q2 (iω)α2 E(ω) = E¯ . 1 + p1(iω)α1 + p2 (iω)α2

(4.5)

with fractional derivatives and 7 parameters is used. The same model but with possible different parameters is used for the Poisson’s ratio

ν (ω) = ν¯

1 + q˜1 (iω)α˜ 1 + q˜ 2 (iω)α˜ 2 . 1 + p˜ 1 (iω)α˜ 1 + p˜ 2 (iω)α˜ 2

(4.6)

Again, the choice of these models is somewhat arbitrary. For the considered frequency range, they have shown to be sufﬁcient. A number of trials with fewer parameters were not successful, as well as models with more parameters, yet without fractional derivatives. Note that in the above equations a special indication of the Fourier transform is skipped for the sake of brevity.

4.4 Homogenization Using Optimization The connection between the material behavior on the microscale and the macroscale is referred to as homogenization. The aim is to ‘replace’ the heterogeneous microstructure by a homogeneous material which describes the material behavior correctly. Here, the homogenization is formulated as an optimization problem, i.e. ‘Find material parameters on the macroscopic scale which describe the micromechanical behavior as good as possible’. Contrary to the static case where the homogenization is performed analytically [13], the consideration of dynamic effects on the microscale makes it necessary to use an optimization algorithm. There are too many unknown parameters in the macroscopic constitutive equation to perform an analytical homogenization. Within the scope of this work, the Sequential Quadratic Programming Method (SQP) is used. It represents a gradient-based optimization technique. As this algorithm is a quite common for details the reader is pointed towards standard books, e.g. [10, 24]. In the following presented examples, the gradient is calculated by ﬁnite differences. Gradient based algorithms often exhibit weaknesses in non-convex

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problems, which means that they run into local minima of the search space. Therefore, also a global optimization technique a Genetic Algorithm is applied to the optimization problem. As expected from literature, tests have shown that the Genetic Algorithm is the robustest (but slowest) technique also compared with Neural Networks as alternative global optimization tool. Genetic Algorithms are stochastic search algorithms based on the mechanisms of nature selection and natural genetics. At the beginning of the algorithm, a population made up of a discrete number of individuals is generated. One individual corresponds to the solution of a problem, and consists of an array of gene values. Just as in nature, the individuals are optimized for their environment by successive modiﬁcation over a number of iterations. In each iteration or generation, the algorithm evaluates, selects, and recombines the members of the population to produce the succeeding generation. Evaluation of each individual which encodes a candidate solution is based on a ﬁtness function. This function is used to select the relatively ﬁtter individuals, i.e. the individuals with the best ﬁtness value. The selected individuals form the parent generation, and they produce offsprings by rules like mutation, crossover, or gene replacement (for details, see, Sch¨oneburg [23]). The algorithm stops after a speciﬁed number of generations or if a certain ﬁtness value is obtained for the best individual. In the present application, one gene corresponds to the value of a material parameter pk , qk , or αk from (4.3) and the whole set of parameters corresponds to one individual. As ﬁtness function, the square difference between the microscopic and macroscopic system response is used. The differences are summed up over the considered frequency range. If a harmonic strain is applied on the unit cell, the minimization function reads f=

ωend

∑

ωstart

2 σmicro (ω) − σmacro (ω, pk , qk , αk ) → 0 .

(4.7)

In (4.7), the stress σmicro (ω) must be calculated only once, whereas σmacro (ω) is calculated for each new individual of the population via Eq. (4.3), using new gene values of pk , qk , and αk . In principle, the procedure can be inverted, i.e. stresses can be applied on the cell and the strain response on the macroscopic scale can be optimized. In case of the SQP, the same ﬁtness function is used. Within the scale consideration in homogenization, a material point on the macroscale is assigned to a volume portion on the microscale. To deﬁne a homogeneous mechanical equivalent to the heterogeneous unit cell, a surface average based approach is used. The stress distribution in the unit cell consisting of the given microstructure is assumed to be equivalent to a stress distribution of the effective medium if 1 1 ti d Γ = ti d Γ (4.8) Γ Γ Γ

Γ

holds, where ti is the traction vector on the surface Γ of the unit cell. Equation (4.8) describes a redistribution of the stress components along the surface of the unit cell

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during the homogenization process. A second equivalence condition interrelates the strain states on the microscopic and macroscopic scale. An average strain state for the unit cell is deﬁned by means of the difference in the displacements of opposite surfaces of the volume element. If the difference in the displacements of opposite surfaces is not constant along the surface, the surface integral of this quantity is taken. Thus, the macroscopic strain can be expressed by εi j =

1 2V

(ui n j + u j ni ) d Γ,

(4.9)

Γ

where V denotes the volume of the unit cell and ni are the components of the outward normal vector on Γ. Thus, the heterogeneous microstructure is smeared over the unit cell (see, Fig. 4.3). With the averaged stress and strain, the effective macroscopic material properties are calculated via σi j = Ci∗jkl εkl ,

(4.10)

with the effective material tensor Ci∗jkl which is deﬁned as the relation between the two averaged quantities σi j and εkl . Here, the material tensor is simpliﬁed due to the assumption of isotropy and the respective equation is given by (4.3), however without denoting the stress and strain as averaged quantities. The advantage of this approach is that it only requires the tractions and deformations on the boundary of the unit cell. No energetic considerations like in the strain energy based homogenization are made [13]. Thus, it can not only be applied in the static case, but also in dynamics. Since the materials discussed in this work are assumed to have a periodic microstructure, it is meaningful to do a dynamic calculation in frequency domain. The reason is that in frequency domain, periodic boundary conditions can be applied to the unit cell. In time domain, this is not possible because wave propagation phenomena take place. Thus, if a wave is running through the unit cell, the tractions and displacements on one edge are different from the tractions and displacements on the opposing edge. It is not possible to fulﬁll the periodicity, which is postulated from the fact that the material consists of equally

s, e

Fig. 4.3 Stress and strain on the micro- and macroscale

·s Ò, ·eÒ

4 Effective Properties of Non-Convex Microstructured Materials

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shaped cells. In a frequency domain calculation, in contrast, the unit cell is situated in a steady state where periodicity can be fulﬁlled. It should be remarked that numerical tests have also been performed based on a volume averaged approach with a map to the boundary. These tests have shown no difference between the surface and volume averaged approach. However, for very high frequencies and soft materials this may change. Some more effort will be put on this aspect in future.

4.5 Numerical Examples Effective properties are calculated in this section. In order to study the effectiveness of different optimization methods in the ﬁrst subsection only a small frequency range up to the ﬁrst structural eigenfrequency of the non-convex micro structure is considered. In the second subsection the more realistic case of a wide frequency range is presented for both auxetic and non-auxetic microstructures. The following two microstructures given in Fig. 4.4 are used for the numerical study. Both cells consist of beams with a quadratic cross section of 0.1 × 0.1. The microstructure material is in the ﬁrst subsection steel and in the second subsection PMMA with a slight damping. The latter material data were measured at the PTB (PhysikalischTechnische Bundesanstalt) in Braunschweig.

4.5.1 Small Frequency Range The non-convex microstructure shown in Fig. 4.4(a) is used for the following considerations. The cell is analyzed in frequency domain. A harmonic strain in one direction is applied to the cell (see, Fig. 4.5).

0.3

0.2 0.1 0.2

0.2

0.5

0.5

(a) Auxetic microstructure Fig. 4.4 Considered microstructures

(b) Non-Auxetic microstructure

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Fig. 4.5 Load case

u = u– .eiωt

u– = u .eiωt

Since the considered microstructure is assumed to be periodic, the opposing nodes of the cell must have the same deformation and the forces of the node pairs must be equal in magnitude but opposite in direction. Due to the symmetry of the cell the load periodicity is implicitly fulﬁlled, e.g. the reaction force of the node on the left hand side corresponds to the applied load on the right hand side. On the macroscale, the one-dimensional version of the constitutive Eq. (4.2) 7

σ(ω) = E(iω) ε(ω) =

∑ pk (iω)k

k=0 7

ε(ω)

(4.11)

∑ qk (iω)k

k=0

with the complex Young’s modulus (4.4) is used. Equation (4.11) has to be fulﬁlled for a frequency range 0–ωmax . The number of parameters is somewhat arbitrary, it depends on the considered frequency range how many parameters have to be used. For the present case, it has turned out that a total of M + N = 14 parameters is sufﬁcient to deliver good results for the considered frequency range. Two optimization methodologies, a soft computing tool – the Genetic Algorithm and a classical technique – a SQP algorithm, are used in the following to ﬁnd the macroscopic material data.

4.5.1.1 Genetic Algorithm The ﬁtness value of the individuals quality is calculated via Eq. (4.7). A total number of M + N = 14 parameters is used, so each individual consists of 14 chromosomes. In a trial and error process, the following parameters of the Genetic Algorithm have shown to be suitable for the present application: • • • •

A population size of 1,000 individuals is chosen The starting population is created randomly New individuals are created via 2-point-crossover The mutation of an individual is done accordingly to the standardized normal distribution • 250 individuals are created newly in every generation, the remaining 750 are created as offspring of the previous generation • The optimization is stopped after a maximum of 50,000 iterations or if the best ﬁtness value satisﬁes f < 10−5.

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45 optimization (1) optimization (2) optimization (3)

40

fitness value f

35 30 25 20 15 10 5 0 0

10000

20000

30000

40000

50000

generation Fig. 4.6 Optimization with genetic algorithm

The proceeding of the optimization is shown in Fig. 4.6. Since the ﬁrst generation is created randomly and also during the optimization, genetic operators are applied randomly, the proceeding of the optimization is different in all the optimization runs. The ﬁtness value of the best individual is plotted versus the number of generations. It can be seen that in all three cases, a relatively small ﬁtness values was obtained, however, after a different number of generations. The obtained values for pk and qk are given in Table 4.1. Contrary to the static case, there is more than one set of parameters which describes the behavior accurately. Figure 4.7 shows the behavior on the micro- and macroscale. The modulus E(ω) is plotted versus the frequency. On the microscale (solid line), E(ω) is E(ω) =

σmicro εmicro

(4.12)

with the stress and strain σmicro , εmicro of the unit cell, whereas for the three macroscale results, Eq. (4.11) was solved for ε(ω) and plotted for the three optimization runs with the respective parameters pk and qk given in Table 4.1.

4.5.1.2 SQP – Sequential Quadratic Programming The Sequential Quadratic Programming method is used to minimize the optimization function (4.7). The following parameters were used: • Random values are used as starting points • The optimization stops if f < 10−5 or a maximum of 1,000 iterations is reached

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Table 4.1 Obtained material parameters by Genetic Algorithm (b) Optimization (2) f = 0.08141 (a) Optimization (1) f = 0.02996 k

pk

qk

1 2 3 4 5 6 7

1.7617 · 101

1.5469 · 107

2.2672 · 102 −1.7471 · 10−14 1.8300 · 100 3.3705 · 10−3 4.2695 · 10−3 1.7815 · 10−19

1.8152 · 108 1.2785 · 105 1.4790 · 106 1.0693 · 104 3.0166 · 103 4.0734 · 101

k

pk

qk

1 2 3 4 5 6 7

4.9398 · 100

4.1763 · 106 3.8899 · 108 4.3765 · 105 2.7570 · 106 6.7831 · 103 4.8445 · 103 2.0826 · 101

4.0007 · 102 −1.6224 · 10−14 2.1568 · 100 1.2825 · 10−16 2.9828 · 10−3 5.0153 · 10−5

(c) Optimization (3) f = 0.02313 k

pk

qk

1 2 3 4 5 6 7

7.6350 · 10−5

6.3904 · 101 1.6060 · 108 4.0609 · 105 1.1117 · 106 5.5493 · 103 1.8973 · 103 1.5891 · 101

2.0017 · 102

−7.0055 · 10−14 1.1501 · 100 4.4045 · 10−16 1.7722 · 10−3 2.6157 · 10−5

100 microscale optimization (1) optimization (2) optimization (3)

E

10

1

0.1

0.01 0

2

4

6

8

10

12

frequency [kHz]

Fig. 4.7 Genetic Algorithm results on the micro- and macroscale

14

16

18

4 Effective Properties of Non-Convex Microstructured Materials

59

• Formulation as unconstrained problem, i.e. no constraints for values of pk and qk are made Figure 4.8 shows typical optimization runs. The ﬁtness value is plotted versus the number of iterations. The SQP method was able to obtain a value f < 1 in two runs. One optimization run was only able to ﬁnd a local minimum of the function. This can be seen from the ﬁnal value of the optimization of f = 31.2730 and also from the results for E(ω) on micro- and macroscale. See Table 4.2 for the obtained values for the parameters pk and qk and Fig. 4.9 for results on both scales. Both optimization procedures were able to ﬁnd adequate material parameters on the macroscopic scale. It can already be seen from the ﬁtness value or ﬁnal value of the optimization function, respectively, how good the parameters ﬁt the microscopic properties. The Sequential Quadratic Programming method has problems with local minima. If a ‘wrong’ starting point is used for the optimization, it may deliver poor results. The Genetic Algorithm is the most reliable of both optimization procedures. No starting point is required, and it always delivers good results. However, it takes much more computing time to obtain results. Hence, in the next test only the Genetic Algorithm is used.

4.5.2 Wide Frequency Range A frequency range from 0 up to 3×103 kHz is chosen. Due to the existence of higher eigenmodes for more than one unit cell (see, Section 4.2), the microstructures are 100 optimization (1) optimization (2) optimization (3)

fitness value f

80

60

40

20

0 0

200

400

600 iterations

Fig. 4.8 Optimization using SQP algorithm

800

1000

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Table 4.2 Obtained material parameters by SQP (a) Optimization (1) f = 31.2730 k

pk

(b) Optimization (2) f = 0.0942

qk

k

pk

1 2.7349 · 108 2 −7.8132 · 108 3 3.0370 · 108 4 3.0801 · 108 5 1.2956 · 108 6 −3.1951 · 106 7 4.4835 · 105

2.3801 · 103 1.7790 · 106 −2.1948 · 104 2.9126 · 106 −3.5378 · 104 9.0371 · 104 −2.0615 · 104 −9.9309 · 105 −6.5232 · 103 −1.7844 · 102 −9.5146 · 102 5.9092 · 103 1.1842 · 104 1.2702 · 100

1 2 3 4 5 6 7

qk 2.3917 · 1011 −2.5963 · 1011 −1.1559 · 1011 −4.0479 · 1010 7.1634 · 1010 −2.2324 · 108 2.8131 · 108

(c) Optimization (3) f = 0.0961 k

pk

1 2 3 4 5 6 7

−1.7145 · 10−1

qk

−6.7364 · 10−1 −9.4824 · 10−1 1.6859 · 100 −1.8562 · 10−1 2.6000 · 10−5 −6.5400 · 10−4

−1.2534 · 102 −1.0430 · 102 4.1414 · 101 1.8603 · 101 −1.0769 · 102 −2.8079 · 10−2 −4.2114 · 10−1

100 microscale optimization (1) optimization (2) optimization (3)

E

10

1

0.1

0.01 0

2

4

6

8

10

12

14

16

18

frequency [kHz] Fig. 4.9 SQP results on the micro- and macroscale

calculated multiple times. Calculations were done for 1, 2 × 2, . . . , 5 × 5 unit cells and the results were averaged. The load case for both cells is given in Fig. 4.5. A displacement amplitude is applied on the left and right hand side of the cell.

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Based on the results in the small frequency range solely a Genetic Algorithm is applied here to solve the optimization problem (4.7). The following parameters were used to determine the material parameters: • • • •

A population size of 1,000 individuals is chosen The starting population is created randomly New individuals are created via 2-point-crossover The mutation of an individual is done accordingly to the standardized normal distribution, mutation probability is 10% • 150 individuals are created newly in every generation, the remaining 850 are created as offspring of the previous generation • The optimization is stopped after a maximum of 50,000 iterations or if the best ﬁtness value f < 10−5 . The results of the optimization are depicted in the following Figures. Figures 4.10 and 4.12 show the Young’s modulus and the Poisson’s ratio for the auxetic microstructure, while in Figs. 4.11 and 4.13, the Young’s modulus and the Poisson’s ratio for the non-auxetic microstructure are shown. In each diagram, the microscopic and macroscopic results are plotted versus the frequency. As it can be seen, the optimization was not able to ﬁt each single eigenfrequency of the microstructure results. However, the general tendency is preserved. In all four result plots, it can be seen that for small frequencies, the ﬁtted parameters converge into the static solution. The obtained material parameters for ν and E are given in Table 4.3 for the auxetic microstructure and in Table 4.4 for the non-auxetic material. 1e+08 microscale macroscale

E[N/m2 ]

1e+07

1e+06

100000

10000

100

10000 frequency [Hz]

Fig. 4.10 Young’s modulus for auxetic microstructure

1e+06

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1e+08 microscale macroscale

E[N/m2 ]

1e+07

1e+06

100000

10000

100

10000 frequency [Hz]

1e+06

Fig. 4.11 Young’s modulus for non-auxetic microstructure

1

microscale macroscale

n [−]

0.5

0

−0.5

−1 100

10000 frequency [Hz]

Fig. 4.12 Poisson’s ratio for auxetic microstructure

1e+06

4 Effective Properties of Non-Convex Microstructured Materials

63

1 microscale macroscale

n [−]

0.5

0

−0.5

−1 100

10000

1e+06

frequency [Hz] Fig. 4.13 Poisson’s ratio for non-auxetic microstructure Table 4.3 Obtained material parameters for auxetic material Young’s modulus k

E¯

pk

qk

αk

1 2

2.560 · 104

2.358 · 10−14

1.842 · 10−8

1.571 · 100 1.099 · 100

k

ν¯

2.432 · 10−8

8.943 · 10−9

Poisson’s ratio 1 2

−0.376

p˜k

q˜k

α˜ k

4.995 · 10−8

1.598 · 10−8

1.497 · 100 1.335 · 100

1.534 · 10−8

1.162 · 10−7

Table 4.4 Obtained material parameters for non-auxetic material Young’s modulus k

E¯

pk

qk

αk

1 2

1.581 · 105

1.057 · 10−6 1.448 · 10−12

2.666 · 10−8 1.856 · 10−8

1.057 · 100 1.389 · 100

Poisson’s ratio k

ν¯

1 2

0.307

p˜k

q˜k

α˜ k

1.869 · 10−11 5.147 · 10−9

1.407 · 10−8 8.443 · 10−10

1.424 · 100 1.573 · 100

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The slightly viscoelastic material on the microscopic scale had virtually no effect on the homogenization process. There were still eigenfrequencies on the microscale which make the search of the material parameters very difﬁcult. For the optimization, it can be observed that there is no difference whether an auxetic or non-auxetic microstructure is used as input data. This could have been expected a` priori, because the same mathematical problem (with slightly different parameters) needs to be solved by the optimization procedure.

4.6 Conclusions In this paper, a dynamical homogenization for a periodic, auxetic material consisting of simple beams has been performed. Due to inertia effects on the microscale, the behavior of the unit cell is frequency dependent. Thus, a viscoelastic constitutive law had to be applied on the macroscale, requiring an optimization procedure for the homogenization process. The SQP and the Genetic Algorithm are used for the homogenization. Both were able to ﬁnd adequate material parameters on the macroscale but the SQP needs, as expected, reliable starting values. On the contrary the Genetic Algorithm was more robust but much slower. For the identiﬁed material parameters pk , qk , and αk , however, no plausibility check was yet made concerning thermodynamic restrictions. Therefore, further investigations have to be carried out assuring the correctness of the found values, i.e. by setting thermodynamic constraints in the optimization problem.

References 1. Alderson K, Kettle A, Neale P, Pickles A, Evans K (1995) The effect of the processing parameters on the fabrication of auxetic polyethylene. Part II: The effect of sintering temperature and time. Journal of Materials Science 30:4069–4075 2. Antes H, Schanz M, Alvermann S (2004) Dynamic analyses of frames by integral equations for bars and Timoshenko beams. Journal of Sound and Vibration 276(3–5):807–836 3. Bagley R, Torvik P (1986) On the Fractional Calculus Model of Viscoelastic Behaviour. Journal of Rheology 30(1):133–155 4. Chan N, Evans K (1997) Fabrication methods for auxetic foams. Journal of Materials Science 32:5945–5953 5. Chen C, Lakes R (1989) Dynamic wave dispersion and loss properties of conventional and negative Poisson’s ratio polymeric cellular materials. Cellular Polymers 8:343–369 6. Choi J, Lakes R (1991) Design of a fastener based on negative Poisson’s ratio foam adapted. Cellular Polymers 10:205–212 7. Christensen R (1971) Theory of Viscoelasticity. Academic Press, New York 8. Dawson B (1968) Rotary inertia and shear in beam vibration treated by the Ritz method. The Aeronautical Journal 72:341–344 9. Gaul L, Klein P, Kempﬂe S (1991) Damping Description Involving Fractional Operators. Mechanical Systems and Signal Processing 5(2):81–88 10. Geiger C, Kanzow C (2002) Theorie und Numerik restringierter Optimierungsaufgaben. Springer, Berlin

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11. Gibson L, Ashby M (1988) Cellular Solids. Pergamon Press, Oxford 12. Hashin Z (1983) Analysis of composite materials – a survey. Journal of Applied Mechanics 50:481–504 13. Hohe J, Becker W (2001) An energetic homogenisation procedure for the elastic properties of general cellular sandwich cores. Computing 32:185–197 14. Hohe J, Becker W (2001) A reﬁned analysis of the effective elasticity tensor for general cellular sandwich cores. International Journal of Solids and Structures 38:3689–3717 15. Hohe J, Becker W (2002) Effective stress-strain relations for two-dimensional cellular sandwich cores: Homogenization, material models, and properties. AMR 55(1):61–87 16. Lakes R (1987) Foam structures with a negative Poisson’s ratio. Science 235:1038–1040 17. Lakes R (1987) Making negative Poisson’s ratio foam. http://silver.neep.wisc.edu/lakes/ PoissonRecipe.html 18. Lakes R (1996) Micromechanical analysis of dynamic behavior of conventional and negative Poisson’s ratio foams. Journal of Engineering Mathematics and Technology, ASME 118:285–288 19. Miehe C (2003) Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at ﬁnite strains based on the minimization of averaged incremental energy. Computational Methods in Applied Mechanical Engineering 192:559–591 20. Nemat-Nasser S, Hori M (1999) Micromechanics: Overall Properties of Heterogeneous Materials, 2nd revised ed. North-Holland, Amsterdam 21. Pickles A, Webber R, Alderson K, Neale P, Evans K (1995) The effect of the processing parameters on the fabrication of auxetic polyethylene. Part I: The effect of compaction conditions. Journal of Materials Science 30:4059–4068 22. Podlubny I (1999) Fractional Differential Equations, Mathematics in Science and Engineering, vol 198. Academic, San Diego New York London 23. Sch¨oneburg E (1996) Genetische Algorithmen und Evolutionsstrategien, 1st edn. AddisonWesley, Bonn 24. Spellucci P (1993) Numerische Verfahren der nichtlinearen Optimierung. Birkh¨auser Verlag, Basel 25. Theocaris P, Stavroulakis G (1998) The homogenization method for the study of variation of Poisson’s ratio in ﬁber composites. Archive of Applied Mechanics 68(3/4):281–295 26. Torquato S, Gibianski L, Silva M, Gibson L (1998) Effective mechanical and transport properties of cellular solids. International Journal of Mechanical Sciences 40(1):71–82 27. Zohdi T, Wriggers P (2005) Introduction to Computational Micromechanics, Lecture Notes in Applied and Computational Mechanics, vol 20. Springer Berlin Heidelberg New York

Chapter 5

Modelling of Diffusive and Massive Phase Transformation Jiˇr´ı Vala

Abstract Diffusion in multi-component alloys can be characterized (a) by the vacancy mechanism for substitutional components, (b) by the existence of sources and sinks for vacancies, and (c) by the motion of atoms of interstitial components. The description of diffusive and massive phase transformation of a multi-component system is based on the principle of maximum dissipation rate by Onsager; the ﬁnite thickness of the interface between both phases can be respected. A new computational model (its one-dimensional, in general non-stationary version is presented here) covers both (a) and (c) and is open to involve (b) in a natural way. The mathematical analysis results in an initial-value problem for a system of partial differential equations of evolution with certain non-local integral term; the unknowns are the mole fractions of particular components (and some additional variables in case (b)). The numerical construction of approximate solutions comes from the method of lines and from the ﬁnite difference and other numerical techniques, namely the numerical integration formulae and the spectral analysis of linear operators. The original software code is supported by MATLAB and partially by MAPLE.

5.1 Introduction The basic principles of diffusion controlled phase transformation have been known for a long time. In general the diffusion in a multi-component system can be characterized by the well-known relation from [11], rewritten in some (more or less general) form in most later monographs or papers, as [2, 5] or [3]. The diffusive ﬂux for every component can be evaluated as a product of the matrix of kinetic coefﬁcients and the gradient of chemical potentials of particular components, understood as Jiˇr´ı Vala ˇ zkova 17, Czech Brno University of Technology, Faculty of Civil Engineering, 602 00 Brno, Ziˇ Republic [email protected] V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

67

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functions of concentrations (or some equivalent characteristics, as mole fractions). In the more general context the diffusive ﬂux may be substituted by some “extensive quantity” and the chemical potential by some “intensive variable”, for illustration of the terminology cf. [14], p. 5, and [16], p. 1370. Onsager showed that such linear phenomenological equation (originally for a heat production in an anisotropic system) can be derived from the requirement of maximum of a functional, having a close relation to the total entropy production of the system, and formulated the “thermodynamic extremal principle” (TEP). Making simplifying assumptions, in case of binary systems it was possible to derive analytic or semi-analytic solutions, useful in the ﬁeld of material development. However, under more general assumptions or in systems including more than two components this approach turned out to be insufﬁcient. The thermodynamics of multi-component systems has its own scientiﬁc history, covering the evolution of the concept of deriving the thermodynamic functions of a system from all available experimental data, the reliable setting of kinetic parameters, etc.; much more information and references to results from the last four decades can be found in [14], p. 3. Both theoretical and experimental works yield that the diffusion in multi-component alloys can be characterized by three attributes: (a) The vacancy mechanism for “slowly” diffusing substitutional components (b) The existence of certain sources or sinks of vacancies (c) The “quick” motion of atoms of interstitial components In the following formal mathematical description of a system only the attributes (a) and (c) will be incorporated properly; the attribute (b) will be left to the concluding remarks. We also limit the discussion to isothermal and isobaric conditions. Let us notice that most papers, as [14], p. 6, or [16], p. 1373, need simplifying assumptions on the equilibrium of vacancies; only [17], p. 3046, attempt to handle the quite general case with non-ideal sources and sinks and vacancies. Another serious problem is connected with the description of the (usually very thin) interface, separating two phases during the phase transformation. Under the assumption of the sharp interface (whose thickness is assumed to be negligible) it is not easy to formulate realistic interface conditions (as boundary conditions for particular phases) [14], p. 9, introduce three types of such conditions: the so-called local equilibrium conditions with partitioning, the local equilibrium conditions with negligible partitioning and the para-equilibrium conditions. Nevertheless, it is possible to avoid all such (rather tricky) conditions, respecting the ﬁnite thickness h of the interface, as explained in [20], p. 3954, where the material behaviour in the centre of the interface corresponds to the ideal liquid phase; consequently the results for the sharp interface can be derived from those for the thick interface, using the limit passage h → 0. Several further simpliﬁcations in our considerations could be removed, but has been accepted here to generate a relatively simple mathematical formulation. Following [14], p. 6, we shall assume that the insertion of an interstitial component does not change the lattice geometry. We shall write all relations for a hypothetical one-dimensional problem, thus the interface motion in other directions will be

5 Modelling of Diffusive and Massive Phase Transformation

69

Fig. 5.1 One-dimensional computational scheme for a two-phase system

ignored. Unlike the “lattice ﬁxed reference frame by Kierkendall” by [14], p. 5 (later completed by certain “number ﬁxed reference frame”), the Cartesian coordinate system x here is ﬁxed with the moving interface (from the right to the left). Moreover, avoiding the attribute b) enables us to preserve the constant interface thickness h; therefore the interface velocity v, related to the ﬁxed reference frame, will be a function of time t ≥ 0, but not of x. Such simple system geometry is documented in Fig. 5.1: a one-dimensional, two-phase system of unit cross-sectional area with two phases α and γ , separated by the thick interface, and with certain integer numbers r ≥ 0 of substitutional components and q ≥ 0 of interstitial ones is considered; the formal third liquid phase β in the interface centre is added, too. Let us remark that most models of binary alloys are formulated for r = 2 and q = 0; cf. [1] where the effect of solute drag on the grain growth of polycrystals, taking into account migration of a grain boundary, is studied. In real conﬁgurations h can be expected to be (unlike Fig. 5.1) much smaller than both −xL and xR , comparable with the size of the whole specimen.

5.2 Gibbs Energy and its Dissipation The formulation of the Gibbs energy and its dissipation presented here is the straight generalization of the approach of [20], p. 3954, where especially the steady-state problem for substitutional alloys is studied. The local chemical composition is described by mole fractions c∗ = (c1 (x,t), . . . , cr (x,t), . . . , cr+q (x,t))T . The corresponding diffusive ﬂuxes will be denoted by j∗ = ( j1 (x,t), . . . , jr (x,t), . . . , jr+q (x,t))T .

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The symbol ∗ here emphasizes that both column vectors c∗ and j∗ with all r + q components are considered; later we shall need also their subvectors c and j with missing terms cr (x,t) or jr (x,t), respectively – cf. (5.1) and (5.2). The same convention will be used for other column vectors, too. For simplicity of notation we shall obey the Einstein summation rule for all indices k, l, m ∈ {1, . . . , r} (related to substitutional components only), a, b ∈ {r + 1, . . . , r + q} (related to interstitial components only), i, g ∈ {1, . . . , r + q} (related to both substitutional and interstitial components) and f from the set of phase identiﬁers {α , β , γ }; an underlined index prohibits summation. Moreover, δ.. means the Kronecker symbol everywhere. We assume negligible vacancy ﬂuxes in the system; this together with the vacancy mechanism leads to the constraint

δkk jk = 0.

(5.1)

The analogous constraint for mole fractions is

δkk ck = 1.

(5.2)

The system moves with the velocity v(t), being the velocity of interface migration to the left, relative to the ﬁxed coordinate system. We assume that no external diffusive ﬂuxes are present at both ends of the specimen; this can be expressed as ji (xL ) = ji (xR ) = 0;

(5.3)

consequently xL and xR are variable in t, too. The dot symbol will denote the partial derivative with respect to t, the prime symbol the partial derivative with respect to x : for any scalar quantity u(x,t) we are able to calculate its total time derivative du = u˙ − vu . dt The chemical potential µi of a component i is supposed to be a function of c∗ and t. One can see immediately that by (5.2) µi can be rearranged as a function of c and t easily. We suppose that the dependencies of the chemical potentials on c∗ are known for both phases α and γ , as well as for the interface – in its centre, referenced by β , we are allowed to assume the ideal liquid behaviour. The well-known Gibbs-Duhem relation, discussed, e.g. in [20], p. 3954, yields ci

d µif = 0, dt

ci µif = 0

(5.4)

The chemical potential µi can be calculated at any point of the specimen as

µi (x, c∗ ) = w f (x)µif (c∗ ),

(5.5)

making use of some reasonably continuous weight functions w f (x), having the properties

5 Modelling of Diffusive and Massive Phase Transformation

wα (x) = 1, wγ (x) = 0 wα (x) = 0, wγ (x) = 1 wβ (x) = 1 − wα (x) − wγ (x)

71

xL < x < h/2, h/2 < x < xR , xL < x < xR .

if if if

f

Let us notice that the formal analogy of (5.4) for µi replacing µi may be not fulﬁlled inside the interface, i.e. in case 0 < x < h. Let Ω be the constant molar volume. The total Gibbs energy G of the system is given by G=

1 Ω

ci µi dx.

If no integral bounds are present then the integration from xL to xR is considered by default. Its time derivative can be expressed as dG d µi dci 1 µ i + ci = dx. dt Ω dt dt Inserting µi into the second additive term from (5.5), we obtain

f d µ dci 1 dG f = µi + ci w f i − vci w f µi dx dt Ω dt dt and consequently (integrating by parts)

f 1 dG dci f d µi f f f f = − vci (w µi ) + vci w µi dx. µi + ci w dt Ω dt dt By the ﬁrst Eq. (5.4) the second and fourth additive terms vanish at all; thus, respecting (5.5), we receive dG 1 dci µi − vci µi dx. = dt Ω dt Since w f ∈ {0, 1} for f ∈ {α , γ } if x ≤ 0 or x ≥ h, the second additive term can be simpliﬁed according to the second Eq. (5.4) with the result dG 1 = dt Ω

dci v µi dx − dt Ω

h 0

ci µi dx.

(5.6)

The mass conservation law reads dci = −Ω ji ; dt

(5.7)

this enables us to reformulate the ﬁrst additive term of (5.6). We obtain dG =− dt

ji µi dx −

v Ω

h 0

ci µi dx.

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Integrating by parts, respecting (5.3), we come to the ﬁnal form of (5.6) dG = dt

ji µi dx −

v Ω

h 0

ci µi dx.

(5.8)

We have to evaluate also the rate Q of dissipation of the total Gibbs energy. Following [20], p. 3955, based on the more extensive analysis of [19], we have Q=

v2 ji2 dx + Ai M

(5.9)

where

ci Di ; (5.10) ΩRT in this expression Di is the tracer diffusion coefﬁcient, R is the gas konstant (8.3143 kmol−1 K−1 ), T is the absolute temperature and M is the interface mobility. Ai =

5.3 Thermodynamic Extremal Principle We shall use the principle of maximum dissipation rate, as it was ﬁrst applied by [13] for a diffusional process. The kinetics of our system corresponds to the variation dG Q (j∗ , v) = 0 + δ dt 2 by (5.8) and (5.9) with respect to the constraint for substitutional components (5.1). Introducing one Lagrange multiplier λ , we can rewrite this equation as dG Q + + λ δkk jk (j∗ , v, λ ) = 0 δ dt 2 without any additional constraints. Performing the variation, step by step, for j1 , . . . , jr , . . . , jr+q , v and λ , we obtain

˜ ˜jk µk + jk jk + j˜k λ dx = 0 Ak

for every j˜k , the same with a instead of k and with the missing last additive term for every j˜a , v˜ h vv ˜ − ci µi dx + = 0 Ω 0 M for every v˜ and formally also (5.1), multiplied by λ˜ , for every λ˜ .

5 Modelling of Diffusive and Massive Phase Transformation

We have derived

µk +

73

jk +λ = 0 Ak

(5.11)

ja =0 Aa

(5.12)

for all substitutional components,

µa + for all interstitial components and v=

Ω M

h 0

ci µi dx

(5.13)

for the interface velocity. From (5.12) the ﬂuxes ja can be expressed directly as ja = −Aa µa . To get the remaining ﬂuxes from (5.11), we need to evaluate λ . Multiplication of (5.11) by Ak and summation of resulting equations gives

δkk Ak λ = −Ak µk − δkk jk . By (5.1) the last additive term is equal to zero, thus

λ =−

Ak µk . δll Al

This enables us to write in general ji = −Ai

Ak µk µi − χ i δll Al

(5.14)

where χi is equal to 1 for i ≤ r, 0 otherwise. Let us introduce the new quantity Ci (x) as the integral of the partial time derivative of ci from xL to any x. Every upper index L will refer to a value in x = xL , every upper index R to a value in x = xR everywhere. Then (5.7) gets the simple integral form Ci − v(ci − cLi ) = −Ω ji . (5.15) cf. [12], p. 1039, where the ﬁrst additive term is neglected. In particular for x = xR we have CiR = v(cRi − cLi )

(5.16)

(and for x = xL formally CiL = 0). For practical calculations we need to express ji by means of (5.14), (5.10) and (5.5); for this purpose it is useful to introduce the decomposition µif (c∗ ) = µ0if + RT ln ci + ϕif (c∗ ) (5.17)

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f

where µ0i are constants for a given temperature T and ϕi are certain functions of f c∗ (usually not dominant, but non-negligible and rather complicated): ϕk depend on f c1 , . . . , cr , ϕa on cr+1 , . . . , cr+q . More detailed thermodynamic analysis is contained in [8]: in the formulae of type (5.17) the ﬁrst right-hand-side additive term in the second relation is assigned to the Gibbs energy of a mechanical mixture of all constituents of the phase, the second one corresponds to the entropy of mixing and the last one (called excess term) includes other physical processes. Differentiating of (5.5) gives µi = w f µif + w f µif , differentiating of (5.17) similarly

µif =

RT f c + ϕi,g cg ci i

where any lower index.,g means the derivative by cg , thus RT f f f f f µi = w µ0i + RT ln ci + ϕi + w c + ϕi,g cg . ci i We know that δ f f w f = 1 and consequently also δ f f w f = 0, therefore RT f µi = w f µ0if + ϕif + c + w f ϕi,g cg . ci i The notation

µˆ i = w f µ0if + ϕif ,

f ϕˆ ig = w f ϕi,g

converts this result into its brief form

µi = µˆ i +

RT c + ϕˆ ig cg . ci i

(5.18)

By means of Di from (5.10) the diffusive ﬂuxes from (5.14), corresponding to both substitutional and interstitial components, can be expressed as c i Di ck Dk µk ji = − µi − χ i , ΩRT c l Dl i.e. for interstitial ones ja = − and for substitutional ones

ca Da µa ΩRT

ck Dk cl Dl µk − µl · jk = − . ΩRT cm Dm

5 Modelling of Diffusive and Massive Phase Transformation

75

The additional notation

Di , η = ζm cm D with some (non-zero) reference value D of the tracer diffusion coefﬁcient enables us to simplify this result into the form

ζi =

ja = − or jk = − in general ji = −

ca ζa D µ , ΩRT a

ck ζk D cl ζl · µk − µl ; ΩRT η

ci ζi D cl ζl · µi − χi µl . ΩRT η

Dividing this equation by −ζi D, according to (5.18) we have ci ji RT = µˆ i + ci + ϕˆ im c m − ζi D ΩRT ci ci RT ζl cl ζl cl ζc χi l l µˆ i + cl + ϕˆ lm c m ; − ΩRT η ci η η using the new notation

µk = µˆ k − extended formally by

ζl cl µˆ , η l

µ¯ a = µˆ a ,

ϕ¯ kl = ϕˆ kl −

ζm cm ϕˆ ml , η

ϕ¯ ab = ϕˆ ab ,

we get (multiplying by Ω) ﬁnally ci ζl ci µ¯ i ci ϕ¯ il Ω ji = c i − χi + cl . − cl + ζi D η RT RT The application of (5.15) in form −

Ci Ω ji v v L − + ci = c ζi D ζi D ζi D ζi D i

now results −

Ci v v L ζl ϕ¯ µ¯ i ci = + c i + ci − χi + il c l + + c ; ζi D η RT RT ζi D ζi D i

(5.19)

Now it is useful to introduce sum indices n ∈ {1, . . . , r − 1}. For substitutional components let us notice that (5.2) implies c r = −δnn c n

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and also

η = ζn cn + ζr (1 − δnncn ) = ζr + (ζn − ζr ) cn without any cr ; this offers a chance to eliminate cr at all. Consequently (5.19) with k < r can be rewritten as Ck ζl ϕ¯ ζr ϕ¯ + c k + ck − + kn c n − ck − + kr δnn c n − ζk D η RT η RT v v L µ¯ k + ck = c . + RT ζk D ζk D k Then the notation Bkn = δkn +

ck (ζr − ζn ) ϕ¯ kn − ϕ¯ kr , + η RT

Kkk =

µ¯ k , RT

Nkk =

1 ζk D

simpliﬁes this result to −NkkCk + Bkl c l + (Kkk + vNkk ) ck = vNkk cLk .

(5.20)

For interstitial components (5.19) degenerates to Ca v L ϕˆ ab µˆ a v − + ca + c a cb + + ca = c . ζa D RT RT ζa D ζa D a The similar notation Bab = δab +

ϕˆ ab , RT

Kaa =

µˆ a , RT

Naa =

1 ζa D

simpliﬁes this result to −NaaCa + Bab cb + (Kaa + vNaa) ca = vNaa cLa .

(5.21)

Let us notice that (5.20) and (5.21) have the same formal structure: they can be rewritten in the common matrix form −NC + Bc + (K + vN) c = vNcL .

(5.22)

Here K and N are two square diagonal matrices of order r − 1 + q, B is a square (in general full) matrix of the same order; their elements undeﬁned by (5.20) and (5.21) are set to zero. The ﬁrst rough estimate B (by the ﬁrst additive term in both deﬁnitions of its elements) could be a unit matrix; B and K are rather complicated functions of c and x, N depends on x only. Let us also remind (5.16), whose vector form is (5.23) CR = v cR − cL , and (5.13), whose vector form, using a scalar product and one new vector y = (µ0 , . . . , µr )T ,

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77

containing functions of c, is Ω v= M

h 0

c · y dx.

(5.24)

5.4 Mathematical Problems and Computational Algorithms Let us recall that our problem is to ﬁnd in any time t ≥ 0 the unknown mole fractions c (and auxiliary variables C) from (5.22) and (5.23) where the interface velocity v comes from (5.24); the initial values of c in time t = 0 are prescribed. From the mathematical point of view, even in the one-dimensional simpliﬁcation we have a system of r − 1 + q partial differential equations of evolution with the same number of unknown functions, located in the vector c. The coefﬁcients of these equations are strongly nonlinear; consequently their practical evaluation (especially in case of B, but often also of K) needs complicated MAPLE-supported symbolic calculations. Moreover, a non-local multiplicative term is present in the problem, thanks to the evaluation of v by means of certain integral containing c from 0 to h. Unfortunately, the analysis of solvability of such problem is far from trivial. Only for very special conﬁgurations the existence and uniqueness of solution has been veriﬁed properly, namely in case r = 2 and q = 0; in particular [10] guaranteed a smooth solution for the diffusion induced grain boundary motion (DIGM) in a polycrystalline material. Nevertheless, the proper mathematical analysis of the general integro-differential problem contains still a lot of open questions. Other difﬁculties are connected with the methodology of measurements and with the reliable identiﬁcation of (rather uncertain) material characteristics, included in B, K and N, as well as in the absence of a reliable mathematical homogenization technique, bridging the microstructural phenomena and the macroscopic behaviour of material. The analytic solution of our problem (even in simpliﬁed cases, because of its integro-differential character) is unknown, moreover no available software package for the numerical analysis of partial differential and integro-differential problems can solve the above formulated system of equations, thus some computational algorithm (not expensive if possible) for the construction of sequences of approximate solutions of the problem is needed. Here we shall only sketch the principal ideas of an original numerical approach, implemented in MATLAB, including standard functions and methods described in [15] and also functions from the MATLAB toolbox symbolic, referring to the core of MAPLE; no other software means are needed. Since in general case most formulae are rather complicated (some of them are just products of MAPLE supported symbolic manipulations), we shall start with intuitive simple schemes, based on the Crank-Nicholson approach, and then we shall mention (without explaining all technical details) necessary and useful generalizations. Let us consider a uniform net, covering the whole moving interval from xL to xR ; the nodes are numbered by integer indices s from 0 to σ , for simplicity xL coincides

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with x0 and xR coincides with xσ . Let cs = c(xs ) denote the value of c for x = xs ; here xs = x0 + s∆ (∆ is some positive length) with s chosen from the set {1, 2, . . . , σ }. For any s > 0 and Z ∈ {B, K, N} we can deﬁne Zs =

1 (Z(cs ) + Z(cs −1 )) . 2

Using the method of lines, in each time t we are able to write the semi-discrete version of (5.22) Bs (cs − cs −1 ) +

∆ (Ks + vNs ) (cs − cs −1 ) = ∆vNs c0 + ∆Cs , 2

(5.25)

obtained by the inaccurate integration of (5.22) from xs −1 to xs , and that of (5.23) Cσ = v (cσ − c0 ) .

(5.26)

Let τ be the constant time step. The rectangular rule enables us to calculate Cs =

∆ × × × c0 − c × 0 + 2(c1 − c1 ) + · · · + 2(cσ −1 − cσ −1 ) + cσ − cσ 2τ

(5.27)

where all symbols c× . refer to values from the previous time step. The similar approach applied to (5.24) gives v=

Ω∆ (cθ · yθ + 2cθ +1 · yθ +1 + · · · + 2cω −1 · yω −1 + cω · yω ) 2M

(5.28)

where θ and ω are integers, 0 < θ < ω < σ , s = θ corresponds (just in the considered time step) to x = 0 and s = ω to x = h. The above sketched numerical analysis makes it possible to suggest the following algorithm, how to evaluate all mole fractions c in each time step: 1. Set all mole fractions c× s by their initial values (in the ﬁrst time step t = τ ) or by values obtained in time t − τ , then estimate cs as c× s , repeat the same for Cs by (5.27) (with zero initial values in the ﬁrst time step if no better information is available) 2. Calculate v from (5.28) 3. Set all Bs , Ks and Ns by cs 4. Set c0 as the vector of r − 1 + q unknown parameters (in the program code this can be simulated using the complex arithmetics) 5. Solve (step by step) each cs with positive s from (5.25) (this is always a linear algebraic system of only r − 1 + q equations with r − 1 + q unknowns) 6. Determine c0 from (5.26), then recalculate all cs 7. Applying some error indicator (incorporating changes in values of v and all cs and Cs ), decide to stop iterations, otherwise return to step 2

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Let us remark that the (seemingly not reasonable) limit case τ → ∞ attempts to ﬁnd the steady-state solution directly (with zero Cs ); however, this works only in special cases, namely for c0 = cσ . This algorithm is rather simple and not expensive. Unfortunately, some other complications occur in practical calculations which can be more evident from the following groups of arguments, forcing substantial generalizations and new software development: 1. The uniform mesh is not realistic because the interface thickness is much less than the specimen size. However, the uniform mesh is acceptable inside the interface; outside the interface the distance between adjacent nodes xs can increase. Moreover, both locations xL and xR must be updated in each time step, thus they are not identical with some nodes in general. Consequently some additional interpolation near xL and xR is needed. 2. The quality of the discretization scheme, presented in (5.25), etc., could be not sufﬁcient. Some possible improvements are intuitive and simple, e.g. the replacement of the rectangular rule by the Simpson one in (5.28), but their effect is not crucial. In practical calculations both in second derivatives of mole fractions c by x and in their mixed derivatives by x and t unpleasant oscillation occur, suppressed only artiﬁcially, by non-physical numerical tricks. Therefore some improvements have been tested: (i) Bs , Ks and Ns constructed as linear splines instead of piecewise constant simple functions (ii) cs constructed as cubic Hermite splines instead of linear ones, with derivatives by x coming from a preceding iteration, using the numerical differentiation technique, similarly also C, obtained from numerical integration (iii) Alternatively to (ii) c constructed seemingly exactly (with piecewise ﬁxed B, K and N) from (5.22) instead of (5.25) For illustration, let us analyze the last improvement in more details. Another (partial) discretization of (5.22) is Bs c s + Ls cs = Fs where Ls = Ks + vNs and Fs = N(vc0 + Cs ). Using the standard spectral analysis of linear operators together with the classical calculus of linear systems of differential equations, we obtain cs = Vs exp (Qs (xs−1 − xs )) V−1 cs−1 − L−1 s s Fs where Vs is the matrix of eigenvectors of the matrix B−1 s Ls and Qs is the matrix of its eigenvalues (exp(.) here is understood as a diagonal matrix, too). This can be useful even in special cases: e.g. for zero Ks (which is true everywhere outside the interface because each µˆ i = 0 and consequently each µ¯ i = 0) and zero Cs (which corresponds to the limit steady-state distribution of c with vanishing time derivatives) we have only L−1 s Fs = c0 .

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3. The rather complicated non-local integro-differential character of the problem does not admit the application of any suitable homogenization technique, as described in [9]. Moreover, the identiﬁcation of material characteristics, included in B, K and N, is difﬁcult; consequently it is not clear how to formulate and study the inverse problems correctly – the mathematical results offered by [6] are insufﬁcient. Most material characteristics can be classiﬁed as semi-empirical ones, based both on some physical considerations and on the extensive experimental study, unfortunately not covering all mole fractions of particular components between 0 and 1. Moreover, no physical barrier is incorporated into the system (5.22), (5.23) and (5.24) to prevent negative or other non-realistic mole fractions. For illustration, a function for evaluation of chemical potentials by (5.17) and their needed derivatives, contains typically several hundreds of lines of the source program code in MATLAB, generated automatically by the original MAPLE-based pre-processing software.

5.5 Illustrative Example The crucial point for numerical simulations is to set reasonable material characteristics. In the following example we shall present the three-component Fe-Cr-Ni system, analyzed in more details in [20], p. 3956. In our notation r = 3 and q = 0, moreover Fe will be dominant, thus the interface mobility (whose proposed values differ dramatically in various research sources) can be derived from that for the pure iron. We have only substitutional components here. The tracer diffusion coefﬁcients can be interpolated using the formula ln Dk = w f ln Dkf , thus it is sufﬁcient to set nine values Dkf . In general we have f Dk

=

Ef E , M = M0 exp − . − RT RT

f D k0 exp

The constants Dk0 (in m2 s−2 ) are for Cr (corresponding to k = 1), Ni (k = 2) and Fe (k = 3) β γ D1 = 0.00022, D1 = 0.00035, Dα1 = 0.00032, f

β

γ

Dα2 = 0.000048,

D2 = 0.000022,

D2 = 0.000035,

Dα3 = 0.00016,

D3 = 0.00011,

D3 = 0.00007;

β

γ

the constants E f and E (in J mol−1 ) are E α = 240,000, E β = 155,000, E γ = 286,000, E = 140,000. −1

It remains to set M0 = 0.00000041 (in m2 s kg ).

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81

Our calculations were performed for T = 1,030 (in K); one exception will be f f mentioned. The complicated MATLAB functions for evaluation of µ0k and ϕk were generated automatically, making use of the experimental results from the Montan University in Leoben (Austria). The initial mole fractions (for t = 0) c1 = 0.001 and c2 = 0.019 (consequently c3 = 0.980) were set everywhere; the interface thickness was h = 5 ·10−10 (in m). The following ﬁgures demonstrate the distributions of mole fractions for Cr and Ni at the interface; outside the interface they are much less interesting, being similar to exponential curves (exactly for the unit matrix instead of B). Figure 5.2 shows the development of both mole fractions from the initial status (solid line) through various distributions in time (dashed lines) to the ﬁnal status (solid line), with no visible changes in time greater then 1 s. All remaining ﬁgures correspond to such ﬁnal status; they may justify the great dispersion of published results in the literature. We know that the values of most material characteristics are more qualiﬁed estimates than exact deterministic numbers. Figure 5.3 demonstrates what happens in case that E γ is multiplied by 1.1 (dashed line) or 0.9 (dash-dotted line): one can see immediately that the difference of mole fractions at both ends of the interface, i.e. c(h) − c(0), is very sensitive to small changes in diffusion characteristics. Figure 5.4 repeats the same analysis for small changes of interface mobility, namely for E multiplied by 1.15 (dashed line) or 0.85 (dash-dotted line). Similar global tendencies in the redistribution of mole fractions as on Figs. 5.3 and 5.4 can be seen also on Fig. 5.5 where alternatively T = 1,010 and T = 1,050 are prescribed. The increasing temperature forces the substantial decrease of interface velocity v (in ms−1 ): for T = 1,010 we obtained v = 3.58563 · 10−7 , for T = 1,030 still v = 2.77768 · 10−7, whereas for T = 1,050 only v = 7.74894 · 10−8. Figure 5.6

Fig. 5.2 Development of mole fractions for Cr and Ni in time

82

Fig. 5.3 Mole fractions for varying diffusion properties

Fig. 5.4 Mole fractions for varying interface mobility

J. Vala

5 Modelling of Diffusive and Massive Phase Transformation

Fig. 5.5 Mole fractions for varying temperature

Fig. 5.6 Chemical potentials of all components

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shows all chemical potentials for the original setting of material characteristics: the ﬁrst additive terms in (5.17) (dashed lines), the sum of the ﬁrst and second additive terms in (5.17), i.e. the potentials with neglected complicated non-linear corrections (dash-dotted lines), and ﬁnally the total potentials by (5.17) (solid lines).

5.6 Conclusions and Generalizations The illustrative numerical results presented above are clearly not sufﬁcient to validate the model and to verify the convergence of computational algorithms, implemented in the MATLAB environment. Both in the experimental analysis (methodology of laboratory measurements, identiﬁcation problems, etc.) and in the theoretical one (existence and uniqueness of solutions in some reasonable sense, quality of convergence of numerical algorithms, etc.), there is a lot of open questions yet. Moreover, the simple way of extension of the results from the micro-scale simulations to the macroscopic analysis is not available. Nevertheless, the created software has been tested for more classes of problems of practical importance. In [20] the stationary solver was applied to the Fe-rich Fe-Cr-Ni substitutional system with various types of chemical potentials and values of material characteristics, which may be rather uncertain in practice, namely in case of the interface mobility and thickness; further numerical simulations has been done also for the similar system with the interstitial C-component and for the binary Al-Mg system yet. For every ﬁxed interface thickness h the numerical simulations show that the interface velocity v decreases with the increasing temperature T ; ﬁnally the phase transformation stops at certain critical temperature. This critical temperature increases with the increasing interface thickness h; the limit case h → 0 returns the (less realistic) results for an idealized sharp interface. The simulation of the massive γ → α transformation shows that the existence of the solute drag in the interface inﬂuences the contact conditions at the interface allowing the massive transformation to occur also in the two-phase region. By choosing α and γ as identical phases and by imposing ﬂuxes to the interface (grain boundary), DIGM was simulated. The interface and grain boundary Gibbs energy were calculated; their realistic values support the responsibility of the model. The MATLAB-supported non-stationary simulations are now performed in the collaboration of researchers from the Institute of Physics of Materials of the Czech Academy of Sciences in Brno (J. Svoboda), from the Brno University of Technology (J. Vala) and from the Montan University of Leoben (F.-D. Fischer, E. Gamsj¨ager). Further generalizations need to incorporate the elastic and creep strain and consequently the shrinking and swelling rate due to the attribute b) in the Introduction; the theoretical background, referring to [4], has been prepared carefully in [17]. Another important research direction is to admit more complicated thermal processes. This leads (from the point of view of the Onsager’s relation) to the coupling of various ﬂuxes, namely the particle ﬂux due to a temperature gradient (Soret effect) and the

5 Modelling of Diffusive and Massive Phase Transformation

85

transport of heat due to a concentration gradient (Dufour effect); more information is contained in [18]. Another direction of possible generalization leads to the two- or even threedimensional models of phase transformation. The one-dimensional formulation covers a wide class of diffusive phase transformations where the interface is planar or nearly planar and has a negligible or constant (a priori known) thickness. If this is not true some more-dimensional formulation is needed. Seemingly it should be not difﬁcult: all equations of diffusion can be rewritten, following [7], in “grad” and “div” notation (replacing the derivatives with respect to x), the simple ﬁnite difference approach can be improved by ﬁnite element or ﬁnite volume techniques, etc. However, the deeper physical analysis shows that some other phenomena, not discusses in this chapter, must be taken into account: namely an additional driving force due to the curvature of the interface and the diffusion along (not only perpendicular to) the interface. The interface migration is moreover allowed to be non-uniform and the assumption on the constant interface thickness may be discreditable. Consequently this type of generalization is rather complicated and cannot be done without additional both theoretical and experimental preliminary research. Acknowledgements The ﬁnancial support from the Grant Agency of the Academy of Sciences of the Czech Republic, Reg. No. IAA200410601, is gratefully acknowledged.

References 1. Fan D, Chen SP, Chen LQ (1999) Computer simulation of grain growth kinetics with solute drag. Journal of Material Research 14: 1113–1123 2. Callen HB (1960) Thermodynamics. Wiley, New York 3. Glicksman ME (2000) Diffusion in solids. Wiley, New York 4. Grinfeld MA (1991) Thermodynamic methods in the theory of heterogenous systems. Longman, New York 5. Groot SR, Mazur P (1962) Non-equilibrium thermodynamics. North-Holland, Amsterdam 6. Isakov V (2006) Inverse problems for partial differential equations. Springer, Berlin 7. Jacot A, Rappaz M (1999) A two-dimensional diffusion model for the prediction of phase transformation: application to austenitization and homogenization of hypoeutectoid Fe-C steels. Acta Materialia 45: 575–585 8. Kattner UR (1997) Thermodynamic modelling of multicomponent phase equilibria. JOM 49: 14–19 9. Marchenko V, Khruslov EYa (2006) Homogenization of partial differential equations. Birkh¨auser, Boston, MA 10. Mayer UF, Simonett G (1999) Classical solutions for diffusion-induced grain boundary motion. Journal of Mathematical Analysis and Applications 234: 660–674 11. Onsager L (1931) Reciprocal relations in irreversible processes. Physical Review 37: 405–426, and 38: 2265–2279 12. Odqvist J, Sundman B, Agren J (2003) A general method for calculating deviation from local equilibrium at phase interfaces. Acta Materialia 51: 1035–1043 13. Onsager L (1945) Theories and problems of liquid diffusion. Annals of the New York Academy of Sciences 46: 241–265

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14. Schneider A, Inden G (2004) Fundamentals and basic methods for microstructure simulation above the atomic scale. In: Raabe D, Roters F, Barlat F, Chen LQ (eds) Continuum Scale Simulation of Engineering Materials. Wiley-VCH, New York, Part I, 36 pp 15. Stanoyevitch A (2005) Introduction to numerical ordinary and partial differential equations using MATLAB. Wiley, New York 16. Svoboda J, Fischer FD, Fratzl P (2002) Diffusion in multi-component alloys with no or dense sources and sinks for vacancies. Acta Materialia 50: 1369–1381 17. Svoboda J, Fischer FD, Fratzl P (20061 ) Diffusion and creep in multi-component alloys with non-ideal sources and sinks for vacancies. Acta Materialia 54: 3043–3053 18. Svoboda J, Fischer FD, Vala J (2007) Thermodynamic extremal principle and its application to Dufour and Sorret effects and plasticity. Atti della Accademia Peloritana dei Pericolanti, to appear, 12 pp 19. Svoboda J, Gamsj¨ager E, Fischer FD, Fratzl P (2004) Application of the thermodynamic extremal principle to the diffusional phase transformation. Acta Materialia 52: 959–967 20. Svoboda J, Vala J, Gamsj¨ager E, Fischer FD (2006) A thick-interface model for diffusive and massive phase transformation in substitutional alloys. Acta Materialia 54: 3953–3960

Chapter 6

The Use of Finite Elements for Approximation of Field Variables on Local Sub-Domains in a Mesh-Free Way V. Sladek, J. Sladek, and Ch. Zhang

Abstract The paper deals with the numerical implementations of local integral equation formulation for the solution of two-dimensional (2-d) problems in linear elastic media with continuously variable Young’s modulus. Two kinds of the ﬁnite element based interpolation are developed for approximation of ﬁeld variables on local sub-domains around nodal points. One of these approximations can be classiﬁed as meshless, since the elements are generated automatically from the predeﬁned nodal points on the analyzed domain. The other one utilizes the predeﬁned mesh of elements. Besides the element based interpolations we present also the meshless Point Interpolation Method. The accuracy, convergence, numerical stability and efﬁciency of the proposed techniques are tested by numerical examples with using the exact benchmark solutions.

6.1 Introducion A rapid progress can be observed in the development of various meshless techniques especially in ﬂuid problems. Simultaneously, a considerable expansion of such techniques can be found also in various applications to engineering and science problems. This can be explained by the fact that there are known certain limitations of standard discretization techniques especially when applied to some classes of problems (e.g. problems in separable media, problems with free or moving boundaries; crack problems; problems with large distortions, degenerate elements, etc.). The question about using either mesh or not mesh has been discussed recently in the paper [1], but it seems that this question can not be answered deﬁnitely because V. Sladek and J. Sladek Institute of Construction and Architecture, Slovak Academy of Sciences, 845 03 Bratislava, Slovakia Ch. Zhang Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

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of advantages and disadvantages on both the sides. Even if the standard discretization techniques are applicable to the numerical solution of complex boundary value problems with including more or less modiﬁcations, it is still reasonable to develop new formulations by proper modelling of essential physical properties and advanced approximations of unknown physical ﬁelds in a consistent way. Simultaneously, the modelling as well as approximations should be as simple as possible without violating previous assumptions. For instance, in continuously non-homogeneous elastic media, the formulations developed for homogeneous media are not applicable directly, since the governing equations are now given by partial differential equations with variable coefﬁcients. There is not a unique classiﬁcation of meshless techniques up to now. Mostly they are classiﬁed according to the employed approximation. Some of the techniques utilize meshless approximation of ﬁeld variables but a background mesh is still required for numerical integration especially in approaches based on global formulations. On the other hand, the local formulations bring a possibility to avoid the mesh completely with using nodes alone for approximation. Then, the physical principles can be formulated in integral forms on local sub-domains. A large group of meshless techniques are denoted as meshless local Petrov-Galerkin methods [2, 3] with bearing in mind that the Petrov-Galerkin weak form idea is applied in a local sense with selecting the trial and test functions independently and approximating the ﬁeld variables in a meshless way. The strong formulations of the problems of mathematical physics and engineering sciences are given by partial differential equations (PDE). Thus, the strong formulations have a local character in contrast to the global character of the standard FEM formulations. Certain globalization is requisite by introducing a ﬁnite number of degrees of freedom via discretization, in order to keep an interaction among the discrete degrees of freedom. In present paper, we shall use local integral equations (LIE) considered on certain sub-domains around each nodal point, with nodal values of the ﬁeld variable being the degrees of freedom. Note that the governing PDE are derived from the integral form of the balance equations valid on an arbitrary small sub-domain. Hence, the consideration of the LIEs on ﬁnite size local sub-domains brings a proper physical interaction among the discrete degrees of freedom. It is well known that the polynomial interpolation used for approximation of ﬁeld variables within ﬁnite elements is reasonable and useful because of convergence of numerical results with increasing the density of nodes. The most of the objections against the mesh based methods consists in the criticism of the complexity of mesh generation including re-meshing (meshes must be conforming), degenerate elements (elements must have reasonable shape) and difﬁculties resulting from respect of boundaries. In this paper, instead of a permanent global mesh of ﬁnite elements, we shall use only nodes on the analyzed 2-d domain and triangular ﬁnite elements of appropriate shape will be generated automatically around particular nodes independently with simultaneous creation of sub-domains formed from these triangles. Since the dimensionality of employed elements is the same as that of the analyzed domain, the gradients of the ﬁeld variable can be approximated as

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gradients of the approximated ﬁeld variable. Thus, both the essential and natural boundary conditions can be involved by collocation at adequate boundary nodes. The present method is applied to two-dimensional problems of linear elasticity. In numerical tests for problems with existing exact solutions, we compare the accuracy and efﬁciency of the LIE formulations implemented by proposed local ﬁnite element meshes and other meshless approximations.

6.2 Governing Equations The elastic processes described within the theory of elasticity are governed by force equilibrium which in the case of static loading is given as

σ ji, j (x) + Xi(x) = 0, in Ω,

(6.1)

where σij is the stress tensor due to cohesive forces as a response to external loadings and Xi (x) are volume forces. The relationship between the inherent forces and deformations of the linear elastic continuum is given by the generalized Hooke’s law

σij (x) = ci jkl (x)uk,l (x),

(6.2)

where uk (x) are Cartesian components of the displacement ﬁeld and the tensor of material coefﬁcients ci jkl is position-dependent in non-homogeneous media. In case of isotropic media, it is deﬁned by only two coefﬁcients, e.g. Young’s modulus E(x) and Poisson’s ratio ν which is usually taken as constant. Then, the constitutive law can be written as

coijkl

ci jkl (x) = E(x)coijkl , 1 2ν¯ δij δkl , = δik δ jl + δil δ jk + 2(1 + ν ) 1 − 2ν¯

where the new parameter ν¯ is deﬁned by the Poisson ratio as ν /(1 + ν ), for plane stress conditions ν¯ = . ν, otherwise

(6.3)

(6.4)

Einstein’s summation convention rule is assumed, δij stands for the Kronecker delta and the subscript following a comma denotes the partial derivative with respect to the corresponding Cartesian coordinate. Inserting (6.3) and (6.2) into (6.1), we obtain the governing PDE for the displacements E(x)coijkl uk,lj (x) + E, j (x)coijkl uk,l (x) = −Xi (x), (6.5) The standard boundary conditions prescribe either the displacements or the tractions on the Dirichlet and Neumann part of the boundary, respectively

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ui (η ) = u˜i (η ) at η ∈ ∂ ΩD , n j (η )ci jkl (η )uk,l (η ) = t˜i (η ) at η ∈ ∂ ΩN .

(6.6)

The fundamental solution for the governing equation is deﬁned as the solution of Eq. (6.5) for point forces Xi (x) = δim δ (x − y), (m = 1, . . . , d) in an inﬁnite space with the dimensionality d. Apparently, the fundamental solution of the PDE with variable coefﬁcients is not available in closed form in the case of general material non-homogeneity. Consequently, the pure boundary formulation based on integral equations is not available. It would appear each numerical solution of BVP in elastic media with arbitrary non-homogeneity should utilize both the boundary and interior degrees of freedom (d.o.f.).

6.3 Domain-Type Approximation of Displacements Let the displacements ui (x) be approximated within a sub-domain Ωs ⊂ (Ω ∪ ∂ Ω). If both Ωs and Ω have the same dimension, then we call the approximation as a domain-type approximation. One of the possibilities to achieve a domain-type approximation is to use standard ﬁnite size domain elements like in FEM.

6.3.1 Finite Element Approximation In standard FEM formulations, a 2-d plane domain Ω is assumed to be subdivided into m conforming serendipity elements Se [4] with polynomial interpolation for the approximation of both the geometry and displacements. Then, m

Ω = ∪ Se , e=1

ui (x)|Se =

xi |Se =

n

∑ xaei N a (ξ1 , ξ2 ),

a=1

n

∑ ui (xae )N a (ξ1 , ξ2 ),

(6.7)

a=1

where xae i are the Cartesian coordinates of the ath nodal point on Se and the interpolation polynomials N a represent the shape functions. Since in local formulations, the shape of local sub-domains is arbitrary, they can be formed as union of elements adjacent to the considered node. The ﬁnite elements can be created either by pre-discretization of the global domain like in FEM formulations or automatically by using an algorithm and predeﬁned mesh of nodes inside and on the boundary of the global domain. In this paper, we shall apply both these approaches. The latter one will be utilized in automatic creation of linear triangular elements around a considered node (linear interpolation, n = 3). The linear triangular elements are created from the nearest nodes to the considered collocation node xc in such a way that their union results in a local sub-domain Ωc xc (Fig. 6.1).

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Fig. 6.1 Local sub-domains around interior and boundary nodes xc and xb , respectively

Then, the approximation on the local sub-domain Ωc is given by linear interpolations within particular elements forming Ωc . Having known the approximation for displacements (6.7), one can express also the gradients of displacements in terms of the same nodal values [5] as n a ∂ ui (x) ae ∂ N (ξ1 , ξ2 ) = (he )−1 , j p ∑ ui (x ) ∂ x j Se ∂ ξp a=1 (he )−1 jp =

ε3 jl ε3pk hekl (ξ1 , ξ2 ) , ε3mn he1m (ξ1 , ξ2 )he2n (ξ1 , ξ2 )

(6.8)

with (he )−1 being the inverse matrix to the matrix he whose matrix elements are given as n ∂ N a (ξ1 , ξ2 ) heα m (ξ1 , ξ2 ) = ∑ xae , (6.9) m ∂ ξα a=1 which are the Cartesian components (m = 1, 2) of two non-collinear vectors heα (ξ1 , ξ2 ) deﬁned for α = 1, 2 on each domain element. Furthermore, εki j is the permutation tensor, ε312 = −ε321 = 1, ε311 = ε322 = 0. Although the automatic generation of local sub-domains around collocation nodes by creation of linear triangular elements is rather simple, one cannot say it in the case of quadratic and higher order interpolation elements. On the other hand, the use of nonlinear elements is desired for efﬁciency of computations under guarantee of reasonable accuracy. In order to compare the accuracy and efﬁciency of computations we shall use also quadratic triangular elements for approximation of displacements. However, instead of automatic generation of quadratic elements, we shall pre-discretize the global analyzed domain into such elements. Then, the sub-domain around a collocation node is formed as a union of elements adjacent to this node. Figure 6.2 illustrates the creation of local sub-domains around collocation nodes in case uniform distribution of nodes within the square domain by using either quadratic or linear triangular elements. The main difference between these two approaches consists in fact that the quadratic elements are taken from the predeﬁned mesh of elements while the linear elements are generated automatically from the mesh of nodes. In order to distinguish these two approaches, we shall refer them as Mesh-Based Finite Element Interpolation (MBFEI) and Mesh-Free Finite Element Interpolation (MFFEI), respectively.

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Fig. 6.2 Creation of local sub-domains around collocation nodes by using: (a) uniform mesh of quadratic triangular elements; (b) uniform distribution of nodes and automatic generation of linear triangular elements

Recall that in all meshless approximation techniques, shape functions have to be deﬁned for the approximation of the ﬁeld variable ui (x) within a sub-domain Ωs ⊂ (Ω ∪ ∂ Ω) using only nodes scattered arbitrarily in the analyzed domain without any predeﬁned mesh to provide a connectivity of the nodes. According to this classiﬁcation of approximation techniques, both the presented approaches for interpolation of displacements are to be considered as element based even if in MFFEI we use only the mesh of nodes and elements are formed automatically during the creation of local sub-domains around each node. There is no correlation among elements utilized for creation of local sub-domains adjacent to different nodes. Since only the mesh of nodes is to be predeﬁned in the MFFEI approach, it resembles closely the other meshless techniques. The shape functions utilized in the MFFEI are the Lagrange interpolation polynomials (although deﬁned in the intrinsic coordinate space of each element) and so they are much simpler than those used in other meshless approximations. Finally, using the elemental interpolation (MBFEI and/or MFFEI) one reduces the problem of ﬁeld unknowns to a ﬁnite number of unknowns which are represented by nodal displacements. Another kind of meshless approximation will be presented in the next sub-section.

6.3.2 Point Interpolation Method (PIM) q

Assuming a ﬁnite series representation of the ﬁeld variable in a sub-domain Ωs surrounding the nodal point xq , the approximated ﬁeld can be written as ui (x)|Ωqs =

N

∑ Ba(x)cai (xq)

a=1

(6.10)

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Fig. 6.3 Selection of N nodal points associated with point xq

where Ba (x) are the basis functions deﬁned in the Cartesian coordinate space, N is the number of nodes in the interpolation domain of the point xq , and cai (xq ) are the expansion coefﬁcients at a given point xq . The interpolation domain is deﬁned as a domain that involves nodes which inﬂuence the approximation in Ωqs . Usually, the interpolation domain is determined by its radius ri (xq ), as showin in Fig. 6.3. Selecting the basis functions as monomials or radial functions, one obtains the polynomial PIM or radial PIM, respectively [6]. The polynomial PIM exhibits many excellent properties with respect to the consistency and accuracy of the method, as long as the moment matrix is invertible [6]. The possible singularity of the moment matrix is dependent on the distribution of nodal points. In [6], several techniques have been proposed to avoid a singular moment matrix. On the other hand, in the case of radial PIM the moment matrix is nonsingular but the consistency is not guaranteed and the accuracy of the approximation is sensitive to the choice of the shape parameters. One way to ensure the reproduction of the linear ﬁeld (C1 consistency) in the radial PIM and to avoid singular moment matrix in the polynomial PIM is to use a radial-polynomial basis [6]. Two different types of modiﬁcations of the polynomial basis by multiquadrics (radial functions) have been proposed in [7] in order to avoid the singularity of the moment matrix and to stabilize the method with respect to properly selected shape parameters involved in the multiquadrics. According to numerical tests [7] of accuracy and stability, we propose to use a combination of M polynomial basis functions and N radial basis functions with N > M. Let n(q, k) be the global number of the k-th nearest nodal point of N supporting n(q,k+1) q q nodes associated with x . Thus, n(q, 1) = q and x − x ≥ xn(q,k) − xq for k = 1, . . . , N − 1. Taking both the polynomial functions and the multiquadrics as the basis functions in Eq. (6.10), we may write ui (x)|Ωqs =

N

(q,k)

∑ Rn(q,k)(x)αi

k=1

M

(q,k)

+ ∑ Pk (x)βi

,

(6.11)

k=1

where the radial basis is given by multiquadrics [8] m/2 , Rn (x) = |x − xn|2 + s2

(6.12)

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with s and m being the shape parameters, and the polynomial basis Pk (x) is given by monomials. For k = 1, . . . , 6 and M = 6 we have for instance (6.13) Pk (x) ∈ 1, r1 , r2 , r1 r2 , r12 , r22 , ri = xi − xqi . The choice M = 6 corresponds to utilization of a complete quadratic polynomial basis. Starting from the approximation given by Eq. (6.11), one can get the interpolation of the ﬁeld variable in terms of its nodal values and the shape functions [7] as ui (x)|Ωqs =

N

∑ ui (xn(q, j))ϕ (q, j) (x).

(6.14)

j=1

Then, the gradients of the ﬁeld variable are approximated on Ωqs as uk,i (x)Ωq = s

N

(q, j)

∑ uk (xn(q, j))ϕ,i

(x).

(6.15)

j=1

The explicit expressions of the shape functions and their derivatives in terms of the basis functions and their nodal values are given elsewhere [7].

6.4 Discretized Boundary Conditions Assuming arbitrary of the presented interpolations, we recast the ﬁeld unknowns into the ﬁnite number of nodal values of displacements being the discrete degrees of freedom (DOF). Certain relationships among the DOFs are given by the discretized boundary conditions resulting from the prescribed boundary conditions collocated at adequate boundary nodes and implemented with the approximation of displacements and their gradients. Thus, in the case of the ﬁnite element interpolations, we may write the discretized boundary conditions (6.6) in view of Eqs. (6.7) and (6.8) as ubi = u˜i (ζ b ),

if ζ b ∈ ∂ ΩD ,

(6.16)

n a n j (ζ b )ci jkl (ζ b ) m e −1 be be ae ∂ N be be (h ) ξ , ξ u (x ) ξ , ξ = t˜i (ζ b ), k ∑ ∑ 1 2 1 2 l p mb ∂ ξp e=1 a=1 b

if ζ b ∈ ∂ ΩN

(6.17)

in which (ξ1be , ξ2be ) are the intrinsic coordinates of ζb ∈ Se , mb is the number of discretization elements adjacent to ζb . The unit normal vector ni (ζb ) is taken as the average of normal vectors evaluated at ζb if mb > 1. Assuming existence of potential

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gradients at corners, one can select arbitrary one of two sides joint at the corner for the deﬁnition of the normal vector ni (ζb ) and the prescribed ﬂux q(ζ ¯ b ) in Eq. (6.17) provided that the Neumann boundary conditions are prescribed on both sides of the corner. Similar, the PIM – discretization of the boundary conditions is given by N

ui (ζ b ) = u˜i (ζ b ), since

∑ ui(xn(b,a))ϕ (b,a) (ζ b ) = ui(ζ b )

(6.18)

a=1

in case of Dirichlet condition, whereas in case of Neumann condition, we have N

(b,a)

n j (ζ b )ci jkl (ζ b ) ∑ uk (xn(b,a) )ϕ,l

(ζ b ) = t˜i (ζ b ) at ζ b ∈ ∂ ΩN

(6.19)

a=1

Another possibility to take the traction boundary conditions into account will be discussed later. In order to obey the interaction among the DOFs properly, the governing equations should be considered appropriately. A direct collocation of the governing PDE at each interior node xc and supplemented with a domain-type approximation of the ﬁeld variable yields usually an ill-conditioned system of algebraic equations and the solution procedure should be supplemented by domain decomposition as well as regularization techniques. In the next section, we present another variant of the governing laws which can be used successfully for the interaction among the DOFs.

6.5 Local Integral Equations In discretized formulations, the governing equations are usually considered in a weak-form sense. In order to exclude physically inadmissible solutions, we shall consider the integral equivalent of the governing equations, i.e. the integral form of the balance equations which are given by an integral form of force equilibrium in case of elasticity problems. Note here that it is not necessary to include any DOF into the mutual interaction with each other. It is sufﬁcient to use a spreading of the interaction throughout the whole body. Thus, a ﬁnite-range interaction might be used in the numerical implementation but a coupling at least between neighboring local sub-domains is required. The use of a ﬁnite range interaction leads to a sparse system matrix for the computation of the unknown DOF. Therefore, we propose to consider the integral form of the force equilibrium on local sub-domains around each interior knot. Basically, the size, shape and composition of the sub-domains can be chosen arbitrarily, since artiﬁcial boundaries are irrelevant to the formulation. Nevertheless, the inﬂuence of the choice of these factors on the numerical results cannot be excluded completely from the numerical point of view.

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The general physical balance principles of a continuum theory take the form of integral equations. The governing equations or ﬁeld equations take the form of differential equations which can be derived from these integral equations because they hold for all arbitrary but small material domains Ωc . Following a reverse path, we can get the integral form of the balance principles by integrating the governing differential equations over Ωc ⊂ Ω. Thus, integration of Eq. (6.1) leads to

σ ji, j (x)dΩ(x) = −

Ωc

Xi (x)dΩ(x)

(6.20)

Ωc

By applying the Gauss divergence theorem to Eq. (6.20) and bearing in mind the Cauchy equations for the tractions, ti = σij n j , we obtain an integral form of the balance principle as −

∂ Ωc

−

∂ Ωc

ti (η )dΓ(η ) =

Xi (x)dΩ(x), or

Ωc

n j (η )ci jkl (η )uk,l (η )dΓ(η ) =

Xi (x)dΩ(x),

(6.21)

Ωc

where n j (η ) denotes the Cartesian components of the unit outward normal vector at η on the boundary ∂ Ωc of the sub-domain Ωc . Equation (6.21) is an expression of the force equilibrium in an arbitrary material domain Ωc . Hence, it is a physically admissible constraint that can be used as a coupling equation in the computation of the unknown DOF of the discretized problem. The unknown ﬁeld variable is involved only in the integral over the boundary of the local sub-domain. In contrast to the integral equations with singular fundamental solutions, Eq. (6.21) is completely nonsingular and moreover, it does not involve the domain integral of unknowns even if the elastic medium is non-homogeneous. In contrast to the global integral formulation, the local one in combination with a meshless approximation of ﬁeld variables enables us to develop truly meshless formulations, since no mesh of elements is needed for modeling the whole analyzed domain or its boundary. Moreover, the resulting system matrix of the discretized equations can be sparse provided that we use a short range interaction among the degrees of freedom. To complete the system of linear algebraic equations for the computation of nodal values of displacements, we propose to utilize integral equations on local sub-domains surrounding interior nodes and supplement the discretized boundary conditions given in Section 6.4. As regards the numerical implementation of the integral equations on local sub-domains, let us consider ﬁrstly the LIE implemented with using the MFFEI. Let mc be the number of linear triangular elements Tec adjamc c c c cent to the interior node x . Then, the local sub-domain Ω = e=1 Te . Denoting the vertex of each element Tec coinciding with xc as the ﬁrst local node, the boundary of c c c the local sub-domain is given as ∂ Ωc = m e=1 Γe (2,3), where Γe (2,3) is the side of c Te between the local nodes #2 and 3 (see, Fig. 6.4).

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Fig. 6.4 Linear triangular elements in: (a) global coordinate system; (b) local coordinate system

Thus,

·

( )dΓ =

∂ Ωc

mc

∑

·

( )dΓ =

e=1 c Γe (2,3)

mc

∑

e=1

2 0

dΓ ( ) ds

·

ds = Γce (2,3)

mc

∑

2

e=1

0

·

( )ge (s)ds, (6.22)

since dΓ = xk |Γce (2,3) = ∂ xk e gk (s) = ∂s

(∂ x1 /∂ s)2 + (∂ x2 /∂ s)2 ds,

3

∑ xaek N a (1 − s/2,

√ 3s/2) ≡ xek (s),

a=1

√ √ √ 1 e 3 e = − h1k (1 − s/2, s 3/2) + h2k (1 − s/2, s 3/2), 2 2 Γce (2,3) ge = (ge1 )2 + (ge2)2 , τk |Γce (2,3) = gek (s)/ge (s), ni |Γce (2,3) = εik3 gek (s)/ge (s).

a

N N,1a N,2a

a=1 √ √ √ ( 3 − 3ξ1 − ξ2)/2 3 −1/2 √ −1/2 3

a=2 √ √ √ ( 3 + 3ξ1 − ξ2 )/2 3 1/2 √ −1/2 3

(6.23) a=3 √ ξ2 / 3 0 √ 1/ 3

Now, assuming the body forces to be absent, we may write the discretized LIE (6.21) as mc

coijkl ε jq3 ∑

3

∑ uk (xae)

e=1 a=1

2

√ E (xe (s)) geq (s) (he )−1 l p 1 − s/2, s 3/2

0

⊗

√ ∂ Na 1 − s/2, s 3/2 ds = 0 ∂ ξp

(6.24)

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Fig. 6.5 Illustration of local sub-domain around a boundary node on ∂ ΩN

In the case of PIM, the discretization of the LIE is much simpler, because of the integration in the global Cartesian coordinate system. Thus, in view of (6.15), the LIE (6.21) becomes coijkl

N

∑ uk (xn(c,a))

a=1

(c,a)

E(x)n j (x)ϕ,l

(x)dΓ(x) = 0.

(6.25)

∂ Ωc

Recall that we can use geometrically simple circular sub-domains around each interior node which is also the center of the sub-domain, since size, shape and composition of sub-domains Ωc can be chosen arbitrarily. Then, the integrations over the boundary as well as over the interior of the sub-domains can be performed very easily. Finally, we brieﬂy discuss an alternative implementation of the traction boundary conditions. Within the concept of local sub-domains, it is straightforward to consider the force equilibrium in the integral form on a local sub-domain Ωc surrounding the boundary node xc ∈ ∂ ΩN as coijkl

N

a=1

Lc

∑ uk (xn(c,a))

E(η )n j (η )ϕ,l (c,a) (η )dΓ(η ) = −

t˜i (η )dΓ

(6.26)

ΓcN

where ΓcN = ∂ ΩN ∩ ∂ Ωc , Lc = ∂ Ωc − ΓcN as shown in Fig. 6.5.

6.6 Numerical Examples In order to test the proposed numerical methods, we consider examples for which analytical solutions are available. The body forces are vanishing in Ω, Poisson’s ratio is constant and taken as ν = 0.25 and the gradation of Young’s modulus is given by a prescribed function E(x) = E0 f (x). In the study of the convergence and accuracy of the numerical results with respect to the increasing density of nodal points, we use the average %-error deﬁned as

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99

Fig. 6.6 Sketch of the analyzed domain with prescribed boundary conditions

GPE = 100

Nt

∑

1/2 ∆uai ∆uai

/

a=1

Nt

∑

1/2 a ex a uex i (x )ui (x )

,

a=1

a (xa ) − uex ∆uai = unum i i (x ),

(6.27)

where Nt is the total number of nodes on the closed domain Ω ∪ ∂ Ω. If we consider an elastic square domain L×L with vertical gradation of the Young modulus and subjected to homogeneous tension load on the top with ﬁxed bottom in vertical direction, the problem is 1-dimensional and we can ﬁnd the exact solution for some kinds of variation of the Young modulus [9]. Then, the tractions on the lateral sides are given by the analytical solution (see Fig. 6.6). In this paper, we shall consider two variations of the Young modulus as well as homogeneous case: (A) Homogeneous medium (simple tension) u2 (x) =

(1 + ν )(1 − ν¯ ) qx2 , E

u1 (x) =

σij = qδi2 δ j2

−ν¯ (1 + ν ) qx1 , E (6.28)

(B) Exponential gradation of Young’s modulus f (x) = (a exp(δ x2 /L) + b exp(−δ x2 /L))2 A2 1 − e−2δ x2 /L u2 (x2 ) = √ , E0 a + be−2δ x2/L A2 =

(1 + ν )(1 − 2ν¯ ) Lq 1 √ , 2(1 − ν¯ ) δ E0 a + b

ν¯ A1 δi1 δ j1 u1 (x2 ) = √ = const, A1 − arbitrary, σij = q δi2 δ j2 + 1 − ν¯ a E0 (6.29)

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(C) Power-law gradation of Young’s modulus f (x) = (1 + δ x2/L)2 A2 (1 + ν )(1 − 2ν¯ ) x2 √ u2 (x2 ) = √ q, , A2 = E0 1 + δ x2 /L (1 − ν¯ ) E0 LA1 2ν¯ (1 + ν ) √ δi1 δ j1 . u1 (x2 ) = √ = const, A1 − arbitrary, σij = q δi2 δ j2 + δ E0 (1 − ν¯ ) E0 (6.30) In the presented numerical results, we have used the parameters: a = 1, b = 0, E0 = 1, δ = 0.5, and the plane strain conditions. As regards the discretization, we shall use regular distributions of nodes in both the horizontal and vertical directions, but with different linear densities (see Fig. 6.7a). The number of nodes is Nt = n1 × n2 in case of the MFFEI approach, while the number of triangular elements with quadratic interpolation is m = e1 × l2 in case of the MBFEI approach when Nt = (e1 + 1) × (2l2 + 1). Finally, in the discretization by the PIM, the radius of the interpolation domain a a is taken as ria = 3.001 ha, the radius of circular sub-domains r = 0.98 h , the shape a a b parameter s = 2, m = 5, M = 10 where h = min x − x . ∀b

Figure 6.8 shows the results of the study of convergence of accuracy as the density of nodes is increased. Since the exact solution is given by the linear dependence of displacements on the coordinates, the increasing number of nodes only increases the numerical error. Nevertheless, the accuracy by any of the employed approach is very good for a wide range of the h parameter. It is seen (Fig. 6.9) that the convergence of accuracy is achieved in nonhomogeneous media by each of the applied approaches. The convergence rate by the mesh based approach with quadratic approximation over triangular elements is comparable with that of meshless PIM approach, though the accuracy by the latter one is several orders better. The convergence rate by the mesh free approach with automatic generation of linear triangular elements is much slower. That is why the efﬁciency of the MFFEI approach is much worse as compared with that of the other two approaches. This can be seen from Fig. 6.10 with bearing in mind both the accuracy and CPU-times. The CPU-times are not affected by the material in-homogeneity. The accuracies shown in Fig. 6.10 corresponds to the exponentially graded Young modulus.

Fig. 6.7 Scheme of the distribution of nodes in: (a) MFFEI approach; (b) MBFEI approach

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Fig. 6.8 Convergence of accuracy in homogeneous square domain with increasing the density of nodes (decreasing h parameter)

Moreover, the efﬁciency of the PIM approach is much better than that of the MBFEI. This can be illustrated in Fig. 6.10, for instance in the case of accuracy 0.02%, the CPU-time by the MBFEI is more than 1,500 times longer than that of the PIM approach. In order to study the numerical stability of the considered meshless techniques with respect to distribution of nodal points, we have made 7 disturbances of the uniform distribution of nodes (Fig. 6.11). Figure 6.12 shows the accuracies obtained for considered nodal point distributions. In the case of homogeneous medium, the total number of nodes is 36 while in the case of exponentially graded Young modulus it is 676. It is seen from Fig. 6.12 that the PIM approach exhibits numerical stability while the MFFEI approach yield results with unacceptable accuracy for any disturbed distribution of nodes.

6.7 Conclusions The paper presents the development and application of two element based approximations to numerical implementation of local integral formulation for solution of elasticity problems in non-homogeneous media. One of these approximations, the

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Fig. 6.9 Convergence of accuracy in non-homogeneous square domain with increasing the density of nodes for: (a) exponential gradation; (b) power-law gradation of Young’s modulus

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Fig. 6.10 Comparison of CPU-times with giving extreme accuracies by three numerical approaches

MFFEI (mesh free ﬁnite element interpolation) needs only predeﬁnition of nodal points distribution in the analyzed domain. Then, the linear triangular elements are generated automatically. The second one, MBFEI (mesh based ﬁnite element interpolation) utilizes the triangular elements with quadratic interpolation. These elements are taken from the predeﬁned mesh of such elements on the analyzed domain, since the automatic generation of non-linear elements is not simple problem. For comparison of numerical results in test examples, we have considered also the other meshless approximation developed previously (PIM). The following conclusions can be drawn • The formulation is quite general without any restriction to continuous variation of the Young modulus as well as the shape and size of the analyzed domain. • The system matrix of the discretized equations is sparse; the numerical integrations are simple and accurate by using standard quadratures, since there are no singularities. • The accuracies obtained by the developed techniques converge to the exact solution with increasing the density of nodes in both cases of gradation of Young’s modulus. • The convergence rate in case of the MFFEI approach is too slow to yield an efﬁcient numerical computational technique. The CPU-time is very long as compared with that by other presented techniques in order to achieve comparable accuracy.

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Fig. 6.11 Uniform distribution of 36 nodal points (A) and its 7 various disturbances (B)–(H)

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Fig. 6.12 Diagrams of accuracies by two meshless techniques obtained for various distributions of nodal points for (a) homogeneous medium; (b) exponentially graded Young’s modulus

• The MFFEI approach does not exhibit the numerical stability with respect to the variations in the distribution of nodal points, though the PIM approach does. Since the Lagrange polynomial interpolations could be effectively utilized in numerical implementation of the solution formulations based on local integral equations, the aim to develop automatic generation of higher order ﬁnite elements from the predeﬁned distribution of nodal points is still attractive. Acknowledgements The authors acknowledge support by the Slovak Science and Technology Assistance Agency (APVV-20-035404, APVV-51-021205), the Slovak Grant Agency (VEGA – 2/6109/6), and the German Research Foundation (DFG).

References 1. Idelson SR, Onate E (2006) To mesh or not to mesh. That is the question. . ., Computer Methods in Applied Mechanics and Engineering 195: 4681–4696 2. Atluri SN, Shen S (2002) The meshless local Petrov-Galerkin (MLPG) method. Tech Science Press, Encino 3. Atluri SN (2004) The meshless method (MLPG) for domain & BIE discretizations. Tech Science Press, Forsyth 4. Hughes TJR (1987) The Finite Element Method. Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs, NJ 5. Sladek V, Sladek J, Zhang Ch (2005) Local integro-differential equations with domain elements for numerical solution of PDE with variable coefﬁcients. Journal of Engineering and Mathematics 51: 261–282 6. LIU GR (2003) Mesh free methods, Moving beyond the ﬁnite element method. CRC, Boca Raton, FL 7. Sladek V, Sladek J, Tanaka M (2005) Local integral equations and two meshless polynomial interpolations with application to potential problems in non-homogeneous media. CMESComputer Modeling in Engineering and Sciences 7: 69–83

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8. Hardy RL (1990) Theory and applications of the multiquadrics-biharmonic method (20 years of discovery 1968–1988). Computers and Mathematics with Application 19: 163–208 9. Sladek V, Sladek J, Zhang Ch (2006) A meshless Point Interpolation Method for Local Integral Equations in elasticity of non-homogeneous media. In: Sladek J, Sladek V (eds.) Advances in the Meshless Method. Tech Science Press, Forsyth, pp 263–290

Chapter 7

Modelling of the Process of Formation and Use of Powder Nanocomposites Alexandre Vakhrouchev

Abstract The methods for the modelling of the processes that accompany obtaining and use of powder nanocomposites are presented. For this purpose, a number of physical-mathematical models, including the models of obtaining of nano-sized powders at ‘up down’ processes, the models of the main compaction steps of powder nanocomposites and the models of deformation of powder nanocomposites under the ambient action were developed. A number of numerical examples of modelling based on the models developed are considered.

7.1 Introduction The use of new composites based on powders, especially powder nanocomposites (PNCs), shows that complete realization of their strength and performance properties requires a signiﬁcantly higher level of prediction of physical-mechanical properties of these materials and the development of new material-engineering methods, which should include a complex analysis of the processes of the formation of composites and structural components [8, 11, 14]. This is dictated by high sensitivity of physical-mechanical properties of PNCs to the composition and structure variations of parent materials and their components and to the parameters of the technological process of product manufacturing. It also should be noted that the processes that accompany obtaining and use of PNCs are very complex and diverse and their space and time scales embrace a wide range [1, 2, 15, 17]. Therefore, the investigation of the above processes using experimental methods only is difﬁcult. Hence, it is necessary to consider the modelling of physical-mechanical processes in PNCs at different structural levels with regard to a complete analysis of the entire “history” Alexandre Vakhrouchev Department of Molecular Mechanics, Institute of Applied Mechanics, Ural Branch of the Russian Academy of Sciences [email protected] V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

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Fig. 7.1 The main steps of the process of formation and use of powder nanocomposites

of their existence, which consists of three main steps (Fig. 7.1): nanopowder obtaining, product manufacturing from nanopowder (presswork and relief) and use of the product [18, 21]. At present, nanosystem simulation at different structural levels is carried out with the use of three main methods: molecular dynamics (MD) that is used to calculate the interaction of atoms and/or molecules [3–5, 7, 12] in nanoelements; mesodynamics (MsD) that is used for calculating nanoelements (nanoparticles) [9–13]; continuum mechanics (CM) that is used for analyzing macro-behaviour of nanosystems [16, 20]. In his previous works [22–26], the author solved some problems on the formation of nanoparticles, problems on static and dynamic interaction of metal nanoparticles and problems on nanoparticle self-organization using manly molecular dynamics and mesodynamics. An object of the present work is to systemize the main theoretical concepts, which allow a simulation of powder nanocomposites formation by continuum mechanics methods.

7.2 Physical Processes and Mathematical Models of Nanopowder Obtaining In the present section, consideration is given to physical processes of nanopowder dispersion at “up down” process at impulsive temperature-force action under isothermal conditions and varying temperature.

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7.2.1 Powder Dispersion Physical Processes The physics and kinetics of the nanopowder dispersion at “up down” process are determined by the following parameters: the particle loading, the ambient pressure and temperature, the rate of the change of the above values with time and the physicalmechanical characteristics of a material particle. Depending on the ratio of the indicated above parameters, one can observe different types of the particle destruction: slabbing destruction of a solid phase of the material particle, fusion of particles, combined destruction of solid and liquid phases and material evaporation from the particle surface, etc. The present section is concerned with the main physical phenomena that accompany the process of destruction of particles.

7.2.1.1 Hard Particle Dispersion Let us consider a hard particle of an arbitrary shape that is in the environment with time-varying pressure and temperature. If the combination of the indicated parameters does not cause a change in the material phase state (its fusion or evaporation) then the analysis of the particle destruction can be carried out within the context of a deformed solid. When there is a change in the material phase state, it is necessary to consider a mutual behavior of solid, liquid and gaseous phases. The main destruction mechanisms of a particle in solid and multi-phase states are presented further. Under the action of the external pressure P0 , the particle under study having a volume Ω0 and a surface S0 is deformed and at the same time accumulates elastic energy (Fig. 7.2). Let us assume that an equivalent stress at a certain point A of the particle reaches the value of its material ultimate stress limit. This causes the division of the particle surface S∗ through the point A into two fragments, which have volumes Ω1 and Ω2 and surfaces S1 and S2 , respectively. In the fragments formed, the pulsing of the stress ﬁelds continues, which in its turn can lead to the destruction of the fragments. The process repeats while maximal equivalent stress exceeds the material ultimate stress limit. The level of the equivalent stress results from the storage of the elastic energy of deformation in each new fragment that is formed after the particle destruction. In the process of dispersion, a parent particle loses a part of energy on inelastic deformations of the destruction and on the dispersion of fragments. It is obvious that a moment should come when there is no energy sufﬁcient for further destruction of particles and the dispersion process stops. The kinetics of the solid-phase dispersion process can be inﬂuenced by the presence of pores and micro cracks, which usually become starting points of destruction. In addition, the interaction of stress waves with different types of defects can essentially change spatial and time distribution of stresses in the particle. However, it does not inﬂuence the physics of the process. Exposure to heat (both the external heat of the environment and the internal heat resulting from the energy dissipation at destruction and inelastic deformations) also

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Fig. 7.2 A hard particle before (I) and after (II) destruction

inﬂuences the kinetics and topography of the particle destruction. An increase or a decrease in temperature leads to the formation of additional temperature stresses and to a change in elastic and strength properties of the material. In particular, Young’s modulus and Poisson’s ratio that determine the speed of stress wave propagation, the ultimate stress limit of material and the beginning of destruction essentially depend on temperature. In addition, with increasing temperature the manifestation of material inelastic properties increases.

7.2.1.2 Particle Dispersion with the Change of the Material Phase State At high-rate loading, intensive heat ﬂows and small thermal capacity of material, the temperature of a particle can exceed the melting temperature of its parent-material, which will cause a “solid-liquid” phase transition. There may be different mechanisms of the destruction of a particle that lead to the formation of a liquid phase on the surface and in the bulk of the material (Fig. 7.3a). The latter is observed when a particle is inhomogeneous and the melting temperature on the surface of a particle is higher than within it. As the result of the “solid-liquid” phase transition, the compressibility and other physical-mechanical characteristics of the material vary and additional stresses appear, which are related to the changes in the material volume. When internal regions of the “solid-liquid” phase transition are formed, additional tensile stresses appear in the particle. Let us consider one after another the mechanisms of the destruction of a particle covered with a liquid layer and a particle that contains liquid within itself.

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Fig. 7.3 Schematic view of the destruction of a particle containing (a, b) solid (1) and liquid (2) phases; (c) solid and liquid phases conﬁned in a hard shell (3)

In the ﬁrst case, after the external pressure has rapidly been removed, the liquid surface layer and the hard nucleus begin to vibrate simultaneously. The destruction of this system caused by the propagation of stress waves in the particle can happen as the result of both the destruction of the hard nucleus and the destruction of the outer layer, the strength of which is conditioned by the action of the surface tension forces of the liquid. It should be noted that the velocities of the waves in the liquid and the hard nucleus are not the same; therefore, they can be reﬂected from the solid-liquid interface. If the liquid phase has formed within the particle, then the ﬁrst stage of the destruction is the split of the solid part of the particle (Fig. 7.3c). Then the destruction process develops as indicated above. Many metals are covered with oxides, the melting temperature and strength of which are much higher than those of the bulk material are. Under the external pressure, the “hard nucleus – oxide layer” system (Fig. 7.3c) is compressed and is warmed up due to the heat exchange with the environment. The surface oxide ﬁlm does not melt. Melting or even evaporation starts under the ﬁlm in the bulk material. This leads to the appearance of additional stresses than compress the nucleus and stresses that stretch the outer oxide layer, which may be destructed if the volume of the melted material changes greatly, and the destruction of the liquid layer can follow. The outer shell of the particle can also be destructed under the external pressure in the process of vibration of the complex system: a hard nucleus, a liquid layer, a gas interlayer and a surface solid ﬁlm. After the shell destruction, the gas

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within it disperses into the environment. The liquid surface layer gets broken forming drops, and the hard nucleus breaks down into smaller parts at a sufﬁcient reserve of potential energy. When the environment is cooling down, evaporated substance is condensed on the hard and liquid particles formed. The latter harden on cooling. Thus, the analysis of the physical processes during the dispersion of powder material shows that in order to analyze it, it is necessary to take into account a number of interrelated phenomena such as dynamic deformation and destruction of a solid, heat exchange, thermal conductivity, melting and evaporation of a substance. A determining parameter is the temperature of the process, which has a signiﬁcant effect on the changes in the material properties and “solid – liquid” and “liquid – gas” phase transitions and inﬂuences the kinetics and topography of destruction in many respects. On this basis, it is practical to consider mathematical models for the dispersion process in isothermal and nonisothermal conditions. Now, let us consider these models for a single elastic particle and for a single inelastic particle, which are presented at constant and varying temperature.

7.2.1.3 Mathematical Model of the Powder Dispersion Process in Isothermal Conditions Let us consider a single particle from inhomogeneous material (a general case), which is in an environment with the initial pressure P0 and the temperature T0 and occupies the region Ω0 (Fig. 7.2). The x(x, ¯ y, z) coordinates determine the position of the particle. Due to the compression over the surface S0 , the particle accumulates elastic energy U0 of deformation; the initial ﬁelds of displacement vectors u¯0 , tensors of deformation εˆ0 and stresses σˆ 0 correspond to this elastic energy of deformation. The value of the elastic energy accumulated by the particle is determined by the work of the pressure forces on the particle surface U0 =

1 2

P0 n¯ ◦ u¯ 0 dS0

(7.1)

S0

that is equal to the work of stresses σˆ for the deformations εˆ in the volume Ω0 . A0 =

1 2

σˆ 0 ◦ εˆ0 dΩ0

(7.2)

Ω0

In the above equations, u¯0 is a displacement vector on the surface S0 , n¯ is a normal to the surface S0 and ◦ is a symbol for the inner composition of tensors. At the moment t0 , the pressure P0 varies and the potential energy U0 accumulated in the particle changes into kinetic energy of its elastic vibrations. At the moment t = t∗ under certain conditions, the destruction of the particle takes place and two new particles are formed with the volumes Ω1 and Ω2 and the surfaces S1 and S2 .

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With time, these particles can also break into smaller ones and the process goes on until there is energy that is sufﬁcient for destruction. Vibrations and destruction of particles can be accompanied by inelastic scleronomous or rheonomous deformations, due to which a part of the energy accumulated at compression dissipates, and by the formation of different kinds of defects as well. Taking into account the above-indicated phenomena makes the dispersion process modeling very complicated. Therefore, in order to obtain the solution that will allow carrying out a preliminary analysis of the dispersion process, at the ﬁrst step we admit that the material is elastic. Non-linear properties of the material are taken into consideration in the statement of the problem of nonisothermal loading. The system of equations that describes a change in the shape of the elastic solid and its destruction with time in isothermal conditions includes: – Differential equations of motion

∂ u¯ ¯ ∗ σˆ = ρ (x) ∇ ¯ ∂ t2 2

(7.3)

– Equations of connection between the deformation tensor εˆ and the displacement vector u¯ 1 ¯ +∇ ¯ ⊗ u¯ εˆ = u¯ ⊗ ∇ (7.4) 2 constitutive equations of elastic body

σˆ = D(x) ¯ εˆ

(7.5)

where D(x) ¯ is the matrix of the material elastic constants. The boundary conditions for the system of Eqs. (7.3)–(7.5) contain: – Force conditions on the surface S0 of the initial particle and the surfaces S1 and S2 of the newly formed particles including the destruction surfaces S1∗ and S2∗ : τn = 0, σn = −P0 , t = 0 (7.6) x¯ ⊂ S0 τn = 0, σn = −P(t) , 0 < t ≤ t∗ where σn and τn are normal and tangential stresses acting on the surfaces of the initial particle and on the surfaces of the particle fragments formed as the result of destruction; t∗ is the moment of the beginning of destruction; – Force and deformation conditions within the initial particle Ω0 and the newly formed particles Ω1 and Ω2 :

σˆ = σˆ 0 , εˆ = εˆ0, t = 0, x¯ ⊂ Ω0 u¯ = u¯∗1 , x¯ ⊂ Ω1 t = t∗ u¯ = u¯∗2 , x¯ ⊂ Ω2

(7.7)

where u¯∗1 and u¯∗2 are displacement vectors within the initial particle at the moment of destruction t∗

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– The condition of destruction

σeq = σb ,

x¯ ⊂ A

(7.8)

where A is a starting point of destruction; σeq is an equivalent stress; σb is an ultimate stress limit of the material of the particle – The condition determining the geometry of the destruction surface S∗ , which can be written with the assumption that the unit vector n¯ S∗ is perpendicular to the destruction surface S∗ and parallel to the maximal principal stress σ¯ 1 n¯σ1 ◦ n¯ S∗ = 1

(7.9)

where n¯ σ1 is a unit vector determining the direction of the principal stress σ¯ 1 . The formation of the surface S∗ starts at the point A, where the condition of destruction (7.8) fulﬁls. In this case, we believe that the time of the surface formation S∗ is small and the particles of the material located on different sides from the surface S∗ move an inﬁnitely small distance away from one another. Thus, the system of Eqs. (7.3)–(7.5), in view of conditions (7.6)–(7.9), determines the statement of the problem on deformation and destruction of an elastic particle in isothermal conditions.

7.2.1.4 Mathematical Model for Powder Dispersion at Varying Temperature In the present section, a mathematical model for destruction of a particle at timevarying temperature is presented having regard to scleronomous and rheonomous properties of a material (here, the “solid-liquid” and “liquid-gas” phase transitions are not taken into consideration). Let us consider a particle from elastoviscoplastic material occupying a region Ω and having a surface S in the environment with time-varying pressure P(t) and temperature T (t). With a change in the above parameters, the particle may deform and break down and at the same time remains in the solid state. The mathematical statement of this problem is formulated based on equations of motion (7.3), Cauchy equation (7.4), constitutive equations for a viscoelastic body with regard to temperature deformations (for the region of viscoelastic deformation of a particle) εˆ = D(x) ¯ −1 σ + R(x) ¯ ∗ σˆ + εˆ T (7.10) constitutive equations for inelastic deformation (in the region of plastic deformations of a particle) ˆ σ, ˆ εˆ , εˆ˙ , D(x), σˆ = Ψ ¯ R(x), ¯ G(x), ¯ T,t, . . .

(7.11)

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and equations of thermal conductivity

λ ∂T aT ∂ t

(7.12)

εˆ T = αT (T (t) − T0 )Iˆ

(7.13)

λ ∇2 T + q p = where

Here, Ψ is a functional (in a general case), which determines the connection between stress tensors σˆ and deformations εˆ and takes into account the inﬂuence of the deformation rate εˆ˙ , the temperature T, elastic D(x), ¯ rheonomous R(x) ¯ and plasticity parameters G(x) ¯ of the material; aT is a temperature conductivity coefﬁcient; λ is a thermal conductivity coefﬁcient; εˆ T is temperature deformation; αT is a coefﬁcient of volume temperature expansion; T0 is an initial temperature; q p is a heat ﬂow from internal heat sources (as the result of plastic deformations); Iˆ is a unit tensor. The system of Eqs. (7.3), (7.4), (7.10)–(7.13) is supplemented with the conditions of plasticity and destruction

σeq = σ p , x¯ ⊂ S p σeq = σb or ω = 1, x¯ ⊂ A,

(7.14) (7.15)

where σ p is a material plastic limit; ω is a parameter of destruction rate; SP is a surface separating the plastic deformation regions from the viscoelastic deformation regions. To calculate the parameter of the destruction rate, the kinetic equation of damage accumulation is used ω = ς (σˆ , εˆ , T, . . .) (7.16) The boundary conditions for this problem includes σˆ = σˆ 0 , εˆ = εˆ0 , T = T0 , x¯ ⊂ Ω t=0 σn = −P0 , τn = 0, x¯ ⊂ S qn = α (TS − T (t)) x¯ ⊂ S, t > 0 σn = −P(t), τn = 0 qn = α (TS − T (t)) x¯ ⊂ S∗ , t > t∗ σn = −P(t), τn = 0 u¯ve = u¯ep ,

σnve = σnep ,

x¯ ⊂ SP ,

(7.17) (7.18) (7.19) (7.20)

where qn is a heat ﬂow normal to the initial surface S of a particle; Ts is a particle surface temperature; T (t) is a temperature of the environment; S∗ is a surface of destruction; u¯ve , u¯ep are displacements on the surface S p ; σnve , σnep are normal stresses perpendicular to the surface S p ; α is a heat-transfer coefﬁcient. The surface S p geometry of a loaded particle is determined by the regularities of the development of plastic deformations in accordance with Eq. (7.11) and the surface S∗ proﬁle of destruction depends directly on the condition of destruction (7.8). Thus, the system of Eqs. (7.3), (7.4), (7.10)–(7.12) together with conditions

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(7.14)–(7.20) determines the statement of the problem on the deformation and destruction of a particle from elastic viscoplastic material at varying temperature.

7.3 Physical and Mathematical Models of the Processes of Powder Nanocomposite Compacting In the present section, physical phenomena are considered and mathematical models of the main steps of the compaction of powder composites are constructed: displacement of a press-tool, removal of pressing force, re-pressing and squeezing a ﬁnished product from a die.

7.3.1 The Physics of the Compaction Process The formation of mechanical and other properties of PNCs is realized in the process of their conversion to ﬁnished products by methods of pressing. In this case, the technological process of the composite preparation is conducted simultaneously with the product manufacturing and is accompanied by a signiﬁcant change in a stressed-strained state (SSS) of the powder being pressed. As an example, Fig. 7.4 shows a typical scheme of the process for preparing PNC: 1 – active pressing (in the present case, double pressing with hydro shaping); 2 – instantaneous unloading at the puncheon exit or die splitting; 3 – a relaxation period observed during the product “rest” or other operations. Note that the powder compaction during pressing is characterized by signiﬁcant volumetric deformations and a continuous change in parameters such as density, elasticity, plasticity,

Fig. 7.4 The stress “time history” σˆ in the powder composites: 0-a, b-c – phases of loading; a-b, c-d-e – unloading; e-f– storing; f-g– loading at operation

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strength and rheonomous properties. Inhomogeneity of the material SSS leads to the nonuniformity of its plastic deformation both in volume and in direction. This results in continuously changing nonuniformity of physical-mechanical characteristics of the material during the process of pressing and in the appearance of residual (technological) stresses in the pressed product. The PNC physical-mechanical characteristics change signiﬁcantly; normally, it is a powder mixture with high porosity and low density before pressing. During a puncheon stroke the mixture consolidates, which results in the formation of an inhomogeneous compact. At the step of the repeated pressing, after some powder has been added to the die, a product is formed, which consists of two parts with different physical-mechanical properties. This is explained by the fact that one part of the material is pressed again and the second one is in the process of formation, i.e. a powdered body is converting into a compact. An adhesive bond appears at the interface of the materials, which provides the connection of successively compacted parts of the powder material into a single whole. In the process of squeezing a ﬁnished product from a die, a signiﬁcant elastic unloading is observed in it, which changes its form and sizes, and in some cases (e.g. at “overpress”) it leads to the destruction of the product. Hydro shaping should be a ﬁnal operation in the technological process because it essentially decreases inhomogeneity of the material and signiﬁcantly decreases or even removes technological stresses. There are some other physical processes accompanying pressing that inﬂuence the parameters of the condition of the material pressed and the product SSS formed. First, it is gas release from a pressed product (which is especially characteristic of super disperse systems containing a great initial amount of gas), which can lead to the growth of internal pressure in the pressed product and the appearance of internal hollows and micro cracks in it. Both the external friction occurring due to the interaction of a powder mixture and a press-tool and the internal friction, which results from the displacement of particles relative to one another, essentially inﬂuence the pressing process as well. Values of frictional forces depend on the sizes and shapes of particles, the tool surface quality, and they are controlled with various technological additives. Thus, pressing is a set of interconnected physical processes taking place in a material on the background of the essential change of its aggregative state from a loose powder-like material to a compacted inhomogeneous body. The selection of an optimal mode of pressing and the investigation of the kinetics of accompanying processes require making a complete mathematical model of the process of pressing.

7.3.2 Mathematical Model of Pressing Figure 7.5 shows schematic views of the above technological steps and the typical regions of the “material being formed – press-tool” system are highlighted. Here, Ω1

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Fig. 7.5 The scheme of a powder compression: (a) pressing and unloading, (b) an ejection of a ﬁnished article

is a region occupied by a PNC; Ω2 , Ω3 and Ω4 are the regions of the die, base and puncheon, respectively; Ω5 is the volume of an resilient member, which is usually present in the system indicated and improves the conditions of pressing. The region Ω1 is subdivided into regions Ω1 1, Ω2 1 and Ωk 1, which are formed during material shaping in “k” press-tool throws. The system of equations describing the problem statement includes the following: equilibrium equations ∇ ∗ σˆ = 0,

(7.21)

geometrical equations

εˆ =

1 u¯ ⊗ ∇ + ∇ ⊗ u¯ + ∇ ⊗ u¯ ∗ u¯ ⊗ ∇ , 2

(7.22)

constitutive equations that describe mechanical properties of the contacting objects, i.e. the press-tool (7.23) and the composite mass compacted during pressing (7.24), and unloading (7.25) ˆ σˆ = 2µn εˆ + λn ε0 I, n = 2, 3, 4, 5,

x¯ ⊂ Ω2 , Ω3 , Ω4 , Ω5 ,

(7.23)

σˆ m = ψˆ (σˆ , εˆ , θ , hi , . . .) , m = 1, 2, . . . , k;

x¯ ⊂ Ω11 , Ω21 , . . . , Ωk1 ,

(7.24)

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where µn and λn are Lam´e constants; ε0 is volumetric deformation; x¯ is coordinates; ˆ is a functional of plasticity of a porous body; θ is current porosity; hi is structural Ψ parameters of a porous material. For writing Eq. (7.24), the theory of plasticity of porous bodies is used [10, 19]. For the regions of the ultimately compacted material, when the nonuniform distribution of density in the compact bulk is taken into account, the functional from (7.24) degenerates into the constitutive equations of an inhomogeneous elastic body ¯ and λm (x): ¯ with variable characteristics µm (x) ˆ σˆ = 2µm (x) ¯ εˆ + λm(x) ¯ ε0 I,

(7.25)

Note that it is sufﬁcient to preserve linear connection for the regions Ω2 , . . . and Ω5 in Eq. (7.23) because the deformations in the press-tool are usually small. Equations (7.21)–(7.25), which determine the problem of the calculation of the system SSS of compacted and elastic bodies, should be supplemented with the equations of thermal conductivity and gas ﬁltration in the mixture being compacted. The thermal conductivity equation for an inhomogeneous porous body pressed, which takes into account heat release during pressing, has the following form: , ∂ ∂ ∂ ∂ ∂ ∂ λ (θ ) ∂ T λ (θ ) (T ) + λ (θ ) (T ) + λ (θ ) (T ) + qP = ∂x ∂x ∂y ∂y ∂z ∂z aT (θ ) ∂ t (7.26) where qP is power of heat sources, which depends on the plastic deformation levels; aT (θ ) and λ(θ ) are coefﬁcients of temperature conductivity and thermal conductivity, which depend on the porosity of a powder material; qP = Φ(εˆ p ),

(7.27)

For the press-tool, the thermal conductivity equation can be written in the form: ∇2 T =

1 ∂T . aT ∂ t

(7.28)

When the equation of gas or liquid ﬁltration through a powder composite is derived, we use Darcy law for a porous medium V¯ = − χ ∇P,

(7.29)

where V is the gas movement velocity; P is the gas pressure; χ is a ﬁltration coefﬁcient depending on the parameters of the gas and the powder medium. The equation of continuity for a porous medium can be written in the form: ¯ ∗ (ρ V) ¯ = − ∂ (θ ρ ) , ∇ ∂t

(7.30)

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Inserting Eq. (7.29) in Eq. (7.30) and taking into account the ideal gas law P = ρ RT,

(7.31)

we obtain a differential equation determining the change in the pressure in the gas medium by the volume of a powder body during the process of pressing ¯ ∗ P χ∇ ¯ ∗P = ∂ θ P , ∇ (7.32) T ∂t T Equation (7.32) is nonlinear because θ and χ change with time and are not the same at different points of the product pressed due to the nonuniform pressing of its separate volumes. The system of the above equations is closed by the boundary conditions, which are an initial state of the regions under study at each technological step, loading conditions, the interaction of the system of bodies on the surfaces S1 , S2 , . . . of contact (non-interpenetration of contacting surfaces, their unilateral connection, the presence of frictional forces) and the conditions of gas and heat exchange with the environment. Varying boundary conditions in parallel with varying mechanical properties of a powder material pressed determine the deformation behavior of the system of bodies at each step of the process of forming. Let us consider the boundary conditions as applied to the schemes of the process of forming displayed in Fig. 7.6.

Fig. 7.6 The gaseous and ﬂuid substance exchange between the powder composite and the environment: Sc – free surfaces; SL – the isolated surfaces; Sk – surfaces, contacting with other bodies

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A downward puncheon throw (Fig. 7.5a): ⎫ I σˆ lm = σˆ lm , θm = (θ )I m , ⎪ ⎪ ⎪ ⎪ ⎬ x¯ ⊂ Ωl m = 1, 2, . . . h=0 ⎪ σˆ k = (σˆ k )I , x¯ ⊂ Ωk ⎪ ⎪ ⎪ ⎭ k = 2, . . . , 5

⎫ u n j = u ni , ⎪ ⎪ ⎪ ⎬ x¯ ⊂ Si , τSi = sign(uSi ) f σni , ⎪ if σn < 0 : ⎪0≤h≤1 ⎪ ⎪ ⎪ ⎭ σ = 0 n τ = 0, Si

σn = 0, τS = 0, x¯ ⊂ Sc

P = P0 /h, x¯ ⊂ S p T = T0 ,

h ↑,

(7.33)

x¯ ⊂ Ωm , m = 1, 2, . . . , k; h = 0; T = Tc , x¯ ⊂ Sc , 0 ≤ h ≤ 1;

where Sc is a free surface of the “pressed product – press-tool” system; Tc is an ambient temperature; h is a parameter of proportional loading. It should also be remembered that if pressing is carried out in different directions then Sp could vary at each step of pressing. Unloading (Fig. 7.5a) ⎫ σˆ 1m = (σˆ 1m )II , θm = (θ )II m, ⎪ ⎪ ⎪ ⎪ ⎬ x ⊂ Ω1 , m = 1, 2, . . . ,

σˆ k = (σˆ k )II , x¯ ⊂ Ωk , k = 2, . . . , 5 ⎫ un j = unl , ⎪ ⎪ ⎪ ⎬ τSi = sign(uSi ) f σni , ⎪

⎪ ⎪ ⎪ ⎪ ⎭

h = 1,

at σn < 0 : τSi = 0 at σn = 0

x¯ ⊂ Si , , h ↓ (0 ≤ h ≤ 1) ⎪ ⎪ ⎪ ⎪ ⎭

σn = 0,

x¯ ⊂ Sc ,

τS = 0,

P = P0 /h, x¯ ⊂ S p

(7.34)

h ↓ (0 ≤ h ≤ 1),

T = T1 , x¯ ⊂ Ωm , m = 1, 2, . . . ; h = 0, T = Tc , x¯ ⊂ Sc , 0 ≤ h ≤ 1, where the index I designates variable parameters of the process, which correspond to the end of the active puncheon throw. The index II designates variable parameters of the system state at many times repeated pressing after unloading. For single pressing and the ﬁrst step of many times repeated pressing, initial stresses in the system are equal to zero and the PNC porosity corresponds to a bulk porosity of the powder composite mixture.

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Squeezing (Fig. 7.5b): ⎫ σˆ 1m = (σˆ 1m )I , θm = (θ )m , ⎪ ⎪ ⎪ x¯ ⊂ Ω1 , m = 1, 2, . . . ⎬ h = 0, ⎪ σˆ k = (σˆ k )I , x¯ ⊂ Ωk ⎪ ⎪ ⎭ k = 2, . . . , 5 ⎫ u n j = un i , ⎪ ⎪ ⎪ τSi = sign(uSi ) f σni , ⎬ x¯ ⊂ Si , , 0≤h≤1 if σn < 0 : ⎪ ⎪ ⎪ σn = 0 ⎭ τSi = 0, σn = 0, τS = 0, x¯ ⊂ Sc (h) , P = P1 /h, x¯ ⊂ S p T = T1 , x¯ ⊂ Ωm

(7.35)

m = 1, 2, . . . ; h = 0,

T = Tc , x¯ ⊂ Sc , 0 ≤ h ≤ 1 where Si is a contacting surface; un is a normal displacement on the surface of the contact; τS and σn are tangential and normal stresses on the surface of the contact; P is a pressure of pressing; P0 is a maximal pressure of pressing; f is a friction coefﬁcient. Here, P1 is a maximal pressure of squeezing. Note that in the process of squeezing, the free surface of pressing essentially increases and the surface of the contact interaction of the PNC and the press-tool decreases. When the product is leaving the instrument, the parameter of loading can increase or decrease within from 0 to 1.

7.4 Physical and Mathematical Models of the Processes Accompanying the Use of Powder Nanocomposites In the present section, the physics of the processes, which take place during the long-term use of the compacted PNCs in an aggressive medium, is considered and the mathematical statement of the problem is formulated.

7.4.1 The Physics of the Processes When in use, PNCs experience the action of various physical and force ﬁelds. The characteristics of these materials are essentially inﬂuenced by temperature as well, the variation of which in the process of the PNC-product use leads to the formation of the ﬁelds of temperature stresses due to the nonuniform heating of the product

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and to restricted conditions of its deformation. In addition, an increase or decrease in the temperature essentially inﬂuences the mechanics of the material and the rate of physical-chemical processes accompanying the storage and use of the products. In the product lifetime assessment, it is necessary to take into account the environmental effect, which provokes aging of a material especially of its surface layers: drying of its surface at low humidity and its swelling at high humidity of the environment. This phenomenon is especially important for the technologies of manufacturing nanopowder products (e.g. high-capacity capacitors) since nanopowder materials readily react with moisture. Some PNCs are gasiﬁed when stored and react with their coatings, which also inﬂuences their properties. The net effect of the service factors leads to changes in the form and geometry of the product, in the properties of the material and in the properties of the product’s armoring with time and causes the degradation of its mechanical characteristics. A complex evaluation of the above phenomena and a reliable prediction of the lifetime of a PNC product require the development of mathematical models, which will allow fast computer-aided evaluation of the inﬂuence of the main service factors such as temperature, humidity, force actions, internal technological stresses, etc. on the physics and mechanics of PNCs.

7.4.2 Mathematical Models Let us consider a powder body occupying a spatial region Ω that is restricted by a surface S (Fig. 7.6). The body is in the space where at the moment t = 0, the temperature is T1 and the concentrations of several gaseous and liquid substances are C11 ,C21 , . . . ,Cn1 . Part of the body surface Sk ⊂ S1 k S2 k S3 k . . . Sm k is ﬁxed (nonmoving) oris in the contact with a solid deformable body. The surface Sc ⊂ S1 c S2 c S3 c . . . Sl c of the body is not ﬁxed and reacts with the environment. The rest of the surface SΠ ⊂ S/Sk /Sc ⊂ S1 L S2 L . . . Si L is covered with a coating, which prevents inward penetration of gaseous or liquid substances from the environment. The initial concentrations of the above-indicated substances inside the body are C10 ,C20 , . . . ,Cn0 . In addition, there is an initial ﬁeld of residual technological stresses σˆ r in the body, to which the ﬁeld of residual deformations εˆ r corresponds. As the result of diffusion processes, heat exchange and heat conductivity, the temperature of the body and the concentrations of liquid and gaseous substances change. This results in the appearance of the volume deformation of the body in the form of swelling or shrinking εˆ w (x,t) ¯ and the gas release within the volume εˆ g (x,t). ¯ This causes the appearance of stresses in the body due to both the heterogeneity of the volume deformation ﬁeld and the restricted conditions of the deformation on the surface Sk . The set of the differential equations describing the physical processes indicated above (diffusion, swelling and the formation of stress ﬁelds with time) includes the following equations:

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Equations of the diffusion of liquid and gaseous substances into the body from the environment ⎧ ∂ ∂ ∂ ∂ ∂ ∂ ∂ C1 ⎪ ⎪ ⎪ ∂ x χ1 (θ ) ∂ x (C1 ) + ∂ y χ1 (θ ) ∂ y (C1 ) + ∂ z χ1 (θ ) ∂ z (C1 ) = ∂ t , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂ ∂ ∂ ∂ ∂ C2 ⎨ ∂ χ2 (θ ) (C2 ) + χ2 (θ ) (C2 ) + χ2 (θ ) (C2 ) = , ∂ x ∂ x ∂ y ∂ y ∂ z ∂ z ∂t ⎪ ⎪ ⎪ ........................ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂ ∂ ∂ ∂ ∂ ∂ Cn ⎪ ⎩ , χn (θ ) (Cn ) + χn (θ ) (Cn ) + χn (θ ) (Cn ) = ∂x ∂x ∂y ∂y ∂z ∂z ∂t (7.36) Equation of thermal conductivity ∂ ∂ ∂ ∂ ∂ ∂ λ (θ ) ∂ T λ (θ ) (T ) + λ (θ ) (T ) + λ (θ ) (T ) = , ∂x ∂x ∂y ∂y ∂z ∂z aT (θ ) ∂ t (7.37) Equation of gas release and gas diffusion

χg ∇2Cg + β =

∂ Cg , ∂t

(7.38)

Equation of body swelling (or shrinking)

∂ ε0w = ϕ (C1 ,C2 , . . . ,Cn , T,t), ∂t

(7.39)

a group of equations determining the stressed-deformed state of the product, differential equations of equilibrium (7.21) and Eq. (7.4). Constitutive equations for the material close the set of Eqs. (7.4), (7.21), (7.36)– (7.39). Let us consider the total effect of the service factors. Taking into consideration the fact that the limiting deformations of the destruction of the compacted powder materials are normally small, we will present the tensor of complete deformations as the sum of components resulting from the action of each separate factor.

εˆ = εˆ F + εˆ w + εˆ g + εˆ T + εˆ r

(7.40)

where εˆ is the tensor of complete deformations; εˆ F is the tensor of deformations due to the action of external loadings; εˆ w is the tensor of “swelling” deformations resulting from the processes of gas absorption from the ambient; εˆ g is the tensor of deformations caused by the processes of gas release within the powder composite; εˆ T is the tensor of temperature deformations; εˆ r is the tensor of deformations due to the action of technological stresses. Let us consider the structure of each tensor individually and write the equations, which will connect each tensor with the appropriate service factor.

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7.4.2.1 Temperature Deformations Temperature is one of the main factors that inﬂuence the physics-mechanics of PNCs since the properties of these materials are determined by its magnitude to a signiﬁcant extent. An increase in temperature intensiﬁes diffusion processes, enhances rheonomous properties of the material, depresses its modulus of elasticity, etc. In addition, a change in temperature directly inﬂuences the volume and shape of a product due to the respective temperature deformation. The latter is determined by a spherical tensor of temperature deformations ˆ εˆ T = αT ∆T I,

(7.41)

where αT is a temperature coefﬁcient of a linear expansion of the material, ∆T is a change in the temperature relative to the temperature of the product manufacturing; Iˆ is a unit spherical tensor. 7.4.2.2 Deformations of Swelling Being porous materials, PNCs have a capability to absorb gases from the ambient. As this takes place, two principal processes occur: adsorption, i.e. the absorption of gases by the product surface and absorption, i.e. the absorption of gases within the bulk. These processes are simultaneous and they lead to the change in the volume and the swelling of the product. In addition, the capillary condensation and capillary contraction of liquid are very essential for porous bodies since they cause signiﬁcant volume deformation. Physical equations describing the above phenomena are rather complicated; therefore, it is practical to evaluate the inﬂuence of separate factors based on phenomenological equations. The tensor of the deformations due to material swelling can be written in the form ˆ (7.42) εˆ w = ε0W I, where ε0W is the swelling of the material volume unit; the rate of the swelling enlargement with time is determined in accordance with (7.39). 7.4.2.3 Deformations Caused by Gas Release For PNCs that are used in aggressive ambient conditions and at elevated temperatures, gas release is observed. The pressure of the gases released can reach a signiﬁcant magnitude, which will result in the product volume deformation. The connection between this deformation and the gas pressure Pg within the material can be written in the form Pg ˆ εˆ g = I, (7.43) 3K where K is the modulus of the material volume compression, which depends on temperature and humidity in the general case.

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7.4.2.4 Deformations Caused by a Force Action Let us present the deformation caused by external forces as the sum

εˆ F = ε e + εˆ p + εˆ c ,

(7.44)

where εˆ e are elastic deformations, εˆ p are plastic deformations and εˆ c are rheonomous deformations of a compacted metal-composite. The elastic deformations are linearly connected with the stresses acting in the product εˆ e = Dσˆ , (7.45) The stress dependence of plastic and rheonomous deformations can be written in the general form

εˆ p = R p ∗ σˆ , εˆ c = Rc ∗ σˆ ,

(7.46) (7.47)

where D is the matrix of the material elasticity; R p and Rc are the matrices-operators determining plastic and rheonomous properties of the material, respectively. In the general case, these matrices and operators are temperature-dependent.

7.4.2.5 Deformations Caused by the Action of Technological Stresses Technological (residual) stresses form at the stage of product manufacturing. Their magnitude can be signiﬁcant, therefore, after the product has been manufactured, its sizes do not match the sizes of a forming tool. During storage and service of the product, the technological stresses relax due to its rheonomous properties. In this case, two processes take place. First, because of the relaxation of the residual stresses and a decrease in their general level, elastic aftereffect that appears right after the product has been manufactured decreases, which leads to a change in the product sizes. On the other hand, the technological stresses cause the material creep, which may result in an increase in the product sizes. The relations determining the magnitude of the residual stresses have the form

εˆ r = −D−1 σˆ r

(7.48)

where σˆ r is the tensor of the residual technological stresses. At the simultaneous action of the above-indicated service factors on the product, the deformation of the latter can be given in the form in accordance with Eqs. (7.41)– (7.48) Pg ˆ I − D−1 σˆ r + D−1σˆ + R p ∗ σˆ + Rc ∗ σˆ εˆ = αT ∆T Iˆ + ε0w Iˆ + 3K

(7.49)

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Combining the summands in Eq. (7.49), which correspond only to volume deformation, we obtain (7.50) εˆ = D−1 (σˆ − σˆ r ) + R p ∗ σˆ + Rc ∗ σˆ + γw Iˆ where

Pg (7.51) 3K is a phenomenological function that takes into account the inﬂuence of temperature, swelling processes and gas release on the product deformation during its service and storage. Let us ﬁnd the stress from Eq. (7.50)

γ = αT ∆T + ε0w +

˜ εˆ − P˜ ∗ σˆ − R˜ ∗ σˆ − γw I) ˆ + σˆ r , σˆ = D(

(7.52)

Equation (7.52) closes the system of Eqs. (7.4), (7.21), (7.36)–(7.39), which is supplemented by the boundary conditions for solving the diffusion and force problems included in it: C1 (0, x) ¯ = C10 (x), ¯

C2 (0, x) ¯ = C20 (x), ¯ ...

Cn (0, x) ¯ = Cn0 (x), ¯

x¯ ⊂ Ω

∂ Ci k ; = 0, x ⊂ Sk ⊂ S1k S2k S3k . . . Sm ∂n

∂ Ci = 0, ∂n

x¯ ⊂ SL ⊂ S1L

S2L . . .

SiL

C1 (0, x) ¯ = C11 (x), ¯ C2 (0, x) ¯ = C21 (x), ¯ . . .Cn (0, x) ¯ = Cn1 (x), ¯ x¯ ⊂ Sc

σˆ (0, x) ¯ = σˆ 0 (x); ¯ εˆ (0, x) ¯ = εˆ0 (x), ¯ un = 0,

τn = σn f ,

x¯ ⊂ Ω; x¯ ⊂ Sk ;

σn (t, x)l ¯ n = 0, x¯ ⊂ Sc ∪ SL ;

(7.53)

where σˆ 0 (x), εˆ0 (x), C10 (x), C20 (x), . . . , Cn0 (x) are initial distributions of tensors, stresses, deformations and gaseous substances in the powder body; ln are the direction cosines of the normal to the surface; un , τn and σn are displacements over the normal, tangential and normal stresses on the contacting surface, respectively; f is a friction coefﬁcient; n is a normal to the surface; Ω is a body volume; Sc is free surfaces; SL is isolated surfaces; Sk is surfaces contacting with other bodies.

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7.5 Numerical Examples Let’s consider the use of the mathematical models offered above for the solution of different problems of a nanopowder pressing. Numerical realization of the problems of pressing is carried out by a ﬁnite element method [6, 27]. In details, the application of the ﬁnite element method in the problem of powder pressing is given in paper [25].

7.5.1 The Calculation of Pressing in an Elastic Die To exemplify the inﬂuence of the die plasticity on the stressed-strained state of the body being compacted, let us consider the case of the compaction of a powder composite using a blind die. In Fig. 7.7a, the scheme (left) and the corresponding ﬁniteelement representation (right) of the object considered are displayed. The calculations have been carried out at the ratio of h/d = 1.5, where h is the height and d is the diameter of the pressing at the end of forming. The sizes of the rest bodies are shown in proportion. The thickness of the contact layer is 0.5 mm. For the material of the puncheon and the die, the value of the elastic modulus is 105 Mpa, and Poisson’s ratio is 0.3. The parameters of the contact layer are determined in process of the numerical experiment [20].

Fig. 7.7 The ﬁnite element scheme of a system (a): 1 – the pressing, 2 – the die, 3 – the puncheon, 4 – contact layer. Diagrams of stresses and displacements in a system (b) (−− σ˜ z −− σ˜ ϕ −◦− σ˜ r − − τ˜rz , −•− u). Stresses are given in fractions of maximal pressing pressure; displacements are given in meters. The scale is indicated in the ﬁgure

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Fig. 7.8 The scheme of the radial-axial pressing (a) and its ﬁnite-element model (b)

The curve of the material strain within the equivalent coordinates [10, 19] has the p following values of the constants: σeq = 7.15 MPa; b1 = 434.5 MPa and b2 = 0.649 p (the strain εeq is given in fractions of unity). The parameter nθ characterizing the shapes of pores of the material [19] is 0.3. The friction coefﬁcient on the “presstool – powder composite” surface was determined by the formula f = 0, 25 − 7 · 10−5 · σ0k , where σ0k is a mean pressure in the elements of the contact layer. The pressure on the puncheon was increased proportionally and at the end of pressing it was 120 MPa. The problem solution for each loading was found by the method of variable parameters of elasticity. At the ﬁrst step over the loading for all the elements constituting the bulk of the powder body, the same initial porosity θ = 0.36 was given. In Fig. 7.8b, the stressed-strained state is shown for the system ‘tool – compacted body” at maximal pressing pressure P = 120 MPa. The epures of stresses are shown in the fractions of P, i.e. σ˜ = σ /P. The suitability of the algorithm offered for the solution of the problem of contact is conﬁrmed by the following results. In the section A–A, the radial stress σ˜ r , which is normal to the contact surface of the bodies, is continuous when it passes through the lateral contact layer, and it changes monotonically in the diametrical direction from practically constant value in the pressing to zero on the outer surface of the die. In the same section, the hoop stress σ˜ ϕ that is compressive in the pressing becomes tensile in the die, when it passes through the contact layer (CL). The distribution of stresses σ˜ r and σ˜ ϕ thus obtained in the die section A–A is in good agreement with the values calculated analytically from Lam´e formula. In the section considered, the change in the axial stress σ˜ z also corresponds to the physical nature of the process, i.e. the compaction of the powder

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body takes place at the compressive stresses σ˜ z ; in the lateral CL, these stresses are close to zero, i.e. the CL does not prevent free relative displacements of the nodes of ﬁnite elements in the pressing and the die. At the left of Fig. 7.8b, the change in the axial stress σ˜ z is shown in the direction of the axis z. Under the puncheon, this stress has a maximal value in the “angular” point of the cylinder pressed. The average value of σ˜ z is close to the pressing pressure value in the layer of the pressing adjacent to the puncheon. In the vicinity of the lower end of the pressing, σ˜ z , on the contrary, has a maximal value in the center of the end, which decreases toward the “angular” points due to the friction on the die surface. When passing through the CL from the pressing to the die, the axial stress is continuous, i.e. the CL transfers the axial pressure to the entire area of the die base. Due to this fact, σ˜ z decreases in the die base since the area taking up the axial pressure increases. In the pressing layer and the die base layer adjacent to the pressing layer (they are separated by the CL), the areas of the epures of the stress σ˜ z differ only by 3.36%. The contact layer also allows modelling real distribution of tangential stresses under the puncheon. In Fig. 7.7, it is seen that in the region of the maximal shear of the material, τ˜rz is maximal, and in an “angular” point and in the center of the product end, τ˜rz is practically zero. It is of interest to trace the change in the article proﬁle (Fig. 7.7). The calculation shows that at pressing with regard for the elasticity of the die, its form is distorted. For example, in the upper part of the cylinder, the diameter becomes larger than in its base. In conclusion, it should be noted that the use of the idea of a contact layer also allows modelling the separation of the bodies on the contact surface. In this case, when positive normal stresses appear on the contact layer surface, the elastic constants of the CL are speciﬁed close to zero. This algorithm was used for solving the problem of unloading and squeezing.

7.5.2 Unloading After Radial-Axial Pressing Figure 7.8 displays a scheme of the radial-axial compaction with the use of a blind die (a) and its ﬁnite-element model (b). The scheme of the compaction allows producing thin-wall tubes from powders. In this case, the most important stage of forming a thin-wall tube is its release from the press-tool since in the process of unloading, residual stresses can cause product fracture. It is possible to remove the press-tool in different ways. Figure 7.10 shows the ﬁnite-element calculations of the change in the stressed state of an insert produced by the radial-axial compacting of the Nb- and Ta-based powder metal-composite at different unloading: (a) the rod is squeezed “upwards” and the die is squeezed “downwards”; (b) both the rod and the die are squeezed “upwards”; (c) the rod is squeezed “downwards”, and the die “immediately” opens; (d) the rod is “immediately” removed, and the die goes “down”.

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For each calculation case, the “line” of the change in the stressed state is traced from the starting point of unloading (the points Sa , Sb , Sc , Sd , respectively) to its end in the most loaded point (the point F in all the cases) according to Mirolubov’s criterion. This point is selected by the comparison of the maximal equivalent stress for all the ﬁnite elements in the calculation scheme at each step of unloading. The analysis of the calculation results shows that the unloading schemes “a” and “b” are the most inappropriate. In these cases, on gradual release of the pressing, in accordance with Mirolubov, the equivalent stresses exceed the ultimate stress of the metal-composite, which leads to the fracture of the pressing even before its complete removal from the press-form. The squeezing scheme “a” is especially unfavorable. Here, the unloading path mostly goes through the region where the equivalent stress is much higher than the ultimate stress (Fig. 7.9a). It should be noted that since for all the schemes, the stressed state of the compacted body is the same at the end of unloading, the analysis of the line of the change in the stressed state under unloading only allows to choose the optimal scheme for the removal of the article from the press-tool.

Fig. 7.9 Trajectories of unloading depend on a mode of clearing of a die after a radial-axial pressing

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7.5.3 Compaction with the Use of the “Active” Frictional Forces Compaction based on the use of “active” frictional forces is one of the promising methods for producing powder metal-composites, especially nano-powder ones. In particular, the investigations of the compaction of the Nb- and Ta-based powders, which are used for compacting high-capacity condensers, show that it is impossible to obtain the desired uniformity of porosity in ordinary dies. Large forces of friction prevent it. Thus, it is necessary to use these forces for compaction, which will allow to create the so called “ﬂoating” dies. The essence of the method lies in the fact that when the frictional forces reach a certain value, additional movements of the press-tool are switched on. As an illustration, let us consider the compaction of a cylinder pressing, the ﬁnite-element scheme of which is displayed in Fig. 7.10. In this scheme, in the beginning of pressing the powder nanocomposite (1) is pressed due to movement of the top puncheon (2). Than, the die (3) begins to move downwards at a certain pressing pressure owing to a resilient member (spring 4) supporting it. In this case, the single-action compaction scheme becomes doubleaction due to the relative movement of the die and the bottom puncheon (5). As the calculations show, when this scheme of compaction is used, the distribution of the residual stresses and physical-mechanical characteristics of the material of the pressing become essentially uniform. Figure 7.11 shows the plots of the porosity change along the cylinder axis in the section a-a (see Fig. 7.11b) at single-action pressing (a) and at pressing with a ﬂoating die (b). In the case of single-action compaction, the porosity distribution is not uniform over the height of the pressing. The top of the article is compacted stronger than its bottom. The experimental researches which have been carried out on NANOTEST600 have shown essential change of an elastic modulus of Young on lateral face of an article along its axis at single-action pressing.

Fig. 7.10 “Floating” die (a) for pressing PNCs and ﬁnite element model (b): 1 – PNC; 2 – top puncheon; 3 – die; 4 – spring; 5 – bottom puncheon

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Fig. 7.11 The change of porosity along the section A–A (Fig. 7.10): a – single-action pressing. b – pressing with the use of a ﬂoating die

Using the active die allows to make the porosity uniform along the axis of the pressing. Experiments have shown that the elastic modulus in this case changes insigniﬁcantly, and there are no cracks on a surface of an article.

7.6 Conclusions Based on the models built, numerical schemes, algorithms and programs were developed for the analysis of evolution processes in powder systems during powder obtaining, PNC compacting and use. Numerical investigations of the evolution of residual stresses in PNCs under compaction were carried out. The inﬂuence of the contact between a powder mixture and a press-tool, external friction, powder properties, the size and form of a product manufactured on the level and distribution of internal stresses was investigated. This allowed determining the regularities of the powder compaction and the unloading and relaxation processes in powders and establishing reasons for possible product failure at any stage of its manufacturing. The results of modelling and numerical analysis were used for the development of new equipment and procedures that allowed decreasing the level of residual stresses, increasing the homogeneity of compaction and enhancing the strength of powder composites. Acknowledgements The Russian Foundation of Basic Research supported this work: project No 07-01-96015Ural a and project No 05-08-50090.

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References 1. Andreeva AV (2001) Fundamentals of physics-chemistry and technology of composites. IPRJR, Moscow 2. Andrievski RA, Ragulia AV (2005) Nanostructural materials. Academia, Moscow 3. Brooks BR, Bruccoleri RE, Olafson BD, States DJ, Swaminathan S, Karplus M (1983) CHARMM: A program for macromolecular energy minimization, and dynamics calculations. Journal of Computational Chemistry 4: 187–217 4. Case DA, Cheatham TE, Darden T, Gohlke H, Luo R, Merz KM, Onufriev A, Simmerling C, Wang B, Woods R (2005) The Amber biomolecular simulation programs. Journal of Computational Chemistry 26: 1668–1688 5. Cagin T, Che J, Qi Y, Zhou Y, Demiralp E, Gao G, Goddard WA (1999) Computational materials chemistry at the nanoscale. Journal of Nanoparticle Research 1: 51–69 6. Connor JJ, Brebbia CA (1977) Finite element techniques for ﬂuid ﬂow. Newnes-Butterworths, London/Boston 7. Diao J, Gall K, Dunn ML (2004) Atomistic simulation of the structure and elastic properties of gold nanowires. Journal of the Mechanics and Physics of Solids 52: 1935–1962 8. Gusev AI, Rempel AA (2001) Nanocrystalline materials. Physical Mathematical Literature, Moscow 9. Geckeler Kurt E (2005) Novel supermolecular nanomaterials: from design to reality Proceedings of the 12th International Conference on Composites/Nano Engineering; CD-ROM edition 10. Green P J (1972) A plasticity theory for porous solids. Journal of Mechanical Science 14: 215–224 11. Hari SN (2002) Handbook of nanostructured materials and nanotechnology. Academic, San Diego, CA 12. Hoare MR (1987) Structure and dynamics of simple microclusters. ACH Models in Chemistry and Physics 40: 49–135 13. Holian BL (2003) Formulating mesodynamics for polycrystalline materials. Europhysics Letters 64: 330–336 14. Koch CC (2002) Nanostructured Materials – Processing, Properties, and Potential Applications. William Andrew, Norwich, NY 15. Kaygorodov AS, Ivanov VV, Paranin SN, Nozdrin AA (2007) The role of adsorbents in pulsed compaction of oxide nanopowders. Russian Nanotechnology 2: 112–118 16. Kompis V, Kompis M, Kaukic M, Hui D (2006) Singular Trefftz functions for modelling material reinforced by hard particles. In: Topping BHV, Montero G, Montenegro (eds) Proceedings of the Fifth International Conference on Engineering Computational Technology, CD-ROM Paper 184, Civil-Comp Press, Stirlingshire 17. Morris DG (1998) Mechanical Behaviour of Nanostructured Materials. Trans Tech Publications, Uetikon-Zurich 18. Ozawa E (1986) Properties, production methods, use and application of ultra dispersed powders. Journal of the Japan Society for Technology of Plasticity 27: 1166–1172 19. Petrosian G.F. (1988) The plasticity deformation of powder materials. Metallurgy, Moscow 20. Ruoff RS, Pugno NM. (2004) Strength of nanostructures. Proceedings of the 21st International Congress of Theoretical and Applied Mechanics, 303–311 21. Thompson RA (1981) Mechanics of powder pressing. I.-III Model for powder densiﬁcation. American Ceramics Society Bulletin 60: 237–243 22. Vakhrouchev AV (2006) Simulation of nano-elements interactions and self-assembling Modelling and Simulation in Materials Science and Engnineering 14: 975–991 23. Vakhrouchev AV (2007) Computer simulation of nanoparticles formation, moving, interaction and self-organization. Journal of Physics. Conference Series 61: 26–30 24. Vakhrouchev AV, Lipanov AM (1992) A numerical analysis of the rupture of powder materials under the power impact inﬂuence. Computer and Structures 44: 481–486 25. Vakhrouchev AV, Vakhroucheva LL (1992) The ﬁnite element analysis of the powder materials compression. Proceedings of NUMINFORM’92, 887–892

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26. Vakhrouchev AV, Vakhrouchev AA.(2006) Computer Simulation of Nanoelements Formation, Interaction and Self-organization. In: Topping BHV, Montero G, Montenegro (eds) Proceedings of the Fifth International Conference on Engineering Computational Technology, CDROM Paper 875, Civil-Comp Press, Stirlingshire 27. Zienkiewicz OC (1971) The ﬁnite element method in engineering science. McGraw-Hill, New York

Chapter 8

Modelling of Fatigue Behaviour of Hard Multilayer Nanocoating System in Nanoimpact Test Magdalena Kopernik, Lechosław Tre¸bacz, and Maciej Pietrzyk

Abstract The chapter presents the basis of the nanoimpact test and the idea of prediction of fracture occurrence, especially fatigue behaviour of nanoimpacted materials. The example of typical nanoimpact test is illustrated. The ﬁrst part of the investigation is focused on the development of the ﬁnite element (FE) model, which is applied to identiﬁcation of the material model of hard nanocoating in the multilayer system. Beyond this, the model is used to describe the friction contact between die and specimen, the shape of the die, the critical and required number of layers and dimension of investigated domain. Searching for the answers on formulated topics is realized using the inverse and the sensitivity analysis. Selected results indicating the problems with computing time in simulation of nanotests for the multilayer systems are presented in the chapter, as well. The FE model of the test with optimized parameters was developed and the results of chosen macrocrack criterion are shown. In the second part of the work the new approach to analysis of fracture phenomena is introduced as the fatigue criteria, which are used in simulation of nanoimpact test. The review of known fatigue criteria and analysis of fatigue behaviour are presented next and the selection of four of the criteria is justiﬁed. Finally, the results of simulations with the fatigue criteria implemented into the FE code for the analysed problem are shown and commented.

8.1 Introduction The impact test performed for mass materials is known as the Charpy impact test. It is a standardized high strain rate test, which determines the amount of energy absorbed by a material during fracture. This absorbed energy is a measure of the considered material toughness and it acts as an aid to study brittle – ductile Magdalena Kopernik, Lechosław Tre¸bacz, and Maciej Pietrzyk Department of Applied Computer Science and Modelling, AGH University of Science and Technology, Mickiewicza 30, 30–059 Krak´ow, Poland [email protected], [email protected], [email protected] V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

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transition. Since it is easy to prepare and conduct, this method is widely applied in industry and results can be obtained quickly and cheaply. But a major disadvantage is that all results are only comparative. The nanoimpact test technique is used in the present work to predict fracture resistance of hard nanocoatings. This test is also called high strain rate indentation. It is not possible to use conventional techniques, such as, for example, tensile testing, to investigate the ductile – brittle behaviour of nanocoatings. The nanoimpact test was developed to extend the capability of depth – sensing nanoindentation [14, 33]/scratch instrumentation [15] and to perform fatigue testing in nanoscale. This instrumented test enables repetitive contact testing of the fatigue properties of coatings and allows investigation of failure mechanisms, as well as comparison of fatigue resistance, depending on the applied deposition process. The objective of the present chapter, which is a continuation of the work [26], is to develop the FE (ﬁnite element) model of nanoimpact test for the multilayer material system with implemented fatigue criteria and to use all Authors previous experiences in numerical modelling of hard coatings to optimize the simulated process. The advantage of the new approach to fracture of hard coatings using fatigue criteria is a more accurate description of material behaviour during cyclic deformation with stress relaxation in unloading stage of each impact.

8.2 Nanoimpact Test 8.2.1 The Idea of the Experimental Nanoimpact Test 8.2.1.1 Basic Principles The principle of nanoimpact test is based on a pendulum set-up. High and low cycle fatigue tests over a wide range of contact stresses, frequencies, test durations and strain rates are possible. High cycle fatigue tests are performed by oscillating the sample, what causes the diamond probe to repetitively impact the surface at high frequency until failure. Low cycle fatigue tests are performed by providing a controlled impulse to the diamond via solenoid (see Fig. 8.1a). Impact testing allows the identiﬁcation of: adhesive failure – failure by a single abrupt jump in probe depth (ﬁlm removal), cohesive failure – failure by a series of several smaller jumps (fracture/partial delamination) and mixed failure modes. Coatings are ranked in terms of: life time (durability), type and energy to failure. The impact test records the evolution of impact induced damage with time (i.e. elastic and plastic deformation, fatigue wear and fracture) during repetitive impact of a test probe. The general response of a brittle material to repetitive contact in the nanoimpact test can be summarized as follows [4, 5]: 1. Initial impact – some plastic deformation occurs and cracks begin to nucleate subsurface

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Fig. 8.1 (a) Schematic illustration of the Nano Test System showing the conﬁguration for impact testing [5], (b) the real view of Berkovich indent in material [15]

2. Fatigue – growth of crack subsurface and further nucleation with little or no change in probe depth 3. Crack coalescence and fracture – rapid change in probe depth 4. Repetition of the fatigue cycle – long periods separated by abrupt fracturing events Initial probe penetration does not involve dramatic ﬁlm fracture. Plastic deformation of the subsurface is common in the initial stages of impact wear. As in nanoindentation [14, 33], the maximum shear stress is rather below than at the surface. Initial plastic deformation of the substrate is followed by a fatigue process that ultimately leads to ﬁlm fracture, without signiﬁcant additional depth change occurring until the onset of fracture. This suggests that fatigue failure occurs by a brittle process (crack formation, growth and coalescence) rather than further plastic deformation. Mechanisms of ﬁlm failure under fatigue conditions include: 1. Transition from stable to unstable crack growth, leading to fast fracture 2. Cracks reaching the ﬁlm–substrate interface and causing spallation 3. The ﬁlm bending stress being exceeded The fracture probability varies during the impact test (number of impacts), depends on impact load and thickness of coating. The ﬁlms resistance to impact induced fracture is compared by: probe depth (i.e. surface position) at the end of the test, presence or absence of clear surface failure/fracture, time at which ﬁrst fracture occurs (time-to-fracture) and probability of fracture within a given time.

8.2.1.2 Materials Investigated in Nanoimpact Test The hard coatings are investigated in experimental nanoimpact test and most of them are compositionally graded coatings. These tests were performed for:

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(a) Super-hard form of DLC–tetrahedral amorphous carbon (ta–C) ﬁlms [5], which are deposited on silicon by the ﬁltered cathodic vacuum arc (FCVA) technique. They have unique properties, such as high hardness, low coefﬁcient of friction, thermal stability, high electrical resistivity and transparency in a wide spectral range. These coatings may be applied in the hard disc industry. (b) The amorphous carbon (a–C), hydrogenated diamond-like carbon (a–C:H, DLC) and CrAlTiN coatings [2, 4], which are deposited on tool steel using the closed ﬁeld unbalanced magnetron sputter ion plating (CFUBMSIP) process. The results are that the performance of the CrAlTiN system depends on the impact load. Due to its graphite-like structure, the a–C showed greatly improved impact resistance compared to the a–C:H coating. (c) The monolayer TiCN, bilayer Al2 O3 –TiCN and trilayer TiN–Al2 O3 –TiCN coatings [3] are deposited on cemented carbide by CVD method. These coatings offer promising opportunities for combining layers with different functionality (e.g. thermal barrier, oxidation resistance) together with larger overall coating thickness and hence load-carrying capability. Particularly, these systems are effective in combining a range of properties including high hardness, wear resistance and toughness, to achieve good performance over a wider range of wear situations. (d) Ti1−x Alx N (x = 0.5, 0.67) [6] are deposited on cemented carbide at 25–500 ◦ C by PVD (physical vapour deposition) method. Life of the cutting tool life made of this material was studied under conditions of machining of the structural AISI 1040 steel. A correlation was found between the results of the rapid nanoimpact test and machining tests. The coating protects the cutting tool surface against the chipping that is typical for cutting operations with intensive adhesive interaction with workpiece materials, such as machining of Ti-based alloys. The results give encouragement that the elevated temperature nanoimpact test can be used to predict the wear and fracture resistance of hard coatings during machining.

8.2.1.3 Specimen The specimen examined in this project is composed of 11 hard nanocoatings. This material system consists of titanium nitride basis, thin mixed elastic (coating 1– 400 nm thick, repeated six times) and elastoplastic (coating 2–40 nm thick, repeated ﬁve times) layers deposited by PVD. Examined material system (Fig. 8.2) has respected length scale.

8.2.1.4 Selected Experimental Example The experimental nanoimpact test, which was simulated in this work, was performed by the Authors of [5] and was analysed for prediction of the fracture properties of tetrahedral amorphous carbon (ta–C) thin ﬁlms deposited on silicon by the ﬁltered cathodic vacuum arc method. The details and conditions of the test are given in [5]

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Fig. 8.2 Investigated system of hard nanocoatings

and are repeated brieﬂy below as the reference point for the used conditions of numerical simulation of the nanoimpact test. The pendulum impulse impact option of the Nano Test System was used for the nanoimpact testing. A solenoid connected to a timed relay was used to produce the probe impacts on the surface, as it is shown schematically in Fig. 8.1a. A blunt (tip radius: 500 nm) Berkovich pyramidal diamond indenter (Fig. 8.1b) [15] was accelerated from a distance of 12 mm from the surface to produce each impact with the force of 300 mN. The experiments were computer controlled, so that repetitive impacts occurred (at the same position) every 4 s. Twenty repeated tests, each of 300 s duration, were performed at different locations on each sample. In addition, the inﬂuence of the impact load on the time to failure was investigated for the 60 nm thick ﬁlm. Ten repeated tests were performed at each of the chosen impact loads (0.1, 0.2, 0.3 and 2.0 mN).

8.2.2 Simulation The present subsection is dedicated to the nanoimpact test, but the experience in FE modelling, as well as inverse and sensitivity analysis was gained mainly in earlier research on more common nanoindentation tests [19, 20]. The typical nanoimpact test is used when conventional fracture and/or fatigue are considered.

8.2.2.1 FE Models – Challenges The FE simulations were performed by the Authors of the present work in earlier projects for nanoindentation [19, 20] and nanoimpact [22, 26] tests. Those papers are focused on overcoming difﬁculties, which occur in the FE simulation of multistage deformation of hard nanocoatings. Therefore, the chosen by the Authors

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FORGE2 code, which can be applied to this task by introducing new procedures and modiﬁed existing ones, is used here as a proper FE platform. The mentioned above numerical problems arise from very small thickness of layers, small scale, necessity of remeshing, very contrasting physical properties and geometrical dimensions in adjacent layers, multi-material and multi-impact character of time-consuming simulations. In the paper [22], the numerical simulations of the nanoimpact tests are described for the two material systems. Three layer TiN–Al2 O3 –TiCN [3] coating deposited on cemented carbide by chemical vapour deposition (CVD) method (system 1) and multilayer TiAlN/Ti [6] deposited on cemented carbide by PVD method (system 2) were investigated [22]. The parameters of material models were such as those observed in literature [3, 6]. The predictive capability of the FE model has been conﬁrmed and possibility of its application to the interpretation of the nanotests was proved. Applying the same test conditions for both cases allowed comparison of the two systems. In the hard coating system 2 stresses are one order of magnitude greater than in the hard coating system 1. In the second system maximum stresses are inside specimen and in the ﬁrst system they are close to the surface. In the hard coating system 2 there is a greater contribution of plastic zones than in the hard coating system 1. Since it was capable to predict realistic stress and strain ﬁelds in the test, the developed FE model was prepared to implementation of the fracture criteria. Large number of coatings can be used in the nanocoating systems. Such a large number of layers in the FE simulation leads to a signiﬁcant increase of number of nodes in the mesh and, in consequence, to high computation costs [24] (Fig. 8.3). It is shown in [25] that increasing number of layers above 7 does not change noticeably the response of the system (Fig. 8.4). Therefore, further simulations are limited to seven layers only. In spite that the real shape of the indenter requires 3D model, assumption of the 2D axisymmetric approximation is common and it does not cause a loss of important information. The nanoindentation test simulations performed by the Authors in [21] additionally proved that the full 3D FE results are comparable to those obtained from the axisymmetric 2D FE model (Fig. 8.4b).

Fig. 8.3 Results for systems with coarse meshes (a) computing time versus number of coatings, (b) number of nodes versus number of coatings

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Fig. 8.4 (a) Work calculated for six cases with the following number of coatings (total/coating 1/coating 2) 1: 1/1/0; 2: 3/2/1; 3: 5/3/2; 4: 7/4/3; 5: 9/5/4; 6: 11/6/5, (b) force versus depth results for 2D FE axisymmetric and 3D FE models

8.2.2.2 Inverse and Sensitivity Analysis The material model of coating 1 of investigated material system was measured in experimental nanoindentation test in [2]. Material model of coating 2 was obtained using inverse algorithm developed by the Authors for the numerical nanoindentation test [19]. The load measured for the eleven layer system is the main output from the experiment (force versus displacement for the whole material system and/or force versus depth for indented coating), which is indirectly used in the objective function of the inverse analysis (Fig. 8.5a) in [19]. Eleven hard nanocoatings are deformed in numerical nanoindentation, multistage test (20 stages) using Berkovich indenter. In this system coating 1 is elastic (E = 368 GPa, ν = 0.177) and coating 2 is an elastoplastic with model: σp = K ε n (E, K, n are the searched parameters). Flow chart of the identiﬁcation of material model of nanocoating 2 is based on hardness and is shown in Fig. 8.5b. Martens hardness HM [19] of coating 1 is deﬁned as: HM =

F F = As (h) 26.43h2

(8.1)

√

where: As (h) = 3 cos3tgαα h2 , As – indent area, h – depth of indent, α – Berkovich tip angle, F – loading force. Depth of indent in simulation is measured on the FE mesh or using AFM (atomic force microscope) in experiment. Optimization variables (E, K, n for the coating 2) are determined by searching for the minimum of the goal function [19] deﬁned by the Martens hardness of coating 1:

! ! 1 N HMc(i) − HMm(i) 2 " φ= ∑ N i=1 HMm(i)

(8.2)

where: HMc(i) , HMm(i) – calculated and measured Martens hardness, respectively, N – number of sampling points.

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Fig. 8.5 (a) General algorithm of the inverse analysis, (b) block diagram for the identiﬁcation of material model of hard nanocoating 2 and mechanical properties of system of 11 coatings

Minimum of function (8.2) is searched with respect to parameters of the elastoplastic material model for coating 2. This layer has no contact with the indenter, and its parameters are obtained using optimization procedures. Stop criterion is reached, when the predicted value of Martens hardness for the coating 1 is close enough to the experimental one. Hardness is determined from the calculated force–depth data.

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Fig. 8.6 Load sensitivity with respect to: (a) the material model parameters E, K, n, (b) friction coefﬁcient µ , geometrical tip parameters R and α for system of seven hard nanocoatings

Finally, determined material model of coating 2 giving minimum of the goal function (8.2) is K = 50 MPa, n = 0.1, E = 25 GPa, HMc = 31 GPa and Φ = 0.01. The sensitivity analysis is realized for the system of seven nanocoatings [24] (coating 1 is repeated 4 times and coating 2–3 times). It shows that the FE predictions are the most sensitive to the Young modulus of the elastic layer, but the parameter K of elastoplastic coating 2, called hardening coefﬁcient, is important, as well (Fig. 8.6a). The parameter n, which controls the slope of the ﬂow curve, is not so signiﬁcant but it also inﬂuences the output. The material model, which contains two parameters, has a disadvantage. It gives the goal function (Eq. (8.2)) in a form of a long valley with a small slope along this valley [23]. Thus, due to numerical errors it is difﬁcult to ﬁnd an unique solution. The sensitivity coefﬁcients ξ [24, 25], which are plotted in Fig. 8.6a and b, are deﬁned as:

ξ=

p F (p + ∆p) − F (p) F ∆p

(8.3)

where: F – force, p – considered parameter (K, n, E, etc.), ∆p – increment of the considered parameter. Sensitivity analysis was also performed for the load with respect to the tip dimensions (radius R and angle α of Berkovich indenter) and friction coefﬁcient µ . The analysis in [21], shown in Fig. 8.6b, proved that in modelling of the nanoindentation with Berkovich indenter, tip shape and Young modulus are the main factors, which inﬂuence the output: force vs. displacement data. The friction coefﬁcient does not require special attention and it can be omitted in the FE model of nanotests. In fact, both considered tests (nanoindentation [14, 33] and nanoimpact) are similar, only the value of indenter energy is different.

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8.2.2.3 Conventional Fracture and Crack Sensitivity Crack sensitivity analysis of the nanoimpact test is based on results of previous work [26], which are discussed brieﬂy below. Numerical test conditions should reproduce those in the laboratory test, but there are few differences, mainly due to some simpliﬁcations. The conditions of the tool are: constant velocity v and initial height h, variable falling mass m and kinetic energy EK . Since impact energy EIMP is equal to potential energy EP = mgh, the falling mass m [22] is calculated as: EIMP = EP ⇒ Fs = mgh ⇒ m =

Fs gh

(8.4)

Several macrocrack criteria were investigated in [26] and the McClintock criterion [30], speciﬁed also in [1, 32], turned out to be the best, due to the highest sensitivity with respect to the impact energy. Examples of its applications and the obtained intervals of its values are presented in Table 8.1. These types of the fracture macrocrack criteria are not recommended to the analysis of nanotest, but the main results and conclusions reached in work [26] are shown below to enable qualitative evaluation. The empirical macrocrack criteria are based on macroscopic parameters. It is assumed that crack occurs when value of crack criterion is higher than certain critical constant. Crack criteria are mostly calculated using integral function [26], which represents cumulative effect of process parameters during deformation: ε (t)

f (process parameters) d ε ≥ C

(8.5)

0

The integrand (8.5) depends on stress, strain and history of deformation. The mentioned criterion [30] was implemented into the FE code in [26] as: ε (t) 0

σm dε ≥ C σ¯

(8.6)

where: C – material constant, σ¯ – effective stress, σm – mean stress. The average values of McClintock crack criterion [26] for the three mean impact energies (1.05 – case I, 10.5 – case II and 105 nJ – case III) of six (1, 1.1, 10, 11, 100

Table 8.1 Review of applications of the McClintock crack criterion and its values Type of the test

Examined material

Value of criterion

Ref.

Tensile Single tension

Al 2024–T351 Two-phase structural steel; SM490YB (ferrite–pearlite) steel

0.38–0.62 0.3–1.5

[17] [18]

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and 110 nJ) during 20 impacts are shown in Fig. 8.7a. Distribution of McClintock crack criterion for the lowest value of energy is presented in Fig. 8.7b. The location of its maximum and minimum is marked by ∆ and ◦, respectively. Similar maximum equivalent strain in the considered system has already been observed in [20, 22]. Sensitivity of crack criteria with respect to the impact energy was investigated for the three mentioned mean impact energies. Sensitivity coefﬁcient (SF ) [26] is deﬁned as: EM ∆C SF = (8.7) CM ∆E where: EM – mean impact energy, CM – mean crack criterion, ∆C – value of crack criterion increment, ∆E – value of energy increment. The sensitivity analysis shows that, as it was mentioned, the highest value of sensitivity coefﬁcient is for McClintock criterion in the case I (Fig. 8.8).

Fig. 8.7 (a) Maximum values of average McClintock crack criterion in impact test for three mean impact energies, (b) distribution of McClintock criterion in the 20th impact of nanotest for the lowest impact energy

Fig. 8.8 Sensitivity of the McClintock crack criterion with respect to the set of impact energies

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8.2.3 Discussion Presented analysis allowed to choose 2D axisymmetric FE model for a specimen composed of seven layer hard coating system, initially covered by 4,643 nodes and 8,941 triangular elements, for the further analysis of nanoimpact test with implemented fatigue criteria. The frictionless contact condition between indenter and specimen is assumed. The Berkovich rigid indenter with tip radius R = 150 nm and angle α = 65.3◦ is used. The material model of coating 1 in the specimen is elastic (E = 368 GPa, ν = 0.177), periodically repeated four times. Model of coating 2, determined in inverse analysis, is elastoplastic: σp = 50ε 0.1 (E = 25 GPa, ν = 0.225), periodically repeated three times.

8.3 Fatigue Fatigue is a failure under a repeated load, which never reaches a high enough level to cause failure in a single load. Since virtually every manufactured product will be worn out or broken down, predicting fatigue life is a critical aspect of the design cycle. The critical issues are whether the product will reach its expected life and, if damaged, whether it will remain safe in service until the damage can be discovered and repaired. One can determine number of cycles to failure by using fatigue criterion. The objective of this work is to evaluate various criteria on the basis of theoretical analysis only, when the experimental data are not available. Four fatigue criteria are implemented into the FE code (Forge2) in this section and their efﬁciency is evaluated.

8.3.1 The Basis of Fatigue Behaviour of Materials The main principles of the material fatigue behaviour are described on the basis of works [8, 28, 34, 35]. Fatigue is the result of the cumulative process consisting of crack initiation, propagation and ﬁnal fracture of a component. During cyclic loading, localised plastic deformation may occur at plane, on which the highest stress is reached. This deformation induces permanent damage to the component and a crack development. As the number of loading cycles increases for the component, the length of the crack increases. After certain number of cycles, the crack causes the component to fail. Fatigue process consists of several stages: crack nucleation, crack growth and ﬁnal fracture. Crack starts on the shear plane in a place, where high stress concentrations occur, e.g. inclusions, persistent slip bands, porosity, discontinuities, etc. After nucleation, crack grows along the plane of maximum shear stress, until the damage of component occurs. Failure occurs by various physical mechanisms: low-cycle fatigue involves larger amounts of plastic deformation in relatively short life, while high-cycle fatigue is

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associated with low loads (stresses and strains are mainly conﬁned to the elastic region) and long life. According to [28] the low-cycle fatigue is associated with fatigue life less than 100,000 cycles and high-cycle fatigue is associated with life greater than 100,000 cycles. The major part of the fatigue life in the high-cycle regime is spent on crack propagation, but in the low-cycle fatigue most of the process is spent on crack initiation [28].

8.3.2 Fatigue Damage Theories The damage of component can be expressed in terms of an accumulation of the crack length (a) from the initial crack length (a0 ) toward the maximum acceptable crack length (a f ). Cumulative damage model [35] is expressed by: # $ n αf a 1 a0 + a f − a 0 D= = af af Nf

(8.8)

where: D – cumulative damage, N f – number of cycles applied to achieve the crack length at ﬁnal fracture, n – number of cycles applied to achieve the crack length, α f = 2/3N 0.4 f – empirical coefﬁcient. When value of D is equal to unity, the fatigue failure occurs. Another approach to prediction of failure, which uses the quotient of number of applied cycles ni to number of cycles Ni, f (i = 1, . . . , m, where m is the number of various stress level) is the linear damage mode. Palmgren created (1924) and Miner popularized (1945) similar solution for the fatigue with the various stress level. The formulated rule, called Miner rule [35], is the linear damage hypothesis. It states that failure occurs when: ni

∑ Di = ∑ Ni, f

≥1

(8.9)

k

where: k – number of different stress levels, ni (Si ) – number of cycles with the same stress level Si , Ni, f (Si ) – number of cycles to failure of a constant stress Si . Failure occurs for processes with various stress levels (see Fig. 8.9), when the sum of ratios at each stress level reaches the critical value. The fatigue life Ni, f is deﬁned as number of cycles of ﬂuctuating stress and strain, which material sustains before failure occurs. Fatigue life is a function of the ﬂuctuating stress, shape of the specimen and test conditions. It can be obtained using the baseline fatigue data generated from constant-amplitude loading tests. There are three methods, which are commonly used to characterise the baseline fatigue data: the stress-life (S − N) method, the strain-life (ε − N) method and the linear elastic fracture mechanics (LEFM) [13, 16, 38]. The methods use different techniques with different accuracy.

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Fig. 8.9 Fatigue life under various stress levels [28]

8.3.3 The Fatigue Criteria The fatigue criteria are divided into uniaxial or multiaxial loading and proportional (orientation of principal axis of loading is ﬁxed) or non-proportional loading (orientation of principal axis of loading changes). These criteria are classiﬁed into three groups, which are based on the critical value of stress, strain and energy. The examples of experimental fatigue tests are tension, torsion, upsetting, mixed tests, etc. The nanoimpact test investigated in the present work can be classiﬁed as a low fatigue tests with uniaxial and proportional loading. The idea of multiaxial fatigue criteria, which is based on the critical plane concept, is shortly presented in this subsection. This concept assumes that the fatigue failure is caused by certain stress or/and strain components acting at the critical plane. It is observed in experiments that in metallic materials fatigue cracks initiate and grow on certain planes. The general aim of the criteria is the reduction of the multiaxial stress state to an equivalent uniaxial one. A critical plane approach predicts both the fatigue life and the critical planes, where cracks are expected to initiate. At the beginning critical plane concept was used in the stress criteria, and further it was introduced into the strain criteria and then to the energy criteria [18, 37].

8.3.3.1 The Implemented Fatigue Criteria Large number of fatigue criteria is available, but only a few of them, which are implemented in the present work, are brieﬂy discussed below. The stress criteria assume that the principal factor deciding about the failure is stress and the amplitude or interval of this stress. For example, the Findley (F) criterion [37] belongs to these

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criteria and it is the linear combination of the amplitude of shear stress (τa ) and the maximum normal stress (σn,max ) at the critical plane:

τa + k1 σn,max ≤ λ

(8.10)

where: k1 (for example, k1 = 0.3 in [31]) and λ (fatigue limit) are experimental material constants. Papadopoulos (P) [37] proposed the criterion based on a linear combination of generalized shear stress amplitude (τa ) on the critical plane and the maximum value of hydrostatic stress (σa,h ):

τa + k2 σa,h ≤ λ

(8.11)

where: k2 and λ are experimental material constants. The Socie-Fatemi (S–F) criterion [37] belongs to the criteria, which are based on strains: σn,max γns,a 1 + k3 =C (8.12) σy This criterion is a combination of maximum shear strain amplitude (γns,a ) and the quotient of maximum normal stress at the critical plane (σn,max ) and yield stress (σy ). The parameters C and k3 (for example, k3 = 0.98 in [16], k3 = 0.1 in [12]) are experimental material constants. The critical plane is the plane of maximum shear strain amplitude. Smith, Watson and Topper (SWT) developed an experimental damage factor, which is based on density of strain energy [37]: SWT = σn,max ∆εa,max

(8.13)

It is a product of maximum normal stress on the critical plane (σn,max ) and amplitude of the maximum normal strain (∆εa,max ). In the FE models of impact test prepared in present work, it is assumed that all experimental constants k1 , k2 and k3 used in the fatigue criteria are equal to 1.

8.3.3.2 The Selected Relations Describing Fatigue Behaviour In literature [9–13, 27, 36, 38] the description of fatigue behaviour of materials is performed using different relations between speciﬁed below physical quantities versus number of cycles and/or time, because of cyclic character of the analysed phenomenon and the reasons of its occurrence. The relations between known, important in analysed problem physical quantities observed in various papers, are classiﬁed into three groups: (a) Stress vs. time, e.g.: tensile stress and stress intensity factor [36], shear stress and normal stress [12], equivalent stress [9, 10, 12], loading (force) [9, 10, 13]. (b) Strain and stress amplitudes vs. number of cycle, e.g. the amplitude of: shear strain [27], equivalent strain [11, 38], equivalent stress [11], shear stress [38].

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(c) Component of strain tensor versus component of stress tensor, e.g. shear strain versus shear stress [12], normal strain versus normal stress [12, 38]. This approach to the fatigue analysis is used in the present work. The number of cycles to failure is the most frequent experimental data and this information is mainly used to determine the fatigue behaviour of materials, their lifetime and critical values of components of stress and strain tensors, which lead to failure.

8.3.3.3 Results The values of implemented four fatigue criteria and selected fatigue relations were computed for the two energies equal to 1 and 10 nJ (case I and II) during 50 impacts. The FE model of nanoimpact test (Section 8.2.3) is used. The focus is on the analysis of impacted nanomaterial behaviour under low cyclic conditions. Popular fatigue relations and selected four theoretical fatigue criteria are used. Contrary to the experiment, they do not specify precisely when and where the specimen is damaged, because the number of cycle to failure or time to failure is inaccessible without experimental data. Therefore, the presented results have theoretical, predictive and prognostic character. The distributions of four selected fatigue criteria (after 25th impact and unloading stage) for the case I are shown in Fig. 8.10. The results presented as distributions of calculated values of S–F, F, P and SWT fatigue criteria in Fig. 8.10 indicate that the most probable location of fatigue is under indenter (in agreement with experimental observations in [9, 10]) and to some extent in the nanocoating 2 in the material system (in agreement with experimental conclusions in [29]). The best prediction of the location of fatigue failure is for S–F, F and SWT criteria, because both locations (under indenter and in the nanocoating 2) are visible. The results of chosen, implemented fatigue criteria are plotted vs. total time in nanoimpact test in Fig. 8.11. The oscillations of fatigue criteria result from the character of curve of each impact. The indenter deforms the material initially weakly, and further stronger, what increases the fracture probability. The value of criteria drops rapidly in the range of oscillation during unloading. The S–F criterion increases during impact–test, but after reaching certain number of impacts, it has a stable value. This stabilization is not seen well for the case II, but it can be concluded from the case I that it begins at 70th impact. The F criterion increases with time, due to the normal stress increase. According to earlier observation the probability of crack nucleation increases with time. For a higher number of cycles the character of curve for the F criterion could differ from the presented one, but in the range of performed simulations the F criterion still grows. Similarly to S–F, the SWT criterion grows with time, but after the 41st impact (what corresponds to the time of 1.54 s) the value of SWT drops, what is caused by the initial crack occurrence before 41st impact. This is conditioned by the SWT criterion, which reaches the critical value. Selected relations of fatigue behaviour for the considered cases are shown in Figs. 8.12–8.14. It is seen in Fig. 8.12 that the stresses and loadings are higher for the case II than for the case I, because the impact energy is higher for the case II. In Fig. 8.12b

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Fig. 8.10 Distributions of values of fatigue in specimen for case I after 25th impact for (a) S–F, (b) F, (c) P, (d) SWT criterion

(case II) the shear stress in the initial stage of process has higher oscillations than in Fig. 8.12a (case I). It is due to higher plastic deformation for the higher impact energy. The increment of equivalent stress (Fig. 8.12c and d) is higher than shear stress. It results from the higher increment of normal stress. It is shown in Fig. 8.12e and f that the loading measured on indenter has an uniform character for both cases of impact energies. For all cases II presented in Fig. 8.13 the values of strain and stress amplitudes are higher than for all plotted cases I, therefore, the impacted material is more subjected to failure in all former cases. The impact energy in all cases II is ten times higher than in all cases I, but the registered value of the amplitude after linear approximation (Fig. 8.13), is only two times higher. The amplitudes of stress and strain have the inﬂuence on material fatigue behaviour and its failure. Presented data show that material impacted by two times higher energy cracks after slightly lower number of cycles (but not two times lower).

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Fig. 8.11 The fatigue criteria versus total time (a) S–F, (b) F, (c) P, (d) SWT

Fig. 8.12 Selected relations of fatigue behaviour as shear stress versus time for case I (a) and II (b), equivalent stress versus time for case I (c) and II (d), loading versus time for case I (e) and II (f)

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Fig. 8.13 The selected relations of fatigue behaviour as (a) amplitude of equivalent strain versus time for case I, (b) amplitude of equivalent stress versus time for case II, (c) amplitude of shear strain versus time for case I, and (d) amplitude of shear stress versus time for case II

Fig. 8.14 The selected relations of fatigue behaviour as (a) shear strain versus shear stress for case I, (b) shear strain versus shear stress for case II

To improve the readability of plots, results for selected impacts only are presented in Fig. 8.14. During following impacts shear strain becomes lower and lower, which requires higher and higher stress. The simulations of nanoimpact test for computing the sensitivity coefﬁcients were performed for two mean impact energies: case I–1.05 nJ (mean of energy 1 and 1.1 nJ) and case II–10.5 nJ (mean of energy 10 and 11 nJ) during 50 impacts. The sensitivity coefﬁcients for all implemented fatigue criteria, calculated according to Eq. (8.7), are presented in Fig. 8.15. The highest sensitivities to impact energy are observed for SWT and S–F criteria. With increasing value of energy the criteria become more sensitive, what is well seen for the S–F criterion. For the lower energy, the sensitivities are unstable for subsequent impacts. Thus, for the higher energy the sensitivity coefﬁcients increase for following impacts. Presented approach helps to ﬁnd parameters of material model

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Fig. 8.15 Sensitivity of the fatigue criteria with respect to the impact energies (a) case I, (b) case II

and other process parameters for known values of fatigue criterion. The observed high sensitivities to impact energies support necessity to perform nanoimpact tests to determine the value of fatigue criteria.

8.4 Conclusions The fatigue criteria are better in prediction of nanocoatings failure than conventional macrocrack criteria, because the latter do not show maximum values under indenter. The main advantage of using the fatigue criteria is more accurate prediction of fracture localisation in comparison to standard criteria [26]. The difﬁculty with the implementation of the fatigue criteria in the commercial FE code (FORGE2) is the disadvantage of the former approach. It is a complicated operation and the computing time is high. It is due to using the amplitudes of strain and stress in each impact, what requires calculation of each stage of the process twice (once for the amplitude and the second time for the criterion). FE model of nanoimpact test with optimal number of layers, frictionless contact conditions, proper shape of indenter (radius and angle of tip, round tip) and identiﬁed material model of coating 2 (model without strain rate sensitivity and realistic

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values of parameters) is the main output of the project. It is characterised by acceptable number of nodes in the mesh, reasonable computing cost and quite good quality of results. Recapitulating, obtained values of fatigue criteria and the character of curves of selected relations of fatigue behaviour are similar to those published in the literature, e.g. three of four considered fatigue criteria localise the crack precisely, directly under indenter and in coating 2. The highest probability of fracture occurs in coating 1, especially in indenter–specimen contact region. The sensitivity analysis indicates that for the case I the highest values of sensitivity are computed for SWT fatigue criterion, but for the case II the highest sensitivity is reached for Socie-Fatemi criterion. This observations lead to the conclusion that the S–F fatigue criterion is the most adequate for the investigated problem. Acknowledgements Financial assistance of MNiSzW, project N507 136 32/396, is acknowledged.

References 1. Bao Y (2005) Dependence of ductile crack formation in tensile tests on stress triaxiality, stress and strain ratios. Engineering Fracture Mechanics 72:505–522 2. Beake BD (2005) Evaluation of the fracture resistance of DLC coatings on tool steel under dynamic loading. Surface and Coatings Technology 198:90–93 3. Beake BD, Ranganathan N (2006) An investigation of the nanoindentation and nano/microtribological behaviour of monolayer, bilayer and trilayer coatings on cemented carbide. Materials Science and Engineering A 23:46–51 4. Beake BD, Smith JF (2004) Nano-impact testing – an effective tool for assessing the resistance of advanced wear-resistant coatings to fatigue failure and delamination. Surface and Coatings Technology 188–189:594–598 5. Beake BD, Lau SP, Smith JF (2004) Evaluating the fracture properties and fatigue wear of tetrahedral amorphous carbon ﬁlms on silicon by nano-impact testing. Surface and Coatings Technology 177–178:611–615 6. Beake BD, Smith JF, Gray A, Fox-Rabinovich GS, Veldhuis SC, Endrino JL (2007) Investigating the correlation between nano-impact fracture resistance and hardness/modulus ratio from nanoindentation at 25–500 ◦ C and the fracture resistance and lifetime of cutting tools with Ti1−x Alx N (x = 0.5 and 0.67) PVD coatings in milling operations. Surface and Coatings Technology 201:4585–4593 7. Bernasconi A, Papadopoulos IV (2005) Efﬁciency of algorithms for shear stress amplitude calculation in critical plane class fatigue criteria. Computational Materials Science 34:355–368 8. Bolotin VV (1999) Mechanics of fatigue. CRC, Boca Raton, FL 9. Bouzakis KD, Vidakis N, Leyendecker T, Lemmer O, Fuss HG, Erkens G (1996) Determination of the fatigue behaviour of thin hard coatings using the impact test and a FEM simulation. Surface and Coatings Technology 86–87:549–556 10. Bouzakis KD, Vidakis N, David K (1999) The concept of an advanced impact tester supported by evaluation software for the fatigue strength characterization of hard layered media. Thin Solid Films 355–356:322–329 11. Chakraborti PC (2005) Room temperature low cycle fatigue behaviour of two high strength lamellar duplex ferrite–martensite (DFM) steels. International Journal of Fatigue 27:511–518 12. Chen X, Song J, Kim KS (2006) Low cycle fatigue life prediction of 63Sn–37Pb solder under proportional and non-proportional loading. International Journal of Fatigue 28:757–766 13. Constantinescu A, Charkaluk E, Lederer G, Verge L (2004) A computational approach to thermomechanical fatigue. International Journal of Fatigue 26:805–818

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14. Doerner MF, Nix WD (1986) A method for interpreting the data from depth–sensing indentation instruments. Journal of Material Research 1:601–609 15. Fisher-Cripps A (2002) Nanoindentation. Springer, New York 16. Jiang Y, Hertel O, Vormwald M (2007) An experimental evaluation of three critical plane multiaxial fatigue criteria. International Journal of Fatigue 29:1490–1502 17. Jindal PC, Santhanam AT, Schleinkofer U, Shuster AF (1999) Performance of PVD TiN, TiCN, and TiAlN coated, cemented carbide tools in turning, International Journal of Refractory Metals and Hard Materials 17:163–170 18. Karolczuk A, Macha E (2005) A review of critical plane orientations in multiaxial fatigue failure criteria of metallic materials. International Journal of Fracture 134:267–304 19. Kopernik M, Pietrzyk M (2007) Rheological model and mechanical properties of hard nanocoatings in numerical simulation of nanoindentation test. In: Cueto E, Chinesta F (eds) 10th ESAFORM Conference on Material Forming. AIP Conference Proceedings, Melville, New York, 907:659–664 20. Kopernik M, Pietrzyk M (2007) 2D numerical simulation of elasto-plastic deformation of thin hard coating systems in deep nanoindentation test with sharp indenter. Archives of Metallurgy and Materials 52:299–310 21. Kopernik M, Szeliga D (2007) Modelling of nanomaterials-sensitivity analysis to determine the nanoindentation test parameters. Computer Methods in Materials Science 7:255–261 ˙ 22. Kopernik M, Pietrzyk M, Zmudzki A (2006) Numerical simulation of elasto-plastic deformation of thin hard coating systems in nano-impact test. Computer Methods in Materials Science 6:150–160 23. Kopernik M, Stanisławczyk A, Kusiak J, Pietrzyk M (2007) Identiﬁcation of material models in hard system of nanocoatings using metamodel. In: Proceedings of 23rd IFIP TC 7 Conference on System Modelling and Optimization, Krakow (CD-ROM) 24. Kopernik M, Stanisławczyk A, Szeliga D (2007) Problems of material models of hard nanocoatings. In: Proceedings of the 17th International CMM Conference, Lodz, Spala (CD ROM) 25. Kopernik M, Szeliga D, Pietrzyk M (2007) Review of application of numerical methods to identiﬁcation material model of hard nanocoating. In: Onate E, Owen DRJ, Suarez B (eds) 9th COMPLAS Conf. on Computational Plasticity Fundamentals and Applications, International Center for Numerical Methods in Engineering (CIMNE) Conference Proceedings, Barcelona, Spain, 667–670 26. Kopernik M, Trebacz L, Pietrzyk M (2007) Simulation of fracture for hard multilayer nanocoating system in impact test. In: Proceedings of the 1st Conference Composites with Micro- and Nano-Structure (CMNS) – Computational Modeling and Experiments, Liptovsky Mikulas, 72–86 (CD-ROM) 27. Kuroda M (2001) Extremely low cycle fatigue life prediction based on a new cumulative fatigue damage model. International Journal of Fatigue 24:699–703 28. Lee Y-L, Pan J, Hathaway RB, Barkey ME (2005) Fatigue testing and analysis. Elsevier, Oxford 29. Li X, Bhushan B (2002) Development of a nanoscale fatigue measurement technique and its application to ultrathin amorphous carbon coatings, Scripta Materialia 47:473–479 30. McClintock FA (1968) A criterion of ductile fracture by growth of holes. Journal of Applied Mechanics 35:363–371 31. Norberg S, Olson M (2005) A fast, versatile fatigue post-processor and criteria evaluation. International Journal of Fatigue 27:1335–1341 32. Ohata M, Toyoda M (2004) Damage concept for evaluating ductile cracking of steel structure subjected to large-scale cyclic straining. Scientiﬁc Technology of Advanced Materials 5:241–249 33. Oliver C, Pharr GM (1992) An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiment. Journal of Material Research 7:1564–1583 34. Qian J, Fatemi A (1996) Mixed mode fatigue crack growth: a literature survey. Engineering Fracture Mechanics 55:969–990

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35. Suresh S (1991) Fatigue of Materials. Cambridge University Press, Cambridge 36. Wang P, Xu LR (2006) Dynamic interfacial debonding initiation induced by an incident crack. International Journal of Solids and Structures 43:6535–6550 37. Wang YY, Yao WX (2004) Evaluation and comparison of several multiaxial fatigue criteria. International Journal of Fatigue 26:17–25 38. Yang X (2005) Low cycle fatigue and cyclic stress ratcheting failure behavior of carbon steel 45 under uniaxial cyclic loading. International Journal of Fatigue 27:1124–1132

Chapter 9

A Continuum Micromechanics Approach to Elasticity of Strand-Based Engineered Wood Products: Model Development and Experimental Validation Reinhard Sturzenbecher, ¨ Karin Hofstetter, Thomas Bogensperger, Gerhard Schickhofer, and Josef Eberhardsteiner

Abstract Engineered wood products enjoy great popularity in structural engineering and are widely used for a variety of applications. To strengthen their competitiveness and to enlarge their range of utilization, in particular in the load-bearing sector, the mechanical properties of these products need to be improved. A validated micromechanical material model is very useful for the required optimization process, since it allows considering the inﬂuences of the relevant (micro-)characteristics, such as strand quality, strand geometry, strand orientation, and compaction during the production process, on the mechanical properties of the panels. The development of such a model, its experimental validation, and its application to selected parameter studies are presented in this contribution.

9.1 Introduction Engineered wood products are widely used in the ﬁeld of structural engineering. Strand- and veneer-based panels are most popular and are primarily used for nonbearing construction elements, such as roof, wall, and ﬂoor sheetings, because of their moderate mechanical properties. Extending the application range of strandbased panels by enabling their utilization for load-bearing purposes requires considerable improvement of their mechanical properties. This motivated research efforts aiming at the development of a homogeneous strand board with increased stiffness and strength by using large-area, ﬂat and Reinhard St¨urzenbecher, Karin Hofstetter, and Josef Eberhardsteiner Institute for Mechanics of Materials and Structures, Vienna University of Technology, Austria [email protected] Reinhard St¨urzenbecher and Gerhard Schickhofer holz.bau.forschungs Ltd., Austria Thomas Bogensperger and Gerhard Schickhofer Institute for Timber Engineering and Wood Technology, Graz University of Technology, Austria V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

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Fig. 9.1 Strand material and veneer strand board (VSB)

slender strands with uniform strand shape and dimensions (see Fig. 9.1). Their consistent and predictable properties render these strands a top quality raw material for panels with load-bearing capacity, which were called veneer strand boards (VSB) [12, 18]. A large surface area of the strands is crucial, since the strength of strand- or particle-based products tends to decrease with decreasing wood element dimensions when compared at the same density [19]. Large-area strands feature a sufﬁciently large contact area between adjacent strands to guarantee optimal bonding quality also at low resin content. Moreover, the homogeneity of the strand shape enables to produce strand boards with practically no pores already at medium densities of 600–650 kg/m3 . With conventional strand material, boards free of pores are only possible at very high density levels [28]. A high slenderness ratio, deﬁned by the ratio of strand length and strand thickness, is advantageous for the stress transfer between contiguous strands as shown by Barnes [1]. In order to optimize the new wood product, the effects of the strand characteristics on the mechanical performance of the veneer strand board have to be revealed. So far, such investigations were mainly performed by mechanical testing. Thereby the positive effect of strand slenderness [22, 32] and of strand length [30, 31] on bending stiffness and strength of strand boards could be conﬁrmed. The strand orientation, which is among the key parameters of strand boards, was addressed in numerous studies [5, 6, 20, 25–27, 38]. The best mechanical characteristics can be achieved with strongly oriented strands, however, at the cost of only moderate properties in the direction orthogonal to the principal one. The relation between the elastic moduli in the principal material directions of a board with aligned strands, E1O and E2O , and the elastic modulus in an arbitrary direction of a board with random strand orientation, E R , has been described either multiplicatively [Geimer [6], Eq. (9.1)] or additively [Zhou [38], Eq. (9.2)]: E1O · E2O = (E R )2 ,

(9.1)

1/2 · (E1O + E2O ) = E R .

(9.2)

Also the impacts of the vertical density proﬁle and of a layered build-up of the board have been intensively investigated [8, 25, 33, 34].

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Experimental work allows for identiﬁcation of parameters with dominant inﬂuence on the mechanical behavior of the boards, but it does not reveal the underlying mechanics. Moreover, it does not allow prediction of the properties of boards for non-tested conditions and conﬁgurations. This motivated the development of empirical and simple micromechanical models to estimate mechanical properties of strand-based materials. Lau [15] presented a regression model for the board stiffness considering the board density, the two in-plane moduli of elasticity of strands, and the strand orientation, characterized by a normal distribution. Mundy and Bonﬁeld [21] applied the rule of mixture, which is a simple volume averaging process, to predict stiffness and strength properties of particle boards for perfect alignment of the strands and for loading parallel to the strand (and ﬁber) direction. Shaler and Blankenhorn [24] applied the Halpin-Tsai equations to predict the bending modulus of wood composite panels with uniform density proﬁle, resulting in an underestimation of measured moduli by 25% on average. The model of Xu and Suchsland [35] does not reproduce the effect of particle size on modulus of elasticity, which was observed in experiments by several other authors [22, 28, 31, 32]. Lee and Wu [17] developed a model to predict the in-plane engineering constants of single-layer and three-layer OSB from orthotropic strand properties, strand alignment, and layer thicknesses, based on the Halpin-Tsai equations and lamination theory. The different strand orientations within a single-layer are considered by numerous thin layers made up of perfectly aligned strands with different orientations. This simpliﬁed modeling approach and the need to back-calculate some input data from macroscopic test results restrict the estimation accuracy and the predictive capabilities of the model. To overcome the limitations of both, experimental and previous simple modeling approaches, a model based on continuum micromechanics was developed to predict the elastic constants of strand boards on the basis of (micro-)structural characteristics. Its theoretical basis, its input values, its application to parameter studies, and its experimental validation are described in the next four sections. A similar model was recently presented by Benabou [2], though with the physically doubtful choice of aligned strands as matrix phase in the Mori-Tanaka scheme (see Section 9.2) and under negligence of the compaction during the production process, which has a substantial effect on the mechanical properties.

9.2 Model Approach The present model is based on a combination of sound mechanical theory and accompanying experiments for parameter identiﬁcation and model validation. The model is intended to enhance insight into the mechanical behavior of the panels and, thus, to contribute to a better understanding of the inﬂuence of the (micro-)structural characteristics, such as strand quality, strand shape, strand size, and strand orientation, on the elastic behavior of veneer strand boards. For a macro-homogeneous strand board, i.e. a strand board with an approximatively homogeneous density proﬁle, two scales of observation can be distinguished: the characteristic dimensions

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Fig. 9.2 Micro- and macroscale of a homogeneous strand board

of the strands on the microscale and the characteristic length of the strand board itself on the macroscale (see Fig. 9.2). The link between the mechanical properties observed at these two length scales is accomplished by means of continuum micromechanics. On the whole, this model enables to predict the macroscopic mechanical performance of homogeneous strand-based panels from microscopic mechanical and morphological characteristics. This can be considered as novel and promising approach in the ﬁeld of strand-based engineered wood products and as valuable tool for product development and optimization.

9.2.1 Fundamentals of Continuum Micromechanics In continuum mechanics a material is considered as a continuous, homogeneous medium. However, in reality also macroscopically homogeneous materials show inhomogenities at smaller scales of observation. The distinctive microstructure of these materials at some macroscopic material point can be described by means of a representative volume element (RVE). The microstructural inhomogenities constitute so-called phases, which are subdomains within the RVE with homogeneous mechanical and morphological properties. The deﬁnition of an RVE requires that the characteristic length of the inhomogenities d is much smaller than the characteristic length of the RVE, l, which in turn has to be much smaller than the typical length of the observed structure and of the applied load, L. In mathematical terms this “separation of scales” condition reads as d l L.

(9.3)

The macroscopic stiffness tensor of the micro-heterogeneous material can be estimated by solving the matrix inclusion problem in the RVE. Based on the work of Eshelby [4] and Laws [16], the estimate for the macroscopic stiffness tensor Chom reads in its most general form as [36, 37]

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−1

C

hom

= ∑ f i Ci : i

[I + P0i

0 −1

: (Ci − C )]

:

∑

f j [I + P0j

0 −1

: (C j − C )]

(9.4)

j

Here, fi denotes the volume fraction of phase i, Ci the stiffness tensor of this phase, and I the fourth-order unit tensor. The sums are taken over all phases of the RVE. The fourth-order tensor C0 is the stiffness tensor of the matrix material of the underlying matrix inclusion problem. Depending on the spatial arrangement of the phases, it is conventionally set to either the stiffness of a contiguous matrix phase in the material (Mori-Tanaka scheme) or – in case of a medium with dispersed phases – the estimated stiffness tensor Chom itself (self-consistent scheme). The fourth-order tensor P0i accounts for the morphology of phase i and depends on C0 as well as on shape and orientation of the corresponding inclusions.

9.2.2 Continuum Micromechanics Model for Strand Boards A macro-homogeneous strand board consists of strands with various orientations, which are bonded together by means of a synthetic adhesive. Within the resulting solid compound, the adhesive does not act as an own material phase but contributes to the mechanical behavior of the compound by two substantial effects: First, the adhesive penetrates the wood tissue and compensates micro-damages in consequence of the strand production. Second, it establishes the crucial bonding between the strands, which is assumed to be perfect in the model. The absence of a contiguous matrix phase in the board motivates the use of the implicit self-consistent scheme where C0 = Chom . The strands constitute the only phase of the model, which is made up of inclusions with different orientations. The orientation distribution of the strands in the plane of the panel is described in terms of a distribution function gs (ϕ ), where the angle ϕ is deﬁned in Fig. 9.3.

Fig. 9.3 Deﬁnition of orientation angle ϕ of the strands

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Under consideration of the mechanically and morphologically justiﬁed periodicity of 180◦ or π, the speciﬁcation of Eq. (9.4) for the strand panel reads as + π 2 hom hom hom −1 = Cs (ϕ ) : [I + Ps (ϕ ) : (Cs (ϕ ) − C )] )gs (ϕ )d ϕ : C π ϕ =− 2

+ π2

[I + Phom s (ϕ ) : π

ϕ =− 2

hom −1

(Cs (ϕ ) − C

)] gs (ϕ )d ϕ

−1 ,

(9.5)

where gs (ϕ ) meets the normality condition +π 2 ϕ =− π2

gs (ϕ )d ϕ = 1.

(9.6)

Cs (ϕ ) denotes the stiffness tensor of the compacted strands (see, Section 9.3.4). In consequence of the different orientations of the inclusions, it is a function of the orientation angle ϕ . The fourth-order tensor Phom s (ϕ ) also depends on ϕ and is evaluated under consideration of the strands as inﬁnitely long cylinders with elliptic cross-section. The suitability of the latter modeling approach is further discussed in Section 9.4.1. The continuum micromechanics model is capable to estimate the entire stiffness tensor of micro-heterogeneous materials and, thus, to deliver all nine independent material constants of an orthotropic veneer strand board, depending on the stiffness tensors of the microstructural components, their volume fractions, their shapes, as well as their orientations.

9.3 Model Validation The validation of the continuum micromechanics model is based on the comparison of model predictions with results of experiments on macro-homogeneous veneer strand boards produced at a laboratory scale. In particular, the two in-plane elastic moduli, Ey and Ez , and the two out-of-plane shear moduli, Gxz and Gxy , were considered.

9.3.1 Production of Veneer Strand Boards The strands were produced from spruce logs of different wood quality, which were sliced to veneers and then cut to veneer strands (VSB strands) with dimensions of 1 mm thickness, 25 mm width, and 210 mm length. The main extension of the strands coincides with the longitudinal material direction. This stranding procedure ensured uniform strand shape and strand size with little divergence. For the strand board production, powder phenol-formaldehyde resin was used as adhesive.

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Table 9.1 Characteristics of produced veneer strand boards Abbreviations

Strand quality

Strand orientation

Density level

Number of boards

R X O GO PO LDO

Medium Medium Medium Good Poor Poor

Random Crosswise Aligned Aligned Aligned Aligned

High High High High High Low

5 5 6 2 3 3

Its content amounted to 8% based on the oven-dry strand mass. The strand orientation was done by hand. The process parameters were chosen as to maximize the uniformity of the density proﬁle. Therefore it was necessary to compact the strand mat in cold state and start heating up the press platens after reaching ﬁnal board thickness. The cooling down of the strand board under moderate pressure prevented the occurrence of damp blisters. In order to identify the inﬂuence of strand quality on the mechanical properties of strand boards and to validate the model in this respect, strands were produced from spruce logs of different wood quality characterized by a different density and a different content of compression wood. Density is well known as key parameter inﬂuencing the elastic properties of wood [3, 14]. Compression wood exhibits higher lignin content, higher microﬁbril angle, and higher density compared to normal wood. These features result in lower mechanical properties in ﬁber direction, but – especially because of the higher microﬁbril angle (MFA) – higher properties perpendicular to grain [10, 13, 29]. The strands were divided into three groups according to the mean elastic modulus measured on clear wood test specimens, representing good (G), medium, and poor (P) quality. In order to study the effect of strand orientation distribution, three distinctive orientation types were considered (see Table 9.1), namely strand boards with mainly parallel strand orientation (type O), with random strand orientation (type R), and with crosswise strand orientation (type X). Moreover, the inﬂuence of the compaction rate on the mechanical properties of the boards was studied by producing and testing boards with different densities [low density level (LD): 450 kg/m3 , high density level: 650 kg/m3 ].

9.3.2 Mechanical Testing of Boards in Tension and Shear All panels were conditioned at 20 ◦ C and 65% relative humidity before cutting them to test specimens and testing them in tension and shear. In particular, the two inplane moduli of elasticity, Ez and Ey , and the two out-of-plane shear moduli, Gxz and Gxy , were measured. One test specimen for each elastic constant was cut out of each of the 24 panels, resulting in 48 tension and 48 shear test specimens.

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Fig. 9.4 Tension and shear testing devices

The tension test specimens had lengths of 650 mm and 500 mm for tension parallel and perpendicular to the strand orientation direction, respectively. The dimension of the testing cross-section amounted to 100 × 20 mm. Clamp-on strain transducers with a basis length of 100 mm were used to measure the corresponding strain component. The dimensions of the shear test specimens were 225 × 100 × 20 mm, and the shear deformation was measured with LVDT’s. Both moduli of elasticity and shear moduli were evaluated from results in the elastic regime within the range of 10–40% of the estimated ultimate load. Figure 9.4 shows the testing devices for both, tension and shear tests.

9.3.3 Stiffness Tensor of Strands The determination of model predictions for the tested boards requires identiﬁcation of the stiffness of the strand material, which is a key input for the micromechanical model. The mechanical potential of the used raw material was investigated by uniaxial tension tests on veneer strands and on clear wood test specimens with a test cross-section of 6 × 20 mm, the latter according to DIN 52 188. The modulus of elasticity of the strands turned out to be slightly lower than that of the clear wood test specimens. This indicates micro-damage during strand production, which has an identiﬁable inﬂuence on thin test specimens, as also reported by Price [23] and Geimer et al. [7] for thin strands. To clarify the effect of bonding and of compaction on the elastic modulus, laminated veneer lumber (LVL) was produced from veneers of the same raw material. It was tested in uniaxial tension along the main extension

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Table 9.2 Experimental characteristics and model parameters of strand material Experimental parameters Strand quality Good Medium Poor

Model parameters

Density [g/cm3 ]

Modulus of elasticity [N/mm2 ]

MFA [◦ ]

Lignin content [%]

0.355 0.385 0.345

11,200 10,100 6,600

14 18 25

18 22 28

of the strands. The experimental results revealed a linear dependence of the modulus of elasticity of LVL on the compaction rate, which is deﬁned as the density ratio of strands and LVL. Further the measured elastic modulus for LVL was approximately equal to the corresponding values for the clear wood test specimens, provided that they had the same density. For the investigated strand material and for the given test conditions, the adhesive-driven bonding and the penetration of the adhesive into the wood tissue seem to level out the damage-caused decrease of stiffness. Together with the lack of more comprehensive experimental data, this observation motivated the use of the modulus of elasticity of clear wood test specimens as estimate for the corresponding value of the strands. The remaining elastic constants of the strands were determined by means of a continuum micromechanics model for solid wood [9–11]. This model allows for estimation of macroscopic stiffness properties of various wood species from the mechanical properties of some basic universal components of wood (cellulose, hemicellulose, lignin, and water) and from a few morphological parameters, such as the MFA, the cell arrangement, and the macroscopic density. The different strand qualities were considered by evaluating the model for different settings of the MFA and the lignin content. Since the latter were not determined experimentally, they were speciﬁed by evaluating the model for spruce wood and for the macroscopic density of the test specimens, and by matching the model predictions to the available corresponding experimental results for the longitudinal modulus of elasticity. The resulting sets of MFA and lignin content for the three different strand qualities are listed in Table 9.2. Their use in the micromechanics model for solid wood resulted in different orthotropic stiffness tensors for the strand material with respect to the principal material and strand directions, Cs (ϕ = 0). These are a key input for the strand board model.

9.3.4 Implementation of Compaction During the production process of a strand board, the strands are extensively compacted to reach the ﬁnal board density. The compaction considerably affects the elastic properties of the strands. The modiﬁed stiffness tensors of the strands were determined by evaluating the micromechanical model for wood described in the previous subsection for the ﬁnal density of the (compacted) strands and for a reasonable

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variation of the cell structure. Since no microscopic investigations of the cell structure were performed, the morphological parameters of the cell structure were chosen such that the model predictions for the elastic constants of the boards show a good agreement with corresponding test results. Evaluation of the micromechanical model for the modiﬁed morphological parameters of the cell structures resulted in suitable estimates for the two in-plane moduli of elasticity and the longitudinal shear modulus Gxz . However, the shear modulus in the cross-sectional plane, Gxy , was overestimated. The variation of the cell structure does not represent the buckling and partial fracture of cell walls in the direction of the panel thickness, which particularly affects the transverse shear behavior. This extraordinary damage-induced decrease of Gxy was considered by a reduction factor of two for the original estimate of the micromechanical model.

9.3.5 Implementation of Strand Orientation The orientation of the strands is among the key characteristics of a strand board and is implemented in the model by the distribution function gs (ϕ ). As for the boards of type O, an exact strand orientation in one single direction is not possible, irrespective of the efforts made to align the strands in the board production. Therefore, the distribution function of the strand orientation for boards of type O is described by a normal distribution function with a mean of 0 ◦ and a standard deviation of 5◦ or π /36, so that gs (ϕ ) reads as: g s (ϕ ) =

1 −1 √ ·e 2 π/36 · 2π

ϕ −0 π/36

.

(9.7)

A random strand orientation does not show any preferred direction and is best characterized by a uniform distribution function, resulting for the range from − π2 to + π2 in 1 g s (ϕ ) = . (9.8) π For crosswise strand orientation, again, no perfect alignment, but a standard deviation of π /36 is assumed for both directions. This leads to two distribution functions with different means but otherwise equal form, which are superposed to result in:

1 − 1 ϕ −0 √ · e 2 π/36 , gs,1 (ϕ ) = π/36 · 2π ϕ −π/2 1 −1 √ · e 2 π/36 , gs,2 (ϕ ) = π/36 · 2π gs (ϕ ) = 0.5 · gs,1(ϕ ) + 0.5 · gs,2(ϕ ).

(9.9) (9.10) (9.11)

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9.3.6 Experimental Validation of In-Plane Moduli of Elasticity and Out-of-Plane Shear Moduli The predictive capability of the model is studied by correlation plots (Figs. 9.5–9.7) and quantiﬁed in terms of mean values and standard deviations of relative prediction errors as well as overall coefﬁcients of determination. Figure 9.5 shows the good agreement of the experimental results for the in-plane elastic moduli and corresponding model predictions. The latter were calculated by evaluating the model for the strand quality, strand orientation distribution, and the apparent density of each test specimen. Strand boards with aligned strand orientation exhibit an outstanding modulus of elasticity in strand orientation direction up to 18,800 N/mm2 . The test results of the boards with aligned strand orientation of type GO, O, and PO impressively show the inﬂuence of strand quality on the modulus of elasticity in the strand direction. Since the strands are very precisely oriented in boards with aligned strand orientation, the elastic modulus perpendicular to the strand orientation direction is only moderate and is illustrated in Fig. 9.5 for all boards. A detailed presentation of the modulus of elasticity perpendicular to the strand orientation, Ey , is shown in Fig. 9.6. The effect of strand orientation distribution, which was revealed by the strand boards of type O, X, and R, has been predicted by the model very well. As shown in the parameter studies (see Section 9.4.2), the modulus of elasticity of type X boards is about 50% and that of

Fig. 9.5 Comparison of experimental data and corresponding model predictions for the in-plane moduli of elasticity Ez and Ey

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Fig. 9.6 Comparison of experimental data and corresponding model predictions for the in-plane modulus of elasticity Ey (detail of Fig. 9.5)

Fig. 9.7 Comparison of experimental data and corresponding model predictions for the out-ofplane shear moduli Gxz and Gxy

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type R boards about 40% of the modulus of elasticity of type O boards in strand orientation direction. The boards of type PO and LDO only differ in board density and so indicate the effect of compaction on the modulus of elasticity. The remarkable predictive capability of the model is pointed out by a coefﬁcient of determination of R2 = 0.98 and by a mean relative deviation of 4.2 ± 12.9% between model predictions and corresponding test results. The test results also underline the great mechanical potential of engineered wood products produced from large-area slender strands, which are remarkably high compared with commercial strand boards. Figure 9.6 provides a detailed presentation of Fig. 9.5 in relation to the in-plane modulus of elasticity Ey perpendicular to the strand orientation of the strand boards with aligned strand orientation. Since the wood quality of the strands does not affect their transverse modulus of elasticity distinctively, also the modulus of elasticity Ey of the veneer boards does not vary much with the strand quality. The compaction rate, though, exhibits a very pronounced inﬂuence. For the modulus of elasticity Ey , the comparison of model predictions with experimental data yields a coefﬁcient of determination of R2 = 0.62 and a mean relative error of 6.6% with a standard deviation of 15.9%. Also regarding the two out-of-plane shear moduli Gxz and Gxy , a very good agreement of model predictions and experimental results was achieved (see Fig. 9.7). Both, experimental results and model predictions, revealed almost the same shear moduli for type X and type R boards, which was not expected. The high shear moduli of type PO boards can be explained by the high compression wood content for the strands of poor wood quality. Compression wood features a higher microﬁbril angle, which contributes to the high shear stiffness. Correlating the experimental data to the model prediction yields a coefﬁcient of determination of 0.97. The mean value of relative prediction errors amount to −0.8% with a standard deviation of 14.7%.

9.4 Model Applications Having conﬁrmed its suitability for prediction of the stiffness of veneer strand boards by experimental validation, the micromechanical model is applicable for studying the effects of different morphological parameters and of different strands orientation distributions on the macroscopic mechanical behavior. The results of these studies are summarized in the following subsections.

9.4.1 Effect of Strand Shape The slenderness ratio is deﬁned as ratio of strand length and strand thickness. For a panel of type X, the two elastic moduli in the board plane, Ez = Ey , strongly increase for small slenderness ratios (see Fig. 9.8). Above a slenderness ratio of

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Fig. 9.8 Effect of slenderness ratio on the in-plane elastic moduli of the board

Fig. 9.9 Effect of aspect ratio on the in-plane elastic moduli of the board

about 100, there is only a very modest effect of this ratio anymore. The estimated moduli approach the respective values obtained for inclusions shaped as inﬁnitely long cylinders then (horizontal line in Fig. 9.8). This underlines the suitability of modeling the strands as inﬁnitely long cylinders instead of considering ellipsoids with a slenderness ratio of 210, which would best ﬁt the strand dimensions. Strand width divided by strand thickness yields the aspect ratio, which has a great inﬂuence on the in-plane moduli of elasticity for the type X boards as shown in Fig. 9.9. There, the model predictions are related to the corresponding value obtained for an inﬁnite aspect ratio, which represents plywood and thus, the maximum of this mechanical characteristic.

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9.4.2 Effect of Strand Orientation Distribution While all strands are mainly oriented in one direction within type O strand boards, in type X boards they are oriented in two perpendicular directions in equal shares. The transition from type O to type X strand boards can be performed in the model by increasing the volume fraction of cross-lying strands from 0% to 50%. Figure 9.10 demonstrates the linear inﬂuence of this volume fraction on the in-plane elastic moduli and shows that the in-plane elastic moduli of type X boards are nearly 50% of the elastic modulus in strand direction of type O boards. As the strand orientation of type O strand boards is characterized by a normal distribution, the transition from type O to type R can be easily achieved in the model by increasing the standard deviation of the strand distribution function compared to perfect alignment of all strands (standard deviation 0◦ ), see Fig. 9.11. Only a small standard deviation of the orientation distribution function already results in a considerable decrease of Ez , whereas Ey increases only slightly. The in-plane moduli of elasticity of type R boards amount to about 40% of the modulus of elasticity in strand direction of type O boards.

9.4.3 Effect of Loading Angle α Since it cannot be assured that the strand boards are only loaded in the direction of their principal axes, the in-plane modulus of elasticity at an arbitrary angle α to these axis is also of interest (see Fig. 9.12). Type O has one main direction with an outstanding modulus of elasticity (chosen as reference value in Fig. 9.12). However, this modulus drops steeply with increasing α. The random strand orientation im-

Fig. 9.10 Effect of strand orientation distribution on the in-plane elastic moduli of the board: transition from type O to type X

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Fig. 9.11 Effect of strand orientation distribution on the in-plane elastic moduli of the board: transition from type O to type R

Fig. 9.12 Dependence of in-plane elastic modulus on angle α relative to the principal material directions

plies equal stiffness of strand boards of type R in all directions of the plane. In the direction of the two principal axes of type X boards, the modulus of elasticity is higher than for type R boards. For most other loading angles not coinciding with the principal board directions, the type R board provides higher stiffness than the type X boards. The modulus of elasticity of type X boards at a 45◦ loading angle amounts to only 25% of the elastic modulus of type O board in its principal direction and is remarkably low.

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Fig. 9.13 Effect of strand board density on the elastic constants

9.4.4 Effect of Veneer Strand Board Density The density of strand boards has a substantial effect on their mechanical properties. The variation of the four investigated elastic constants Ey , Ez , Gxy , and Gxy for board densities between 400 kg/m3 and 800 kg/m3 is depicted in Fig. 9.13, where each elastic constant is related to its respective value at a typical strand board density of 600 kg/m3 . For the longitudinal modulus of elasticity Ez and the longitudinal shear modulus Gxz , an almost linear change with the board density is observed. Both transverse moduli Ey and Gxy show a nonlinear dependence on the board density. An increase of the board density by one third results in doubling of these moduli. While the variations of the elastic constants are more pronounced for the transverse properties than for the longitudinal ones for relative values (cf. Fig. 9.13), the contrary holds if absolute values are considered.

9.5 Conclusion Both, the elastic properties of the veneer strand board and the predictive capability of the micromechanical model are outstanding. The experimental results remarkably exhibit the high potential of the veneer strand board for application as load bearing construction member. The good agreement of model predictions and experimental results underlines that the material model includes all relevant characteristics in order to describe the mechanical behavior of a homogeneous strand-based engineered wood panel. As future research activity, the presented model will be extended for consideration of non-homogeneous density proﬁles and distinctive stacking sequences of

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different single-layers by means of lamination theory. Such a model for multi-layer strand boards will then be used for further product development and optimization and will enable to create strand boards with custom-designed properties. Acknowledgements This study is part of the research project P04 “strand products” of holz.bau forschungs Ltd., which represents a joint work of the Institute for Mechanics of Materials and Structures at Vienna University of Technology and the Institute for Timber Engineering and Wood Technology at Graz University of Technology. The funding by the Federal Ministry of Economics and Labor of the Republic of Austria (BMWA), by the Styrian Business Promotion Agency (SFG), by the Federal State of Styria, and by the municipality of Graz is gratefully acknowledged.

References 1. Barnes D (2001) A model of the effect of strand length and strand thickness on the strength properties of oriented wood composites. Forest Prod J 51(2):36–46 2. Benabou L, Duchanois G (2007) Modelling of the hygroelastic behaviour of a wood-based composite for construction. Compos Sci and Tech 67:45–53 3. Bodig J, Jayne B (1982) Mechanics of Wood and Wood Composites. Van Nostrand Reinhold, New York 4. Eshelby JD (1957) The determination of the elastic ﬁeld of an ellipsoidal inclusion and related problems. Proc Roy Soc Lond, Ser A/241:376–396 5. Geimer RL (1976) Flake alignment in particleboard as affected by machine variables and particle geometry. Res Pap FPL-275, USDA Forest Serv, Forest Prod Lab Madison WI, pp 1–16 6. Geimer RL (1986) Mechanical property ratios – a measure of ﬂake alignment. Res Pap FPL– 468. USDA Forest Serv, Forest Prod Lab Madison WI, pp 1–10 7. Geimer RL, Mahoney RJ, Loehnertz SP, Meyer RW (1985) Inﬂuence of processing-induced damage on strength of ﬂakes and ﬂakeboards. Res Pap FPL 463, USDA Forest Serv, Forest Prod Lab Madison WI, pp 1–15 8. Geimer RL, Montrey HM, Lehmann WF (1975) Effects of layer characteristics on the properties of three-layer particleboards. Forest Prod J 25(3):19–29 9. Hofstetter K, Hellmich Ch, Eberhardsteiner J (2005) Development and experimental veriﬁcation of a continuum micromechanics model for wood. Eur. J. Mech. A/Solids 24:1030–1053 10. Hofstetter K, Hellmich Ch, Eberhardsteiner J (2006) The inﬂuence of the microﬁbril angle of wood stiffness: a continuum micromechanics approach. Comput Assist Mech Eng Sci 13:523–536. 11. Hofstetter K, Hellmich Ch, Eberhardsteiner J (2007) Micromechanical modeling of solid-type and plate-type deformation patterns within softwood materials. A review and an improved approach. Holzforschung 61(4):343–351 12. Katzengruber R, Schickhofer G (2000) Eingrenzung von Baustoffkenngr¨oßen und Einsatzbereichen f¨ur massive lastabtragende Platten großer Abmessungen aus Holzausgangsprodukten mit unterschiedlichem Zerlegungsgrad am Beispiel des plattenf¨ormigen, massiven Bauproduktes VSB (in German). Res. Rep. LR 9909/02, Lignum Research, Graz University of Technology 13. Kollmann F (1982) Technologie des Holzes und der Holzwerkstoffe, vol 1, 2nd edn. Springer, Berlin/Heidelberg/New York 14. Kollman F, Cˆot`e W 1968 Principles of Wood Science and Technology, vol 1, Springer, Berlin/Heidelberg/New York 15. Lau PWC (1981) Numerical approach to predict the modulus of elasticity of oriented waferboard. Wood Sci 14(2):73–85 16. Laws N (1977) The determination of stress and strain concentrations at an ellipsoidal inclusion in an anisotropic material. J Elasticity 7:91–97

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17. Lee JN, Wu Q (2003) Continuum modeling of engineering constants of oriented strandboard. Wood Fiber Sci 35(1):24–40 18. L¨ocker HJ (2001) Zero waste production technique and technology for a new engineered wood product. In: L¨ocker HJ EU-F¨orderung von “Forschung und technologische Entwicklung.” Analyse und Eliminierung von St¨oreinﬂ¨ussen im Verfahren. Master thesis, Graz University of Technology 19. Marra AA (1992) Technology of wood bonding principles in practice. Van Nostrand Reinhold, New York 20. McNatt JD, Bach L, Wellwood RW (1992) Contribution of ﬂake alignment to performance of strandboard. Forest Prod J 42(3):45–50 21. Mundy JS, Bonﬁeld PW (1998) Predicting the short-term properties of chipboard using composite theory. Wood Sci Technol 32:237–245 22. Nishimura T, Amin J, Ansell MP (2004) Image analysis and bending properties of model OSB panels as a function of strand distribution, shape and size. Wood Sci Technol 38:297–309 23. Price EW (1976) Determining tensile properties of sweetgum veneer ﬂakes. Forest Prod J 26(10):50–53 24. Shaler SM, Blankenhorn PR (1990) Composite model prediction of elastic moduli for ﬂakeboard. Wood Fiber Sci 22(3):246–261 25. Shigehiko S (2000) Effects of strand length and orientation on strength properties of OSB made from Japanese cedar. In: Proceedings of the World Conference of Timber Engineering, pp 1.3.8.1–1.3.8.8 26. Shupe TF, Hse CY, Price EW (2001) Flake orientation effects on physical and mechanical properties of sweetgum ﬂakeboard. Forest Prod J, 51(9):38–43 27. Tabarsa T, Yadollahi S (2004) Effects of strand orientation and press cycle on properties of OSB made from Iranian beech strands. In: Proceedings of the 3rd International Conference of the European Society for Wood Mechanics, pp 317–325 28. Thole V, Sch¨afer K (2001) Spankultur (in German). Holz- und Kunststoffverarbeitung 36(5):46–52 29. Timell TE (1986) Compression Wood in Gymnosperms, vol 1. Springer, Berlin/Heidelberg/ New York/Tokyo 30. Udulft I (2002) The effects of strand length, strand angle and press closing time on surface compression and strength properties of oriented strand board. Master thesis, University of Hamburg 31. USDA Forest Products Service (1999) Wood Handbook, Wood as an Engineering Material. USDA Forest Products Service, USA 32. Wang K, Lam F (1999) Quadratic RSM models of processing parameters for three-layer oriented ﬂakeboards. Wood Fiber Sci 31(2):173–186 33. Wong ED, Zhang M, Wang Q, Kawai S (1999) Formation of the density proﬁle and its effects on the properties of particleboard. Wood Sci Technol 33:327–340 34. Xu W (1999) Inﬂuence of vertical density distribution on bending modulus of elasticity of wood composite panels: a theoretical consideration. Wood Fiber Sci 31(3):277–282 35. Xu W, Suchsland O (1998) Modulus of elasticity of wood composite panels with a uniform vertical density proﬁle: a model. Wood Fiber Sci 30(3):293–300 36. Zaoui A (1997) Structural morphology and constitutive behaviour of microheterogeneous materials. In: Suquet P (ed) Continuum Micromechanics. CISM Courses and Lectures No. 377, Springer, Wien, New York, pp 291–347 37. Zaoui A (2002) Continuum micromechanics: Survey. J Eng Mech (ASCE) 128 (8):808–816 38. Zhou D (1990) A study of oriented structural board from hybrid poplar. Physical and mechanical properties of OSB. Holz Roh-Werkstoff 48:293–296

Chapter 10

Nanoindentation of Cement Pastes and Its Numerical Modeling Jiˇr´ı Nˇemeˇcek, Petr Kabele, and Zdenˇek Bittnar

Abstract This paper focuses on experimental investigations and numerical modeling of micromechanical behavior of cement paste. Cement paste is taken as a fundamental representative of building materials with heterogeneous microstructure. The main studies are devotes to experimental nanoindention and its implications to evaluation of material properties. The paper concerns the appropriateness of conventional methods used for evaluation of micromechanical properties and investigates possibilities of the use of enhanced methods for better description of the nanoindentation process. Limitation of traditional elastic solution is shown on the unique experimental program. Better descriptions of indentation based on analytical viscoelastic solution and ﬁnite element model with general visco-elasto-plastic constitutive relation are proposed. These models are used for simulation of indentation and for estimation of material parameters at micrometer scale.

10.1 Introduction Cement is the widespread building material whose impact to the environment is enormous. Therefore, investigation of its behavior and proper numerical modeling on all scales is desirable. The overall behavior of any material is directly dependent on the microstructure and its individual phase properties. Modern constitutive models for composite materials try to respect this fact and link the overall material response with its micro-level. The development of various experimental techniques in the past decades made possible to access mechanical properties of various materials at submicron length scales. Nanoindentation plays an important role among them. This technique is based on the direct measurement of the load-displacement Jiˇr´ı Nˇemeˇcek, Petr Kabele, and Zdenˇek Bittnar Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Prague, Czech Republic [email protected], [email protected], [email protected] V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

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relationship. The depth of penetration starts from the level of nanometers. Although, nanoindentation was originally developed and used mainly for studying homogeneous materials like metals, coatings, ﬁlms, glass, and crystal materials, the evolution of this method allows us to use it also for materials like concrete and cement. The major studies can be found, e.g. in [1, 2, 8]. However, the interpretation of measured data is more complicated due to the large heterogeneity of concrete and cement as well. In contrary to classical macroscopic tests, numerous microscale phenomena can occur during indentation tests. Size-dependent indentation results are commonly obtained and reported mainly for metals by many researchers [3, 5, 9]. At our studies, creep of the material was found to be the main factor contributing to such interpretations. Ignoring creep in the evaluation of results can lead to spurious size effect on elastic properties. Simulation of indentation process and comparison with experimental data can answer the question on the appropriateness of different constitutive relations and the underlying material behavior.

10.2 Nanoindentation of Cement Pastes In contrast to usual indentation on metals, for example, cement paste is much more complex and also time-dependent material. From the microstructural point of view, cement paste is a heterogeneous material with several material phases. The most important are the hydrated phases (C-S-H gels, Portlandite and other minor phases), unhydrous phases (rest of clinker minerals) and porosity. Cementitious materials exhibit also signiﬁcant time-dependent behavior that can affect evaluation of even elastic properties using standard procedures. The material properties in this study are assessed for hydrated phases which occupy majority of the specimen volume. Nanoindentation was performed so that a large number of indents were carried out in the well hydrated areas of the specimen. Subsequently, separation of indents into groups corresponding to distinct material phases was done with the aid of electron microscope (ESEM). An example of indented cement paste sample, which is usually produced in a rectangular area, is shown in Fig. 10.1. Typical pyramidal shape of indenter imprints is formed by the shape of Berkowich indenter tip. Micromechanical properties can be evaluated for each indent in the matrix.

Fig. 10.1 ESEM image of indented cement paste

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Holding Load, P

Fig. 10.2 Nanoindentation load vs. depth of penetration diagram (P-h curve)

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S Loading Unloading

Fig. 10.3 Example of multicycle loading with increasing load and no holding period at the peaks

Load [mN]

Depth of penetration, h 20 10 0 0

500

1000

1500

Penetration depth [nm]

10.3 Experiments White cement paste samples (CEM-I 52, 5 White, Holcim, SK) mixed in water/ cement ratio w/c = 0.5 were prepared and stored in water for 28 days. Before testing, a 2 mm thick slice from the bulk material was cut and polished on coarse to very ﬁne emery papers to achieve very smooth and ﬂat surface [4]. Specimens were washed in ultrasonic bath to remove all the dust. The resultant surface had the roughness about several tens of nm as checked by AFM. Cement paste samples were prepared and tested by means of nanoindentation. Only hydrated phases were considered for the evaluation of results. Several types of loading were prescribed: (i) One-cycle loading (loading, holding and unloading) (Figs. 10.2 and 10.6b) (ii) Multi-cycle loading to increasing loads without holding periods at peaks (Fig. 10.3) (iii) Cyclic loading with long holding periods at the peak (Fig. 10.6c) Experiments were carried out in a large load range to cover also wide ranges of penetration depths.

10.4 Analysis of Indentation Data and Results Experimental results from nanoindentation include the load vs. depth of penetration diagram (the P-h curve). This diagram contains loading and unloading branch and may contain also holding (dwelling) period at the peak of the diagram (Fig. 10.2).

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10.4.1 Elastic Solution Commonly, two elastic properties, the hardness and the elastic modulus are extracted from indentation data. The most popular method was elaborated by Oliver and Pharr [6]. The elastic properties are evaluated from an unloading part of the P-h curve. The analysis is based on the analytical solution known for rotational bodies punched into the elastic isotropic half-space. Hardness and indentation modulus are than deﬁned as follows: Pmax A √ S π Er = √ 2 A H=

(10.1) (10.2)

where Pmax is the peak load, A is the projected contact area at peak load and S is the contact stiffness evaluated as the initial slope of unloading curve (Fig. 10.2). The effect of non-rigid indenter can be accounted for by the following equation: 1 1 − ν 2 1 − νi2 + = Er E Ei

(10.3)

where E and ν are tested material elastic modulus and Poisson’s ratio, respectively. Ei and νi are indenter’s parameters (for diamond: Ei = 1,141 GPa and νi = 0.07). However, the material behavior does not always fulﬁll such a strong assumption of elasticity even for the unloading. It can be seen in Fig. 10.3 that the P-h curve contains a bulge at the beginning of unloading in each cycle. It shows the role of creep that is present even on the unloading branch. It leads to spurious size effect on the evaluation of elastic properties using standard procedures based on elasticity [6] as shown in Fig. 10.4 (thin line). On the other hand, using long holding periods signiﬁcantly reduces this kind of spurious size effect (Fig. 10.4, thick line). Generally, material response may contain also inelastic deformations and more precise material models are needed for its description.

Fig. 10.4 Size effect on the elastic modulus evaluated with method of Oliver and Pharr [6] ignoring creep effects (thin line) and using long holding periods (thick line)

E [GPa]

45 35 25 15 0

1000 2000 Penetration depth [nm]

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10.4.2 Visco-Elastic Solution According to experimental evidence cement paste is a time-dependent material whose behavior cannot be characterized only by elasticity. Elastic modulus can be extracted from P-h curve in case of a special loading with long dwell periods where the effect of creep on the unloading branch is minimized. However, simulation of the indentation process cannot be achieved. Several models can be proposed to solve such phenomena. Recent work of Vandamme and Ulm [7] assumes that the indentation response is dominated by visco-elastic behavior. Multiple formulations can be constructed in such a case. Vandamme and Ulm [7] derived closed form analytical solution based on the assumption of linear visco-elastic material with deviator creep. The best results are obtained using 4-parameters combined Kelvin-Voigt-Maxwell model as shown in Fig. 10.5. Viscous deformation in this model is given by G0 nv nM 2ε¨d (t) + G0 Gv nM 2ε˙d (t) = nv nM σ¨ d (t) + (G0 nv + G0 nM + Gv nM )σ˙ d (t) + G0Gv σd (t)

(10.4)

where εd and σd are deviator part of strain and stress tensor, respectivelly. Dots over the tensors have the meaning of time derivatives. G0 and Gv are elastic stiffness parameters, nM and nv are viscosity parameters as also can be seen in Fig. 10.5. For a speciﬁc load-time function an analytical solution was derived for loading and holding periods. Unloading cannot in general be described by this solution since the derivation of the analytical form was based on the assumption of monotonically increasing contact area that is not the case of unloading. Visco-elastic model parameters can be obtained by nonlinear ﬁtting of the experimental data from holding period where the force is kept constant over the time. This analytical viscolelastic solution was applied to the simulation of experimental P-h curves. Material parameters were obtained by nonlinear least-square ﬁtting (standard nonlinear Levenberg-Marquardt procedure solved in Matlab) of material creep during holding period (Fig. 10.6a). Such set of parameters was used for simulation of the loading curve. The results are quite satisfactory as can be seen in Fig. 10.6b. The same procedure was applied for more complex type of experiment with cyclic loading (Fig. 10.6c). Since the loading path was too complicated for the analytical solution simpliﬁed loading history without intermediate unloading was used. It is the reason why no cyclic loading is obtained in the simulation in

Fig. 10.5 Schematic representation of 4-parameters linear visco-elastic deviator creep model based on combined Kelvin-VoigtMaxwell chain

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Time [s] Load [mN]

(a) 10 5 0 0

300

600

Load [mN]

Depth [nm]

(b)

60 40 20 0 0

1000 Depth [nm]

2000

(c)

Fig. 10.6 Experimental curves (thin lines) and numerical responses (thick lines): (a) holding period (creep), (b) one-cycle loading, and (c) multi-cycle loading modeled by visco-elasticity

Fig. 10.6c. However, the numerical response does not wrap the experimental curve as it was expected. Using the same material parameters lead to underestimation of deformation in this experiment. It motivated us to construct the FE model with more complex constitutive laws and with the possibility of setting an arbitrary loading history.

10.4.3 Finite Element Analysis It follows from the previous paragraphs, that analytical solutions of the indentation problem are presently available only for some simple constitutive models and special loading histories. It also appears that neither the elastic, nor the visco-elastic model fully represent the behavior of cement paste. Therefore, performance of more complex material models was examined in conjunction with the ﬁnite element (FE) method. The employed constitutive model utilizes the decomposition of the strain tensor:

εi j = εiEj + εiCj + εiPj

(10.5)

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where εiEj is the time-independent and fully recoverable elastic component of strain, εiCj is the time-dependent viscous component of strain (creep strain), and εiPj is timeindependent plastic strain. The elastic strain is related to stress through isotropic elastic compliance tensor. The evolution of creep strain is governed by a creep ﬂow rule:

ε˙iCj = γ

∂ J2 ∂ σi j

(10.6)

where a dot symbol above indicates time rate and J2 is the creep potential expressed by the second invariant of deviatoric stress tensor. Scalar γ depends on the current value of equivalent stress 1 σ¯ = (3J2 ) 2 (10.7) and the equivalent creep strain, which is expressed by the power creep law:

ε¯ C = a0 σ¯ a1 t a2

(10.8)

Here a0 , a1 , a2 are material constants and t stands for time. In contrary to creep strains, plastic strains develop only upon satisfaction of the von Mises yield condition 1 J2 − σY = 0 (10.9) 3 where σY is a material parameter (uniaxial yield strength). The evolution of plastic strain then follows the associated ﬂow rule and perfect plasticity. In general, the constitutive model is characterized by two elastic parameters (E, ν ), three creep parameters (a0 , a1 , a2 ) and one plastic parameter (σY ). The geometry of the nanoindentation experiments was simpliﬁed in FE model by assuming axial symmetry of both the indenter and the material specimen. The height of the modeled domain was equal to that of the real specimen (4 mm), while the diameter was reduced to one half (15 mm – which is still much larger than the zone affected by any indent). The FE mesh consisted of 1,800 isoparametric four-node elements and it was signiﬁcantly reﬁned in the proximity of the indent. The indenter was modeled as perfectly rigid because intrinsic deformations of the indenter itself are negligible compared to approximately forty times softer sample of the cement paste. Such assumption also signiﬁcantly reduces computational complexity of the model. Variable contact of the indenter with the specimen was identiﬁed and imposed in each loading step. The analysis was performed with consideration of large strains and large displacements. In order to clarify the inﬂuence of different mechanisms of deformation on the response of cement paste in nanoindentation experiments, FE analyses were carried out with some or none of the strain components in Eq. (10.5) assumed equal to zero. In all calculations, ﬁve-cycle experiment with maximum force 50 mN, attained in ﬁve equal steps was used. Comparison of the experimental result and the response calculated with elastic perfectly-plastic material model is shown in Fig. 10.7. In this case, the material

J. Nˇemeˇcek et al. Load [mN]

188 50 40 30 20 10 0

Experiment Model 0

500

1000

1500

2000

Depth [nm]

Load [mN]

Fig. 10.7 Force vs. depth diagram for elastic-plastic material

50 40 30 20 10 0

Experiment Model

0

500

1000

1500

2000

Depth [nm]

Fig. 10.8 Force vs. depth diagram for elastic-creep material

parameters (E = 23.25 GPa and σY = 170 MPa) were determined so as to match the maximum force at maximum depth in the experiment. It is obvious from the Fig. 10.7, that due to excluding the viscous effects, the model cannot capture the creep of the material during load-dwell periods. The model also does not produce any hysteresis, which is typical for the unloading-reloading periods of the experiment. In another numerical experiment, we excluded plastic strains while creep strains were accounted for. The results are shown in Fig. 10.8. It is seen that even though material parameters were determined so as to ﬁt well the ﬁrst loading-dwellunloading cycle, the subsequent cycles do not match. In particular, the model does not capture the signiﬁcant reduction of tangential stiffness that occurs during loading periods of second, third, and fourth cycle. The model used for the last analysis included all strain components according to Eq. (10.5). In this case, the number of parameters was too high to be determined by a simple trial and error approach. The main difﬁculty consisted in the fact, that even during the dwell periods when the loading force was constant, creeping of the material under the indenter resulted in changing contact area and consequently in changing stress ﬁeld. Thus, it was not possible to separate the response due to plastic yielding and creeping. However, Fig. 10.9 shows that this model qualitatively matches all the major features of the experimental response curve: the creep during load-dwell periods as well as variation of tangential stiffness during the loading phases. FE model also gives information on the deformations and stress ﬁeld under the indenter probe as shown in Fig. 10.10.

Load [mN]

10 Nanoindentation of Cement Pastes and Its Numerical Modeling 50 40 30 20 10 0

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Experiment Model

0

500

1000 Depth [nm]

1500

2000

Fig. 10.9 Force vs. depth diagram for elastic-plastic-creep material

Fig. 10.10 FE model. Effective stress under indenter probe

10.5 Concluding Remarks An extensive experimental study was performed on cement paste which is an heterogeneous and also creeping material. The work was motivated by the fact that simple evaluation procedures lead to overestimation of elastic properties (size effect on elastic modulus). Experiments showed that the reason of this fact is the presence of creep during unloading in an indentation P-h curve. In such a case, simple evaluation procedures do not give satisfactory results and different loading and/or different evaluation methods need to be used. The comparison of several approaches showed that classical elastic solution [6] can be used for the estimate of elastic parameters only in connection with a special loading path. A simple visco-elastic solution [7] can capture the loading and holding periods of the P-h curve for one cycle experiment. However, using of the same material parameters does not lead to satisfactory results for the case of cyclic loading. Thus, a more general FE model was proposed. The FE analyses showed that in description of the micromechanical behavior of cement paste, both time-independent plastic strains and time-dependent creep stains appear to play an important role. However, parameters of the qualitatively most suitable elastic-plastic-creep model are difﬁcult to obtain. Presently, the possibility of using a more sophisticated method of parameter identiﬁcation based on genetic algorithms is being researched.

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Acknowledgements This work has been supported by Czech Grant Agency (GACR 103/05/0896) and by the Ministry of Education of the Czech Republic (project MSM 6840770003). Their support is gratefully acknowledged.

References 1. Constantinides G, Ulm FJ, Van Vliet K (2003) On the use of nanoindentation for cementitious materials, Materials and Structures, 36, 191–196 2. Constantinides G, Ulm FJ (2004) The effect of two types of C-S-H on the elasticity of cementbased materials: Results from nanoindentation and micromechanical modeling, Cement and Concrete Research, 34 (1), 67–80 3. Choi Y, Van Vliet KJ, Li J, Suresh S (2003) Size effect on the onset of plastic deformation during nanoindentation of thin ﬁlms and patterned lines, Journal of Applied Physics, 94 (9) 4. Detwiler RJ et al. (2001) Preparing Specimens for Microscopy, Concrete International, 23 (11) 5. Elmustafa AA, Stone DS (2002) Indentation size effect in polycrystalline F.C.C. metals, Acta Materialia, 50 (14), 3641–3650 6. Oliver WC, Pharr GM (1992) An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments, Journal of Material Research, 7, 1564–1583 7. Vandamme M, Ulm FJ (2006) Viscoelastic solutions for conical indentation, International Journal of Solids and Structures, 43, 3142–3165 8. Velez K et al. (2001) Determination of nanoindentation of elastic modulus and hardness of pure constituents of Portland cement clinker, Cement and Concrete Research, 31, 555–561 9. Wei Y, Wang X, Zhao M (2004) Size effect measurement and characterization in nanoindentation test, Journal of Material Research, 19 (1), 208–217

Chapter 11

Ductile Crack Growth Modelling Using Cohesive Zone Approach Vladislav Koz´ak

Abstract The paper studies the prediction of the crack growth of the ductile fracture of forged steel 42CrMo4. Crack extension is simulated by means of element extinction algorithms and two approaches have been compared. The ﬁrst one is based on the damage model Gurson-Tvergard-Needleman (GTN) (see [12]), the second on the cohesive zone model with the exponential traction separation law. The bulk of the paper is concentrated on the cohesive zone modelling. Determination of micro-mechanical parameters is based on the combination of static tests, microscopic observation and numerical calibration procedures. The attention is paid on the inﬂuence of initial value of J-integral and the slope of R-curve (J-∆a) which is modelled by 3D FEM. Based on tensile test the static elastic-plastic characterization of metals consist of the determination of the curve expressing the equivalent von Mises stress as a function of equivalent plastic strain. For ductile materials capable of undergoing large post-necking deformations, the exact material curve determination requires exact approach. The approximation suggested by Mirone [18] has appeared to be promising and valid for the structural steels.

11.1 Introduction An important issue when considering failure is the observation that most engineering materials are not perfectly brittle in the Grifﬁth sense, but display some ductility after reaching the strength limit, in which small-scale yielding, micro-cracking and void initiation, growth and coalescence take place. If the process zone is sufﬁciently small compared to structural dimension, linear elastic fracture mechanics can apply. If not cohesive forces that exist in the fracture zone must be taken into account. Vladislav Koz´ak Institute of Physics of Materials, Academy of Sciences of the Czech Republic, 616 62 Brno, ˇ zkova 22, Czech Republic Ziˇ [email protected] V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

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The most powerful and natural way is to use cohesive zone model, which was introduced by Dugdale [8] and modiﬁed by Barenblatt [2] for elastoplastic fracture in ductile metals. For ductile fracture the most important parameters of the cohesive zone model appear to be tensile strength and the work of separation or fracture energy. The cohesive model is a phenomenological model and most authors take their own formulation for dependence of traction on the separation. The crack tip, the term used very often in the fracture mechanics, is a mathematical idealization. In reality, a region of material degradation exists in some process zone. In this zone the microbehaviour becomes important for constitutive modelling. Most of the newer models developed and proposed are a bit different from Barenblatt’s model in that they deﬁne the traction acting on the ligament as a function of the opening and not on the crack tip distance as Barenblatt did. The material separation and, thus, damage of the structure is described by interface elements in FE method. Using this technique, the behaviour of the material is split into two parts: the damage free continuum with arbitrary material law, and the cohesive interfaces between the continuum elements, which specify only the damage of the material. Principally there are four approaches how to model damage, predict separation of materials and fracture process, e.g. [3] and [19]: (a) The process zone is small, its inﬂuence is neglected and than only special fracture criteria are used, e.g. K, J and C∗ etc. (b) We admit the creation of the new surfaces (separation) in the local area and the rest of the body is modelled using standard continuum element. Only local area is described using cohesive elements. (c) Damage is implemented in constitutive models and accumulation of damage is processed. (d) The advance approaches, e.g. [22] enable to combine mechanics of continuum with the local behaviour of collection of atoms on the base of molecular dynamics [14]. Micromechanical models based on the damage mechanics showed themselves as a perspective way how to avoid to numerical problems connected with the dependence of characteristic parameters used in the classical fracture mechanics on the size and geometry of body. The principal question is if the obtained parameters can be used without problems on the real component and predict its integrity. From it follow procedures which can be applied on the strategy in modelling of the crack growth. In contrast to crack growing of elastic materials, where the crack initiation is directed to the catastrophic damage of component, the presence of the ductile area leads to the stable crack growth. Crack growth can be simulated by the following way: (i) using node release techniques, where the control of the crack growth is based on the J, CTOD, CTOA, e.g. [5, 23], (ii) using cohesive elements, e.g. [17, 21, 26], and (iii) using constitutive equations based on the damage mechanics, e.g. [4, 16, 21]. Within the framework of the damage mechanics the GTN [24] model is thought of as a micromechanical process like initiation, growth and coalescence of voids. The constitutive equations which are used to describe ductile fracture processes are based on the relatively simple models which are dependent on many micromechanical parameters obtained experimentally.

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11.2 Determination of Cohesive Parameters Since the cohesive model is a phenomenological model, there is no evidence, which form is to be taken for the cohesive law, T (δ ) (see Fig. 11.1). Cohesive models can be used for normal and tangential separation as well for mixed mode loading. Experiment in this paper were made only on a pure mode I crack and therefore cohesive law for mode I was applied. Cohesive law has to be assumed independently of speciﬁc material as a model of the separation process. The exponential model is used by many authors for both the ductile and the cleavage fracture [6]. An exponential relationship between the effective traction (T ) provides a decohesion model. The T -δ response follows an irreversible path with unloading always direct to origin. This model represents all the features of the separation process by: (1) the shape of the cohesive traction/separation curve (T -δ ), and (2) the local material strength by the peak traction (T0 ), and, the local ductility deﬁned by the work of separation (Γ0 ) given by the area under (T -δ ) curve. For the determination of the cohesive stress, T0 in the case of normal fracture a hybrid technique has been developed. Using conventional elastic-plastic analysis, the distribution of the axial stress across the notch section of the specimen geometry is determined for the instant of the crack initiation in the centre of specimen. At that event, the axial stress exhibits a maximum in the centre of specimen, which is supposed to be equal to T0 . The case of the standard computation by Abaqus [1] can be seen in the Fig. 11.2a and in the Fig. 11.2b (used load steps are in the Table 11.1). Approximately twelve specimens have been used and mean value 2000 MPa was determined as the T0 value (T0 = 2,000 ± 50 MPa). The standard CT specimens were used for J-integral determination according the ASTM 1820–99a procedure. The experimentally determined value of Ji was

Fig. 11.1 Cohesive laws proposed for various materials

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Fig. 11.2a Stress distribution for notched tensile specimen 2400 step step step step step step step step step step step

axial stress [MPa]

2000 1600 1200 800

45 43 37 34 31 24 16 13 10 6 3

400 0

0

0.5

1

1.5

2

2.5

3

distance [mm] Fig. 11.2b Axial stress distribution Table 11.1 Load steps used in Fig. 11.2b Step

3

6

10

13

16

24

F [N] Step F [N]

2504.2 31 32,502

8000.53 34 33675.4

16003.4 37 33217.3

22003.6 43 31149.1

25001.9 45 29999.7

29002.4

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195

found to be Ji = 115 ± 5 MPa.mm and this value was calibrated using numerical procedure in WARP3D because of absence cohesive element carrying plasticity in Abaqus [1]. FE mesh for CT specimen can be seen in Fig. 11.3. Set of computations were applied to the calibration of the cohesive parameters used for J − R curve prediction (T0 , Ji ). Various combinations for T0 and Ji were tested. In the Fig. 11.4 the values Ji are marked as Cohe 1, Cohe 2 a Cohe 3 (Ji = 110, 120, 130) and

Fig. 11.3 FE mesh for CT specimen

300

J-integral [MPa.mm]

250 200 150 100

Cohe_1 Cohe_2 Cohe_3 Experiment

50 0 0

0.1

0.2

0.3

∆a [mm] Fig. 11.4 J-R curve in the initial phase of the crack propagation

0.4

0.5

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J-integrál [MPa.mm]

250

200

150

100

Cohe_1 Cohe_2 Cohe_3 Experiment

50

0 0.5

1

1.5

CTOD [mm] Fig. 11.5 J-CTOD curve in the initial phase of the crack propagation (data obtained from CT specimen)

received data were compared with the experimental values. The best correlation was found for ﬁrst value of Ji· = 110 MPa.mm. The calibration process can be seen further in the Fig. 11.5. After this procedure the input data for stable crack growth modelling have been received. For simulation at given material curve the cohesive parameters seem to be: T0 = 2,000 MPa and Ji = 110 MPa.mm.

11.3 Stable Crack Growth Modelling Using Cohesive Elements The experimental results of the SE(B) specimens were available in the form of the J-R diagram (J-∆a). The characteristic mesh size of the cohesive element was 0.2 × 1.4 × 0 mm. Some material curves received by the standard material tests for the same material show necessity of the diligent approach and accurate methods for the material curve determination. The best coincidence with the experimental data was in case of the material curve obtained from the tensile specimens and where the Mirone approach [18] was applied. Using the cohesive parameters received on the notched specimens (T0 ) and Ji on the CT specimens the numerical simulation of the stable crack growth was simulated and J − R curve was predicted. By utilization of two symmetry planes (plane xy and yz) for SE(B) specimen with a/W = 0.5 only one quarter of the real body was modelled. The FE mesh consists from 8,560 nodes and 7,155 element C3D8 (Abaqus 2005). For application of nonlinear cohesive element, the package Warp3D [10] was

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197

necessary to use. In the same mesh generated in Abaqus [1] the next 180 cohesive elements inter 8 were added. Owing to numerical instability the loading increment from one step was decreased from 0.025 mm to 0.0025 mm (therefore more than 1,000 loading steps were applied). Numerical modelling found strong dependence on the mesh size, especially on the mesh size in the direction in the thickness of the body. Therefore 15 various meshes for the FE modelling were applied. In Fig. 11.6 one can see detail of the crack tip for FE mesh used for the application of the cohesive elements.

Fig. 11.6 The FE mesh and detail of the crack tip for SE(B) specimen

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Fig. 11.7 Standard tensile specimen

Fig. 11.8 Modiﬁed tensile specimen – waisted

To determine material curve for characterization of the elastoplastic behaviour of tested material the standard tensile experiments have been using for many years. The relation σekv (εekv ) is found but the problem is the validity of this relation after necking. According to many experimental observations in [18] the new approximate curve in Eq. (11.1) was received for standard tensile specimens (see, Fig. 11.7) and modiﬁed tensile specimens – waisted specimens (Fig. 11.8). This Eq. (11.1) was used in our prediction of the material curve. At least 10 specimens were used in both cases; the third curve [25] was determined by standard procedure without MLR elongation. Material curves used for modelling can be seen in the Fig. 11.9 and corresponding J-R curves in the Fig. 11.10. The higher values of deformation are the ﬁtted values. MLRσ (εekv − εN ) = 1 − 0, 6058 (εekv − εN )2 + 0, 6317 (εekv − εN )3 −0, 2107 (εekv − εN )4

(11.1)

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199

1400

equivalent stress [MPa]

1200 1000 800 600

according to MLR model from standard tensile specimens according to MLR model from tensile specimens - waisted from (Vlcek 2004)

400 200 0

0

0.5

1

1.5

2

2.5

equivalent deformation [-] Fig. 11.9 Material curves used for modelling 700 600

J [MP amm]

500 400 300 experiment FEM - material curve 1

200

FEM - material curve 2

100

FEM - material curve 3

0

0

0.5

1

1.5

∆a [mm]

2

2.5

Fig. 11.10 J-R curve for various material curves

11.4 Stable Crack Growth Modelling Using GTN Model The base micromechanical parameters are determined from the real tensile test combined with the ﬁnite element calculations. Determination of micro-mechanical parameters is based on the combination of tensile tests and microscopic observation.

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Fig. 11.11 Void distribution in the neck area of the round tensile bar

The standard cylindrical specimens with a diameter of 6 mm have been tested at room temperature at crosshead velocity of 2 mm min−1 . From the reason for the numerical simulation the following data have been measured: force vs. elongation and force vs. contraction using optical method. The tensile specimens have been analysed as a ﬁrst. The methodology for the assessment of the micromechanical parameters requires the metallographic observation not only in the area of the local change of the diameter but in the area of non-affected by the plastic deformation [15]. Half of the tensile specimen has been bisected and a die head has been separated. The two samples for optical microscopy have been prepared. Received photos have been analyzed using image analysis. Void distribution in the neck area of the round tensile bar can be seen in Fig. 11.11. f

0.0049

0.0073

0.0073

0.0083

0.0126

0.0131

0.0349

On the base of the received photos recording the contracted area it is possible to submit that in this case of ductile process the coalescence of cavities is minimal. Then the value of fc is very close to value of fF and the critical conditions for ductile damage is being received only by combination of growth and nucleation. One can see in Fig. 11.11 that the fF close to fracture surface nearly reaches the value equal 0.034 and f0 the value 0.005. Major item of GTN model is the yield criterion which extends von Mises yield criterion and can be expressed as:

2 S i j Si j 3 q2 σm ∗ Φ= + 2q1 f cosh − 1 + q3 f ∗2 = 0 (11.2) 2 3 σY 2 σY f fc ≥ f ∗ (11.3) f = f ∗− f fc − fFu − fcc ( f − fc ) fc ≤ f The parameters q1 , q2 , q3 are used to adjust the model, σm is hydrostatic stress, σY is yield stress, f∗ is void fraction, fc is the critical void fraction for coalescence, fF is the ﬁnal value of f, fu ∗ = 1/q1 . The void volume fraction, f, which is deﬁned as the total volume of all cavities to the volume of the body, is introduced as an internal variable to characterize the damage. Its equation consists of two terms due to nucleation and growth:

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df = dfgrowth + dfnucl with f(t0 ) = f0 ,

(11.4)

with f0 as the initial void volume fraction. The void growth rate is proportional to the plastic volume dilatation rate and an empirical approach for nucleation of void was proposed by [7, 11], suggested a normal distribution for void nucleation (for strain-controlled nucleation) and then we can write: dfnucl = Adεp ,

(11.5)

where A represents the intensity of nucleation and εp is equivalent plastic deformation and A is given by fN 1 εp − εN 2 A = √ exp − (11.6) 2 SN SN 2π εN is the mean strain for nucleation, SN is its standard deviation, fN is the volume fraction of void nucleating particles. The results following from the metallographic observation in question of the absence of the coalescence stadium have been veriﬁed on the base of the ﬁnite element calculation too. An important query for the applicability of the model is whether or not the micro-mechanical parameters are dependent on the mesh size. Material affection was given by the true stress-strain curve, fN = 0.04 and the statistical model of nucleation with recommended values εN = 0.3 and SN = 0.1 was chosen see [20] for all computations. As can be seen in Fig. 11.12, where the number of elements in the neck area is varying, one can observe the discrepancy between Fig. 11.12a and b. Next computations for more elements than 14 are giving the same curves as in Fig. 11.12b. The inﬂuence of the initial volume fraction on the slope of elongation-contraction curve has been tested; it can be seen in Fig. 11.13. It is evident that the increasing value of f0 implicates the increasing of the value of the plastic deformation in the neck area. Reciprocally the volume fraction of void nucleating particles, fN , has been tested and it is presented in [25]. In the framework of the stable crack growth modelling the 3D model of the standard specimen for three point bend test was created. Using two planes of symmetry only one fourth of the real body was modelled. The problem of determination of the proper mesh size ahead the crack tip was solved on the base of comparison experimentally determined force-COD curve and numerically received one (see [25]). The characteristic mesh size was initially determined to 0.5 mm, but the coincidence between experiment and the numerical data was not good. Having been used the mesh where the characteristic mesh size for crack growth area had been selected to 0.1 mm the agreement between experiment and numerical model has been markedly increased. Using above mentioned micromechanical data and FEM software WARP3D [10] the dependence of J-integral on the ductile crack length has been received. As can be seen in Fig. 11.14 the computed curve and the experimental curve are nearly the

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V. Koz´ak 8 GT model fN = 0,04

∆ L [mm]

6

4

number of elements: 10 f0 = fN = 0

2

f0 = 0,001 f0 = 0,0001

0 0

0.5

1

1.5

2

2.5

∆ d [mm] Fig. 11.12a The inﬂuence of the mesh size on the curve elongation-contraction 8 GT model fN= 0,04

∆ L [mm]

6

4

number of elements: 14 f0 = fN = 0

2

f0 = 0,001 f0 = 0,0001

0 0

0.5

1

1.5

2

2.5

∆ d [mm] Fig. 11.12b The inﬂuence of the mesh size on the curve elongation-contraction

same, but the good correlation has been found due to varying of q2 parameter. The values q1 = 1.5 and q2 = 1 have been accepted for a long time as constants nondependent on the material behaviour. Faleskog [9] as the ﬁrst referred to difﬁculties following from the consideration of the independence of these parameters on the

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10

GT model fN = 0.004

∆ L [mm]

8

6

4 EXP f0=fN=0 2

f0=0,005 f0=0,001 f0=0,008

0 0

1

2

3

∆ d [mm] Fig. 11.13 The dependence of the elongation-contraction curve on the varying value f0 800

J [MPamm]

600

400

EXP q2 =0,8

200

q2 =0,82 FEM q2 =0,85

0

0

0.5

1

1.5 ∆a [mm]

2

2.5

Fig. 11.14 J-R – curve

material. The new method of calibration q1 and q2 parameters is being discussed in the latest work presented by Kim [13]. In our calibration the q1 parameter is ﬁxed and the dependence of q2 (representing the local triaxiality) on the slope of the R-curve is tested. In connection of the slope the attention has been paid on the ductile crack initiation. From

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physical-mechanical point of view the value of Ji is appeared to be independent on the geometry of the body. This value can be regarded as a material characteristic and it has been determined numerically as the value when the ﬁrst element vanished. This value can be denominate as a pseudo-physical and its dependence on the q2 parameter (in Fig. 11.15) and on the triaxiality factor h (in Fig. 11.16) has been

120

Ji FEM [MPamm]

100

80

60

40 0.75

0.8

0.85

0.9

0.95

1

q2 [-] Fig. 11.15 The dependence of Ji FEM on the q2 parameter 120

Ji FEM [MPamm]

100

80

60

40 0.9

0.95

1

1.05

1.1

h [-] Fig. 11.16 The dependence of Ji FEM on the stress triaxiality factor

1.15

1.2

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205

determined. As can be seen the decreasing of the local stress triaxiality leads to the increasing of the initial value of Ji FEM and for our experimental steel this initiation value was markedly higher then for recommended value of q2 . The value of Ji could be dependent on the geometry of the body (from the engineering approach) and this ﬁnding in necessary to take into account in the case of transferability of the experimentally determined data to the real components.

11.5 Conclusions Using of the “vanishing elements” the ductile crack growth was simulated and the R-curve (J-∆a) was predicted using damage and cohesive elements in case of a pure mode I. The coincidence of the predicted curve and experimental curve is very good; – To receive these results is necessary to calibrate parameters q1 and q2 in GTN model. – For modelling using cohesive elements the hybrid experimental and numerical procedure is used for the calibration T0 and Γ0 . – Stable crack growth modelling is strongly dependent on the quality of the FE mesh. For the damage modelling the blunting crack tip is used; for the modelling using cohesive elements the thickness of these separate elements is zero and the exponential traction separation law (see Fig. 11.1b) was used. Obtained results can be seen in Figs. 11.17 and 11.18.

700 600

J [MPamm]

500 400 300 experiment FEM - cohesive model FEM - GTN model

200 100 0 0

0.5

1

1.5

2

∆a [mm] Fig. 11.17 J-R curve generated by cohesive model and by GTN model

2.5

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V. Koz´ak

Fig. 11.18 The reconstruction of the crack path

– The shape of the J-R curve is more determined by the material curve than by the shape of the traction separation law. The exact determination of the material curve is the key point for proper application of the cohesive zone model used for the ductile fracture. – Three separate tensile geometries were investigated, including waisted and notched specimens. Waisted samples and standard tensile specimens were used to determine the material curve. The notched specimens have been found very suitable for the veriﬁcation of the measured material curves. The shallow notch of the waisted tensile specimen allows monitoring of the diametral contraction during testing, so that the true stress could be accurately monitored. Agreement of the computed curve (using Abaqus [1]) force – deﬂection and force – contraction for notched specimen was excellent. Notched specimens were used for T0 determination and calibration. – The standard CT specimens were used for J-integral determination according the ASTM 1820–99a procedure and Ji value has been calibrated through the best ﬁt of the calculated and measured J-R curve. Acknowledgements This research was supported by the grant 101/05/0493 of the Grant Agency of the Czech Republic.

References 1. ABAQUS 6.5.4 (2005) User’s manual, ABAQUS Inc. 2. Barenblatt GI (1962) The mathematical theory of equilibrium cracks in brittle fracture, Adv. Appl. Mech. 7: 55–129 3. Brocks W, Cornec A, Schneider I (2003) Computational Aspects of Nonlinear Fracture Mechanics, GKSS 2003/30: 129–203 4. Brocks W, Sun DZ, Hoeing A (1995) Veriﬁcation of the transferability of micromechanical parameters by cell model calculations with visco-plastic materials, Int. J. Plasticity 11(8): 971–989 5. Brocks W, Eberle A, Fricke S, Veith H (1994) Large stable crack growth in fracture mechanics specimens, Nucl. Eng. Design, 151(2–3): 387–400

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6. Cornec A, Schneider I, Schwalbe KH (2003) On the practical application of the cohesive model, Eng. Fract. Mech. 70: 1963–1987 7. Chu C, Needleman A (1980) Void nucleation effects in biaxially stretched sheets, J. Eng. Mater. Technol. 102: 249–256 8. Dugdale DS (1960) Yielding of steel sheets containing slits, J. Mech. Phys. Solids 8: 100–104 9. Falescog J (1995), Effects of local constraint along three-dimensional crack fronts – a numerical and experimental investigation, J. Mech. Phys. Solids 43(3): 447–493 10. Gullerud A, Koppenhoefer K, Roy A, RoyChowdhury R, Walter M, Bichon B, Cochran K, Carlyle A, Dodds RH Jr. (2005) WARP3D – Release 15.4 – 3D Dynamic Nonlinear Fracture Analysis of Solids Using Parallel Computers and Workstation, University of Illinois, 1–433 11. Gurland J (1972) Observation on the fracture of cementie particles in spheroidized 1.05 C steel deformed at room temperature, Acta Metal 20: 735–741 12. Gurson AL (1997) Continuum Theory of Ductile Rapture by Void Nucleation and Growth: Part 1-Yield Criteria and Flow Rules for Porous Ductile Media, J. Eng. Mater. Technol. 99: 2–15 13. Kim J, Gao X, Srivatsan TS (2004) Modelling of void growth in ductile solids: effects of the stress triaxiality and initial porosity, Eng. Fract. Mech. 71: 379–400 14. Klein PA et al. (2001) Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods, Theor. Appl. Fract. Mech. 37: 99–166 15. Koz´ak V, Vlˇcek L (2005) Parameters identiﬁcation for GTN model and their veriﬁcation on 42CrMo4 steel, Mater. Sci. Forum 482: 335–338 16. Lemaitre JA (1985) Coupled elasto-plasticity and damage constitutive equations, Comput. Methods Appl. Mech. Eng. 51(1–3): 31–49 17. Li YN, Bazant Z (1997) Cohesive crack model with rate-dependent opening and viscoelasticity: II Numerical algorithm, behaviour and size effect, Int. J. Fract. 86: 267–288 18. Mirone G (2004) A new model for elastoplastic characterization and the stress-strain determination on the necking section of a tensile specimen, Int. J. Solid Struct. 41: 3545–3564 19. Moes N, Belytschko T (2002) Extended ﬁnite element method for cohesive crack growth, Engng. Fract. Mech. 69: 813–833 20. Needleman A, Tvergaard V (1987) An analysis of ductile rupture at a crack tip, J. Mech. Solids 35: 151–183 21. Needleman A (1990) An analysis of tensile decohesion along an interface, J. Mech. Phys. Solids 38: 289–324 22. Shilkrot LE, Miller RE, Curtin WA (2004) Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics, J. Mech. Phys. Solids, 52(4): 755–787 23. Siegele D, Schmidt W (1983), Comput. Struct. 17: 697 24. Tvergaard V, Needleman A (1984), Analysis of the cup-cone fracture in a round tensile bar, Acta Metal. 32(1): 157–169 25. Vlˇcek L (2004) Numerical 3D analysis of cracked specimen: Constraint parameter computation and stable crack growth modelling, Ph. D. thesis, Brno University of Technology, Brno, Czech Republic 26. Yang B, Ravi-Chandar K (1996) On the role of the process zone in dynamic fracture, Journal of the Mechanics and Physics of Solids 44, Issue 12: 1955–1976

Chapter 12

Composite (FGM’s) Beam Finite Elements ˇ s Just´ın Mur´ın, Vladim´ır Kutiˇs, Michal Masn´y, and Rastislav Duriˇ

Abstract The composite structures (e.g. laminate, sandwich structures, or FGM’s) are often used in engineering applications. Their FE analyses require creating very ﬁne mesh of elements even for relatively small sized bodies, what increases computational time, particularly in nonlinear analyses. Macro-mechanical modelling of the composites is based on material properties homogenisation. The homogenisation of the material properties is made for three layers sandwich bar with constant material properties of middle layer and polynomial variation of effective elasticity modulus and volume fraction of ﬁbre and matrix at the top/bottom layer. In derivation of the bar element matrices the effective longitudinal elasticity modulus have been considered. The uni-axially polynomial variation ﬁbre elasticity modulus E f and the matrix elasticity modulus Em are given as polynomials. In the numerical examples/analyses we assume a three-layers two-node sandwich bar with double symmetric rectangular cross-section. As a typical example of geometric nonlinear behaviour the three-hinge mechanism was analysed. Two different approaches have been considered for calculation of the effective longitudinal elasticity modulus of the composite (FGM’s) bar with both polynomial variation of constituent’s volume fraction and polynomial longitudinal variation of the elasticity modulus. Stiffness matrix of the composite bar contains transfer constants, which accurately describe the polynomial uni-axially variation of effective Young’s modulus.The obtained results are compared with 3D analysis in the ANSYS simulation program. Findings show good accuracy and effectiveness of this new ﬁnite element. The results obtained with this element do not depend on the mesh density. Just´ın Mur´ın, Vladim´ır Kutiˇs, and Michal Masn´y Department of Mechanics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkoviˇcova 3, 812 19 Bratislava, Slovakia [email protected], [email protected], [email protected] ˇ s Rastislav Duriˇ Department of Applied Mechanics, Faculty of Material Sciences and Technology, Slovak University of Technology, Paul´ınska 16, 917 24 Trnava, Slovak Republic [email protected] V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

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12.1 Introduction Composite structure elements, like the laminate, sandwich, or functionally graded material (FGM’s) links and beams with the simple or double symmetric crosssections are most important in engineering applications. Materials that are made by mixing two or more different materials (evenly or unevenly) can acquire much better properties than their single components. These new materials (composite or FGM’s) [1–3] are characterised with a continuous or discontinuous variation of material properties. Macro-mechanical modelling and analysis of such new materials is based on the material properties homogenisation. Micro-mechanical modelling leads to correlation between the constituent properties and the average effective composite properties. Mixture rules have been used very often in the engineering applications for the material properties derivation. These rules of mixture are based on the statement that the composite property (Young’s modulus, Poisson’s ratio, coefﬁcient of thermal and electrical conduction, thermal expansion coefﬁcient, . . .) is the sum of the properties of each constituent multiplied by its volume fraction. To increase the accuracy of the composite material properties calculation, the new homogenisation techniques and the improved mixture rules have been established [4, 5]. The most important trend in the homogenisation is the multiscale computation [6, 7]. In many publications the constant volume fractions and material properties of the composite constituents in the whole composite beam have been considered. The similar consideration is made for sandwich beam, where the constant properties in each layer were assumed. In several papers only a transversal variation of the material properties has been considered. In this contribution a longitudinal polynomial continuous variation of the stiffness properties will be considered in the stiffness matrix of the beam element [8, 9]. Our new FGM’s link and beam ﬁnite elements are presented for the analysis of the electric, thermal and structural ﬁeld with above mentioned material properties variations [10]. The transversal and longitudinal variation of the material properties will be considered by the symmetric three-layer sandwich beam ﬁnite element as well [11]. The straight sandwich composite bar ﬁnite element with transversal and longitudinal variation of Young’s modulus is established, which is intended to perform non-incremental full geometric non-linear analysis. The effectiveness and accuracy of these elements will be evaluated and discussed.

12.2 FGM’s and Beam Finite Elements Composites and functionally graded materials will play an important role in the future in design of the mechanical structures. These materials provide the designer with an opportunity to engineer the material’s microstructure at the micro level in order to enhance their structural performance. This is accomplished by spatially varying the micro structural details through non-uniform distribution of the

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reinforcement phase(s), by using reinforcement with different properties, sizes and shapes, as well as by interchanging the roles of reinforcement and matrix phases in a continuous manner. The result is a microstructure that produces continuously changing thermal, electrical and mechanical properties at the macroscopic or continuum level, thereby blurring the traditional distinction between a material and a structural component. At present, FGMs (originally developed in Japan for high-thermal gradient applications such as protective thermal coatings) are rapidly progressing beyond their originally intended use to a wide range of structural, bio-mechanical and energy conversion applications, opening up new and exciting areas of research. In the analysis section, Aboudi et al. [12] combine the previously developed higher-order theory for FGMs (known in the literature as HOTFGM), that explicitly couples the material’s microstructure and macrostructure, with an optimization algorithm in order to identify optimal ﬁber distributions in unidirectional composites subjected to a through-thickness thermal gradient that minimize the in-plane moment resultant, and thus the tendency to bend about an axis. For the analysis of FGM structure are developed and improved current numerical method such as ﬁnite element method, meshless methods, meshfree methods, ﬁnite difference methods, ﬁnite volume methods, etc. Analysis of FGM models by the commercial ﬁnite elements is approached due to discrete material properties of each element of the model. Preparing of input dates for such modeling and simulation is very time consuming, and the solution results are strong depended on the mesh ﬁneness. In the recent years the new beam and link ﬁnite elements have been developed, that are able to model the beams with varying material properties through the crosssection (through the depth of the element, for example, in [12, 13]). A new beam and link ﬁnite elements with varying material properties have been developed [8–11]. These elements can be used in the multidisciplinary analysis: electric-thermal, thermal-structural and electric-thermal-structural weak coupled ﬁelds. The effective material properties are calculated by the extended mixture rules [11]. Of course, these elements can be also used, when the varying material properties are calculated using other homogenization techniques. These new ﬁnite elements will be presented in the following parts of this contribution.

12.3 Beam and Link Elements of FGM’s with Uniaxial Variation of Material Properties Figure 12.1 shows beam element with variation of cross-section and with variation of effective material properties. Variation of effective material properties is deﬁned as: – Effective longitudinal elasticity modulus

l

E (x) = Ei ηE (x) = Ei 1 + ∑ ηEk x k=1

k

,

(12.1)

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Fig. 12.1 Beam element with varying material properties

– Effective longitudinal thermal expansion coefﬁcient

m

αT (x) = αTi ηα T (x) = αTi 1 + ∑ ηα T k x

k

,

(12.2)

k=1

– Effective longitudinal thermal conductivity

n

λ (x) = λi ηλ (x) = λi 1 + ∑ ηλ k xk ,

(12.3)

k=1

– Effective longitudinal electrical conductivity

p

σ (x) = σi ησ (x) = σi 1 + ∑ ησ k x

k

,

(12.4)

k=1

where ηE (x) is the polynomial for variation of elasticity modulus, ηα T (x) is the polynomial for variation of thermal expansion coefﬁcient, ηλ (x) is the polynomial for variation of thermal conductivity, ησ (x) is the polynomial for variation of electrical conductivity. Values at node i (Ei , αTi , λi and σi ) and also the orders of polynomials η (x) (l, m, n and p) in previous expressions depend on material properties of ﬁbre and matrix (constituents of FGM) and also on their volume fraction in ﬁnal mixture. We assume that the variation of parameters, which describe the double axial symmetrical cross-section, can be expressed in polynomial forms: – Cross-section area

q

A (x) = Ai ηA (x) = Ai 1 + ∑ ηAk x

k

k=1

,

(12.5)

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– Area moment of inertia around the y-axis

r

Iy (x) = Iyi ηIy (x) = Iyi 1 + ∑ ηIyk x

k

,

(12.6)

,

(12.7)

k=1

– Area moment of inertia around the z-axis

s

Iz (x) = Izi ηIz (x) = Izi 1 + ∑ ηIzk x

k

k=1

– Torsion constant of the cross-section

t

IT (x) = ITi ηIT (x) = ITi 1 + ∑ ηIT k xk ,

(12.8)

k=1

where ηA (x) is the polynomial for variation of cross-section area, ηIy (x) is the polynomial for variation of moment of inertia around the y-axis, ηIz (x) is the polynomial for variation of moment of inertia around the z-axis, ηIT (x) is the polynomial for variation of torsion constant of the cross-section. Values of cross-section characteristics at node i (Ai , Iyi , Izi and ITi ) and the orders of polynomials (q, r, s and t) in previous expressions depend on the type of crosssection variation.

12.3.1 3D Structural Beam Finite Element Figure 12.2 shows a straight beam element with two nodes, i and j, with local static and kinematic structural quantities. Local kinematic quantities can be summarized in vector form u = [ui , vi , wi , ϕxi , ϕyi , ϕzi , u j , v j , w j , ϕx j , ϕy j , ϕz j ]T

(12.9)

and local static quantities F = [Ni , Qyi , Qzi , Mxi , Myi , Mzi , N j , Qy j , Qz j , Mx j , My j , Mz j ]T .

(12.10)

Local stiffness matrix equation can be expressed in form Ku = F, where local stiffness matrix K has form

(12.11)

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Fig. 12.2 Beam element with varying cross-sectional area

⎡ E A /b i i

2N

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 cz b 2z

0 0 cy b 2y

−Ei Ai /b 2N 0 0 0 0 −cz b 2z 0 0 0 −cy b 3y 0 0 −cy b 2y 0 Gi IiT /b 2T 0 0 0 0 0 −Gi IiT /b 2T cy (Lb3y − b3y ) 0 0 0 cy b3y 0 cz (Lb 3z − b3z ) 0 −cz b 3z 0 0 S Ei Ai /b2N 0 0 0 Y cz b 2z 0 0 cy b2y 0 M Gi IiT /b 2T 0 0

0 0

0 cz b 3z 0

0 0 −cy b2y 0 cy b3y 0 0 0 cy b2y 0 cy (Lb2y − b3y )

0 cz b2z 0 0 0 cz b3z 0 −cz b2z 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

cz (Lb2z − b3z )

(12.12) b 2N is transfer constant for axial loading, b 2z , b2z , b 3z , b3z are transfer constants for bending in xy plane, b 2y , b2y , b 3y , b3y are transfer constants for bending in xz plane, b 2T is transfer constant for torsion. cy and cz have form cy = EIiy (b2y b 3y − b3y b 2y ). cz = EIiz (b2z b 3z − b3z b 2z ).

(12.13) (12.14)

All mentioned transfer constants are numerically solved quantities, which can be described as follows: a j (x) , (12.15) b j+2 (x) = η (x) where a j (x) =

xj . j!

(12.16)

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The goal is to solve the ﬁrst and second integral of expression (12.15) along the length of element. Because the expression (12.15) is usually too complicated to be solved analytically, we have to solved it numerically 0, 0 and 0. Numerical values of this integrations are transfer constants b j+2 =

(L)

a j (x) dx, b j+2 = η (x)

(L)

a j (x) dxdx. η (x)

(12.17)

For example, transfer constant for axial loading represents the ﬁrst integral of expression a0 (x) 1 a0 (x) = = . (12.18) b 2N (x) = ηA (x)ηE (x) ηAE (x) ηAE (x) All other transfer constants in this contribution have a similar meaning, that can be solved using simple numerical algorithm described in Appendix A.

12.3.2 Multiphysical Beam Element for Weak Coupled Analysis Proposed multiphysical beam ﬁnite element considers three coupled ﬁelds: electrical, thermal and structural. The one-way coupling of the ﬁelds is assumed, it means that the solution method for the coupled ﬁeld problems is sequential, and the FEM equations are coupled by the right hand side (load vector). Figure 12.3 shows the thermal link element and the electrical link element with two nodes, i and j, varying cross-sectional area A(x) and continuous longitudinal generated heat p(x) (for thermal link) and continuous longitudinal electric current i(x) (for electrical link). The FEM equations for the electric and thermal link elements are $# $ # $ # # $ Ai σi 1 −1 Vi Iekvi Ii = + , (12.19) Iekv j Ij b 2Aσ −1 1 V j $# $ # $ # # $ # $ Ai λi 1 −1 Ti Pi Pekvi PekvJi = + + . (12.20) Tj Pj Pekv j PekvJ j b 2Aλ −1 1

Fig. 12.3 Thermal and electrical link element with varying parameters

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Thermal ﬁeld is coupled to the electrical ﬁeld. Coupling between the electric and thermal ﬁeld is performed by the equivalent nodal Joule heat, which is deﬁned by the term # $ # $ Ai σi (Vi − V j )2 1 PekvJi . (12.21) = PekvJ j 1 2b 2Aσ The FEM equations of the structural beam element (12.3) are coupled with thermal ﬁeld through the equivalent nodal forces caused by thermal load. The equivalent nodal thermal load vector is represented by #

$ # $ L Fε T xi −1 Ai Ei αTi Ti ηα T (x)ηT (x)dx. = Fε T x j 1 b 2AE

(12.22)

0

Transfer constants are calculated according to the terms (12.7). Distribution of temperature along the thermal link element is approximated with the polynomial function T (x) = Ti ηT (x).

12.3.3 2D Beam Element for 2nd Order Beam Theory Special situation arises when the 2nd order linearized beam theory is considered. The stiffness matrix (12.12) has formally the same form, but the transfer constants which are in the stiffness matrix have different meaning. For the sake of simplicity, we will assume only 2D beam-column with variation of stiffness along the length (variation of cross-section, or variation of material properties, or variation of both cross-section and material properties) and the prime interest is focused on the solution of critical forces. These variations are deﬁned at the beginning of Section 12.3. Figure 12.4 shows 2D beam-column element with variation of cross-section and material properties in undeformed and deformed conﬁguration. As we can see, there are two possible conﬁgurations, M − R and M − Q (R is the transversal force, Q is the shear force and M is bending moment). The basic beam equations are derived in M − Q conﬁguration, but the ﬁnal FEM equations are transformed to M − R conﬁguration. The transformation between both conﬁgurations can be found in 0. In the case of the 1st order beam theory, both conﬁgurations coincide. As was mentioned, the ﬁnal local stiffness relation is formally the same as in the 1st order theory, i.e. Ku = F, (12.23) where the displacement vector u and local load vector F have form

T u = ui , wzi , ϕyi , uk , wzk , ϕyk ,

(12.24)

T F = Ni , Rzi , Myi , Nk , Rzk , Myk .

(12.25)

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Fig. 12.4 Beam-column element with variation of cross-section

Stiffness matrix K has form ⎤ ⎡ E i Ai E i Ai 0 0 − 0 0 ⎥ ⎢ b b 2N ⎥ ⎢ 2N ¯ ¯ ¯ ⎥ ¯ ⎢ −cy b 3y 0 −cy b 2y cy b2y cy b 2y ⎥ ⎢ ⎥ ¯ ¯ ⎢ ¯ ¯ cy (Lb 3y − b3y ) 0 cy b 3y cy b3y ⎥, ⎢ ⎥ ⎢ Ei Ai ⎥ ⎢ S 0 0 ⎥ ⎢ b2N ⎥ ⎢ ⎦ ⎣ cy b¯ 2y Y cy b¯ 2y M cy (Lb¯ 2y − b¯ 3y) where cy = Ei Iyi b2y b 3y − b3yb 2y .

(12.26)

Transfer constant b 2N is the same as in the section 12.3.1, but the transfer constants b¯ 2y , b¯ 3y , b¯ 2y and b¯ 3y have different meaning as b 2y , b 3y , b2y and b3y in section 12.3.1. In these new transfer constants (b¯ 2y , b¯ 3y , b¯ 2y ) and b¯ 3y , there is also internal axial force Ni included in solution process (not only the variation of stiffness). The critical force of the system is force, which causes that determinant of global structure stiffness matrix is zero. The process how to ﬁnd this force is described in 0.

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12.4 Sandwich Beam Finite Element Let us consider a sandwich straight beam with double symmetric cross-section A that is predominantly rectangular. The beam is loaded orthogonally to the plane of lamination. Debonding of layers is not considered. If the lamination is symmetric, the elementary theory of homogeneous isotropic beams can be used for all solutions, but the elasticity modulus has to be replaced by its effective value 0. The single layers are built from composite layers with longitudinal variation of the volume fractions and elasticity modulus of the constituents. The two-node beam element (Fig. 12.5) with three composite layers has the following geometric and material properties: – A1 = A3 , A2 are cross-sections of the layers, where v1 = v3 = A1 /A, v2 = A2 /A are layer volume fractions of the sandwich beam.

−E1 (x) = E1i ηE1 (x) = E1i 1 + ∑ ηE1k xk

= E3 (x)

k

= E3i ηE3 (x) = E f (x)

(12.27)

is an effective longitudinal elasticity modulus of layers 1 and 3 (faces).

−E2 (x) = E2i ηE2 (x) = E2i 1 + ∑ ηE2l xl

= E c (x)

(12.28)

l

is an effective longitudinal elasticity modulus of layer 2 (core). – E1i = E3i , and E2i is an effective longitudinal elasticity modulus of layers 1 and 3, and layer 2 at node i, respectively. – ηE1 (x) = ηE3 (x) and ηE2 (x) is the polynomial variation of the effective longitudinal elasticity modulus of layers 1 and 3, and layer 2, respectively.

Fig. 12.5 Symmetric sandwich (three layers) composite beam

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Layers 1 and 3 (faces) have the same thickness h f and they are of the same material; layer 2 (core) has thickness hc . Parameter d = hc + h f . Using the mixture rule, the effective longitudinal elasticity modulus for stretching is 3

ELN (x) =

∑ vn En (x) = ELNi ηELN (x) ,

(12.29)

n=1

where ELNi = E1i v1 + E2i v2 + E3i v3 is an effective longitudinal elasticity modulus for stretching at node i, ηELN (x) is the polynomial variation of the effective longitudinal elasticity modulus for stretching of the sandwich beam. The advanced effective longitudinal elasticity modulus for ﬂexural loading of the sandwich beam, according to the laminate theory, is

2 3 E f (x) h f hc + h f E c (x) (hc )3 12 E f (x) h f M EL (x) = 3 + + h 6 2 12 = ELMi ηELM (x) .

(12.30)

Here, ELMi is an effective elasticity modulus for ﬂexural loading at node i, ηELM (x) is its polynomial variation, h = hc + 2h f is the total beam depth. Elasticity moduli (12.29) and (12.30) affect the stiffness matrix parameters of the new beam element with varying stiffness. The stiffness matrix of this sandwich beam K (with the classical six degrees of freedom) has the form: ⎡E A ELNi A LNi 0 0 − 0 b2ELN ⎢ b 2ELN ⎢ 0 0 −cM b 2ELM cM b 2ELM ⎢ c M b3ELM ⎢ 0 c b c − b Lb 0 −c M 3ELM M M b3ELM 3ELM 3ELM ⎢ ELNi A ⎢ ELNi A ⎢− 0 0 0 ⎢ b2ELN b 2ELN ⎣ 0 −c b 0 c b −c b M 2ELM

0

cM b2ELM

M 3ELM

cM b3ELM

0

⎤ 0 cM b2ELM cM b3ELM 0

⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

M 2ELM −cM b2ELM −cM b2ELM cM Lb 3ELM − b3ELM

(12.31) ELMi I is the bending stiffness parameter. I is a quadratic where cM = b 2ELM b3ELM −b3ELM b2ELM area moment of inertia of the whole cross-sectional area A. The transfer constants b 2ELN , b 2ELM , b 3ELM , b2ELM and b3ELM depend on the cross-sectional characteristics and the effective longitudinal elasticity modulus variation for stretching (marked with the index N) and ﬂexural bending (marked with the index M), respectively. The strains due to in-plane loading are constant over the whole cross-section but non-linear along the beam length axis. The stretching stresses are constant over each layer of cross-section, but there is a jump in the stresses at the face/core interface. All mentioned transfer constants are formally evaluated according to (12.5) and have forms 1 b 2ELN (x) = , (12.32) ηELN (x)

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1 , ηELM (x) x b 3ELM (x) = . ηELM (x)

b 2ELM (x) =

(12.33) (12.34)

The axial displacement at the point x could be expressed using the new shape functions φ for the two nodes beam with varying stiffness 0: # $ b b (x) (x) u j, (12.35) u (x) = φi ui + φ j u j = 1 − 2ELN ui + 2ELN b2ELN b 2ELN where k = 1 for one-dimensional element. From (12.32) we get the expression for the axial strains (due to in-plane loading):

ε N (x) =

u j − ui du (x) . = dx ηELN (x) b 2ELN

(12.36)

The longitudinal variation of stretching stresses in the layers will be non-linear along the beam length axis

σ1N (x) = ε N (x) E1 (x) = σ3N (x), σ2N (x) = ε N (x) E2 (x) .

(12.37)

The bending strains vary linearly with y over the whole cross-section and nonlinearly along the beam length axis. The advanced ﬂexural sandwich beam rigidity, according to the laminate theory, is

3 2 E f (x) h f E f (x) h f hc + h f E c (x) (hc )3 D (x) = b + + (12.38) 6 2 12 and the bending strain is:

ε M (x, y) =

M (x) y, D (x)

(12.39)

where M (x) is the bending moment at point x. The bending stress vary transversal linearly (with the y position) within each layer, but there is a jump in the stresses at the face/core interfaces E1 (x) y = σ3M (x, y) , D (x) E2 (x) y. σ2M (x, y) = M (x) D (x)

σ1M (x, y) = M (x)

(12.40)

The longitudinal variation of the layers bending stress will be non-linear as usual. The shear stress for the core and the faces, according to the laminate theory, can be calculated using the following expressions (Q (x) is the shear force at point x)

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Q (x) E f (x) h f d E c (x) (hc )2 , + − y2 D (x) 2 2 4

f 2 Q (x) E f (x) (hc )2 f c f 2 +h h + h −y . τ (x, y) = D (x) 2 4

τ c (x, y) =

(12.41)

The maximum shear stress appears at the neutral axis and it is described by the function

f (x) h f d c (x) (hc )2 Q (x) E E τmax (x) = τ c (x, y = 0) = + . (12.42) D (x) 2 8 The shear stress in the core/face interface is described by the function: Q (x) E f (x) h f d c f τmin (x, y = hc /2) = τmax (x, y = hc /2) = . D (x) 2

(12.43)

There is no jump in the shear stresses at the interfaces, and the shear stress are zero at the outer ﬁbres of the faces.

12.5 Geometric Nonlinear Bar Element with Varying Stiffness A new approach to evaluation of geometric non-linear problems is presented in this part of contribution suggested by Mur´ın 0. This solution is based on principle of virtual work Sij δ Eij dV = Fi δ ui dA + Fˆk δ qk , (12.44) V0

A0

where Si j is the II Piola-Kirchhoff stress tensor, Ei j is Green-Lagrange strain tensor, Fi are the surface tractions, δ ui is the variation of displacement ﬁeld and δ qk are the virtual displacements of points in which the nodal forces Fˆk are applied. In our approach no linearization of the variation of Green-Lagrange strain tensor is used. Thus we can obtain the full non-linear non-incremental formulation of the element stiffness matrices. When total Lagrangian formulation is used, non-linear equation derived from the equilibrium of internal and external work (12.44) has the form

Cijkl ekl δ eij dV +

V0

=

A0

Cijkl (ηkl δ eij + ekl δ ηij + ηkl δ ηij )dV

V0

Fi δ ui dA + Fˆk δ qk ,

(12.45)

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where Cijkl is a tensor of material properties deﬁning constitutive law of the II. Piola-Kirchhoff stress tensor Sij and Eij = ei j + ηij is the Green-Lagrange strain tensor, where eij = 12 (ui, j + u j,i ) and ηij = 12 uk,i uk, j is its linear and non-linear part, respectively. After implementation of correspondent approximation of the displacement functions ui = φik qk and their derivatives, we can modify Eq. (12.45), for FEM requirements, to the form 1 4

V0

+

1 4

Cijkl (φkm,l + φlm,k )(φin, j + φ jn,i )qm dV

Cijkl φ pm,k φ pr,l (φin, j + φ jn,i )qm qr dV

V0

+ +

1 2 1 2

Cijkl φ pr,i φ pn, j (φkm,l + φlm,k )qm qr dV

V0

Cijkl φ pm,k φ pv,l φrq,i φrn, j qm qv qq dV =

Fi φin dA + Fˆn,

(12.46)

V0

where qk are the nodal degrees of freedom. Equation (12.46) represents a basic relation, which can be used for an arbitrary ﬁnite element derivation. If the concept of transfer functions and constants, published by Rubin 0, is used in the derivation of the new shape functions, we obtain local non-linear stiffness matrix of the element for elastic composite material. Stiffness matrix of the composite bar contains transfer constants, which accurately describe the polynomial uni-axial variation of the effective Young’s modulus ELN (x) ≡ Cijkl . In following, the shape functions φik for the two node bar element will be marked as φi and φ j for a starting (i) and ending ( j) point according to Fig. 12.1. Similarly, displacements of nodes qk will be marked as ui (u j ) for a starting (ending) point. The effective longitudinal elasticity modulus of the three layer composite bar is described by Eq. (12.29). After substitution of the new straight bar shape functions (12.35) into (12.46) and corresponding modiﬁcations, the non-linear stiffness matrix of the bar element with varying stiffness has the form # $ b b 1 3 Ai ELNi 1 −1 2 2ELN 2ELN 0 0 2 + (λ − 1) (L ) , 1 + (λ − 1) L K= b2ELN 2 (b2ELN )2 2 (b2ELN )3 −1 1 (12.47) where b 2ELN ,

b 2ELN =

0 L +

0

(b 2ELN (x))2 dx,

b 2EL =

0 L +

0

(b 2ELN (x))3 dx are transfer

constants the composite bar. The base for calculation of the transfer constants is the transfer function (12.32). These constants can be computed by using algorithm u −u mentioned in appendix A. Parameter λ = jL0 i + 1 is the stretching of the bar. Then the internal force in the bar element can be calculated using the formulae

12 Composite (FGM’s) Beam Finite Elements

Ni =

223

Ai ELNi 1 3 2 0 2 b2ELN 0 b2ELN ( ( λ − 1) L + λ − 1) (L ) 1 + (λ − 1)L0 . b 2ELN 2 (b 2ELN )2 2 (b 2ELN )3 (12.48)

Final, the resulting system of non-linear equations is usually solved using an iterative method. For this solution process, the full tangent stiffness matrix was expressed by KT =

∂F ∂u

# $ 3 Ai ELNi 1 −1 2 0 2 b2ELN 0 b2ELN + (λ − 1) (L ) = . 1 + 3 (λ − 1) L b2ELN (b2ELN )2 2 (b2ELN )3 −1 1 (12.49)

12.6 Examples 12.6.1 Multiphysical Analysis of an Actuator Electric-thermal-structural actuator with variation of a rectangular cross-section and material properties has been considered (Fig. 12.6). The actuator is made from functional graded materials (FGMs). The actuator is loaded by an electric current and has an ideal thermal insulation. The goal is to perform an electric-thermal-structural analysis using the new beam element. The results of this analysis will be compared with the results obtained using the coupled analysis with classical electro-thermal link and structural beam elements of code ANSYS 0. The FEM-model has been created with only six new beam elements. Geometry, material properties and boundary conditions are represented in the following boxes (Figs. 12.7–12.9).

Fig. 12.6 Electric-thermal-structural actuator from FGMs materials

224

Fig. 12.7 Geometry properties of the actuator

Fig. 12.8 Material properties of the actuator

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225

Fig. 12.9 Boundary conditions of the actuator

Fig. 12.10 Deformed and undeformed shape of the actuator

Each part of the actuator is made from FGM. The parts consist of two different materials, namely matrix and ﬁbers. The extended rule of mixture for the volume fraction of each material 0 has been used to determine the effective constitutive properties (CP) of each part. Individual parts have the following ﬁber, matrix and resulting effective material properties, which are shown in the Fig. 12.8. Boundary conditions are shown in the Fig. 12.9. This coupled problem has been solved as a sequential coupling. In the ﬁrst step the electric analysis has been performed according to Eq. (12.19). The output from this analysis is the Joule heat according to Eq. (12.21). The second analysis is the thermal analysis – Eq. (12.20), where the determined Joule heat has been used as the loading. The output from this analysis is thermal forces which are computed according to Eq. (12.22). In the last step the structural analysis has been performed according to Eq. (12.11), where the determined thermal forces act as the loads. The ﬁnal outputs are the structural displacements and rotations and also the reaction forces and moments. The deformed and undeformed shape is shown in Fig. 12.10. The same problem has been analyzed by a sequential method using the multiphysical element LINK68 and beam element BEAM3 0. To show the effectiveness and accuracy of our element each of the six parts of the actuator was divided to 1 and 25 elements. LINK68 element is used for the electro-thermal analysis and the BEAM3 element is used for the structural analysis. The comparison of the two solution results is given in Tables 12.1 and 12.2. The last line in these tables contains the results obtained using only one our link/beam element for each part of the actuator. Comparison of these results shows very good effectiveness of the new link/beam ﬁnite element.

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Table 12.1 Comparison of the nodal variables NOE

Electric potential [V] V1

V2

V3

V4

V5

V6

1 25

20.0360 20.0356

20.0698 20.0714

20.0700 20.0716

20.0702 20.0718

20.1040 20.1075

20.1400 20.1431

1

20.0356

20.0714

20.0716

20.0718

20.1075

20.1433

Temperature [◦ C]

NOE T1

T2

T3

T4

T5

T6

1 25

36.6916 37.1214

34.3761 34.6131

27.9994 28.4347

27.0886 27.4994

26.1753 26.5612

15.6806 15.4821

1

37.1195

34.8210

28.8313

27.8961

26.9579

15.6906

Table 12.2 Comparison of the nodal variables Displacements Uxi [×10−6 m]

NOE Ux2

Ux3

Ux4

Ux5

Ux6

1 25

4.9047 5.02142

2.24937 2.46304

2.42084 2.63814

2.58616 −1.05899 2.8059 −0.97073

1

4.98519

2.46033

2.63824

2.80881 −0.93455

Displacements Uyi [×10−6 m]

NOE Uy2

Uy3

1 25

−1.4252 −4.6715 −0.94808 −4.2661

1

−0.9347

Uy4 −4.8434 −4.3807

Uy5 −4.2550 −3.7385

Uy6 −2.0734 −1.5108

−4.24733 −4.35789 −3.71962 −1.49722 Rotation Rotzi [×10−6 rad]

NOE Rotz2

Rotz3

Rotz4

Rotz5

Rotz6

1 25

−148.41 −147.63

−112.59 −102.35

−42.729 −54.287

−191.53 −201.71

1

−145.735 −100.732 −54.3004 −200.119 −208.689

−201.45 −210.58

12.6.2 Buckling of a Beam-Column Figure 12.11 shows 2D beam-column structure with variation of stiffness (crosssection and material properties). Structure comprises three columns, stiffness of column 1 and 3 varies along the column length and column 2 has constant stiffness along the length. Stiffness parameters: Column 1: E1 (x)I1y (x) = 1.03084 × 1010 (0.05 + 0.005x2)4 [Nm2 ] Column 2: E2 (x)I2y (x) = 1.03084 × 1010 [Nm2 ] Column 3: E3 (x)I3y (x) = 1.03084 × 1010(0.05 + 0.00294118x2)4 [Nm2 ]

12 Composite (FGM’s) Beam Finite Elements

227

Fig. 12.11 2D beam-column structure Table 12.3 The convergence of ANSYS solution to our one element solution NOE

Fcr(exact) [N]

1 2 4 6 8 10 20 50 100

20435.1 24079.5 21554.8 20928.9 20712.0 20612.1 20479.3 20442.2 20436.9

Geometry parameters are: L1 = 1 m, L2 = 4 m, L3 = 5 m, H1 = 3 m, H2 = 4 m. The goal is to determine the ﬁrst critical force Fcr . The problem has been solved by our beam element with transfer constants, as well as with FEM program ANSYS 0. In our program, each of three beam-column members was modelled with only one new beam element (Fig. 12.11) is identical with FEM model. In program ANSYS, element BEAM54 has been used and the number of elements in each of three beam-column members was gradually increased. The number of elements (NOE) in Table 12.3 represents the number of elements in each member of structure. As we can see from Table 12.3 and Fig. 12.12, ANSYS solution with increasing number of elements converges to our one element solution, but when the mesh is very coarse, the error is approximately 18%.

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Fig. 12.12 The convergence of ANSYS solution to our one element solution

12.6.3 Structural Analysis of a Sandwich Beam The following academic parameters have been chosen for the sandwich beam (Fig. 12.5): – The cross-section is square (b × h ≡ 0.01 × 0.01m), its area is A = 0.0001 m2, I = 8.3334 × 10−8 m4 is the quadratic area moment of inertia, the length of the beam is L = 1 m – The cross-section area of layers is: A1 = A2 = A3 = A/3 = 3.3334 × 10−5 m2 – The effective longitudinal elasticity modulus of layers is: E1 (x) = E3 (x) = E f (x) = 1 × 1010 (1 + x) [Pa] E2 (x) = E c (x) = 2 × 1010 (1 + 2x) [Pa]. The whole beam was modelled with only one of our sandwich beam element. The transversal unit force (F = 1 N) has been applied at the free end of clamped sandwich beam. A linear – elastic analysis has been done. The deﬂection curve and distribution of normal stresses along the beam layers and over the beam depth have been examined. In this case, the effective longitudinal elasticity modulus for ﬂexural bending (12.30) of the sandwich beam changes linearly and has form: ELM (x) = 1.037037 × 1010 (1 + 1.071428571x) [Pa],

(12.50)

where ELMi = 1.037037 × 1010 Pa and ELM j = 2.148148 × 1010 Pa (Fig. 12.13).

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229

Fig. 12.13 Effective longitudinal elasticity modulus [Pa] for ﬂexural bending

Fig. 12.14 Bending strain distribution at the top of layer 3

Fig. 12.15 Longitudinal distributions of the bending normal stress

Using this elasticity modulus, the deﬂection and rotation at the free beam end is v j = −0.031188 m and ϕ j = −0.044056 rad respectively. As can be shown, the reactions at node i satisfy the equilibrium equations exactly. The bending strain (12.39) has been obtained as the following function at the top of the layer 3:

ε M (x) = 0.005 (1 − x) / (8.6419 + 9.2592x) ,

(12.51)

This function is depicted in Fig. 12.14. Its value at node i is 0.0005785 [−]. Longitudinal distribution of the bending normal stress (12.40) is shown in Fig. 12.15. Figure 12.15a shows the bending normal stress at the top of the layers 3 and Fig. 12.15b shows the bending normal stress at the top of the layer 2 and bottom of the layer 3.

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The same problem has been solved using the SOLID45 ﬁnite elements. By this very ﬁne mesh the ANSYS solution converged to our sandwich beam element solution. For example, the deﬂection at the free beam end coincided with our value; the maximal normal stress at the top of the layers 1 and 3 (at the clamped beam end) has the value of 5.7689 MPa. The shear stress has been calculated using the expressions (12.41)–(12.43). In Fig. 12.16 distribution of the maximal core shear stress (12.42) (shear stress at the middle of the layer 2) and the shear stress at the face/core interfaces (shear stress at the top of the layer 2 and shear stress at the bottom of the layer 3) is shown along the beam length. Maximal core shear stress in the both ends are: τmax (x = 0, y = 0) = 16071.4 Pa; τ c (x = L, y = 0) = 17068.9 Pa and the values of stress at the face/core interfaces at the same nodal points are: τmin (x = 0) = 12857.1 Pa; τmin (x = L) = 12413.7 Pa. The transversal distribution of the normal and shear stress across the beam depth at the node i is shown in Fig. 12.17. As we can see, there is a jump in the normal stresses at the face/core interfaces (this stress jump cannot be reached using the SOLID45 element because of averaging of the nodal stress), but in shear stress there is no jump in this location. The normal stress at individual points has these values: point 1 (top of the core): 3.8571 MPa, point 2 (bottom of the face): 1.9285 MPa, point 3 (top of the face): 5.7857 MPa and between these points, there is a linear distribution of the normal f stress in core σcNODEi (y) and in face σNODEi (y) (Fig. 12.17).

Fig. 12.16 Longitudinal distribution of the maximal core shear stress and the shear stress at the face/core interfaces

Fig. 12.17 Transversal distribution of the normal and shear stress across the beam depth

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231

The shear stress at the individual points has these values: point 4 (middle of the core): 16.0714 kPa, and point 5 (top of the core and bottom of the face): 12.8571 kPa. The shear stress distribution in core is described by c τNODEi (x = 0, y ∈ 0, hc /2) = 16071.4285 − 1.1571 × 109y2 [Pa]

(12.52)

and in the face by f τNODEi (x = 0, y ∈ hc /2, h/2) = 14464.2857 − 5.7857 × 108y2 [Pa].

(12.53)

12.6.4 Deﬂection of Mises Bar Structure In these numerical experiment the accuracy and effectiveness of our new nonincremental geometric non-linear bar element equations with varying of effective material properties was examined. We assume a three layered two-node sandwich bar with double symmetric rectangular cross-section (Fig. 12.5). As a typical example of geometrically non-linear behavior the von Mises bar structure was analysed (Fig. 12.18). Two different approaches have been considered for calculation of the effective longitudinal elasticity modulus of the composite sandwich (FGMs) bar with both polynomial variation of constituent’s volume fraction and polynomial longitudinal variation of the elasticity modulus. In presented numerical experiments the elasticity modulus of the faces layers (Fig. 12.5) is described by polynomials E1 (x) and E3 (x). Elasticity modulus of the core E2 (x) is constant. These material properties were used to compare our results with the solution by ANSYS. There were developed two reference models in ANSYS. Firstly, one-dimensional beam model with division to 20 tapered elements and secondly, the solid model with 4,500 quadratic brick elements (Fig. 12.19). There was created a code in software MATHEMATICA for our new element. To obtain variation of the effective longitudinal elasticity modulus we have used two different approaches for calculation of the effective longitudinal elasticity modulus of the composite sandwich (FGM’s) bar with both polynomial variation of

Fig. 12.18 Von Mises bar element structure

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Fig. 12.19 ANSYS model with 4,500 solid elements to compare against our approach

constituent’s volume fraction and polynomial longitudinal variation of the elasticity modulus. First, we used new extended mixture rules (labeled as MR) published in 0, 0 and shortly described in section 12.4 of this contribution. In second approach, improved homogenization techniques described by Love and Batra (LB) 0 were used. The following geometrical parameters for the bar have been chosen for numerical examples (Figs. 12.5 and 12.18): L0 = 0.1 m, α 0 = 10o , A = 0.01 × 0.01 m2, h f = 0.001 m. The material properties have been considered following: elasticity modulus of ﬁbers E f = 400 GPa, elasticity modulus of matrix Em = 255 GPa and the change of volume fraction of ﬁbers v f have been considered in range 0.5; 0.8 along the bar length. Substitution of chosen parameters into above mentioned equations (Section 12.4) gave effective elasticity modulus as function of the ﬁber volume fraction in the form ELMR (v f ) = v f E f + (1 − v f )Em = 255 + 145v f .

(12.54)

Effective elasticity modulus for the identical composite according to 0 can be expressed by (12.55) ELLB (v f ) = 241.76 + 155.03v f + 4.53v2f . In addition, the polynomial variations of the component’s volume fraction have been considered. The variation of the ﬁber volume fraction is given by expression v f (x) = 0.5(1 + 6x)

(12.56)

and of the matrix volume fraction by expression vm (x) = 1 − v f (x) = 0.5(1 − 6x).

(12.57)

By implementation of these polynomial variations (12.27) – (12.29) we get both the effective longitudinal modulus for faces (for ANSYS solid analysis) and the effective longitudinal modulus of sandwich bar used in new bar element and for the ANSYS beam model, respectively. Resultant variation of elasticity moduli are summarized in Table 12.4.

12 Composite (FGM’s) Beam Finite Elements

233

Table 12.4 Variation of effective longitudinal elasticity moduli used in numerical examples

Solid element model (ANSYS) Beam element model (ANSYS) New bar element (MATHEMATICA)

Extended mixture rules (Mur´ın 0 – MR) [GPa]

Improved mixture rules (Love and Batra 0 – LB) [GPa]

E1 (x) = E3 (x) = 327.5 + 435x

E1 (x) = E3 (x) = 320.4075 + 478.68x + 40.77x2 E2 (x) = 255

E2 (x) = 255 ELN (x) = 269.5 + 87x

ELN (x) = 268.0695 + 95.736x + 8.154x2

50000 0

new bar element (LB) new bar element (MR) ANSYS - solid analysis (LB) ANSYS - solid analysis (MR) ANSYS - 20 beam elements (LB) ANSYS - 20 beam elements (MR)

axial force N [N]

−50000 −100000 −150000 −200000 −250000 −300000 −350000 −400000 −450000 0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

displacement uy [m] Fig. 12.20 Axial force-displacement response of common hinge

To examine the accuracy of the new bar element, the software Mathematica was used. The results obtained by this new element were compared with results of solid and beam element analysis in the ANSYS simulation programme. Results of both, ANSYS and new bar element solutions are presented in the following graphs. First graph shows relation between common hinge displacement vs. axial force (Fig. 12.20). Second graph shows global reaction – displacement response for common hinge (Fig. 12.21). Intensity of both axial forces and global reaction forces obtained from numerical analyses are given in the Table 12.5. In the Table 12.6 the absolute percentage errors of the new bar solutions against the ANSYS reference solutions are presented.

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global reaction F [N]

40000 30000 20000 10000 0

−10000 −20000 −30000 −40000 −50000 −60000 0.000

new bar element (LB) new bar element (MR) ANSYS - solid analysis (LB) ANSYS - solid analysis (MR) ANSYS - 20 beam elements (LB) ANSYS - 20 beam elements (MR)

0.005

0.010

0.015

0.020

0.025

0.030

0.035

displacement uy [m] Fig. 12.21 Global reaction-displacement response of common hinge

Table 12.5 Results of maximum forces for new bar and ANSYS solutions Axial force N [N]

Global reaction F [N]

New bar element

ANSYS 20 beam elements

ANSYS solid model

New bar element

ANSYS 20 beam elements

ANSYS solid model

Extended mixture rules – MR

−406570.2

−419200

−425980

−55183.7

−56343.6

−57046.8

Improved mixture rules – LB

−405126.9

−414559.7

−424468

−54987.8

−56143.8

−56844.2

Table 12.6 Percentage differences between the new bar analysis and ANSYS solutions [%] Difference of axial force N

Difference of global reaction F

(new bar – ANSY S beam) ANSY S beam

(new bar – ANSY S solid) ANSY S solid

(new bar – ANSY S beam) ANSY S beam

(new bar – ANSY S solid) ANSY S solid

Extended mixture rules – MR

3.01

4.56

2.06

3.27

Improved mixture rules – LB

2.28

4.56

2.06

3.27

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235

12.7 Conclusions New composite link/beam ﬁnite elements have been presented in this contribution. The ﬁrst part of the contribution deals with multiphysics link/beam element with variation of material and cross-section properties suitable for the ﬁrst and the second order beam theory. Stiffness matrix of a new sandwich beam element is presented in the second part. The polynomial longitudinal and transversally layer-wise symmetric variation of the sandwich beam stiffness has been taken into the account. Elastic behavior of the sandwich beam was modeled by laminate theory. The numerical experiments have been performed. The obtained results have been evaluated, discussed and compared with those obtained using the common link/beam elements. All the numerical results proved the high accuracy and effectiveness of the new link/beam elements. In our future work an extension of the proposed FGM’s beam elements for geometric nonlinearity will be done. The FGM’s sandwich beam will be established for more symmetric and nonsymmetrical layers as well. Very important task is the establishing of a FGM’s beam with spatial variation of material properties for solution of the coupled multiphysical problems, e.g. the electro-thermo-structural problem.

Appendix A – Transfer Constants Determination of the transfer functions and transfer constants occurring in the previous sections is based on the expression (12.15) b j+2 (x) = where a j (x) =

a j (x) , η (x)

xj for j ≥ 0 j!

(12.58)

and for j ≤ 0, a0 = 1, a j = 0. Closed solutions for the 1st and 2nd integrals of function b j+2 (x) are known only for lower degree polynomials η (x). For their numerical solution, which is more general, a recurrence rule was derived (n)

m

b j (x) = a j−n (x) − ∑ ηk k=1

( j − 2 + k)! (n) b (x), for j ≥ 2, n = 0 and 1. ( j − 2)! j+k

(12.59)

After some manipulation we get (n)

∞

b j (x) = a j−n (x) ∑ βt,0 (x), k=1

(12.60)

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where βt,0 (x) is expressed by m

βt,0 (x) = − ∑ ηk βt,k (x) k=1

−1

∏ (s − 1 + r)

(12.61)

r=−k

with parameters s = 1 + t, e =

x , βt,k = eβt−1,k−1 for k = 1, . . . m s−n

(12.62)

and initial values are

β0,0 = 1, β0,k = 0, for k = 1, . . . m.

(12.63)

Acknowledgements The authors gratefully acknowledge the support of this research by the Slovak Grant Agency for Science (Grant VEGA no. 1/4122/07 and no. 1/3092/06).

References 1. Y. Bansal, M.J. Pindera, Efﬁcient Reformulation of the Thermoelastic Higher-Order Theory for FGMs, NASA, 2002. 2. M. Koizumi, FGM activity in Japan, Composites, Part B(28B), 1997. 3. M. Koizumi, M, Niino, Overview of FGM Research in Japan, MRS Bulletin (20), 19–21, 1995. 4. J.C. Halpin, J.L. Kardos, The Halpin-Tsai equations. A review. Polymer Engineering and Science 16(5), 344–352, 1976. 5. T. Mori, K. Tanaka, Average stress in matrix and average elastic energy of materials with misﬁtting inclusions. Acta Metallurgica 21, 571–574, 1973. 6. J. Fish, W. Chen, Y. Tang, Generalized mathematical homogenisation of atomistic media at ﬁnite temperatures. International Journal of Multiscale Computational Engineering 3(4), 393–413, 2005. 7. W.K. Liu, E.G. Karpov, H.S. Park, Nano Mechanics and Materials: Theory, Multiple Scale Analysis, and Applications. Wiley, 2005. 8. J. Mur´ın, V. Kutiˇs, Beam element with continuous variation of the cross-sectional area, Computers and Structures 80, 329–338, 2002. 9. V. Kutiˇs, J. Mur´ın, Bar element with variation of cross-section for geometric non-linear analysis, Journal of Computational and Applied Mechanics 6, 83–94, 2005. 10. J. Mur´ın, V. Kutiˇs, M. Masn´y, An effective solution of electro-thermo-structural problem of uni-axially graded material, Structural Engineering and mechanics, 28, 695–713, 2007. 11. J. Mur´ın, V. Kutiˇs, Extended mixture rules for the composite (FGM’s) beam ﬁnite elements, prepared for publication, 2007. 12. J. Aboudi, M.J. Pindera, S.M. Arnold. Higher-order theory for functionally graded materials. Composites Part B: Engineering 30(8): 777–832, December 1999. 13. A. Chakraborty, S. Gopalakrishnan, J.N. Reddy. A new beam ﬁnite element for the analysis of functionally graded materials. International Journal of Mechanical Sciences, 45, 519–539, 2003. 14. J. Mur´ın, Implicit non-incremental FEM equations for non-linear continuum. Strojn´ıcky cˇ asopis 52(3), 2001. 15. H. Rubin, Analytische L¨osung linearer Differentialgleichungen mit ver¨anderlichen Koefﬁzienten und baustatische Anwendung. Bautechnik, 76, 1999.

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16. V. Kutiˇs, Beam element with variation of cross-section satisfying local and global equilibrium conditions, Ph.D. thesis, Slovak University of Techno1ogy, Bratislava, 2001. 17. H. Altenbach, J. Altenbach, W. Kissing, Mechanics of Composite Structural Elements, Springer, 2004. 18. J. Mur´ın, V. Kutiˇs, Geometrically non-linear truss element with varying stiffness, Engineering Mechanics 13(6), 435–452, 2006. 19. Ansys Theory Manual 2004. 20. J. Mur´ın, V. Kutiˇs, Improved mixture rules for the composite (FGM’s) sandwich beam ﬁnite element. In: IX International Conference on Computational Plasticity COMPLAS IX, E. Onate and D.R.J. Owen (eds.), CIMNE, Barcelona, 2007. 21. B.M. Love, R.C. Batra, Determination of effective thermomechanical parameters of a mixture of two elastothermoviscoplastic constituents. International Journal of Plasticity 22, 1026–1061, 2006.

Chapter 13

Computational Modal and Solution Procedure for Inhomogeneous Materials with Eigen-Strain Formulation of Boundary Integral Equations Hang Ma, Qing-Hua Qin, and Vladimir Kompiˇs

Abstract In the present study a novel computational modal and solution procedure are proposed for inhomogeneous materials with the eigenstrain formulation of the boundary integral equations. The model and the solution procedure are both resulted intimately from the concepts of the equivalent inclusion of Eshelby with eigenstrains to be determined in an iterative way for each inhomogeneity embedded in the matrix with various shapes and material properties via the Eshelby tensors, which can be readily obtained beforehand through either analytical or numerical means. As unknowns appeared in the ﬁnal equation system are on the boundary of the solution domain only, the solution scale of the inhomogeneity problem with the present model is greatly reduced. This feature is considered to be signiﬁcant because such a traditionally time-consuming problem with inhomogeneities can be solved most cost-effectively with the present procedure in comparison with the existing numerical models such as ﬁnite element method (FEM) and boundary element method (BEM). Besides, to illustrate computational efﬁciency of the proposed model, results of overall elastic properties are presented by means of the present eigenstrain model and the newly developed boundary point method for particle reinforced inhomogeneous materials over a representative volume element. The inﬂuences of scatted inhomogeneities with a variety of properties and shapes and orientations on the overall properties of composites are computed and the results are compared with those from other methods, showing the validity and the effectiveness of the proposed computational modal and the solution procedure.

Hang Ma Department of Mechanics, College of Sciences, Shanghai University, Shanghai 200444, China Qing-Hua Qin Department of Engineering, Australian National University, Canberra, ACT 0200, Australia Vladimir Kompiˇs Department of Mechanical Engineering, Academy of Armed Forces of General M.R. Stefanik, 03119 Lipt. Mikulas, Slovakia V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

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13.1 Introduction The determination of mechanical behaviour of an embedded inclusion is of considerable importance in a wide variety of physical and engineering problems. Since the pioneer work of Eshelby [1, 2], inclusion and inhomogeneity problems have been a focus of solid mechanics for several decades. In the terminology of Eshelby and Mura [3], an inclusion denotes a sub-domain subjected to an eigenstrain or a transformation strain in a solid, the ﬁrst problem of Eshelby. On the other hand, an inhomogeneity is a region with properties distinct from those of the surrounding material and subjected to an applied stress, the second problem of Eshelby. However, it is noted that in the literature, the term ‘inclusion’ has been referred to as ‘inhomogeneity’ in the sense of Eshelby. In the present report, the original terminology of Eshelby is employed. Following Eshelby’s idea of equivalent inclusion and eigenstrain solution, quite a diverse set of research work has been reported analytically [4–7] and/or numerically [8–16], to name a few. The eigenstrain solution can represent various physical problems where eigenstrain may correspond to thermal strain mismatches, strains due to phase transformation, plastic strains or ﬁctitious strains arising in the equivalent inclusion problems, overall or effective elastic, plastic properties of composites, quantum dots, microstructural evolution, as well as the intrinsic strains in the residual stress problems [17]. The analytical solutions of inclusion problems available in the literature can be taken as a basis for understanding the process of predicting stress/strain distribution either within or outside an inhomogeneity and for further research of such a category of problem. However, these solutions usually apply only for cases of simple geometries such as ellipsoidal, cylindrical and spherical in an inﬁnite domain. Therefore, numerical simulation using FEM, BEM, or volume integral methods (VIM) have been conducted in the analysis of inhomogeneity problems with various shapes and materials. In general, FEM may yield results for the whole composite materials including those inside and outside the inhomogeneity [10]. Consequently, the solution scale might be quite large since both matrix and inhomogeneities need to be discretized and the corresponding unknowns are solved simultaneously. In contrast, VIM and BEM seem more suitable for analyzing inhomogeneity problems. With the VIM [11–13], the ﬁelds inside inhomogeneities are expressed in terms of volume integrals, which will simplify the construction of ﬁnal matrix of the system of linear algebraic equations to which unknowns of the problem are signiﬁcantly reduced after discretization. However, as the interfaces between matrix and inhomogeneities also need to be discretized and identiﬁed by additional unknowns, only those problems with a few inhomogeneities have been studied in the literature. Regarding applications of BEM to inclusion problems, it is often coupled with VIM [14, 15] in practice and the procedure is more or less similar to that in VIM. It is usually used to solve problems with simple arrays of inclusions [15] as ﬁeld continuity condition across interfaces will signiﬁcantly increase the number of unknowns. Recently, the fast multipole expansion scheme [18] is introduced into BEM to improve computational efﬁciency in solving large- scale inclusion-matrix problems [16].

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It appears that the Eshelby’s idea of equivalent inclusion and eigenstrain solution has not yet been sufﬁciently followed in the regime of numerical study on inhomogeneity problems [19]. Motivated by this situation, a novel computational modal and solution procedure are proposed for inhomogeneous materials in this paper. The eigen-strain formulation of boundary integral equations (BIE) in the proposed model is developed based on the concept of equivalent inclusion of Eshelby with eigen-strains, which can be determined in an iterative way for each inhomogeneity embedded in the matrix with various shapes and material properties via the Eshelby tensors. As unknowns appeared in the ﬁnal linear equation system are on the boundary of the solution domain only, the solution scale of the inhomogeneity problem can be greatly reduced. In this paper, the second problem of Eshelby is addressed in Section 13.2 with the BIE model while the ﬁrst problem of Eshelby is discussed in Section 13.3 on the solution procedure with determination of Eshelby tensor. Section 13.4 describes the newly developed boundary point method for particle reinforced inhomogeneous materials over a representative volume element. The numerical results are presented in Section 13.5 for particle reinforced inhomogeneous materials over a representative volume element. The inﬂuence of scatted inhomogeneities with a variety of properties and shapes and orientations on the overall properties of composites are studied and compared with the results from other approaches. The results conﬁrm the validity and effectiveness of the proposed computational modal and the solution procedure. Finally, some conclusions are presented in Section 13.6.

13.2 The Boundary Integral Equations In the present model, the interface between inhomogeneity and matrix is assumed to be perfectly bonded together so that displacement continuity and traction equilibrium remain along their interface. All materials considered are isotropic. Let the domain of the problem to be a bounded region Ω surrounded by the outer boundary Γ, and inhomogeneous zones be ΩI surrounded by the interface ΓI and ΓI = ΩI ∩ Ω. This constitutes the second problem of Eshelby. The displacement ﬁeld of the problem can be described by the basic BIE as follows: C (p) ui (p) =

τ j (q) u∗i j (p, q) dΓ (q) −

Γ

Γ

u j (q) τi∗j (p, q) dΓ (q) + ∑ II

(13.1)

u∗i j

where is the Kelvin solution, or the displacement in j direction at the ﬁeld point q under a unit force acting in i direction at the source point p. τi∗j is the corresponding traction kernel and C is a boundary constant depending on the position of the source point and the boundary geometries at the source point [21]. The last term in Eq. (13.1) represents the effect of all inhomogeneities summed by the boundary integrals along the interfaces, namely the inhomogeneity integral: II =

ΓI

τ j (q) u∗i j (p, q) dΓ (q) −

ΓI

u j (q) τi∗j (p, q) dΓ (q)

(13.2)

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It is noted that each inhomogeneity integral II describes locally an exterior problem. For the completeness of the problem, there should be complement equations corresponding to each of the inhomogeneity integral (13.2) as follows: 0=

τ Ij (q) uIi j (p, q) dΓ (q) −

ΓI

uIj (q) τiIj (p, q) dΓ (q), p ∈ Ω

(13.3)

ΓI

where superscript I denotes the quantity is associated with inhomogeneity I and ΓI represents the boundary of the domain ΩI . In contrast to Eq. (13.2), each of the complement Eq. (13.3) describes locally an interior problem. The basic BIE (13.1) together with Eq. (13.3) can be solved simultaneously with the conventional BEM. However, this will make the solution scale very large since the unknowns on both the outer boundary Γ and the interfaces ΓI are included. To reduce the number of unknowns in the ﬁnal equation system, following effort is made based on Eqs. (13.10) and (13.3). As the interfaces are bounded perfectly by assumption, there are following relations on displacement continuity and traction equilibrium: u j = uIj , τ j = −τ Ij , τi∗j = −τiIj

(13.4)

Using the interface relations (13.4) and Eq. (13.3), the exterior problem of inhomogeneity integral (13.2) can be transformed into an equivalent interior problem as follows: (13.5) II = − τ Ij (q) u∗i j (p, q) dΓ (q) + τ Ij (q) uIi j (p, q) dΓ (q) ΓI

ΓI

It is noted that the ﬁrst integral at the right-hand side of Eq. (13.5) represents the domain ΩI ﬁlled with the matrix material while the second integral represents the domain ΩI ﬁlled with the inhomogeneity material both of them describes an interior problem. For the sake of convenience, following integral relations are introduced, i.e. for any homogeneous domain Ω with boundary, we have Cui +

u j τi∗j dΓ (q) =

Γ

Γ

τ j u∗i j dΓ =

ε jk σi∗jk dΩ =

Ω

σ jk εi∗jk dΩ

(13.6)

Ω

where σi∗jk and εi∗jk are, respectively, the corresponding stress and strain kernels to the Kelvin solution. The integral relations can be viewed as an integral form of the reciprocity theorem in elasticity. By invoking the integral relation (13.6), each of the inhomogeneity integral (13.5) can be rewritten as: II = −

ΩI

ε Cjk (q) σi∗jk (p, q) dΩ +

ε Ijk (q) σiIjk (p, q) dΩ (q)

(13.7)

ΩI

where ε Cjk denotes the ﬁctitious matrix strain in domain ΩI , ﬁlled with the matrix material rather than the inhomogeneity, induced by the action of interface tractions, while ε Ijk represents the strain of inhomogeneity under the same interface tractions.

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By introducing the concept of eigenstrain ε 0jk = ε Ijk − ε Cjk and noticing that the kernel relation σi∗jk = σiIjk exists, the inhomogeneity integral II in Eq. (13.7) can be written as II = ε 0jk (q) σi∗jk (p, q) dΩ (q) (13.8) ΩI

Therefore Eq. (13.1) becomes C (p) ui (p) =

Γ

−

τ j (q) u∗i j (p, q) dΓ (q) Γ

u j (q) τi∗j (p, q) dΓ (q) + ∑

ε 0jk (q) σi∗jk (p, q) dΩ (q)

(13.9)

ΩI

It can be seen that Eq. (13.1) together with the complement Eq. (13.3) have been reduced to a single BIE (13.9) with a sum of domain integrals for inhomogeneities, namely the eigenstrain formulation of the BIE developed here. Each domain integrals describes an equivalent inclusion problem made of the identical matrix material containing eigenstrains. This is, in fact, a equivalent inclusion method which was initiated by Eshelby [1, 2] and developed by Mura [3] and many others. The eigenstrains are deﬁned as the stress-free strains of the equivalent inclusion, which will cause the same traction state at interfaces and the stress state in the matrix as the inhomogeneity. If ε 0jk = 0, Eq. (13.9) describes an elastic state of homogeneous problem. Therefore, it can be seen that the key step for the solution of Eq. (13.9) is to determine the eigenstrains in each of the inhomogeneities. This issue is discussed in Section 13.3. The stress equation, which is indispensable in the analysis, can be derived directly from Eq. (13.9) as: C (p) σi j (p) =

τk (q) u∗i jk (p, q) dΓ (q) −

Γ

+∑

uk (q) τi∗jk (p, q) dΓ (q)

Γ

εkl0 (q) σi∗jkl (p, q) dΩ (q) + εkl0 (p) O∗i jkl

(13.10)

ΩI

where O∗i jkl (p, q) = lim

Ωε →0 ∆Γε

xl τi∗jk (p, q) dΓ (q)

(13.11)

and Ωε with its boundary Γε is an inﬁnitesimal zone within ΩI located at point p, when the distance between q and p approaches zero [22, 23] and xl = xl (q) − xl (p). The detailed derivation can be found from [21]. In certain cases such as dealing with crack problems with inhomogeneity, it is desirable to use the traction equation, which can be derived with the Cauchy relation τi = σi j n j directly from the stress equation (13.10) by putting p ∈ Γ:

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Cτi (p) = ni (p)

τk (q) u∗ijk (p, q) dΓ (q) − ni (p)

Γ

+ ni (p) ∑

∗ uk (q) τijk (p, q) dΓ (q)

Γ ∗ εkl0 (q) σijkl (p, q) dΩ (q) + ni (p) εkl0 (p) O∗ijkl

(13.12)

ΩI

where n j is the unit outward normal at the point p. The traction Eq. (13.12) is a hypersingular BIE. The domain integrals in (13.9), (13.10) and (13.12) can be evaluated by using the corresponding boundary type integrals [22, 23] which is discussed in Sections 13.3 and 13.4. It should be pointed out that the proposed matrix-inclusion algorithm is mainly based on the eigenstrain formulation of BIE. With this formulation, the system unknowns are deﬁned on the outer boundary Γ only, which signiﬁcantly reduces the number of system unknowns. This is the major advantage over the conventional FEM and BEM.

13.3 Solution Procedures In this section, outline of the solution procedure and related issues to the proposed eigenstrain formulation of BIE is described. While the numerical solution features such as treatment of domain integrals and application of boundary point method (BPM) are discussed in Section 13.4. As mentioned in the previous section, determination of the eigenstrain in each of the inhomogeneities is the crucial step for the solution of Eq. (13.9) or (13.12). The eigenstrain in each inhomogeneity depends on applied stresses or strains, geometries as well as material constants of inhomogeneity and matrix. Following the idea of Eshelby [1, 2], the eigenstrains, or the stress-free strains, in an inclusion with the material identical to matrix is related to the constrained strains εijC by a tensor Sijkl , namely the Eshelby tensor as follows:

εijC = Sijkl εkl0

(13.13)

The Eshelby tensor is only geometry dependent and generally in the form of integrals. For some simple geometry, the components of Sijkl can be found in the literature [4]. For an inhomogeneity under applied strains εij to be replaced by an equivalent inclusion without altering its stress state, the following relation can be given: C C νI µI εij + εij + δij εkk + εkk 1 − 2 νI C C ν 0 0 = µ εij + εij − εij + (13.14) δij εkk + εkk − εkk 1 − 2ν

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where µ and ν are the shear modulus and Poisson’s ratio, respectively. The subscript I denotes the variable is associated with the inhomogeneity. By deﬁning the ratio of modulus β = µI / µ and supposing νI = ν and eliminating εijC by using Eqs. (13.13) and (13.14), the eigenstrains εij0 can be obtained from the applied strains εij by the following equations:

εij0 =

1−β εij 1 − 2 (1 − β ) Sijij

(i = j)

(i = j)

ν ν 0 0 ε 0 + εij0 (1 − β ) Skkmn εmn + (1 − β ) Sijmn εmn + 1 − 2ν 1 − 2ν kk ν (i = j) εkk + εij = − (1 − β ) 1 − 2ν

(13.15a)

(13.15b)

Obviously, the applied strains or the applied stresses over each inhomogeneity will be disturbed by other inhomogeneities, especially those close to the concerned inhomogeneity. As a result, the applied strains and then the induced eigenstrains need to be evaluated in an iterative way. After discretization and incorporated with the boundary conditions, either Eq. (13.9) or (13.12) can be written in matrix form as: Ax = b + Bε (13.16) where A is the system matrix, B the coefﬁcient matrix related to the kernels in the domain integrals in Eq. (13.9) or (13.12), b the loading vector related to the known quantities with the outer boundary and the corresponding kernels, x the unknowns deﬁned on the outer boundary. ε is the eigenstrains of all inhomogeneities to be evaluated in an iteration way. It should be pointed out that all components of A, B and b are constants in the whole process so that they need to be computed only once. It begins with assigning the vector ε with an initial value. Then the applied strains can be evaluated via the Eq. (13.15) at any position of the inhomogeneities. Having determined the vector ε, the unknown vector x can then be computed by the following simple iterative formulae: x(k+1) = A−1 b + Bε(k) (13.17) where k is the iteration count. Deﬁne the maximum iteration error as , - εmax = max ε(k) − ε(k−1)

(13.18)

which is the maximum difference of the eigenstrain components between the two consecutive iterations. The convergence criterion in the present study is chosen as follows: E εmax ≤ 10−3 (13.19)

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where E is Young’s modulus of the matrix material. If the criterion (13.19) does not meet, then the stress states at each inhomogeneity are computed using Eq. (13.10) with the renewed vector x then the applied strains to renew the values of the eigenstrains. It should be addressed here that in the use of Eq. (13.10) the domain integral for the current inhomogeneity must be excluded from the computation because the stress states at the due place are the disturbance of all the other inhomogeneities in the solution domain under the prescribed boundary conditions. If the criterion (13.19) meets, go to the next step such as computation of the stresses interested or the overall properties, etc. The principal steps in the solution procedure can be summarized as follows: (a) Compute the constant coefﬁcients in A, B and b in Eq. (13.16). (b) Assign the vector ε with an initial value with the applied strains via the Eq. (13.15) at each position of the inhomogeneities at the elastic state computed without inhomogeneity. (c) Compute the unknown vector x using the iterative formulae (13.17) with the current eigenstrain vector ε. (d) Check the convergence criterion (13.19). (e) If the criterion (13.19) does not meet, perform the following computing and then return to the step (c). Otherwise go to the next step (f). (i) Compute the current stresses at each place of inhomogeneity using Eq. (13.10) with the current inhomogeneity at the due place being excluded. (ii) Compute the current applied strains at each place of inhomogeneity via Hooke’s Law. (iii) Renew the eigenstrain vector ε using Eq. (13.15). (f) Compute the stresses interested or the overall properties, etc. It can be seen that the determination of the eigenstrains in each of the inhomogeneities is the crucial step where the Eshelby tensor plays an essential role. Although some explicit expressions exist in the literature for the simple cases, in general the Eshelby tensor can always be obtained conveniently via the numerical means. Consider a single equivalent inclusion with eigenstrains in an inﬁnite medium under stress-free condition, the ﬁrst problem of Eshelby, Eqs. (13.9) and (13.10) are, respectively, reduced to the following form:

σij (p) =

∗ ε 0jk (q) σijk (p, q) dΩ (q)

(13.20)

∗ εkl0 (q) σijkl (p, q) dΩ (q) + εkl0 (p) O∗ijkl

(13.21)

ui (p) =

ΩI

ΩI

Assuming the case of uniform distribution of all the components of eigenstrains in the equivalent inclusion, both Eqs. (13.20) and (13.21) can be transformed into boundary-type integrals as [22, 23]

13 Computational Modal and Solution Procedure for Inhomogeneous Materials

ui (p) = ε 0jk

247

xk τij∗ (p, q) dΓ (q)

(13.22)

∗ xl τijk (p, q) dΓ (q)

(13.23)

ΓI

σij (p) = εkl0

ΓI

Equations (13.20) and (13.21) describe, respectively, the displacement and the stress ﬁelds, in matrix (when p ∈ Ω) and in equivalent inclusion (when p ∈ ΩI ). It should be mentioned that the displacement ﬁelds are continuous when the source point p go across the interface ΓI , while the value of stresses have a jump when ∗ are discontinacross the interface because the kernels xk τij∗ are continuous and xl τijk uous when across the interface. Noting the deﬁnition of eigenstrains ε 0jk = ε Ijk − ε Cjk in the derivation of Eq. (13.9) and comparing with that of Eshelby’s, one can see clearly that ε Cjk corresponds to the constrained strains in the equivalent inclusion in the sense of Eshelby. This holds true for the case of single inclusion in an inﬁnite matrix under stress-free condition. Now put the source point p inside the domain ΩI , σij becomes the constrained stresses in the equivalent inclusion and the boundary-type integral (13.23) becomes regular, which can be evaluated numerically. Combining Eqs. (13.13) and (13.23) with Hooke’s law εijC = Cijkl σkl where Cijkl represents the elastic compliance tensor of the matrix, the Eshelby tensor can then be expressed as follows: Sijkl = Cijmn

∗ xk τmnl (p, q) dΓ

(13.24)

ΓI

It can be seen from Eq. (13.24) that the Eshelby tensor is geometry dependent only. It can also be shown that Eq. (13.24) is equivalent to that of Eshelby’s expression for Sijkl but is more suitable for numerical evaluations. In the case of nonuniform distribution of eigenstrains, the domain integrals (13.20) and (13.21) can also be transformed into the corresponding boundary-type integrals without any difﬁculty by means of the complete polynomial expansion of eigenstrains [22, 23]. However, this case is not considered in the present study for the sake of conciseness.

13.4 The Boundary Point Method The inhomogeneity problems using the present model are solved with the boundary point method (BPM). The BPM is a newly developed boundary-type meshless method [20] based on the indirect or direct formulations of conventional and hypersingular BIE but has some features of the method of fundamental solution (MFS) [24–26]. In the BPM, the boundary of the solution domain is discretized only by boundary nodes, each node having a territory or support where the ﬁeld variables are deﬁned. By making use of the properties of fundamental solutions, the coefﬁcients of the system matrix are computed according to the distance between

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the source and the ﬁeld points. In cases when the distance between the two points is not very small in comparison with the dimension in any one coordinate direction, the integrals of kernel functions are evaluated by one-point computing. This is similar to that carried out in the MFS, which constitute the most off-diagonal terms of the system matrix. Conversely, in cases when the distance is very small the integral of kernel functions is evaluated by Gauss quadrature over territories. If the two points coincide, the integrals are treated by well-established techniques in the BEM, which constitute the principal diagonal terms of the system matrix. Noting that the combined use of adjacent nodes can describe the local features of a boundary such as position, curvature and direction, the ‘moving elements’ are introduced by organizing the relevant adjacent nodes tentatively, over which the treatment of singularity and Gauss quadrature can be carried out for evaluating the integrals in the latter two cases abovementioned, namely coincidence or small distance between the source and ﬁeld points. It should be mentioned that as the source points are placed on the real boundary in the BPM, the main difﬁculties of coincidence of the source and ﬁeld points and the inconvenience of using a ﬁctitious boundary encountered in the MFS are removed. Compared to the BEM, the BPM can improve efﬁciency by both reducing the burden of data preparation and taking advantage of the one-point computing for most of the integrals of kernel functions while maintaining reasonable accuracy. It is suggested that the evaluation of kernel functions by Gauss quadrature over boundary elements in the BEM would be necessary only for cases of coincidence and small distance between the two points. For details of BPM please refer to [20]. As mentioned previously, the domain integrals in Eqs. (13.9), (13.10) and (13.12) can be evaluated by the following boundary-type integrals [22, 23] ΩI

ΩI

σi∗jk dΩ =

xk τi∗j dΓ

ΓI

σi∗jkl dΩ + O∗i jkl =

(13.25) xl τi∗jk dΓ

(13.26)

ΓI

then all the techniques in the BPM become applicable at the interfaces ΓI . That is, the interfaces are discretized by interface nodes, each node having a territory and the integrals in Eqs. (13.25) and (13.26) are evaluated by boundary-type integrals in closed loops while the one-point computing can be used in most of the cases when the two-point distances are not too small. However, there are two special aspects to be addressed here for the proposed computational model. The ﬁrst is that the purpose of evaluating domain integrals is for computing the inﬂuences of eigenstrains in inhomogeneities on the outer boundary unknowns but not for the system matrix itself. The second is that the use of boundary-type integrals in closed loops is necessary only for a few limited cases when the distances between the source and the ﬁeld points are relatively small. In other words, the domain integrals can be evaluated simply by the one-point computing as follows with reasonable accuracy if the two-point distance is not too small:

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σi∗jk dΩ ≈ σi∗jk AI

(13.27)

σi∗jkl dΩ ≈ σi∗jkl AI

(13.28)

ΩI

ΩI

/ ΩI in the range where AI is the area of ΩI and O∗i jkl = 0 in Eq. (13.28) because p ∈ that the one-point computing is applicable. It is well known that the kernel functions of integrals in the BIE are two-point functions and behave singularity to various extents when the two-point distance tends to zero. However, there are properties of the kernel functions related to the two-point distances just opposite to the singularities, which have not been fully utilized yet. For example, if a kernel has a singular order of O (r−s ), where s is an integer, then the decaying order of the integral values of this kernel function is also s with the increase of the distance r.Furthermore, the varia tion of the kernel functions will have the order of O r−s−1 if r is not too small. In other words, the values of the kernel function can be well represented by the value at the center point of the integration domain since the variation of the kernel functions in the domain becomes negligibly small. Comparison of integral results from one-point computing using Eqs. (13.27) and (13.28) with those from boundary-type quadrature along a closed loop using Eqs. (13.25) and (13.26) is displayed in Fig. 13.1, where AI is the area of a circular zone and r is the distance from the source point to the center of the circular zone. It can be seen from Fig. 13.1 that the difference of integral results between the two computing methods decreases along with an increase in the dimensionless 1/2 1/2 distance r/AI . In the present study, r/AI = 5 serves as the criterion to change the computing method between the one-point computing and the quadrature along closed loops.

By quadrature σ*111

0.2

Integral values

σ*122 σ*222 σ*1111 0.1

σ*2222

0.0

1

10

Dimensionless distance, r /AI1/2 Fig. 13.1 Comparison of integral results between one point computing and boundary type quadrature along a closed loop

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13.5 Numerical Examples The purpose of the ﬁrst numerical example is to assess the validity of the algorithm over a simple square domain with an elliptical inhomogeneity under the uniform tension of unit traction in x2 direction as shown in Fig. 13.2. a and b are, respectively, the half lengths of the long and short axes and 2a/w = 0.1 where w is the side length of the square domain. The aspect ratio b/a of the inhomogeneity is taken to be 0.5. The outer boundary and the interface are discretized by N = 60 and NP = 120 nodes, respectively. The problem is ﬁrstly solved by combined use of Eqs. (13.1) and (13.3) as well as BPM. The solution procedure is usually known as ‘domain decomposition’ as there is two (or more) areas in the solution domain with distinct material properties to be decomposed. The two areas are described by the two boundary integral equations (13.1) and (13.3) which are coupled at the interface. As displacements and tractions are unknown at the interface, the degrees of freedom of the problem is as large as 2 (N + NP ) and the size of the system matrix is 360 × 360 even for such a simple problem. The problem is then solved with the BPM combined with the eigenstrain formulation (13.9). In the present study, the Eshelby tensor required for evaluating the eigenstrains in elliptical inhomogeneities is computed numerically using Eq. (13.24) beforehand and some results are shown in Fig. 13.3. Because the unknowns are deﬁned only on the outer boundary in the proposed computing model, the degrees of freedom of the problem is 2N only and the size of the system matrix is reduced to 120 × 120, which is only 11.1% of that in the domain decomposition method. The computed stresses along x2 = 0 in the matrix with β = µI /µ = 0.1 and 10 are listed in Fig. 13.4 and comparison is made those obtained using the domain decomposition. It is evident from Fig. 13.4 that the results obtained from the present model are in good agreement with those from the domain decomposition. The second numerical example is carried out for the computation of overall properties of particle reinforced materials over a representative volume element (RVE) shown in Fig. 13.5 with NI = 100 inhomogeneities scattered in the RVE with loadings in either single tension mode or in pure shearing mode. The RVE is discretized

x2

x1

2b 2a

Fig. 13.2 A square domain with inhomogeneity under uniform tension in x2 direction

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S2222 S1212

1.0

Eshelby tensor

0.8 0.6 0.4

S1111 S1122 S2211

0.2 0.0 0.1

1

10

Aspect ratio, b /a Fig. 13.3 Eshelby tensor for elliptical inhomogeneities with various aspect ratios

Dimensionless stresses

5

By domain decomposition σ11( b = 0.1)

4

σ22( b = 0.1) σ11( b = 10)

3

σ22( b = 10) b /a =0.5

2 1 0 −1 1.0

1.2

1.4

1.6

1.8

2.0

x1/a Fig. 13.4 Veriﬁcation of the algorithm by comparison with domain decomposition

by N = 100 nodes on the outer boundary, giving a size of the system matrix of 200 × 200 using the proposed model of the eigenstrain formulation. It is noted that discretization is necessary only for those inhomogeneities when the distance between source and ﬁeld points are small so that the domain integrals need to be evaluated by the quadrature of boundary type integrations over closed loops. Otherwise, the domain integrals are evaluated by the one-point computing so that the discretization of inhomogeneities is unnecessary in most cases. In our analysis, circular inhomogeneity is discretized by NP = 20 and elliptical inhomogeneity is modeled by NP = 40. Obviously, the size of system equation using the proposed model is 0.25% of that using the traditional domain decomposition method if circular inhomogeneity is considered.

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Fig. 13.5 The representative volume element for the particle reinforced composite materials

Overall properties

1.0 Ec /E (by tension) uc (by tension) mc /E (by tension)

0.8 0.6

b /a =1

mc /E (by shearing)

0.4 0.2 10−3

10−2

10−1

100

101

102

103

Ratio of modulus, b

Fig. 13.6 The overall properties of the RVE as a function of the ratio of modulus

The second example is solved using the BPM described in Section 13.4 and the eigenstrain formulation (13.12) of the hypersingular BIE. The eigenstrain is evaluated using the iterative scheme described in Section 13.3 in which the convergent tolerance is deﬁned in Eq (13.19). Figure 13.6 shows the computed overall properties of the RVE with circular inhomogeneities as a function of the ratio of modulus β , where EC , νC and µC stand for Young’s modulus, Poisson’s ratio and shear modulus of the composite material. The area of each inhomogeneity and the volume fraction (=7.07%) are kept constant in the calculation. It is seen from Fig. 13.6 that when the range of ratio of modulus is around unit the computed modulus varies comparatively large. When the ratio of modulus is very small (corresponding to empty holes) or very large (corresponding to rigid inhomogeneities) the computed modulus varies stagnantly. It is noticed that there is some difference in the computed shear modulus between the single tension and the pure shearing loading modes.

13 Computational Modal and Solution Procedure for Inhomogeneous Materials

253

1.6

Overall properties

1.4

b /a =1 b =10

1.2 Ec /E

1.0

uc mc /E

0.8 0.6 0.4 0.2 0.0 0.1

0.2

0.3

0.4

0.5

0.6

Particle radius, a Fig. 13.7 The overall properties of the RVE as a function of the particle radius 1.6

E L /E

Overall properties

E T /E

1.2

0.8

m L /E m T /E

uL uT

m /E(By shearing) 0.4

b=10 p ab =0.377

0.0 2

3 4 6 Aspect ratio, b /a

5

7 8 9 10

Fig. 13.8 The overall properties of the RVE as a function of the aspect ratio of inhomogeneities

Figure 13.7 shows that the overall properties of the RVE, the Young’s modulus EC and shear modulus µC increase monotonically with the particle radius when the number of particles and the ratio of modulus are kept constant. Figure 13.8 shows the overall properties of the RVE as a function of the aspect ratio of inhomogeneities with the constant ratio of modulus and the constant area π ab of the particles while the axes of the elliptic zone are aligned in parallel with the coordinate axes. It shows from Fig. 13.8 the anisotropic behaviors of the overall properties with the variation of the aligned inhomogeneities where the subscripts L and T denote the properties in longitudinal and transverse directions, respectively. It can be seen that there is no limitations such as the shape, the material, etc., to solve the inhomogeneity problems with the proposed model. In the computation, the convergence can be achieved

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in general two to ﬁve times. It is found that the iterations will increase with the proportion of the inhomogeneity in the matrix, that is, the quantity, the size as well as the shape of the particles. However, the maximum number of iteration is eleven. It seems that the improvement of iteration remains a topic to be studied.

13.6 Conclusions The novel computational model and solution procedure are proposed for inhomogeneous materials in the present study. The method is based on the eigenstrain formulation and BPM. As the unknowns of the ﬁnal system equation are deﬁned on the outer boundary of the solution domain only, the solution scale of the inhomogeneity problem with the present model remains fairly small in comparison with those with the traditional computation model using FEM and BEM. Both the model and the solution procedure are developed intimately from the concept of equivalent inclusion of Eshelby with eigenstrains to be determined in an iterative way via the Eshelby tensors, which can be readily obtained through either analytical or numerical approach. The convergence of the algorithm can be achieved within a few iterations. The validity and the effectiveness of the proposed computational model and the solution procedure are demonstrated by two numerical examples.

References 1. Eshelby JD. The determination of the elastic ﬁeld of an ellipsoidal inclusion and related problems. Proceedings of the Royal Society of London A241 (1957) 376–396. 2. Eshelby JD. The elastic ﬁeld outside an ellipsoidal inclusion. Proceedings of the Royal Society of London A252 (1959) 561–569. 3. Mura T, Shodja HM, Hirose Y. Inclusion problems (part 2). Applied Mechanics Review 49(10) (1996) S118–S127. 4. Federico S, Grilloc A, Herzog W. A transversely isotropic composite with a statistical distribution of spheroidal inclusions: a geometrical approach to overall properties. Journal of the Mechanics and Physics of Solids 52 (2004) 2309–2327. 5. Cohen I. Simple algebraic approximations for the effective elastic moduli of cubic arrays of spheres. Journal of the Mechanics and Physics of Solids 52 (2004) 2167–2183. 6. Franciosi P, Lormand G. Using the radon transform to solve inclusion problems in elasticity. International Journal of Solids and Structures 41 (2004) 585–606. 7. Shen LX, Yi S. An effective inclusion modal for effective moluli of heterogeneous materials with ellipsoidal inhomogeneities. International Journal of Solids and Structures 38 (2001) 5789–5805. 8. Kompis V, Kompis M, Kaukic M. Method of continuous dipoles for modeling of materials reinforced by short micro-ﬁbers. Engineering Analysis with Boundary Elements 31 (2007) 416–424. 9. Doghri I, Tinel L. Micromechanics of inelastic composites with misaligned inclusions: Numerical treatment of orientation. Computer Methods in Applied Mechanics and Engineering 195 (2006) 1387–1406.

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10. Kakavas PA, Kontoni DN. Numerical investigation of the stress ﬁeld of particulate reinforced polymeric composites subjected to tension. International Journal for Numerical Methods in Engineering 65 (2006) 1145–1164. 11. Kanaun SK, Kochekseraii SB. A numerical method for the solution of thermo- and electrostatic problems for a medium with isolated inclusions. Journal of Computational Physics 192 (2003) 471–493. 12. Lee J, Choi S, Mal A. Stress analysis of an unbounded elastic solid with orthotropic inclusions and voids using a new integral equation technique. International Journal of Solids and Structures 38 (2001) 2789–2802. 13. Dong CY, Cheung YK, Lo SH. A regularized domain integral formulation for inclusion problems of various shapes by equivalent inclusion method. Computer Methods in Applied Mechanics and Engineering 191 (2002) 3411–3421. 14. Dong CY, Lee KY. Boundary element analysis of inﬁnite anisotropic elastic medium containing inclusions and cracks. Engineering Analysis with Boundary Elements 29 (2005) 562–569. 15. Dong CY, Lee KY. Effective elastic properties of doubly periodic array of inclusions of various shapes by the boundary element method. International Journal of Solids and Structures 43 (2006) 7919–7938. 16. Liu YJ, Nishimura N, Tanahashi T, Chen XL, Munakata H. A fast boundary element method for the analysis of ﬁber-reinforced composites based on a rigid-inclusion model. ASME Journal of Applied Mechanics 72 (2005) 115–128. 17. Ma H, Deng HL: Nondestructive determination of welding residual stresses by boundary element method. Advances in Engineering Software 29 (1998) 89–95. 18. Greengard LF, Rokhlin V. A fast algorithm for particle simulations. Journal of Computational Physics 73 (1987) 325–348. 19. Nakasone Y, Nishiyama H, Nojiri T. Numerical equivalent inclusion method: a new computational method for analyzing stress ﬁelds in and around inclusions of various shapes. Materials Science and Engineering A285 (2000) 229–238. 20. Ma H, Qin QH: Solving potential problems by a boundary-type meshless method – the boundary point method based on BIE. Engineering Analysis with Boundary Elements 31 (2007) 749–761. 21. Brebbia CA, Telles JCF, Wrobel LC. Boundary element techniques—theory and applications in engineering. Berlin: Springer (1984). 22. Ma H, Kamiya N, Xu SQ: Complete polynomial expansion of domain variables at boundary for two-dimensional elasto-plastic problems. Engineering Analysis with Boundary Elements 21 (1998) 271–275. 23. Ma H, Kamiya N: Boundary-type integral formulation of domain variables for threedimensional initial strain problems. JSCE Journal of Applied Mechanics 1 (1998) 355–364. 24. Golberg MA. The method of fundamental solution for Poisson’s equation, Engineering Analysis with Boundary Elements 16 (1995) 205–213. 25. Fairweather G, Karageorghis A. The method of fundamental solutions for elliptic boundary value problems. Advances in Computational Mathematics 9 (1998) 69–95. 26. Wang H, Qin QH, Kang YL. A meshless model for transient heat conduction in functionally graded materials. Computational Mechanics 38 (2006) 51–60.

Chapter 14

Studies on Damage and Rupture of Porous Ceramics Ioannis Doltsinis

Abstract Results on the reliability of porous ceramics are highlighted on the background of experimentation and analysis. The industrial interest in the research is related to ceramic elements employed in nanoﬁltration. The diametral compression (Brazilian) test is utilized in the laboratory as a substitute for the burst test to characterize the strength of the structural components. Damage prior to rupture is registered, and aging because of corrosive activity in the porous material is discussed. Remarks on a continuum approach to fracturing are followed by the description of a computational procedure that operates on the material structure. The fracturing processes in the material that result to rupture under pore pressure are studied on the microscale by the numerical simulation model. Statistical sampling aims at exploring the rupture statistics.

14.1 Introduction The paper addresses modelling of the damage of porous ceramics on the microstructural level and tests on the failure of structural components. The research has been occasioned by the industrial interest in the strength of ceramic ﬁlter supports used in nanoﬁltration, where internal pressure is of importance. Porous ceramics subjected to ﬂuid pressure in the pores are prone to brittle fracturing on the microscale which may progress to brittle or quasi-brittle rupture of the component. Apart from physical and numerical modelling on the microscale, subtle laboratory testing of components has been of paramount importance; brittleness suggests statistical evaluation. The study focuses on mechanical issues, but conceptual thoughts on stress-enhanced corrosion are included. The phenomenon is signiﬁcant to aging which determines the life-time of parts exposed to chemically aggressive media. Ioannis Doltsinis Faculty of Aerospace Engineering and Geodesy, University of Stuttgart, Pfaffenwaldring 27, D-70569 Stuttgart [email protected] V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

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In Section 14.2, laboratory measurements on the structural parts of interest (circular cylinders with longitudinal channels) refer to the diametral compression (Brazilian) test which adequately replaces the condition of channel pressure, as justiﬁed by ﬁnite element stress analysis. The impact of the two different loading cases is formally explained by Weibull statistics. Effects of basic material and doting on component rupture are investigated, the damage tolerance conﬁrmed, corrosion discussed. Synthesis incorporates material strength statistics in the ﬁnite element model for stress analysis and estimates critical locations in the structural part under internal pressure [8]. In Section 14.3, a brief discussion on the formalism for the fracturing elastic continuum is followed by a numerical scheme simulating cracking in the microstructure by separation of grain boundaries. In Section 14.4, the fracturing algorithm is applied to the analysis of rupture in artiﬁcial microstructures generated in the computer for given material characteristics, and determines strength statistics for material specimens in dependence of various structural parameters. The present account highlights results of co-operative research performed at the Universities of Stuttgart and Caen, reported previously in [5, 8]. The laboratory experiments are the work of F. Osterstock with O. Vansse [11, 13], the micromechanical modelling bases on [3] and has been elaborated by R. Dattke [2].

14.2 Laboratory Tests and Observations 14.2.1 The Diametral Compression (Brazilian) Test The rupture of ceramic parts is due to the action of the stress ﬁeld on cavity defects in the microstructure of the brittle material. The stress is sensitive to the specimen geometry and the boundary conditions, while nature and distribution of the defects can be considered material properties ascribed to the microstructure. It is noticed, however, that the material structure is largely determined by the entire process of manufacturing such that it is speciﬁc to the product. For this reason tests on rupture should apply to the structural part under investigation and to the loading conditions in the use environment. The parts of interest here are ceramic ﬁlter supports, essentially circular cylinders with interior channels (Fig. 14.1). The conﬁguration allows circular disks cut from the supports to be used as specimens in the so-called Brazilian test. The Brazilian (diametral compression) test refers to cylindrical specimens of circular cross-section which are subject to compressive loading along a diameter. The test is widely used in material testing for measuring brittle resistance to tensile stress. Its suitability is due to a constant maximum tensile stress induced in the elastic cross-section perpendicular to the direction of the compressive force along the loading diameter. In the case of the ﬁlter supports the presence of the interior holes implies a stress ﬁeld that differs from the compact disk, but the tensile stress causes likewise rupture along the loading diameter (Fig. 14.2, right). An other point regards the actual loading

14 Damage and Rupture of Porous Ceramics

259 F σ2

σ1

σ1 = F / π R t = const. y

σ2 = − [1 + 2R / (R - y )]σ1 x

R

Disc radius R thickness t

F

Fig. 14.1 Ceramic rod with channels. Diametral compression of circular disk

Fig. 14.2 The diametral compression (Brazilian) test. Test set-up showing Teﬂon load transmitters in white (left); disk specimen at rupture (right)

which is rather by internal pressure in the channels. The diametral compression test is a nevertheless useful alternative to the burst test under channel pressure. In both cases the highest tensile stress from elastic ﬁnite element analysis is seen to appear at homologous positions such that the compression test is considered representative of the burst pressure test. There are more locations of high stress values in the burst test, however, and on account of the random defects in the material, the likelihood of meeting rupture conditions is higher in this case. The assessment of the Brazilian test as a substitute for the burst test under internal pressure in the channels will be based on the weakest link hypothesis in conjunction with the Weibull statistics [14]. Under uniaxial conditions the latter gives the probability for rupture under a stress with magnitude less or equal to σ as # $ σ − σu m . (14.1) Pr = 1 − exp − σ0 The quantity σu is a threshold value such that the rupture probability is zero for σ ≤ σu , and σ0 is known as the characteristic stress associated with the rupture probability Pr = 1 − 1/e. The exponent m > 0 is the Weibull modulus. For m = 0 the rupture probability would be independent of the stress level. The higher the value

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of m the less the scatter and the more homogeneous the material. For m = ∞ rupture would occur uniquely at σ0 . For the transition from uniaxial test data specifying m and σ0 at specimen volume V0 to the rupture probability of structural components subject to multiaxial stress, a simpliﬁed interpretation of the approach developed in [12] gives the probability of rupture for a stressed part of volume V as ⎡ ⎤ # $ m 1 V σN m σ 1 ⎣ ⎦ dV = 1 − exp − Σ . (14.2) PrV = 1 − exp − V0 σ0 V0 σ0 V

The above assumes statistically homogeneous distribution of the material strength, and dominance of the maximum principal stress σ1 with regard to rupture. The quantity σ1 m 1 Σ= dV, (14.3) V σN V

is known as the stress volume integral (SVI). The integral accounts in a dimensionless manner for the distribution of the principal stress σ1 in the component of volume V , and can be computed by post-processing numerical results from ﬁnite element stress analysis. The reference stress σN characterizes the magnitude of the applied loading. The formalism may be utilized for a comparison between rupture tests that differ with respect to loading conditions. Evaluation of Eq. (14.2) for the Brazilian test as applied to the ceramic ﬁlter support gives for a compressive force F, # $ σNF m Pr (F) = 1 − exp − ΣF . (14.4) σ0 Here, the probability of rupture Pr (F) is that obtained by the test series as is also the modulus m. A suitable measure for the magnitude of the loading is the quantity σNF = F/2Rt, where R denotes the radius of the disk and t the thickness. Next, in the case of the burst test under pressurized channels Eq. (14.2) becomes # $ σN∆ p m Σ∆ p , (14.5) Pr (∆ p) = 1 − exp − σ0 and ∆ p denotes the overpressure, the pressure difference between the channels and the outer surface of the ﬁlter support. It can be taken directly as the reference stress: σN∆ p = ∆ p. Equations (14.5) and (14.4) refer to the same specimen geometry but different loading conditions. Given the data of the Brazilian test, it is of interest to estimate the pressure that corresponds to the same probability of rupture. From Eqs. (14.5) and (14.4) one obtains:

σN∆ p ∆p = = (F/2Rt) σNF

ΣF Σ∆ p

m1

.

(14.6)

14 Damage and Rupture of Porous Ceramics

261

14.2.2 Load at Rupture The series of the Brazilian test refer to lots of 50 specimens (one out of two from 100 disks cut along each of the ﬁlter supports). Figure 14.3 depicts the load at rupture obtained in the sequence of tests appertaining to one support. The large scatter, typical of polycristalline ceramics, amounts up to 30–40% of the maximum value thus suggesting statistical description of the registered data. Despite the scatter a systematic variation of the rupture load is visible along the support, and applies to all the components tested. In the ﬁgure, the strength is seen to be highest at the left end, attaining lowest values in the middle region. The smoothed variation of strength ranges to about 20% of the corresponding maximum value. In the middle of the support the level of the rupture load is lowest, the scatter largest. The scatter at the left-hand end region is larger than at the right end, but the coefﬁcients of variation (standard deviation/mean value) are apparently comparable. The individual member may be considered indicative of the rupture behaviour in a homogeneous group of components. In order to explore the rupture statistics within single supports, the frequency distribution of the load at rupture is contrastet to the Weibull failure statistics. In analogy to Eq. (14.1) the latter gives the probability of rupture Pr under forces less than or equal in magnitude to F by the distribution function $ # F − Fu m Pr (F) = 1 − exp − . (14.7) F0 Rupture does not occur for F ≤ Fu , the characteristic force F0 is associated with the rupture probability Pr = 1 − 1/e. Figure 14.4 shows representation of the test data in Weibull diagrams: ln(ln(1/(1 − Pr ))) vs ln F. In such a diagram the probability distribution function given by Eq. (14.7) would appear as a straight line with slope m that crosses the strength axis at F − Fu = F0 . Taking Fu = 0 for the threshold value the statistics is determined by two parameters: the Weibull modulus m and the characteristic

Fig. 14.3 Variation of the load at rupture along the ceramic ﬁlter support

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Fig. 14.4 Weibull plots of the load at rupture from the diametral disk compression test on assintered titania and alumina based components. Upper: 30% porosity. Lower: 40% porosity

strength F0 . When plotting the test data, the rupture probability ascribed to the measured force was estimated by Pr (F ≤ Fi ) =

i , N +1

(14.8)

with i being the rank of the rupture force with magnitude Fi and N the total number of specimens in the test. The diagrams in Fig. 14.4 offer comparisons in strength for as-sintered components with different material composition and porosity. The upper frame refers to a porosity of 30%, the lower frame to 40%. Both comprise alumina and titania based supports. Consideration within each ﬁgure shows the inﬂuence of the material composition. Comparison between the frames reveals the inﬂuence of porosity, which is seen to diminish the strength. At the lower porosity of 30% the alumina based, though rare earth doted support is slightly less resistant than the titania dominated material. Similarly at the higher porosity of 40% when considering titania and doted alumina, left in the diagram. The other titania based support is stronger because of

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263

a lower porosity (<40%). It is worth noticing the superiority of the alumina in the absence of doting, right in the diagram. The experimental data obviously do not ﬁt a straight line as for a Weibull distribution with modulus m and characteristic force F0 . Several approaches to an evaluation on the background of Weibull statistics are feasible, two of them have been explored. The one proposal regards representation of the entire test data by a multimodal distribution, the other, alternative interpretation relies on the unimodal distribution and excludes extreme values that disturbe. For a conservative description the highest values of the rupture force are consecutively discarded until a straight line is approached. Removal of only a few values indicates adequacy of the unimodal distribution.

14.2.3 Damage Preceding Rupture Prior to rupture the ceramic material may undergo damage by microcracking processes that take place on the structural level of the material. The capability to damage is an important property because rupture does not occur suddenly in this case, but appears retarded. The initiation and progress of microcracking during the course of the diametral compression test has been detected in the laboratory by registering acoustic emission signals. Figure 14.5 demonstrates the variation of the force as a function of time for a disk subject to the compression test. In the diagram one distinguishes the initial stage of slowly applying the load via smoothing Teﬂon transmitters, the subsequent stiffer linear elastic response of the compressed disk, and ﬁnally the very short range (encircled) just before destruction, that deviates from linearity as a consequence of damage. The effect of various parameters on the extent of the damage interval has been obtained for different ﬁlter supports via the Brazilian test on the disk specimens. A documentation of the test series is given in Fig. 14.6. The ﬁgure shows probability plots for two quantities: the force at damage initiation and the force at rupture. The left frame compares titania dominated supports at two different porosities. Here,

Fig. 14.5 Simultaneous registration of applied force and acoustic signals in the diametral compression test detects damage

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Fig. 14.6 Probability plots for titania (left) and alumina based ﬁlter supports (right). Damage initiation and rupture at 30% and 40% porosity

the porosity value is seen to not appreciably modify the force interval between ﬁrst appearance of damage and rupture. The right frame refers to alumina based ﬁlter supports, doted. For this material the interval between damage initiation and rupture is larger for lower porosity (30%, curves at higher force level on the right). The higher porosity (40%, curves on the left) diminishes both, the force level and the damage capacity of the specimens. A comparison with tests on alumina based ﬁlter supports without doting reveals that the effect of doting is similar to that of porosity: the level of rupture strength is lowered and the damage capacity reduced such that brittleness is enhanced.

14.2.4 Aging Due to Stress-Enhanced Corrosion Operation with corrosive media limits the life-time of ceramics by fatigue due to stress-enhanced corrosion. Cracks undergo stable extension at stress intensity factors less than the critical one because the action of stress facilitates chemical processes at the crack tip that lower the surface energy. The stable growth of subcritical ﬂaws progresses up to the critical size. In contrast to dense ceramics where usually rupture occurs from the extension of a single defect, in ceramics with open porosity as investigated here the corrosive medium is present throughout the whole specimen, provoking stable extension of microcracks from any pore. Alumina-based components that differ only by the time of use have been subjected to the diametral compression test, and are compared to as-sintered one (Fig. 14.7). Time of use of R-2 is longer than that of R-1; no further details were known; R-0 marks the unused, as sintered component. A simple resonance method evealed that the elastic modulus of the used supports was lower than that of the as-sintered support R-0. Since weighting indicated no marked change in apparent density, it can be argued that the loss in elastic stiffness results from the creation and/or extension of microcracks.

14 Damage and Rupture of Porous Ceramics

265 Strength

Rupture force R-0 Beginning of damage

Start-up

R-1 R-2 Destruction zone

Time

Fig. 14.7 Dependence of rupture probability on service time (alumina-based components; from the right: unused R-0, used R-1, longer in service R-2). Life-time concept (right)

From the three data sets on damage initiation and rupture, one concludes that the damage tolerance diminishes during use; the mean strength also decreases as in the case of dense ceramics. These observations are summarized in Fig. 14.7 to a concept regarding the change of strength during the course of use in corrosive environment: the ability for damage decreases at the same time as the remaining strength. The rate of decrease depends on the kinetics of the chemical reactions taking place at the microcrack tips. However, the gradual ceasing of microcracking processes as ultimate rupture approaches implies a smooth transition from quasi-brittle to brittle failure.

14.3 Modelling Damage and Rupture 14.3.1 Fracturing Continuum The approach considers an otherwise elastic material that undergoes damage as a consequence of microcracks created during the course of the loading. Let the degradation of the elastic stiffness due to homogeneously distributed microcracks be characterized by a number of parameters collected in the column matrix D. Then the relation σ (ε , D) = κ (D)ε , (14.9) between the stress vector σ and the strain vector ε describes linear elastic behaviour as long as D remains constant. Progressing fracturing affects the elasticity matrix κ (D) of the material which reﬂects on the time rate

σ˙ = κ (D)ε˙ + κ˙ (D)ε .

(14.10)

A criterion for the onset resp. the progress of fracturing can be arrived at by considering energy changes in a unit volume element of the material. The difference in mechanical energy between loading and elastic unloading is

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G=

ε 0

1 σ t dε − σ t ε , 2

(14.11)

and it vanishes if fracturing does not occur. The time rate 1 G˙ = − ε t κ˙ (D)ε , 2

(14.12)

gives the energy release due to the progression of damage in the element. If an area A is assigned collectively to the system of distributed microcracks, the energy request per unit fractured area is 2γ , where γ denotes the speciﬁc surface energy of the material. Comparison of energy release and consumption for a virtual increment δ A suggests that fracturing will not progress as long as

δG =

dG 1 d2 G 2 δA+ δ A < 2γδ A. dA 2 dA2

(14.13)

By virtue of the ﬁrst order term, fracturing depends on the energy release rate G=

1 dκ −1 (D) 1 dκ (D) dG ε = σt σ. = − εt dA 2 dA 2 dA

(14.14)

In view of the possibility of softening the fracturing/non-fracturing condition is stated preferably in strain space:

Ψ (ε , D) = G(ε , D) − 2γ ≤ 0.

(14.15)

The fracture surface Ψ (ε , D) = 0 limits the domain of strain states that can be achieved elastically, before damage progresses. During stable fracturing the consistency condition observed on the surface controls the second order term of Eq. (14.13): dΨ dG d2 G = = 2 = 0. (14.16) dA dA dA Loading/unloading criteria for fracturing refer to incremental changes of the function Ψ (ε , D) at Ψ = 0, and are related to the energetic stability of the microcrack state as by the second order term of Eq. (14.13). In particular, fracturing is spontaneous if d2 G/dA2 > 0 at constant stress or strain. The microcrack system affects the elasticity matrix of the material via the parameters in D. Analytical developments base on the contribution ε c of cracks to the overall strain

−1 −1 ε = κ −1 (14.17) o σ + ε c = κ o + M(D) σ = κ (D)σ . The elasticity matrix κ o refers to the material without cracks, while the matrix operator M(D) furnishes at state D the excess strain ε c for an imposed stress σ . The energy release rate of Eq. (14.14) thus becomes

14 Damage and Rupture of Porous Ceramics

267

1 dM(D) 1 dM(D) G = σt σ = ε tκ κε . 2 dA 2 dA

(14.18)

Evaluation for a virtual progress of fracture presumes speciﬁcation of the actual state D, availability of the functional dependence M(D), and determination of dD/dA along the trial process under exploration. A formal completion of the constitutive description for the microcracking continuum requires, in addition, an expression of the loading criterion and an evolution law for D. Any release of stored energy of eigenstrains as emanating from the manufacturing process adds to G or egually diminishes 2γ in the criterion of Eq. (14.15). For an elastic solid with homogeneous porosity the strain can be written as −1 ε = κ −1 0 σ + εp = κ p σ ,

(14.19)

where ε p denotes the contribution of the pores. The second expresssion introduces the elasticity matrix κ p of the porous material used in place of κ 0 in Eq. (14.17) when microcracks are present in addition. This might be sufﬁcient in establishing the operator M(D) for existing microcracks [9], but not necessarily for determining the process of fracturing. If the pores are subject to internal pressure the macroscopic stress σ in the solid is controlled by both the strain ε and the applied pressure p:

σ = κ p ε − (α p)e.

(14.20)

The pressure p carries part of the normal stress components, the incidence vector e effecting the proper allocation. The factor α is considered a material parameter. From a micromechanics point of view, averaging the stress ﬁeld in the elementary volume gives for the macroscopic stress

σ + χ pe = κ 0 (ε − ε p ) = κ p ε − κ 0 ∆ ε p .

(14.21)

The factor χ deﬁnes the porosity as the volume fraction of the porous phase, ε p denotes the contribution of the pore displacements to the macroscopic strain. The elasticity matrix κ p refers to the porous continuum in the absence of pressure as in Eq. (14.19), the corrective term κ 0 ∆ ε p reﬂects the difference in pore deformation. In Eq. (14.20) this has been incorporated in the modiﬁed pressure α p. Microcracks add the quantity ε c to the macroscopic strain. Apart from the above introductory remarks, it is not the ambition of the present treatise to complete a constitutive model for progressive microcracking. A continuum damage approach would limit the strength by bifurcation along the deformation path resp. stability of the loading process [4]. Figure 14.8 illustrates the essential behaviour of a tensile bar of quasi-brittle (damage tolerant) material. Simple localization analysis considers two cross-sections of the bar: A2 at the apparently localized position and A1 , in the undisturbed region. Equilibrium demands that (σ A)2 = (σ A)1 ,

δ (σ A)2 = δ (σ A)1 .

(14.22)

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I. Doltsinis σA

σ δε1

O

δε2 E0

E(D)

δε1 σA

ε

Fig. 14.8 Uniaxial behaviour of quasi-brittle material

For an isochoric material

δ (σ A) = A

dσ − σ δ ε, dε

(14.23)

and therefore disturbance of a homogeneous deformation state must comply with the condition dσ − σ (δ ε2 − δ ε1 ) = 0. (14.24) A dε Formally, bifurcation will become possible when dσ = σ, dε

(14.25)

which is the classical result marking the maximum sustainable force, at the same time. Physically, the bifurcated solution implies elastic unloading of the undisturbed part of the rod while the deformation localizes within a narrow region where microcracking intensiﬁes until rupture occurs. For the uniaxial stress-strain law

σ = E(D)ε = (1 − D)E0 ε ,

0 ≤ D ≤ 1,

(14.26)

the bifurcation condition of Eq. (14.25) can be interpreted in terms of a diminishing elastic modulus E(ε ) as 1 dE 1−ε ∼ 1 =− (14.27) =− , E dε ε ε or in terms of the augmenting damage variable D(ε ): 1 dD 1 − ε ∼ 1 = = . 1 − D dε ε ε

(14.28)

The last expressions in Eqs. (14.27) and (14.28) account for the fact that brittle rupture occurs at small strain ε 1; equivalently dσ /dε ∼ = 0 in Eq. (14.25). The applicability of a constitutive framework for the fracturing continuum is limited by the phenomenon of localization. Microcracks cease to be uniformly distributed and form patterns of distinct cracks that initiate rupture by separation. The

14 Damage and Rupture of Porous Ceramics

269

σ

ε

Fig. 14.9 Lattice under vertical extension. Damage localization at maximum load

continuum approach is the more useful the larger the interval of damage prior to rupture. Figure 14.9 shows results for a two-dimensional lattice model of fracturing material subject to vertical extension. The strength of the brittle elastic spring elements that form the triangular cells of the lattice follows a truncated Gaussian probability distribution. The damage pattern at maximum load, visualized by eliminating in the plot broken elements, shows localization of damage to a macroscopic crack normal to the imposed extension. The stress-strain diagram demonstrates that localization and rupture occur rather suddenly, damage is appreciable only within a narrow interval. Similarly for shear and other loading conditions. The behaviour resembles the test results reported in Section 14.2.3.

14.3.2 Numerical Modelling on the Microscale The limitations of the fracturing continuum accounting for damage by degradation of global material properties demands alternative approaches. In particular, investigation of rupture mechanisms in conjunction with the material structure suggests numerical models operating on the microscale; for a survey see [10]. The present approach focuses on a representation of the material structure. The algorithm operates on two-dimensional geometrical models of the microstructure generated in the computer under observance of actual morphological data like the area fraction and speciﬁc perimeter of the participating phases. The artiﬁcial structure consists of grains appertaining to the different material phases and pores of various shapes, Fig. 14.10. Electronic scanning fractographs of samples investigated in the laboratory reveal that fracture is intergranular, Fig. 14.11. Therefore, the model assumes the grain interfaces to constitute a network of potential cracks that may nucleate under loading. Elementary stress solutions for cavities compiled from the literature supplement the stress disturbances emanating from interacting cracks in order to estimate the straining of grain interfaces. At each incremental step during the loading programme the algorithm computes the tractions along grain interfaces while the

270

I. Doltsinis 180 160 140

µm

120 100 80 60 40 20 0

50

100

µm

150

200

Fig. 14.10 Left: SEM of an aluminium-oxide-based ﬁlter support with 28% porosity. Right: computer generated model of microstructure. Black areas refer to pores, grey-coloured grains represent the Al2 O3 phase, white grains the TiO2 phase

Fig. 14.11 SEM of an aluminium-oxide-based ﬁlter support showing intergranular fracture

kinematic activity of cracks is considered with regard to crack closure and frictional sliding [6]. Separation of grain interfaces is based on the Grifﬁth energy criterion employed here for discrete microcracking in the material: it is assumed that the jth grain interface of length l j fails completely and forms a crack if the energy released by virtual separation is not less than the increase in surface energy: j

Gdl ≥ (2γ + ε ) j l j .

(14.29)

In the two-dimensional system the elastic energy G is released per unit crack length during virtual separation of the grain interface. Evaluation is simple: it is based on the average tractions along the interface actually investigated at the state of loading under consideration. The speciﬁc surface energy γ may vary among grain interfaces, and can be adapted to the inﬂuence of corrosive media. It is reduced by the diminution ε of stored energy due to eigenstrains, usually of thermal origin from manufacturing.

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Computer simulations underline that the process of brittle microcracking and rupture reﬂects the antagonism between structural disorder and the steering action of the applied forces [3]. From the micromechanics point of view, localization is seen to be a consequence of synergy between separating grain interfaces. The effect disappears if crack interactions are suppressed in the simulation, in which case the microcracking progresses at random [7]. The effect of crack interaction is seen to be signiﬁcant for rupture in contrast to the minor inﬂuence it has on the overall elastic properties of a solid with randomly distributed microcracks. Structural disorder promotes the quasi-brittle mode of rupture that is preceded by distributed damage. Disorder can be introduced in the microstructure by the nature and size of grain phases, eigenstrains, and cavity defects. The presence of pores comparable in size was found to promote distributed damage in an otherwise regularly structured material. If loading is by internal pressure in the pores, however, the mode of rupture can be brittle or quasi-brittle depending on topological details of the microstructure. Fracturing under internal pore pressure modiﬁes the concept with respect to the local criterion of fracture. Superposition of elementary solutions still gives an estimate for the stress distribution prior to the initiation of cracking and identiﬁes critical locations (Fig. 14.12). But, the particular loading condition favours that cracks emanate from pores boundaries. Such a pore/crack conﬁguration suggests computation of the energy release rate as from the stress intensity factor instead from the stress ﬁeld. Recalling the relation G=

K2 E

(14.30)

between the energy release rate G, the stress intensity factor K and the elastic modulus E, the energy release integral for the virtual separation of individual grain interfaces in Eq. (14.29) is evaluated with expressions for K available in the literature [1]. For the straight crack emanating from an elliptical pore (Fig. 14.13) there is

K πt

2

l (1 + l/a)2 − 1 (1 + l/a) + 1 , = a+ 2 (1 + l/a)2 − m (1 + l/a) + m

a−b a+b

(14.31)

100

100

80 µm

80 µm

m=

60

60 40

40

20

20 20

40

60

80 µm

100

120

20

40

60 80 µm

100

120

Fig. 14.12 Triangular pore under pressure. Grain boundaries prone to fracture at a certain level

I. Doltsinis stress - intensity factor K in units of [t ( π a)0.5]

272

2

1.5

a

b

1

0.5

P

C l

b=0 b=a b→∞

0 0 1 2 3 4 crack length l (in units of the ellipse semiaxis length a)

Fig. 14.13 Crack/pore conﬁguration and stress intensity factor

where t denotes the intensity of remote biaxial tension equivalent to the pressure acting in the pore. A graphical representation of the above relationship is included in the ﬁgure. Modiﬁcations necessary for more general pore/crack geometries have been performed [6]. The local nature of stress intensity factors resulting from isolated solutions does not support consideration of interactions in the present case, but the effect is not signiﬁcant here because cracking is essentially driven by the pressure in the pores.

14.4 Signiﬁcance of Microstructure 14.4.1 Variability of Strength Under Internal Pore Pressure Brittleness is known to enhance the variability of strength among specimens from the same material because of ﬂuctuations in the microstructure. Numerical experiments show that the mode of rupture of porous ceramics under internal pressure can be brittle or quasi-brittle depending on topological details of the material structure; Fig. 14.14 depicts computational results. In the quasi-brittle mode of rupture (upper row) damage by distributed microcracking preceds ultimate failure by separation of the specimen. In contrast, in the brittle mode (lower row) the cracks propagate from one boundary to the opposite in a rather straightforward manner. The associated ﬂuid pressure–overall strain diagrams along the horizontal and the vertical sides of the specimen indicate some anisotropy inherent to the material structure. More important, it is seen from the diagrams that quasi-brittle damage delays rupture to approx 50% beyond the strength of the brittle mode. For transparency, the following investigations refer to simpler conﬁgurations. Figure 14.15 demonstrates the variability of strength with the topology of circular pores, all other parameters kept constant. Positioning the pores of equal size at random within the microstructure leads to separation patterns that differ among

14 Damage and Rupture of Porous Ceramics

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200

int. fluid pressure [MPa]

120

µm

150

100

50

100 80 60 40 εxx εyy

20 0

0

50

100

150

200

250

0

µm

200

0.4 0.6 0.8 strain ε [ × 10−3]

1

int. fluid pressure [MPa]

120

150

µm

0.2

100

50

0

50

100

µm

150

200

250

100 80 60 40 εxx εyy

20 0 0

0.2

0.4

0.6

0.8

1

−3

strain ε [ × 10 ]

Fig. 14.14 Porous material subject to internal pore pressure (pores in black, ﬁne grains in white, coarse grains in light grey/yellow; rupture patterns of pore/crack networks highlighted in dark grey/red). Quasi-brittle (upper) and brittle mode of rupture (lower) under pressure in the pores

Fig. 14.15 Variability of fracture pattern and strength with pore topology. At specimen separation the pressure values in the pores relate as 1:1.197:1.015:1.091 (left to right from above)

274

I. Doltsinis

individual realizations. Despite the throughout brittle mode of rupture, the value of the respective pressure varies up to 20%. The observed ﬂuctuations suggest rigorous statistical treatment that can be based on random sampling of the topology of the pores. Figure 14.16 summarizes results regarding the effects of porosity and pore shape. The two upper rows refer to microstructures with equal speciﬁc perimeter of the porous phase but with different porosity. The higher porosity is seen to lower the mean pressure for specimen separation. At constant porosity, a higher speciﬁc perimeter associated with ﬁner pores is found to raise the strength. In the two lower rows porosity and speciﬁc perimeter are the same, but the pore shape changes from circular to elliptical. The elongated pores imply rupture at a reduced mean pressure while the scatter increases because pore orientation constitutes an additional random variable. The effect of the grain size is also of importance. Reﬁning the grains of the solid while maintaining the characteristics of the porous phase was found to increase the mean critical pressure and to reduce scatter [2]. The above investigations refer to specimens of equal dimensions. Numerical experiments indicate that with increasing specimen size the ultimate pressure is lowered. At the same time the sensitivity to the topology of the pores diminishes. The former phenomenon is attributed to a higher likelihood of encountering critical constellations, the latter to an increasing homogeneity of the specimen. In Fig. 14.17, polydisperse structures that contain circular pores with varying diameter are contrasted to monodisperse structures with pores of unique diameter equal to the polydisperse mean. In both cases mean value and standard deviation of the critical pressure diminish with increasing specimen size. Pores with unique diameter imply a higher strength because polydisperse porosity exhibits a higher speciﬁc perimeter and is coarser [6]. Beyond mean value and standard deviation, knowledge of the probability distribution of the strength in brittle rupture allows to estimate the number of parts to fail up to a certain load level. The following explores suitability of Weibull statistics [14]. The probability of rupture Pr of the material specimen by separation at pore pressure p is then assumed given by # $ V p − pu m . (14.32) Pr (p) = 1 − exp − V0 p0 The probability function in Eq. (14.32) accounts for the effect of the actual volume V with respect to the test volume V0 . The lowest pressure for rupture is pu , the characteristic strength p0 refers to V = V0 and is associated with the rupture probability Pr = 1 − 1/e. From Eq. (14.32) the characteristic strength pV for specimens of volume V = V0 is 1 V0 m pV = p0 . (14.33) V It is noticed that with increasing specimen volume the characteristic strength diminishes, while the Weibull modulus m remains constant.

14 Damage and Rupture of Porous Ceramics

275

180

AA(pores): 0.22

LA(pores): 94 mm / mm2

160 0.35

140

0.3 rel. frequency

µm

120 100 80 60

0.25 0.2

mean: 188.5 ± 1 MPa standard deviation: 4.8 MPa

0.15 0.1

40

0.05

20 0

50

100 µm

150

0

200

100

180

150 200 250 critical fluid pressure P / MPa AA(po)=0.385 ; LA(po)=94 mm / mm2

160 140

0.4 rel. frequency

µm

120 100 80 60 40

mean: 165.3 ± 0.6 MPa standard deviation: 4.5 MPa

0.3 0.2 0.1

20 0

0

50

100 µm

150

200

180

100

150 200 250 critical fluid pressure P / MPa 2

0.4

160

AA(po)=0.382 ; LA(po)=94 mm/mm

140 0.3 rel. frequency

µm

120 100 80 60

mean: 160.1 ± 1.2 MPa standard deviation: 8.3 MPa

0.2

0.1

40 20

0

0

50

100 µm

150

200

100

150 200 250 critical fluid pressure P / MPa

Fig. 14.16 Studies on rupture. Above and middle: specimens with equal speciﬁc perimeter of the porous phase, but different porosity (above: 0.22, middle: 0.385). Middle and below: Specimens with equal porosity (0.385) and speciﬁc perimeter of the porous phase, but different ratio of the pore semiaxes (middle: b/a = 1, below: b/a = 0.73); a higher eccentricity of the pores results in a lower average failure load and causes a higher scatter of the results. The average grain size is 5.4 µ m throughout

Projection of the numerical results on the Weibull probability distribution of the strength values is summarized in Figs. 14.18–14.20. The evaluation extends to the microstructures with both polydisperse and monodisperse circular pores as in Fig. 14.17. The diagrams in Fig. 14.18 compile test series with polydisperse porosity at different specimen dimensions assuming pu = 0. When the specimen size is increased the strength is shifted to lower values according to a diminishing

276

I. Doltsinis polydisperse pore size distribution, 21044 µm2

monodisperse pore size distribution, 21044 µm2 0.25

mean: 167.2 ± 0.4 MPa standard deviation: 11.3 MPa

0.15

0.1

rel. frequency

rel. frequency

0.2

0.2 mean:

168.1 ± 0.9 MPa standard deviation: 9.4 MPa

0.15 0.1

0.05 0.05 0 100

0 100

120 140 160 180 200 critical fluid pressure P / MPa

monodisperse pore size distribution, 124838 µm2

polydisperse pore size distribution, 124838 µm2

0.8

0.15

mean: 152.4 ± 1 MPa standard deviation: 7.3 MPa

rel. frequency

rel. frequency

0.25 0.2

0.1

0.4

0 100

120 140 160 180 200 critical fluid pressure P / MPa

300

300

250

250

200

200

150

100

50

50 100

200 µm

300

140

160

180

200

150

100

0

120

critical fluid pressure P / MPa

µm

µm

0 100

mean: 152.4 ± 0.6 MPa standard deviation: 1.8 MPa

0.6

0.2

0.05

0

120 140 160 180 200 critical fluid pressure P / MPa

0

0

100

200 µm

300

Fig. 14.17 Inﬂuence of specimen size on rupture. Left: Varying pore diameter. Right: Unique pore diameter. For increasing specimen size (from above to below) mean value and standard deviation of the strength diminish. The higher speciﬁc perimeter of the monodisperse porous phase augments the strength, variability is lower. The porosity is 0.33, the average grain size 5.4 µ m

characteristic strength for both, the polydisperse and the monodisperse porosity. From the ﬁgure, the constancy of the modulus m in a series can hardly be supported by the results since they deviate from the straight line as for the Weibull distribution. Although the situation is improving for larger specimens with a polydisperse porous phase (lower frame in Fig. 14.18), a multimodal approximation appears suitable or disturbing values to be discarded. Fitting the Weibull function nevertheless with the numerical results furnishes the characteristic strength p0 and the modulus m shown

14 Damage and Rupture of Porous Ceramics polydisperse pore size distribution

2

ln ln ( 1 / (1−Pf (p)) )

277

0

−2 specimen size / µm2 21044 46023 69757 92276 124838

−4

−6 4.9

5

5.1

5.2

5.3

ln (p / MPa)

3

ln ln ( 1 / (1−Pf (p)) )

2

specimen size: 69757 µm2

1 0 −1 −2 −3 −4 −5 −6 4.8

4.85

4.9

4.95

5

5.05

5.1

5.15

ln (p / MPa)

Fig. 14.18 Pores of varying size (polydisperse). Upper: Frequency of rupture strength in Weibull plot. Lower: Strength of larger specimen tends to reproduce straight line of Weibull distribution

Weibull char. strength p0 / MPa

monodisperse pore sizes polydisperse pore size distrib.

175 170 165 160 155 150

0

5 10 specimen size [ µm2]

15 x 104

Fig. 14.19 Variation of characteristic strength p0 with specimen size

in dependence of the specimen size in Figs. 14.19 and 14.20, respectively. For circular pores of unique diameter (monodisperse) the augmenting m-value reﬂects a diminution of scatter as a result of increasing statistical homogeneity. In contrast, the modulus m is seen to assume a stationary value if the pore diameter varies within

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I. Doltsinis 65

Weibull shape parameter m

60 55

monodisperse pore sizes polydisperse pore size distrib.

50 45 40 35 30 25 20 15 10 0

5 10 specimen size [ µm2]

15 x 10

4

Fig. 14.20 Variation of modulus m with specimen size

the specimen (polydisperse). The latter situation is more likely to encounter in actual materials. The above observations signify that in the case of monodisperse porous structures the specimen dimensions may be increased up to a deterministic representation of the strength; for the variable pores a certain specimen size is required in order that the statistical description of the strength becomes stationary.

14.5 Conclusions The particular interest of the present study has been in porous ceramics subject to internal pressure. The industrial objective of the research is the strength of ﬁlter supports used as members of ﬁltration equipment. Experimentation methods, mathematical modelling and computer simulation techniques have been applied for an exploration of the rupture behaviour of the porous ceramic components, which is statistical in nature. In the laboratory preference has been given to the better controlled diametral compression (Brazilian) test as a substitute for the the burst test under channel pressure. Experimentation characterizes the rupture strength, and revealed damage capability. Testing of ﬁlter supports used in corrosive environment led to a life-time concept. The tests and observations in the laboratory supplied information for a pragmatic background in the modelling issue. In discussing a continuum damage approach to the fracturing solid, it was pointed out that utility is restricted because of narrow damage intervals prior to localization and rupture. Numerical modelling has been employed for an elucidation of the fracturing processes in the microscale. The comptational approach simulates microcracking by separation of grain interfaces, and helps draw conclusions regarding the signiﬁcance of structural material parameters on macroscopic rupture. The numerical model has been employed in conjunction with synthetic sampling of the material structure in order to explore the statistics of rupture. The results tend

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to support the Weibull approach to the rupture probability for the brittle material, as do the laboratory tests on specimens from components.

References 1. Berezhnitskii LT (1966) Soviet Materials Science 2:16–23 2. Dattke R (2003) Modelling the microstructure and simulation of progressive fracturing in brittle porous ceramics. Doctoral Thesis, University of Stuttgart, Stuttgart 3. Doltsinis I (1998) Issues in modelling distributed fracturing in brittle solids with microstructure. In: Idelsohn SR et al. (eds) Computational Mechanics – New Trends and Applications (CD-ROM). CIMNE, Barcelona 4. Doltsinis I (2004) Critical states in deformation processes of inelastic solids. In: Topping BHV, Mota Soares CA (eds) Progress in Computational Structures Technology. SaxeCoburg Publications, Stirling, Scotland 5. Doltsinis I (2006) Strength of porous ceramics – Mechanical testing and numerical modelling. In: Mota Soares CA et al. (eds) Computational Solid and Structural Mechanics. Springer, Berlin 6. Doltsinis I, Dattke R (2001) Comput Meths Appl Mech Engng 191:29–46 7. Doltsinis I, Dattke R (2005) Solids and Structures 42:565–579 8. Doltsinis I, Osterstock F (2005) Arch Comput Meth Engng 12:303–336 9. Kachanov M, Tsukrov I, Shaﬁro B (1994) Appl Mech Rev 47:151–174 10. Krajcinovic D (1996) Damage Mechanics. North-Holland, Elsevier, Amsterdam 11. Osterstock F, Doltsinis I, Vansse O (2004) The Brazilian reliability test and micromechanical modelling for channelled cylinders of multiphase porous ceramics. In: High-Performance Ceramics II, Trans. Tech. Publications 12. Stanley P, Fessler H, Sivill A (1973) An engineer’s approach to the prediction of failure probability of brittle components. In: Proceed. British Ceramic Society 22 13. Vansse O (2000) Fiabilite thermomecanique de ceramiques poreuses multiphases. Doctoral Thesis, University of Caen, Caen 14. Weibull W (1951) ASME, J Appl Mech:293–297

Chapter 15

Computation of Effective Cement Paste Diffusivities from Microtomographic Images K. Krabbenhoft, M. Hain, and P. Wriggers

Abstract A computational framework for extracting effective diffusivities from microtomographic images is presented. As an example of the capabilities of this framework, the effective diffusivity of a cement paste whose microstructure has been digitized to a resolution of 1 µm is derived. Besides presenting a consistent homogenization procedure, the importance of statistical testing is also highlighted. Indeed, for the problem at hand, it appears that statistical testing and subsequent interpretation of the results in terms of statistical quantities is a necessity for obtaining quantitative information on the property of interest.

15.1 Introduction The transport properties of cement pastes are of crucial importance to the durability of much of our civil infrastructure. Due to its corrosive effects and abundance in many natural environments, considerable efforts have been made to quantify particularly the transport of chloride. To date most research has been concerned with the diffusive transport through fully saturated cement pastes and a number of specialized testing methods for evaluating the effective diffusivity have been developed [1–3]. The available experimental data shows both a signiﬁcant scatter for pastes with the same speciﬁcations, a dramatic inﬂuence on the water/cement ratio as well as a number of apparent anomalies. One of the most well known of these manifests itself by negatively charged ions being signiﬁcantly less mobile than positively charged K. Krabbenhoft Centre for Geotechnical and Materials Modelling, University of Newcastle, NSW, Australia [email protected] M. Hain and P. Wriggers Institute for Mechanics and Computational Mechanics, University of Hannover, Hannover, Germany V. Kompiˇs (ed.), Composites with Micro- and Nano-Structure – Computational Modeling and Experiments. c Springer Science + Business Media B.V., 2008

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ions (relative to what should be expected on the basis of the respective free diffusion coefﬁcients). This discrepancy tends to increase with decreasing pore diameters, i.e. with decreasing water/cement ratios [4–6] and has been sought explained by the presence of internal surface charges giving rise to electric double layers in the interface between the cement matrix and the pore solution [4–8]. So far, however, there is not much in the way of quantitative data that can substantiate these speculations. This is primarily due to the fact the observed effects necessarily must depend strongly on the exact geometry and structure (morphology) of the material. Such information has not traditionally been available. However, with the recent advances in non-destructive testing methods, particularly microtomography and related technologies, there is some cause for optimism that many of the questions relating to the mechanisms of ion transport in cement pastes can be settled, or at least investigated more thoroughly, in the near future. The present paper can be seen as a step in this direction. More speciﬁcally, we describe a computational framework for deriving the effective diffusivity (or conductivity) of porous materials on the basis of microtomographic images. Fick’s law is here assumed valid on both the macroscopic level as well as on the microscopic (pore-scale) level. Using the proposed computational homogenization procedure it is then possible to compute the macroscopic, or effective, diffusivity. We note, however, that physically, such diffusivities must necessarily be thought of as concerning the transport of a neutral species that does not react with the cement matrix, for example oxygen diffusing through a water saturated cement paste [5, 6]. The work presented in this paper has a number of similarities with other recent work. Bentz et al. [9] and Manwart et al. [10] use microtomographic images of Fontainebleau sandstone to create ﬁnite difference meshes for the solution of the Stokes equation. In this way, the effective ﬂuid permeability is obtained. In addition, Bentz et al. also determine effective diffusivities using similar numerical methods. The resolution of the images (19 µm, enhanced to 6.65 µm by optical magniﬁcation) was, however, signiﬁcantly coarser than the ones used in the present study (1 µm) and the ‘solid phase’ was therefore not considered impermeable but ascribed a not insigniﬁcant microscopic diffusivity. A different approach was taken by Koster et al. [11] who constructed an equivalent network of cylindrical tubes and used this to derive effective ﬂuid permeabilities and water vapour diffusivities. It is encouraging that all these studies resulted in effective properties that compare well with experimental data. In the present study we take an approach similar to that of Bentz et al. [9] and Manwart et al. [10]. In contrast to these studies, however, we will only deal with the problem of determining effective diffusivities. In addition, and also contrasting previous studies, we present a detailed statistical analysis of the computed effective diffusivities. Thus, when computing effective transport properties from representative morphological data it is quite natural, and in principle straightforward, to include an analysis of the statistical variation of the quantities of interest. Assuming that enough data is available, a large number of virtual experiments can be carried out and the results of these can then be treated as any other set of experimental data. In fact, in the present context of effective diffusivities for cement pastes, such analyses appear to be crucial for extracting quantitative material data.

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15.2 Materials and Microtomographic Images Computed tomography (CT) imaging was originally conceived for medical applications. The basic idea is to scan section by section and then post-process the resulting 2D images to obtain full 3D representations of the material. Whereas most medical tomographs have a resolution in the range of 1–3 mm, microtomography provides the possibility of scanning 3D objects with a resolution of 1 µm or less (a number of systems offering a resolution of 50 nm or less are now commercially available). In the last few years such microtomographic images have been created for a number of porous materials. The NIST Visible Cement Data Set [12], for example, contains 3D images of hydrating cement, plaster and clay brick, all with a resolution of 1 µm. Other studies have been concerned with sandstone [13], soils [14], bone [15], cellular ceramics [16], calcium aluminate cements [17], and polyurethane foams [18]. A broad overview of the possibilities offered by microtomography for these and similar materials has been given by Maire et al. [19]. In this paper a new data set consisting of 3D images of hardened ordinary Portland cement paste with a water/cement ratio of w/c = 0.45 and a capillary porosity of φ = 0.142 is used. This data set was compiled by the Bundesanstalt f¨ur Materialforschung und -pr¨ufung in Berlin. As with the NIST data set, the resolution is 1 µm and cubic specimens consisting of 10243 voxels (‘volume pixels’) were eventually obtained after processing the raw data. Each voxel can then be represented by an 8-node ﬁnite element to obtain digitized specimens such as the one shown in Fig. 15.1. In this ﬁgure the pores appear as voids, the gray material is the hydrated cement paste and the black material in the unhydrated cement paste. Further details on materials and the processing of the raw micro-CT data are given in [20].

64 µm

64 µm 64 µm

Fig. 15.1 Representative volume element of cement paste obtained from a micro-CT scan with 1 µm resolution. The volume consists of 643 = 264, 144 voxels, each of which is represented by an 8-node 3D ﬁnite element

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15.3 Governing Equations 15.3.1 Microscale Equations On the microscale Fick’s law is assumed valid. The ﬂuxes are thus given by j = −D∇c

(15.1)

where j = ( j1 , j2 , j3 )T are the ﬂuxes, c is the concentration and D is the diffusivity matrix (tensor) which is symmetric and positive deﬁnite and which in the general case can be represented by six independent diffusivities: ⎡ ⎤ D11 D12 D13 D=⎣ D22 D23 ⎦ (15.2) sym D33 For future reference we deﬁne the inverse of the diffusivity matrix: S = D−1

(15.3)

The requirement of mass balance in the steady state is expressed as ∇ · j = 0 in Ω

(15.4)

Finally, combining the mass balance equation with the constitutive law (15.1) leads to the classical steady state diffusion equation: ∇ · (D∇c) = 0 in Ω

(15.5)

which is subject to the essential (Dirichlet) and natural (Neumann) boundary conditions (15.6) c(x) = c0 (x) on Γ e , n · j(x) = jn on Γ n .

where Ω is the domain under consideration and the boundary Γ = Γ e Γ n is divided into two parts: Γ e on which the concentration c0 (x) is prescribed and Γ n on which the ﬂux jn is prescribed. The outward directed unit normal to the boundary is given by n.

15.3.2 Macroscale Equations On the macroscale Fick’s law is again assumed valid: j = Deff ∇c

(15.7)

15 Effective Cement Paste Diffusivities from Microtomographic Images

285

where Deff is the effective diffusivity and the volume average of a quantity ρ is given by 1 ρ (x) dΩ (15.8) ρ = |Ω | Ω For porous media where the microscopic diffusivity of the solid phase is zero, the effective diffusivity is often expressed as Deff = φ τ D

(15.9)

where φ is the porosity and τ is the tortuosity (of the same rank as D). In the following, however, we will make reference only to Deff , thus lumping the combined effects of porosity and tortuosity into a single quantity.

15.4 Computational Homogenization Procedure The procedure used for extracting effective diffusivities follows the one developed by Hain and Wriggers [20] for deformation problems (linear elasticity, damage, etc.). In the following, the basic elements of this procedure, as relating to the current application, are brieﬂy described.

15.4.1 Boundary Conditions and Size of the RVE Any homogenization procedure begins with deﬁning a representative volume element (RVE), i.e. a volume element of material whose macroscopic behaviour is representative of the macroscopic behaviour of the porous material under consideration. In setting up such a procedure, two questions are of fundamental importance: (i) which boundary conditions should be imposed on the RVE? and (ii) is the size of the RVE sufﬁcient to be considered representative of the macroscopic material behaviour? These two questions, which in fact are intimately linked, are discussed in the following. For a statistically homogeneous medium we may deﬁne three characteristic length scales: the microscopic length scale d, the length scale of the considered volume element l, and the macroscopic length scale L. In the current application the microscopic length scale may be taken as the resolution of the micro-CT scans, i.e. 1 µm, whereas the macroscopic length scale L typically would be measured in centimeters. The characteristic length scale of the RVE must on one hand be much larger than microscopic length scale d while remaining well below the macroscopic length scale L. To sum up, these considerations lead to well known requirement that dlL

(15.10)

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To gauge the necessary size of the RVE in a more quantitative manner, we may use the framework of Ostoja-Starzewski and Schulte [21] who considered the problem of deriving effective thermal conductivities, which mathematically is identical to the problem considered in this paper. We consider a statistically homogeneous medium and introduce the length scale

δ=

l d

(15.11)

so that for δ = 1 corresponds to the side lengths of an RVE consisting of a single voxel. In addition, we deﬁne the ensemble average [X]δ as the average of a quantity X over all possible realizations of elementary volumes with characteristic dimension δ . Next, we consider two sets of characteristic boundary conditions, namely Natural boundary conditions: n · j = n · J, on Γ

(15.12)

with J being an arbitrary nonzero ﬂux type vector, and Essential boundary conditions: c = G · x, on Γ

(15.13)

where G is an arbitrary nonzero vector. Ostoja-Starzewski and Schulte [21] then show that the effective diffusivity Deff is bounded by [Snδ ]−1 Deff [Deδ ] (15.14) where denotes negative deﬁniteness1 and [Snδ ]−1 and [Deδ ] are, respectively, the ensemble average effective diffusivities that result from imposing natural and essential boundary conditions on RVEs with characteristic dimension δ . It is instructive to note that for δ = 1, i.e. for an RVE consisting of a single voxel, we recover the Reuss and Voigt bounds: −1 = 0 = DReff [Sn1 ]−1 = (φ D−1 pore + (1 − φ )Dsolid )

(15.15)

[De1 ] = φ Dpore + (1 − φ )Dsolid = φ Dpore = DVeff

(15.16)

Finally, it should be noted that the bounds (15.14) refer to the case where the exact solutions to the RVE boundary value problems are available. If such solutions are not available (which they of course never are in practice), rigorous bounds may still be obtained from approximate solutions. In these cases the strong form of the balance equations (15.5) are satisﬁed in the natural boundary condition case whereas standard weak ﬁnite element formulations in conjunction with essential boundary conditions lead to rigorous upper bounds. A B means that all eigenvalues of C = A − B are non-positive or, alternatively, that xT (A − B) x ≤ 0 for all vectors x.

1

15 Effective Cement Paste Diffusivities from Microtomographic Images

287

15.4.2 Window Boundary Conditions As shown by Ostoja-Starzewski and Schulte [21], the upper and lower bounds in (15.14) eventually coalesce in the limit of δ → ∞ to produce a unique macroscopic effective diffusivity. Obviously, this limit is not practically possible to consider and we have to settle on some ﬁnite size of the RVE. However, for materials with large contrasts between the properties of the phases, the adequate size of the RVE may still be far to large to handle using standard numerical methods such as the ﬁnite element or ﬁnite difference methods – and may in fact even be of such a magnitude that the l L no longer holds. This problem has previously been addressed (in the setting of deformation problems) by the formulation of special boundary conditions known as window, mixed, or natural boundary conditions [20]. The idea is to impose boundary conditions that fall somewhere in between the natural and essential ones and thereby obtain an effective diffusivity which, though not rigorously bounded, should be closer to the actual macroscopic effective diffusivity. The procedure is roughly as follows. The sample is ﬁrst embedded in a homogeneous matrix of a considerable thickness and with the same constitutive behaviour as the effective material (Fig. 15.2). Starting from a matrix with a high diffusivity, essential boundary conditions of the type (15.13) are imposed on its surface. The effective diffusivity of the RVE is then computed. Next, the embedding matrix is ascribed this diffusivity and a new effective diffusivity of the RVE is computed. This iterative procedure continues until “convergence”, i.e. until the change in the computed effective diffusivity from one step to the next is sufﬁciently small. Since the boundary conditions that effectively are imposed fall somewhere in between the natural and essential ones, the effective diffusivity thus computed will also fall in between the ones resulting from each of these

Fig. 15.2 Sample embedded in homogeneous matrix (window boundary conditions)

288

K. Krabbenhoft et al. Deff Essential (Dirichlet) b.c.

Window b.c.

Natural (Neumann) b.c. RVE size (δd )

Fig. 15.3 Effect of boundary conditions and RVE size on computed effective diffusivity

types of boundary conditions. In principle we thus obtain a convergence behaviour as sketched in Fig. 15.3. The effective diffusivities resulting from the imposition of natural and essential boundary conditions converge from below and above respectively. The effective diffusivities resulting from the imposition of the window boundary condition falls somewhere in between these two results, and with convergence both from above and below being possible.

15.4.3 Effective Properties Once the properties of the embedding matrix have been established along the lines described above, a set of microscopic state variables follows from the solution of the RVE boundary value problem. From the resulting micro-scale variables a set of relevant macroscopic quantities are deﬁned as ∇c = j =

1 |Ω |

Ω

1 |Ω |

Ω

j(x) dΩ = −

∇c(x) dΩ 1 |Ω |

Ω

D(x)∇c(x) dΩ

(15.17) (15.18)

where c(x) and j(x) are the local (microscopic) concentration and ﬂuxes respectively and D(x) is the local diffusivity matrix. Next, on the macroscale the ﬂuxes and concentration gradients are assumed to be related by j∗ = −Deff ∇c

(15.19)

where Deff is the effective constitutive modulus. Following Hain and Wriggers [20], this quantity is determined by minimizing the difference between the two expressions for the macroscopic ﬂuxes given above:

15 Effective Cement Paste Diffusivities from Microtomographic Images

minimize Π = || j − j∗ ||, j∗ = −Deff ∇c Deff

289

(15.20)

In the case of macroscopic isotropy the effective diffusivity matrix is of the type Deff = Deff I

(15.21)

where I is the unit matrix. The effective diffusivity then follows as j · ∇c dΠ = 0 =⇒ Deff = − dDeff ∇c · ∇c

(15.22)

More general cases can be treated in a similar manner although a number of independent tests must then be considered for each RVE (up to six in the most general case of full orthotropy).

15.4.4 Statistical Testing As an essential part of the homogenization procedure used in the present paper, we compute ensemble averages of the effective diffusivity. The procedure follows that of Hain and Wriggers closely [20]. From a large RVE consisting of, say, 1,0003 voxels, a number of smaller RVEs are selected at random. For each of these, the relevant effective material constants are determined as described above. In order to derive reasonable estimates of relevant statistical quantities such as mean value and variance, a total of perhaps 100 to 1,000 different RVEs should be considered. Next, a series of larger RVEs are considered and the statistical distributions of the effective properties are recorded. This procedure continues until the mean value of the diffusivity converges to a ﬁnal unique value. At this stage, the size of the RVE is deemed to be sufﬁciently large. As a by-product of such a procedure, the ﬁnal probability distribution determined may be viewed as representing the natural variability of the porous material at that length scale.

15.4.5 Solution of RVE Boundary Value Problems All boundary value problems were solved using a standard weak form ﬁnite element formulation. Each voxel of the RVE was represented by an 8-node trilinear ﬁnite element. The microscopic diffusivity of the pore space was set equal to unity while the microscopic diffusivity of the solid matrix was set equal to an appropriately small number (typically of order 10−7). The main advantage of this strategy is generality and ease of implementation. Disadvantages include potential ill-conditioning of the resulting conductivity (stiffness) matrix and an overestimation of the computed effective diffusivities. The RVE boundary value problems (incorporating the embedding matrix) were solved using the ﬁnite element program FEAP [22].

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15.5 Analytical Estimates Over the years, numerous analytical expressions for the effective diffusivity (or conductivity) of heteronegeous media have been proposed. For two-phase media where the phases are not too dissimilar these estimates are often relatively accurate. On the other hand, for media such as the one considered here, where the diffusivity of the solid phase is zero and where the porosity (φ = 0.142) is well below the site percolation threshold of all the common lattices [23], we can expect relatively little from these estimates. However, for completeness a number of the more common analytical estimates are brieﬂy summarized below. For general two-phase porous media the effective diffusivity is bounded rigorously by the Voigt [24] and Reuss [25] estimates: DReff ≤ Deff ≤ DVeff

(15.23)

In the case where one of the phases is impermeable we have DReff = 0, DVeff /Dpore = φ = 0.142

(15.24)

which in most cases, including the one considered here, can be expected to be of little value, i.e. the lower bound is much too low and the upper bound is much too high. Another commonly used estimate is due to Maxwell [26]: DM eff /Dpore =

2φ = 0.099 3−φ

(15.25)

For the special case considered here (Dsolid = 0) this estimate is identical to the Hashin-Shtrikman upper bound [27] which again can be expected to overestimate the effective diffusivity signiﬁcantly. Finally, on the basis a self-consistent effective medium theory, Sen et al. [28] derive the following approximation DSeff /Dpore = φ m , m ≥ 1.5 ⇒ DSeff /Dpore ≤ 0.054

(15.26)

where the limiting value of m = 1.5 corresponds to an assembly of impermeable spherical particles.

15.6 Numerical Results The ﬁrst task of the numerical homogenization procedure is to determine an appropriately small solid matrix diffusivity, i.e. small enough to have a negligible effect on the computed effective conductivity, but large enough to maintain a relatively well conditioned system of ﬁnite element equations. This value was established by simple trial and error with the results shown in Fig. (15.4). On the basis of these

15 Effective Cement Paste Diffusivities from Microtomographic Images

291

0.035

Deff/Dpore

0.030 0.025 0.020 0.015 0.010 0.005 102

103

104

105

106

107

Dpore/Dsolid Fig. 15.4 Effect of solid matrix diffusivity: mean value based on 150 tests (64 µm3 RVEs) 0.045 0.040

Deff/Dpore

0.035 0.030 0.025 0.020 0.015 0.010 0.005

0

5

10

15

20

25

Window width (µm)

Fig. 15.5 Effect of window width: mean value based on 150 tests (64 µm3 RVEs)

results an artiﬁcial solid matrix diffusivity of Dsolid = 10−7Dpore was chosen and used in all subsequent calculations. Next, the necessary width of the window (see, Section 4.2) was established. This was again done via trial and error, starting from a zero width (corresponding to usual essential boundary conditions) and progressively increasing the width until the effect becomes negligible. The results of this procedure are shown in Fig. 15.5 where we observe the expected decrease in effective diffusivity as the window width increases. A width of 24 µm was eventually chosen and used in all subsequent calculations. With the artiﬁcial solid matrix diffusivity and the window width established the necessary size of the RVE must be determined. This was done by carrying out tests on RVEs ranging from 83 to 643 voxels. For each size a total of 150 RVEs were selected at random and the effective diffusivities were computed. The results of these tests in terms of the mean effective diffusivity and its standard deviation are shown in Figs. 15.6 and 15.7.

292

K. Krabbenhoft et al. 0.0105

mean(Deff)/Dpore

0.0100 0.0095 0.0090 0.0085 0.0080 0.0075 0.0070 8

16

32

64

Side length of RVE (mm)

Fig. 15.6 Effect of RVE size: mean value based on 150 tests 0.030

std(Deff)/Dpore

0.025

0.020

0.015

0.010

0.005 8

16

32

64

Side length of RVE (mm)

Fig. 15.7 Effect of RVE size: standard deviation based on 150 tests

Surprisingly, the mean diffusivity appears to be relatively insensitive of the size of the RVE, decreasing only by some 25% from the smallest size (83 voxels) to the largest (643 voxels). It thus appears that as, far as the mean value is concerned, an RVE consisting of 643 voxels is sufﬁcient. However, the standard deviation follows a somewhat different trend, decreasing by approximately a factor of four from the smallest to the largest RVE considered. Furthermore, the same degree of ‘convergence’ as observed for the mean value is not obtained. Indeed, for the largest RVE the standard deviation is of the same order of magnitude as the mean value. This is more than what can be ascribed to the natural variability of samples used in typical laboratory tests (of order 1–3 cm) and must necessarily be attributed to the size of the RVE. The mean value, on the other hand, does converge as the size of the RVE is increased and it can be expected that

15 Effective Cement Paste Diffusivities from Microtomographic Images

293

a further increase would not affect the mean value while it would continue to lead to a decrease in standard deviation. This trend can be explained in terms of an analogy involving one-dimensional diffusion elements. Consider N such elements, each of which can either be fully conducting (with a diffusivity equal to unity) or completely impermeable. Let φ be the probability that an element is fully conducting. If the N elements are connected in parallel the mean and variance of the effective diffusivity of the assembly are given by µ p − µ p2 (15.27) µ p = φ , σ p2 = N In contrast, for N one-dimensional elements placed in series we have

µs = φ N , σs2 = µs − µs2

(15.28)

Thus, in the former case the mean value is constant and independent of N while the variance is inversely proportional to N. In the latter case, both the mean value and the variance decrease rapidly to zero as N increases (for the values of φ that would be relevant to cement pastes). The diffusion in three-dimensional porous media can to a certain extent be seen as a combination of these two types of one-dimensional transport. The results obtained for the three-dimensional cement paste RVEs are quite consistent with this analogy, particularly if the transport is seen to be tending more to the parallel mode of diffusion. In conclusion, although the variance for the largest RVE considered is signiﬁcant, the mean value obtained using this size may well be representative of what one would ﬁnd in physical experiments involving samples with dimensions several orders of magnitude larger, for example, in the centimeter range. Having settled on an RVE size of 643 µm3 , a total of 9216 tests were carried out on randomly selected RVEs of this size. The ﬁnal mean value and standard deviation were found to be µ (Deff /Dpore ) = 0.0081 (15.29) σ (Deff /Dpore ) = 0.0078 The distribution of the effective diffusivities thus computed is shown in Fig. 15.8. Also shown in this ﬁgure is the Weibull distribution: f (x; k, λ ) =

k x k−1 −(x/λ )k e , x = Deff /Dpore λ λ

(15.30)

where the parameters k and λ can be determined from

µ = λ Γ (1 + 1/k), σ 2 = λ 2Γ (1 + 2/k) − µ 2

(15.31)

with Γ being the Gamma function. The Weibull parameters corresponding to the computed mean value and standard deviation (15.29) are given by k = 1.04, λ = 0.0082

(15.32)

294

K. Krabbenhoft et al. 1 Weibull Lognormal

Frequency

0.8

0.6 0.4

0.2

0

0

0.01

0.02

0.03

0.04

0.05

Deff /Dpore

Fig. 15.8 Results of 9,216 tests on 64 µm3 RVEs 0.014 OPC (exp., Ngala et al. [6]) OPC / 30%PFA (exp., Ngala et al. [6]) Computed (mean value)

0.012

Deff /Dpore

0.010 0.008 0.006 0.004 0.002 0

0

5

10

15

20

25

30

Capillary porosity

Fig. 15.9 Computed effective diffusivity and experimental results of Ngala et al. [6]. The dashed line represents a curve ﬁt of the experimental data

Also shown is the lognormal distribution (again with parameters corresponding to the computed mean value and standard deviation of the effective diffusivity). This distribution is widely used to model the spatial variability of hydrological data such as permeability of porous media [29]. In the present application, however, the Weibull distribution provides a somewhat better ﬁt to the data. Nevertheless, it is encouraging to observe the well established trend of transport parameters following a right-skewed probability distribution. The computed mean value (15.29) can be compared to the experimental results of Ngala et al. [6] (Fig. 15.9) who determined effective oxygen diffusivities for various cement pastes with different water/cement ratios and hence different capillary porosities. Compared to these, the computed effective diffusivity appears to be overestimated somewhat. It is, however, much closer to the experimental data than any

15 Effective Cement Paste Diffusivities from Microtomographic Images

295

of the analytical estimates presented in Section 5. The most obvious source of deviation between the present results and those of Ngala et al. is the fact that slightly different materials, curing conditions, etc. were used. In addition, some error could also be attributed to the overestimation inherent in the weak form of the governing equations used as a basis for homogenization in this study. Thus, it is conceivable that the use of alternative methods of discretization could lead to a somewhat lower estimate of the effective diffusivity. Nevertheless, taking also the signiﬁcant scatter in the experimental results into account, the magnitude of the computed effective diffusivity is quite satisfactory.

15.7 Conclusions A computational framework for extracting effective diffusivities from microtomographic images has been presented. This framework is quite general and is applicable to many other problems than the one dealt with in this paper. Concerning the particular problem of diffusion in cement pastes we have highlighted the importance of statistical testing. Indeed, such a strategy appears to be absolutely essential in order to extract quantitative data. The combination of computational homogenization and statistical testing presents some interesting prospects for ‘upscaling’ of the microscopic governing equations. Such an upscaling could be performed by considering a volume consisting of, say, 643 cubes with side lengths of 64 µm. Each of these elements would then be ascribed a diffusivity according the relevant statistical distribution found by direct treatment of micro-CT images. The statistical distribution of the effective diffusivity for a ∼4 mm3 volume would then be obtained. One should here ﬁnd the same mean value as for the smaller volume, but a different (i.e. smaller) variance which would then represent the actual variance in samples of this dimension. Such a procedure, which clearly involves a number of points that require further scrutiny, is currently being developed. Regarding ﬁnite element and related techniques, it appears that there is still some room for improvement. Thus, the weak form of the governing equations used as a basis of discretization consistently overestimates the effective diffusivity. Alternative methods based on the strong form of the governing equations, and leading to consistent underestimates, are currently the subject of further investigation.

References 1. L. Tang and L. O. Nilsson. Rapid determination of the chloride diffusivity in concrete by applying an electric ﬁeld. Cement and Concrete Research, 89:49–53, 1992. 2. C. Andrade. Calculation of chloride diffusion coefﬁcients in concrete from ionic migration measurements. Cement and Concrete Research, 23:724–742, 1993.

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K. Krabbenhoft et al.

3. H. Friedmann, O. Amiri, A. Ait-Mokhtar, and P. Dumargue. A direct method for determining chloride diffusion coefﬁcient by using migration test. Cement and Concrete Research, 34: 1967–1973, 2004. 4. S. Goto and D. M. Roy. Diffusion of ions through hardened cement pastes. Cement and Concrete Research, 11:751–757, 1981. 5. S. W. Yu and C. L. Page. Diffusion in cementitious materials: 1. Comparative study of chloride and oxygen diffusion in hydrated cement pastes. Cement and Concrete Research, 21:581–588, 1991. 6. V. T. Ngala et al. Diffusion in cementitious materials: II. Further investigations of chloride and oxygen diffusion in well-cured OPC and OPC/30% PFA pastes. Cement and Concrete Research, 25:819–826, 1995. 7. J. Z. Zhang and N. Buenfeld. Presence and possible implications of a membrane potential in concrete exposed to chloride solution. Cement and Concrete Research, 27:853–859, 1997. 8. S. Chatterji. Colloid electrochemistry of saturated cement paste and some properties of cement based materials. Advances in Cement Based Materials, 7:102–108, 1998. 9. D. P. Bentz et al. Microstructure and transport properties of porous building materials. II: Three-dimensional X-ray tomographic studies. Materials and Structures, 33:147–153, 2000. 10. C. Manwart et al. Lattice-Boltzmann and ﬁnite-difference simulations for the permeability for three-dimensional porous media. Physical Review E, 66(016702), 2002. 11. M. Koster, J. Hannawald, and W. Brameshuber. Simulation of water permeability and water vapor diffusion through hardened cement paste. Computational Mechanics, 37:164–172, 2006. 12. D. P. Bentz et al. The Visible Cement Data Set. Journal of Research of the National Institute of Standards and Technology, 107:137–148, 2002. URL http://visiblecement.nist.gov. 13. W. B. Lindquist et al. Pore and throat size distributions measured from sychrotron x-ray tomographic images of fontainebleau sandstones. Journal of Geophysical Research, 105B: 21508–21528, 2000. 14. P. Lehman et al. Tomographical imaging and mathematical description of porous media used for the prediction of ﬂuid distribution. Vadose Zone Journal, 5:80–97, 2006. 15. P. Weiss et al. Synchrotron and non synchrotron x-ray microtomography three dimensional representation of bone ingrowth in calcium phosphate biomaterials. European Cells and Materials, 9:48–49, 2005. 16. E. Maire et al. Characterization of the morphology of cellular ceramics by 3D image processing of X-ray tomography. Journal of the European Ceramic Society, 27:1973–1981, 2007. 17. T. J. Chotard et al. Characterisation of early stage calcium aluminate cement hydration by combination of non-destructive techniques: acoustic emission and X-ray tomography. Journal of the European Ceramic Society, 23:2211–2223, 2003. 18. M. A. Knackstedt et al. Elastic and transport properties of cellular solids derived from threedimensional tomographic images. Proceedings of the Royal Society A, 462:2833–2862, 2006. 19. E. Maire et al. On the application of X-ray microtomography in the ﬁeld of materials science. Advanced Engineering Materials, 3:539–546, 2001. 20. M. Hain and P. Wriggers. On the numerical homogenization of hardened cement paste. Compational Mechanics, 2007. Submitted. 21. M. Ostoja-Starzewski and J. Schulte. Bounding of effective thermal conductivities of multiscale materials by essential and natural boundary conditions. Physical Review B, 54:278–285, 1996. 22. FEAP. A Finite Element Analysis Program (R. L. Taylor). URL http://www.ce.berkeley. edu/rlt/feap/. 23. D. Stauffer and A. Aharony. Introduction to Percolation Theory. Taylor and Francis, London 1992. 24. W. Voigt. Lehrbuch der Kristallphysik. Teubner, Leipzig, 1928. 25. A. Reuss. Berechnung der ﬂeissgrense von Mischkristallen auf Grund der Plastizit¨atsbedingung f¨ur einkristalle. Zeitschrift f¨ur Angewandte Mathematik und Mechanik, 9: 49–58, 1929.

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26. J. C. Maxwell. A Treatise on Electricity and Magnetism. Dover, New York 1954. 27. Z. Hashin and S. Shtrikman. A variational approach to the theory of the effective magnetic permeability of multiphase materials. Journal of Applied Physics, 33(10):3125–3131, 1962. 28. P. Sen, C. Scala, and M. Cohen. A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads. Geophysics, 46:781–795, 1981. 29. J. P. Chiles and P. Delﬁner. Geostatistics, Modeling Spatial Uncertainty. Wiley, New York 1999.

Index

2nd order beam theory, 216 Anisotropic elastic properties, 167 Atomic-scale ﬁnite element, 2

Equivalent inclusion, 241 Eshelby tensor, 244 Experimental Validation, 171

Berkovich pyramidal diamond indenter, 141 Binary systems, 68 Boundary integral equations, 241 Boundary point method, 247 Brownian forces, 14

Factor triaxiality, 204 Failure work, 21 Fatigue, 148 Fatigue criteria, 150 Fatigue damage, 149 Fatigue life, 149 FE simulations, 141 Fe-Cr-Ni system, 80 Findley criterion, 150 Finite element method, 128 Finite elements, 90 FORGE2, 142 Fracturing continuum, 265 Functionally graded material, 210

Cement paste, 182 Cohesive element, 196 Cohesive law, 193 Composite beam, 210 Composite materials, 28 Compositionally graded coatings, 139 Computational modelling, 23 Continuous source functions, 30 Continuum micromechanics, 164 Coulomb’s law, 14 Coupled ﬁeld problems, 215 Creep, 184 Critical torsional angle, 1 CT specimen, 192 Cyclic loading, 185 Damage, 263 Diffusion coefﬁcient, 72 Diffusion induced grain boundary motion, 77 Distinct Element Method (DEM), 11 Effective diffusivity, 285 Effective properties, 288 Eigen-strain, 241 Elasticity, 89

Genetic algorithm, 56 Geometric nonlinear bar element, 221 Gibbs energy, 69 Gibbs-Duhem relation, 70 GTN damage model, 199 Hard coatings, 139 Hertz Law, 12 Homogenization, 52, 285 Impact test, 137 Inhomogeneity, 240 Interaction of ﬁbres, 34 Interface migration, 70 Interface mobility, 72 Interstitial components, 68 Inverse algorithm, 143

299

300 J − R curve, 194 J-integral, 193 Johnson, Kendall and Roberts model (JKR), 12 Langevin equation, 11 Local integral equations, 95 Low cycle fatigue tests, 138 Martens hardness, 143 Mass conservation law, 71 Material curve, 198 Material non-homogeneity, 90 Material properties of copper, 15 MATLAB functions, 81 McClintock criterion, 146 Mechanical strength, 19 Microtomographic images, 283 Miner rule, 149 Modelling, 107 Modelling damage and rupture, 265 Mole fractions, 69 Multi-component systems, 68 Multiphysical beam element, 215 Nanocomposite, 10 Nanoimpact test, 138 Nanoindentation, 182 Non-convex microstructure, 47 Numerical efﬁciency, 100 Numerical Modeling, 181 Pair correlation function, 17 Papadopoulos criterion, 151 Point Interpolation Method, 92

Index Porous ceramics, 257 Powder nanocomposites, 107 Principle of maximum dissipation rate, 72 Probability of rupture, 274 Pull-off force, 12 PVD (physical vapour deposition) method, 140 Representative volume element, 250 Rupture, 261 Sandwich beam ﬁnite element, 218 Self-assembly, 16, 23 Sensitivity analysis, 145 Sequential quadratic programming, 57 Short ﬁbres, 30 Smith, Watson and Topper criterion, 151 Socie–Fatemi criterion, 151 Sources or sinks of vacancies, 68 Strain energy, 2 Strand-based engineered wood products, 164 Stress tensor, 19 Stress volume integral, 260 Substitutional components, 68 Tensile specimen, 198 Thermodynamic extremal principle, 68 Titanium nitride, 140 Torsional buckling, 1 Unconﬁned yield stress, 19 Unit cell, 49 Van der Waals force, 13

Computational Methods in Applied Sciences 1. 2. 3. 4. 5. 6.

7. 8. 9.

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