composites with micro- and nano-structure

Composites with Micro- and Nano-Structure

Computational Methods in Applied Sciences

Series Editor

E. OñateInternational Center for Numerical Methods in Engineering (CIMNE)Technical University of Catalunya (UPC)Edificio C-1, Campus Norte UPCGran Capitán, s/n

[email protected]

08034 Barcelona, Spain

Volume 9

Composites with Micro-and Nano-Structures

Computational Modeling and Experiments

ˇVladimir Kompis

123

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Vladimir Kompi

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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Torsional Buckling of Single-Walled Carbon Nanotubes . . . . . . . . . . . 1A.Y.T. Leung, X. Guo, and X.Q. He

2 Equilibrium and Kinetic Properties of Self-AssembledCu Nanoparticles: Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . 9Roberto Moreno-Atanasio, S.J. Antony, and R.A. Williams

3 Method of Continuous Source Functions for Modelling of MatrixReinforced by Finite Fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Vladimır Kompis, Mario Stiavnicky, Marian Kompis, ZuzanaMurcinkova, and Qing-Hua Qin

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

5 Modelling of Diffusive and Massive Phase Transformation . . . . . . . . . 67Jirı Vala

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

7 Modelling of the Process of Formation and Use of PowderNanocomposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Alexandre Vakhrouchev

8 Modelling of Fatigue Behaviour of Hard Multilayer NanocoatingSystem in Nanoimpact Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Magdalena Kopernik, Lechosław Trebacz, and Maciej Pietrzyk

v

vi Contents

9 A Continuum Micromechanics Approach to Elasticityof Strand-Based Engineered Wood Products:Model Development and Experimental Validation . . . . . . . . . . . . . . . . 161Reinhard Sturzenbecher, Karin Hofstetter, Thomas Bogensperger,Gerhard Schickhofer, and Josef Eberhardsteiner

10 Nanoindentation of Cement Pastes and Its Numerical Modeling . . . . 181Jirı Nemecek, Petr Kabele, and Zdenek Bittnar

11 Ductile Crack Growth Modelling Using Cohesive Zone Approach . . . 191Vladislav Kozak

12 Composite (FGM’s) Beam Finite Elements . . . . . . . . . . . . . . . . . . . . . . 209Justın Murın, Vladimır Kutis, Michal Masny, and Rastislav Duris

13 Computational Modal and Solution Procedure for InhomogeneousMaterials with Eigen-Strain Formulation of Boundary IntegralEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239Hang Ma, Qing-Hua Qin, and Vladimir Kompis

14 Studies on Damage and Rupture of Porous Ceramics . . . . . . . . . . . . . 257Ioannis Doltsinis

15 Computation of Effective Cement Paste Diffusivitiesfrom Microtomographic Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281K. Krabbenhoft, M. Hain, and P. Wriggers

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

Introduction

This book contains selected, extended papers presented (with three exceptions) inthe thematic ECCOMAS conference on Composites with Micro- and Nano-Structure (CMNS) – Computational Modelling and Experiments held inLiptovsky Mikulas, Slovakia, in May 28–31, 2007 and sponsored by the SlovakMinistry of Education.

Composite materials play important role in all mechanical, civil as well as inelectrical engineering structures especially in the last two decades in connectionwith the nano-materials and nano-technologies. Of course the chemical engineeringis a connecting element for all these applications.

Computational methods and experiments are a basis for the Simulation-BasedEngineering Science (SBES). SBES can play a remarkable role in promoting thedevelopments vital to health, security and technological competitiveness of nations.

Recent experimental and computational results demonstrate that materials rein-forced with stiff particles and fibres can obtain substantial improvements in stiffness,thermal conductivity and electro-magnetic properties. Materials reinforced with fi-bres can have very different properties in different directions. The material can havevery good conductivity in one direction and can be isolator in other directions. Mostimportant reinforcing materials discovered less than 20 years ago, which are in thecentre of interest in many universities and research institutions are the carbon nano-tubes.

With these new materials also importance of computational simulations and ex-perimental 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 depen-dence of critical torsion angle on the length is discussed and compared with conven-tional shell theory. Strain energy and morphologies of the SWNT are discussed.

In the second paper, the authors study the structure and mechanical strength prop-erties 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 andcoupled Johnson, Kendall and Roberts (JKR) and Brownian force models. The as-semblies considered have poly-size distribution of particles.

In the third paper the interaction of matrix with fibres and fibre with other fibresin the fibre-reinforced composites (FRC) is numerically simulated using 1D contin-uous source functions along the fibre axis and 2D source functions inside the endsof fibres. Numerical examples show that the interaction of the end parts of fibresis crucial for evaluation of the mutual interaction of fibres in the composite. Cor-rect simulation of all parts is important for evaluation of stiffness and strength ofthe FRC.

In the fourth paper the effective dynamic material properties for materials withmicrostructure are assumed. Micro-scale inertia are taken into account and numer-ical homogenization is performed. The frequency dependent macroscopic materialparameters are found for frequency range from 0 up to 1 MHz.

The fifth paper describes modelling of diffusive and massive phase transforma-tion of a multi-component system based on the principle of maximum dissipationrate by Onsager; the finite thickness of the interface between both phases can be res-pected. The mathematical analysis results in an initial-value problem for a systemof partial differential equations of evolution with certain non-local integral term; theunknowns are the mole fractions of particular components.

Paper number six deals with numerical implementation of local integral equationformulation of 2D linear elastic media with continuous variable Young’s modulus.Two meshless presentations of the formulation are described and accuracy, conver-gence and numerical stability are investigated.

The seventh paper presents methods for the modelling of the processes that ac-company obtaining and use of powder nano-composites. For this purpose, a numberof physical-mathematical models, including the models of obtaining of nano-sizedpowders at “up down” processes, the models of the main steps of powder nano-composites compaction and the models of deformation of powder nano-compositesunder the ambient action were developed.

The eighth paper presents the basis of the nano-impact test and the idea of predic-tion of fracture occurrence, especially fatigue behaviour of nano-impacted materials.Finite element (FE) model is applied to identification of the material model of hardnano-coating in the multilayer system. In the second part of the chapter a new ap-proach to analysis of fracture phenomena is introduced as the fatigue criteria, whichare used in simulation of nano-impact test. Results of simulations with the fatiguecriteria implemented into the FE code for the analysed problem are discussed.

The ninth paper contains a description of a continuum micromechanics modeldevelopment and experimental validation of strand-based engineering wood prod-ucts aimed to improve the mechanical properties of wood products. The model al-lows considering of the relevant (micro-) characteristics of the wood on the mechan-ical properties of panels.

The tenth paper focuses on experimental investigations and numerical modellingof micromechanical behaviour of cement paste taken as a fundamental represen-tative 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 andfinite element model with general visco-elasto-plastic constitutive relation are pro-posed. These models are used for simulation of indentation and for estimation ofmaterial parameters at micrometer scale.

Paper number eleven studies the prediction of the crack growth of the ductilefracture of forgets steel 42CrMo4. Crack extension is simulated in sense of elementextinction algorithm based on the damage model Gurson-Tvergard-Needleman andon the cohesive zone model. Determination of micro-mechanical parameters isbased on the combination of static tests, microscopic observation and numericalcalibration procedures by FEM.

In the twelfth paper a longitudinal polynomial continuous variation of the stiff-ness properties is considered in the stiffness matrix of the beam element and pre-sented for the analysis of the electric, thermal and structural field. The transversaland longitudinal variation of the material properties is considered.

In the thirteenth paper a novel computational modal and solution procedureare proposed for inhomogeneous materials with the eigenstrain formulation of theboundary integral equations. Each inhomogeneity embedded in the matrix with vari-ous shapes and material properties described via the Eshelby tensors can be obtainedthrough either analytical or numerical means. As the unknowns appear only on theboundary of the solution domain, the solution scale of the inhomogeneity problemwith the present model is greatly reduced. To enhance further the computationalefficiency, the overall elastic properties using the present model with eigenstrainformulation are solved using the newly developed boundary point method for parti-cle reinforced inhomogeneous materials over a representative volume element.

The fourteenth paper deals with reliability of porous ceramics based on exper-imentation and analysis. Damage prior to rupture and aging because of corrosiveactivity in porous materials is discussed. Fracturing processes in the material thatresult to rupture under pore pressure are studied on the micro-scale by the numeri-cal simulation model.

Finally, in the last paper a computational framework for extracting effective dif-fusivities from micro-tomographic images is presented. Effective diffusivity of acement paste whose microstructure has been digitized is derived, consistent homog-enization and statistical testing and interpretation of results are highlighted.

I would like to thank my colleague Dr. Stiavnicky for his help in preparation ofthe book.

Vlado Kompis

Chapter 13Computational Modal and Solution Procedurefor Inhomogeneous Materials with Eigen-StrainFormulation of Boundary Integral Equations

Hang Ma, Qing-Hua Qin, and Vladimir Kompis

Abstract In the present study a novel computational modal and solution procedureare proposed for inhomogeneous materials with the eigenstrain formulation of theboundary integral equations. The model and the solution procedure are both resultedintimately from the concepts of the equivalent inclusion of Eshelby with eigenstrainsto be determined in an iterative way for each inhomogeneity embedded in the ma-trix with various shapes and material properties via the Eshelby tensors, which canbe readily obtained beforehand through either analytical or numerical means. Asunknowns appeared in the final equation system are on the boundary of the solu-tion domain only, the solution scale of the inhomogeneity problem with the presentmodel is greatly reduced. This feature is considered to be significant because sucha traditionally time-consuming problem with inhomogeneities can be solved mostcost-effectively with the present procedure in comparison with the existing numer-ical models such as finite element method (FEM) and boundary element method(BEM). Besides, to illustrate computational efficiency of the proposed model, re-sults of overall elastic properties are presented by means of the present eigenstrainmodel and the newly developed boundary point method for particle reinforced in-homogeneous materials over a representative volume element. The influences ofscatted inhomogeneities with a variety of properties and shapes and orientations onthe overall properties of composites are computed and the results are compared withthose from other methods, showing the validity and the effectiveness of the proposedcomputational modal and the solution procedure.

Hang MaDepartment of Mechanics, College of Sciences, Shanghai University, Shanghai 200444, China

Qing-Hua QinDepartment of Engineering, Australian National University, Canberra, ACT 0200, Australia

Vladimir KompisDepartment of Mechanical Engineering, Academy of Armed Forces of General M.R. Stefanik,03119 Lipt. Mikulas, Slovakia

V. Kompis (ed.), Composites with Micro- and Nano-Structure – Computational Modeling 239and Experiments.c© Springer Science + Business Media B.V., 2008

240 H. Ma et al.

13.1 Introduction

The determination of mechanical behaviour of an embedded inclusion is of consid-erable importance in a wide variety of physical and engineering problems. Sincethe pioneer work of Eshelby [1, 2], inclusion and inhomogeneity problems havebeen a focus of solid mechanics for several decades. In the terminology of Eshelbyand Mura [3], an inclusion denotes a sub-domain subjected to an eigenstrain or atransformation strain in a solid, the first problem of Eshelby. On the other hand, aninhomogeneity is a region with properties distinct from those of the surroundingmaterial and subjected to an applied stress, the second problem of Eshelby. How-ever, 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 terminol-ogy of Eshelby is employed. Following Eshelby’s idea of equivalent inclusion andeigenstrain solution, quite a diverse set of research work has been reported analyti-cally [4–7] and/or numerically [8–16], to name a few. The eigenstrain solution canrepresent various physical problems where eigenstrain may correspond to thermalstrain mismatches, strains due to phase transformation, plastic strains or fictitiousstrains arising in the equivalent inclusion problems, overall or effective elastic, plas-tic properties of composites, quantum dots, microstructural evolution, as well as theintrinsic strains in the residual stress problems [17].

The analytical solutions of inclusion problems available in the literature can betaken as a basis for understanding the process of predicting stress/strain distributioneither within or outside an inhomogeneity and for further research of such a categoryof problem. However, these solutions usually apply only for cases of simple geome-tries such as ellipsoidal, cylindrical and spherical in an infinite domain. Therefore,numerical simulation using FEM, BEM, or volume integral methods (VIM) havebeen conducted in the analysis of inhomogeneity problems with various shapes andmaterials. In general, FEM may yield results for the whole composite materialsincluding those inside and outside the inhomogeneity [10]. Consequently, the solu-tion scale might be quite large since both matrix and inhomogeneities need to bediscretized and the corresponding unknowns are solved simultaneously. In contrast,VIM and BEM seem more suitable for analyzing inhomogeneity problems. With theVIM [11–13], the fields inside inhomogeneities are expressed in terms of volume in-tegrals, which will simplify the construction of final matrix of the system of linearalgebraic equations to which unknowns of the problem are significantly reduced af-ter discretization. However, as the interfaces between matrix and inhomogeneitiesalso need to be discretized and identified by additional unknowns, only those prob-lems with a few inhomogeneities have been studied in the literature. Regarding ap-plications of BEM to inclusion problems, it is often coupled with VIM [14, 15] inpractice and the procedure is more or less similar to that in VIM. It is usually usedto solve problems with simple arrays of inclusions [15] as field continuity conditionacross interfaces will significantly increase the number of unknowns. Recently, thefast multipole expansion scheme [18] is introduced into BEM to improve computa-tional efficiency in solving large- scale inclusion-matrix problems [16].

13 Computational Modal and Solution Procedure for Inhomogeneous Materials 241

It appears that the Eshelby’s idea of equivalent inclusion and eigenstrain solutionhas not yet been sufficiently followed in the regime of numerical study on inho-mogeneity problems [19]. Motivated by this situation, a novel computational modaland solution procedure are proposed for inhomogeneous materials in this paper.The eigen-strain formulation of boundary integral equations (BIE) in the proposedmodel is developed based on the concept of equivalent inclusion of Eshelby witheigen-strains, which can be determined in an iterative way for each inhomogeneityembedded in the matrix with various shapes and material properties via the Eshelbytensors. As unknowns appeared in the final linear equation system are on the bound-ary of the solution domain only, the solution scale of the inhomogeneity problem canbe greatly reduced.

In this paper, the second problem of Eshelby is addressed in Section 13.2 withthe BIE model while the first problem of Eshelby is discussed in Section 13.3 on thesolution procedure with determination of Eshelby tensor. Section 13.4 describesthe newly developed boundary point method for particle reinforced inhomogeneousmaterials over a representative volume element. The numerical results are presentedin Section 13.5 for particle reinforced inhomogeneous materials over a represen-tative volume element. The influence of scatted inhomogeneities with a variety ofproperties and shapes and orientations on the overall properties of composites arestudied and compared with the results from other approaches. The results confirmthe validity and effectiveness of the proposed computational modal and the solutionprocedure. 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 assumedto be perfectly bonded together so that displacement continuity and traction equilib-rium remain along their interface. All materials considered are isotropic. Let the do-main 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 prob-lem can be described by the basic BIE as follows:

C (p)ui (p) =∫Γ

τ j (q)u∗i j (p,q)dΓ(q)−∫Γ

u j (q)τ∗i j (p,q)dΓ(q)+∑ II (13.1)

where u∗i j is the Kelvin solution, or the displacement in j direction at the field pointq under a unit force acting in i direction at the source point p. τ∗i j is the correspond-ing traction kernel and C is a boundary constant depending on the position of thesource point and the boundary geometries at the source point [21]. The last termin Eq. (13.1) represents the effect of all inhomogeneities summed by the boundaryintegrals along the interfaces, namely the inhomogeneity integral:

II =∫ΓI

τ j (q)u∗i j (p,q)dΓ(q)−∫ΓI

u j (q)τ∗i j (p,q)dΓ(q) (13.2)

242 H. Ma et al.

It is noted that each inhomogeneity integral II describes locally an exterior prob-lem. For the completeness of the problem, there should be complement equationscorresponding to each of the inhomogeneity integral (13.2) as follows:

0 =∫ΓI

τ Ij (q)uI

i j (p,q)dΓ(q)−∫ΓI

uIj (q)τ I

i j (p,q)dΓ(q), p ∈ Ω (13.3)

where superscript I denotes the quantity is associated with inhomogeneity I andΓI represents the boundary of the domain ΩI . In contrast to Eq. (13.2), each of thecomplement 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 boththe outer boundary Γ and the interfaces ΓI are included. To reduce the number of un-knowns 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 follow-ing relations on displacement continuity and traction equilibrium:

u j = uIj, τ j = −τ I

j , τ∗i j = −τ Ii j (13.4)

Using the interface relations (13.4) and Eq. (13.3), the exterior problem of inho-mogeneity integral (13.2) can be transformed into an equivalent interior problem asfollows:

II = −∫ΓI

τ Ij (q)u∗i j (p,q)dΓ(q)+

∫ΓI

τ Ij (q)uI

i j (p,q)dΓ(q) (13.5)

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

Cui +∫Γ

u jτ∗i jdΓ(q) =∫Γ

τ ju∗i jdΓ =

∫Ω

ε jkσ∗i jkdΩ =

∫Ω

σ jkε∗i jkdΩ (13.6)

where σ∗i jk and ε∗i jk are, respectively, the corresponding stress and strain kernels to

the Kelvin solution. The integral relations can be viewed as an integral form of thereciprocity theorem in elasticity. By invoking the integral relation (13.6), each of theinhomogeneity integral (13.5) can be rewritten as:

II = −∫ΩI

εCjk (q)σ∗

i jk (p,q)dΩ+∫ΩI

ε Ijk (q)σ I

i jk (p,q)dΩ(q) (13.7)

where εCjk 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 I

jk represents the strain of inhomogeneity under the same interface tractions.


By introducing the concept of eigenstrain ε0jk = ε I

jk − εCjk and noticing that the ker-

nel relation σ∗i jk = σ I

i jk exists, the inhomogeneity integral II in Eq. (13.7) can bewritten as

II =∫ΩI

ε0jk (q)σ∗

i jk (p,q)dΩ(q) (13.8)

Therefore Eq. (13.1) becomes

C (p)ui (p) =∫Γ

τ j (q)u∗i j (p,q)dΓ(q)

−∫Γ

u j (q)τ∗i j (p,q)dΓ(q)+∑∫ΩI

ε0jk (q)σ∗

i jk (p,q)dΩ(q) (13.9)

It can be seen that Eq. (13.1) together with the complement Eq. (13.3) have beenreduced 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 inte-grals describes an equivalent inclusion problem made of the identical matrix ma-terial containing eigenstrains. This is, in fact, a equivalent inclusion method whichwas initiated by Eshelby [1, 2] and developed by Mura [3] and many others. Theeigenstrains are defined as the stress-free strains of the equivalent inclusion, whichwill cause the same traction state at interfaces and the stress state in the matrix asthe inhomogeneity. If ε0

jk = 0, Eq. (13.9) describes an elastic state of homogeneousproblem. Therefore, it can be seen that the key step for the solution of Eq. (13.9) isto determine the eigenstrains in each of the inhomogeneities. This issue is discussedin Section 13.3.

The stress equation, which is indispensable in the analysis, can be derived di-rectly from Eq. (13.9) as:

C (p)σi j (p) =∫Γ

τk (q)u∗i jk (p,q)dΓ(q)−∫Γ

uk (q)τ∗i jk (p,q)dΓ(q)

+∑∫ΩI

ε0kl (q)σ∗

i jkl (p,q)dΩ(q)+ ε0kl (p)O∗

i jkl (13.10)

whereO∗

i jkl(p,q) = lim

Ωε→0

∫∆Γε

xlτ∗i 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 isdesirable to use the traction equation, which can be derived with the Cauchy relationτi = σi jn j directly from the stress equation (13.10) by putting p ∈ Γ:

244 H. Ma et al.

Cτi (p) = ni (p)∫Γ

τk (q)u∗ijk (p,q)dΓ(q)−ni (p)∫Γ

uk (q)τ∗ijk (p,q)dΓ(q)

+ ni (p)∑∫ΩI

ε0kl (q)σ∗

ijkl (p,q)dΩ(q)+ ni (p)ε0kl (p)O∗

ijkl (13.12)

where n j is the unit outward normal at the point p. The traction Eq. (13.12) is a hy-persingular BIE. The domain integrals in (13.9), (13.10) and (13.12) can be evalu-ated by using the corresponding boundary type integrals [22, 23] which is discussedin Sections 13.3 and 13.4.

It should be pointed out that the proposed matrix-inclusion algorithm is mainlybased on the eigenstrain formulation of BIE. With this formulation, the system un-knowns are defined on the outer boundary Γ only, which significantly reduces thenumber of system unknowns. This is the major advantage over the conventionalFEM and BEM.

13.3 Solution Procedures

In this section, outline of the solution procedure and related issues to the proposedeigenstrain formulation of BIE is described. While the numerical solution featuressuch as treatment of domain integrals and application of boundary point method(BPM) are discussed in Section 13.4. As mentioned in the previous section, deter-mination of the eigenstrain in each of the inhomogeneities is the crucial step for thesolution of Eq. (13.9) or (13.12). The eigenstrain in each inhomogeneity dependson applied stresses or strains, geometries as well as material constants of inhomo-geneity and matrix. Following the idea of Eshelby [1, 2], the eigenstrains, or thestress-free strains, in an inclusion with the material identical to matrix is related tothe constrained strains εC

ij by a tensor Sijkl , namely the Eshelby tensor as follows:

εCij = Sijklε0

kl (13.13)

The Eshelby tensor is only geometry dependent and generally in the form ofintegrals. For some simple geometry, the components of Sijkl can be found in theliterature [4]. For an inhomogeneity under applied strains εij to be replaced byan equivalent inclusion without altering its stress state, the following relation canbe given:

µI

(εC

ij + εij)+

νI

1−2νIδij(εC

kk + εkk)

= µ(

εCij + εij − ε0

ij

)+

ν1−2ν

δij(εC

kk + εkk − ε0kk

)(13.14)


where µ and ν are the shear modulus and Poisson’s ratio, respectively. The sub-script I denotes the variable is associated with the inhomogeneity. By defining theratio of modulus β = µI/µ and supposing νI = ν and eliminating εC

ij by using

Eqs. (13.13) and (13.14), the eigenstrains ε0ij can be obtained from the applied strains

εij by the following equations:

ε0ij =

1−β1−2(1−β)Sijij

εij

(i = j) (i = j) (13.15a)

ν1−2ν

(1−β)Skkmnε0mn +(1−β)Sijmnε0

mn +ν

1−2νε0

kk + ε0ij

= −(1−β)(

ν1−2ν

εkk + εij

)(i = j) (13.15b)

Obviously, the applied strains or the applied stresses over each inhomogeneitywill be disturbed by other inhomogeneities, especially those close to the concernedinhomogeneity. As a result, the applied strains and then the induced eigenstrainsneed to be evaluated in an iterative way. After discretization and incorporatedwith the boundary conditions, either Eq. (13.9) or (13.12) can be written in matrixform as:

Ax = b+ Bε (13.16)

where A is the system matrix, B the coefficient matrix related to the kernels in thedomain integrals in Eq. (13.9) or (13.12), b the loading vector related to the knownquantities with the outer boundary and the corresponding kernels, x the unknownsdefined on the outer boundary. ε is the eigenstrains of all inhomogeneities to beevaluated 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 onlyonce. It begins with assigning the vector ε with an initial value. Then the appliedstrains 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 thefollowing 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 twoconsecutive iterations. The convergence criterion in the present study is chosen asfollows:

Eεmax ≤ 10−3 (13.19)

246 H. Ma et al.

where E is Young’s modulus of the matrix material. If the criterion (13.19) does notmeet, 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 eigen-strains. It should be addressed here that in the use of Eq. (13.10) the domain integralfor the current inhomogeneity must be excluded from the computation because thestress states at the due place are the disturbance of all the other inhomogeneitiesin 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 orthe 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 computedwithout inhomogeneity.

(c) Compute the unknown vector x using the iterative formulae (13.17) with thecurrent 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 viaHooke’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 inhomo-geneities is the crucial step where the Eshelby tensor plays an essential role. Al-though some explicit expressions exist in the literature for the simple cases, ingeneral the Eshelby tensor can always be obtained conveniently via the numer-ical means. Consider a single equivalent inclusion with eigenstrains in an infi-nite medium under stress-free condition, the first problem of Eshelby, Eqs. (13.9)and (13.10) are, respectively, reduced to the following form:

ui (p) =∫ΩI

ε0jk (q)σ∗

ijk (p,q)dΩ(q) (13.20)

σij (p) =∫ΩI

ε0kl (q)σ∗

ijkl (p,q)dΩ(q)+ ε0kl (p)O∗

ijkl (13.21)

Assuming the case of uniform distribution of all the components of eigenstrainsin the equivalent inclusion, both Eqs. (13.20) and (13.21) can be transformed intoboundary-type integrals as [22, 23]


ui (p) = ε0jk

∫ΓI

xkτ∗ij (p,q)dΓ(q) (13.22)

σij (p) = ε0kl

∫ΓI

xlτ∗ijk (p,q)dΓ(q) (13.23)

Equations (13.20) and (13.21) describe, respectively, the displacement and thestress fields, in matrix (when p ∈ Ω) and in equivalent inclusion (when p ∈ ΩI). Itshould be mentioned that the displacement fields are continuous when the sourcepoint p go across the interface ΓI , while the value of stresses have a jump whenacross the interface because the kernels xkτ∗ij are continuous and xlτ∗ijk are discontin-uous when across the interface.

Noting the definition of eigenstrains ε0jk = ε I

jk −εCjk in the derivation of Eq. (13.9)

and comparing with that of Eshelby’s, one can see clearly that εCjk corresponds to

the constrained strains in the equivalent inclusion in the sense of Eshelby. This holdstrue 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 constrainedstresses in the equivalent inclusion and the boundary-type integral (13.23) becomesregular, which can be evaluated numerically. Combining Eqs. (13.13) and (13.23)with Hooke’s law εC

ij = Cijklσkl where Cijkl represents the elastic compliance tensorof the matrix, the Eshelby tensor can then be expressed as follows:

Sijkl = Cijmn

∫ΓI

xkτ∗mnl (p,q)dΓ (13.24)

It can be seen from Eq. (13.24) that the Eshelby tensor is geometry dependentonly. It can also be shown that Eq. (13.24) is equivalent to that of Eshelby’s ex-pression for Sijkl but is more suitable for numerical evaluations. In the case of non-uniform distribution of eigenstrains, the domain integrals (13.20) and (13.21) canalso be transformed into the corresponding boundary-type integrals without any dif-ficulty 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 bound-ary point method (BPM). The BPM is a newly developed boundary-type mesh-less method [20] based on the indirect or direct formulations of conventional andhypersingular BIE but has some features of the method of fundamental solution(MFS) [24–26]. In the BPM, the boundary of the solution domain is discretizedonly by boundary nodes, each node having a territory or support where the fieldvariables are defined. By making use of the properties of fundamental solutions, thecoefficients of the system matrix are computed according to the distance between

248 H. Ma et al.

the source and the field points. In cases when the distance between the two points isnot very small in comparison with the dimension in any one coordinate direction, theintegrals of kernel functions are evaluated by one-point computing. This is similarto that carried out in the MFS, which constitute the most off-diagonal terms of thesystem matrix. Conversely, in cases when the distance is very small the integral ofkernel functions is evaluated by Gauss quadrature over territories. If the two pointscoincide, the integrals are treated by well-established techniques in the BEM, whichconstitute the principal diagonal terms of the system matrix.

Noting that the combined use of adjacent nodes can describe the local featuresof a boundary such as position, curvature and direction, the ‘moving elements’are introduced by organizing the relevant adjacent nodes tentatively, over whichthe treatment of singularity and Gauss quadrature can be carried out for evaluatingthe integrals in the latter two cases abovementioned, namely coincidence or smalldistance between the source and field points.

It should be mentioned that as the source points are placed on the real boundary inthe BPM, the main difficulties of coincidence of the source and field points and theinconvenience of using a fictitious boundary encountered in the MFS are removed.Compared to the BEM, the BPM can improve efficiency by both reducing the burdenof data preparation and taking advantage of the one-point computing for most of theintegrals of kernel functions while maintaining reasonable accuracy. It is suggestedthat the evaluation of kernel functions by Gauss quadrature over boundary elementsin the BEM would be necessary only for cases of coincidence and small distancebetween 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

σ∗i jkdΩ =

∫ΓI

xkτ∗i jdΓ (13.25)

∫ΩI

σ∗i jkldΩ+ O∗

i jkl =∫ΓI

xlτ∗i jkdΓ (13.26)

then all the techniques in the BPM become applicable at the interfaces ΓI . Thatis, the interfaces are discretized by interface nodes, each node having a territoryand the integrals in Eqs. (13.25) and (13.26) are evaluated by boundary-type inte-grals in closed loops while the one-point computing can be used in most of the caseswhen the two-point distances are not too small. However, there are two special as-pects to be addressed here for the proposed computational model. The first is thatthe purpose of evaluating domain integrals is for computing the influences of eigen-strains in inhomogeneities on the outer boundary unknowns but not for the systemmatrix itself. The second is that the use of boundary-type integrals in closed loopsis necessary only for a few limited cases when the distances between the sourceand the field points are relatively small. In other words, the domain integrals can beevaluated simply by the one-point computing as follows with reasonable accuracyif the two-point distance is not too small:

13 Computational Modal and Solution Procedure for Inhomogeneous Materials 249∫ΩI

σ∗i jkdΩ ≈ σ∗

i jkAI (13.27)

∫ΩI

σ∗i jkldΩ ≈ σ∗

i jklAI (13.28)

where AI is the area of ΩI and O∗i jkl = 0 in Eq. (13.28) because p /∈ ΩI in the range

that the one-point computing is applicable. It is well known that the kernel functionsof integrals in the BIE are two-point functions and behave singularity to various ex-tents when the two-point distance tends to zero. However, there are properties of thekernel 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 orderof O(r−s), where s is an integer, then the decaying order of the integral values of thiskernel 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 atthe center point of the integration domain since the variation of the kernel functionsin 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 usingEqs. (13.25) and (13.26) is displayed in Fig. 13.1, where AI is the area of a circu-lar zone and r is the distance from the source point to the center of the circularzone. It can be seen from Fig. 13.1 that the difference of integral results betweenthe two computing methods decreases along with an increase in the dimensionless

distance r/A1/2I . In the present study, r/A1/2

I = 5 serves as the criterion to changethe computing method between the one-point computing and the quadrature alongclosed loops.

101

0.0

0.1

0.2

By quadratureσ*

111

σ*122

σ*222

σ*1111

σ*2222

Inte

gral

val

ues

Dimensionless distance, r /AI1/2

Fig. 13.1 Comparison of integral results between one point computing and boundary type quadra-ture along a closed loop

250 H. Ma et al.

13.5 Numerical Examples

The purpose of the first numerical example is to assess the validity of the algorithmover a simple square domain with an elliptical inhomogeneity under the uniformtension 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 lengthof 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 wellas BPM. The solution procedure is usually known as ‘domain decomposition’ asthere is two (or more) areas in the solution domain with distinct material proper-ties to be decomposed. The two areas are described by the two boundary integralequations (13.1) and (13.3) which are coupled at the interface. As displacementsand tractions are unknown at the interface, the degrees of freedom of the problem isas large as 2(N + NP) and the size of the system matrix is 360×360 even for sucha simple problem.

The problem is then solved with the BPM combined with the eigenstrain for-mulation (13.9). In the present study, the Eshelby tensor required for evaluating theeigenstrains in elliptical inhomogeneities is computed numerically using Eq. (13.24)beforehand and some results are shown in Fig. 13.3. Because the unknowns are de-fined only on the outer boundary in the proposed computing model, the degrees offreedom of the problem is 2N only and the size of the system matrix is reduced to120×120, 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 and10 are listed in Fig. 13.4 and comparison is made those obtained using the domaindecomposition. It is evident from Fig. 13.4 that the results obtained from the presentmodel are in good agreement with those from the domain decomposition.

The second numerical example is carried out for the computation of overall prop-erties of particle reinforced materials over a representative volume element (RVE)shown in Fig. 13.5 with NI = 100 inhomogeneities scattered in the RVE with load-ings in either single tension mode or in pure shearing mode. The RVE is discretized

x1

x2

2a

2b

Fig. 13.2 A square domain with inhomogeneity under uniform tension in x2 direction


1010.1

0.0

0.2

0.4

0.6

0.8

1.0 S2222S1212

Esh

elby

tens

or

Aspect ratio, b /a

S1111 S1122 S2211

Fig. 13.3 Eshelby tensor for elliptical inhomogeneities with various aspect ratios

1.0 1.2 1.4 1.6 1.8 2.0−1

0

1

2

3

4

5

Dim

ensi

onle

ss s

tres

ses

x1/a

By domain decompositionσ11(b = 0.1)

σ22(b = 0.1)

σ11(b = 10)

σ22(b = 10)

b /a=0.5

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 of200×200 using the proposed model of the eigenstrain formulation. It is noted thatdiscretization is necessary only for those inhomogeneities when the distance be-tween source and field points are small so that the domain integrals need to beevaluated by the quadrature of boundary type integrations over closed loops. Other-wise, the domain integrals are evaluated by the one-point computing so that the dis-cretization of inhomogeneities is unnecessary in most cases. In our analysis, circularinhomogeneity is discretized by NP = 20 and elliptical inhomogeneity is modeledby NP = 40. Obviously, the size of system equation using the proposed model is0.25% of that using the traditional domain decomposition method if circular inho-mogeneity is considered.

252 H. Ma et al.

Fig. 13.5 The representative volume element for the particle reinforced composite materials

10−3 10−2 10−1 100 101 102 103

0.2

0.4

0.6

0.8

1.0

Ratio of modulus, b

Ove

rall

prop

ertie

s Ec/E (by tension)

uc (by tension)mc/E (by tension)

mc/E (by shearing)b /a=1

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 theeigenstrain formulation (13.12) of the hypersingular BIE. The eigenstrain is evalu-ated using the iterative scheme described in Section 13.3 in which the convergenttolerance is defined in Eq (13.19). Figure 13.6 shows the computed overall proper-ties of the RVE with circular inhomogeneities as a function of the ratio of modulusβ , where EC, νC and µC stand for Young’s modulus, Poisson’s ratio and shear mod-ulus of the composite material. The area of each inhomogeneity and the volumefraction (=7.07%) are kept constant in the calculation. It is seen from Fig. 13.6 thatwhen the range of ratio of modulus is around unit the computed modulus variescomparatively large. When the ratio of modulus is very small (corresponding toempty holes) or very large (corresponding to rigid inhomogeneities) the computedmodulus varies stagnantly. It is noticed that there is some difference in the computedshear modulus between the single tension and the pure shearing loading modes.


0.1 0.2 0.3 0.4 0.5 0.60.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Ove

rall

prop

ertie

s

Particle radius, a

Ec/E

mc/E

b /a=1b=10

uc

Fig. 13.7 The overall properties of the RVE as a function of the particle radius

2 3 4 56 7 8 9 100.0

0.4

0.8

1.2

1.6

mL/E

mT/E

m /E(By shearing)

EL/E

ET/E

Ove

rall

prop

ertie

s

Aspect ratio, b /a

uL

uT

b=10 pab=0.377

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 modulusEC and shear modulus µC increase monotonically with the particle radius when thenumber of particles and the ratio of modulus are kept constant. Figure 13.8 showsthe overall properties of the RVE as a function of the aspect ratio of inhomogeneitieswith the constant ratio of modulus and the constant area πab of the particles whilethe axes of the elliptic zone are aligned in parallel with the coordinate axes. It showsfrom Fig. 13.8 the anisotropic behaviors of the overall properties with the variationof the aligned inhomogeneities where the subscripts L and T denote the propertiesin longitudinal and transverse directions, respectively. It can be seen that there isno limitations such as the shape, the material, etc., to solve the inhomogeneity prob-lems with the proposed model. In the computation, the convergence can be achieved

254 H. Ma et al.

in general two to five times. It is found that the iterations will increase with the pro-portion of the inhomogeneity in the matrix, that is, the quantity, the size as well asthe shape of the particles. However, the maximum number of iteration is eleven. Itseems that the improvement of iteration remains a topic to be studied.

13.6 Conclusions

The novel computational model and solution procedure are proposed for inhomo-geneous materials in the present study. The method is based on the eigenstrain for-mulation and BPM. As the unknowns of the final system equation are defined onthe outer boundary of the solution domain only, the solution scale of the inhomo-geneity problem with the present model remains fairly small in comparison withthose with the traditional computation model using FEM and BEM. Both the modeland the solution procedure are developed intimately from the concept of equiva-lent inclusion of Eshelby with eigenstrains to be determined in an iterative way viathe Eshelby tensors, which can be readily obtained through either analytical or nu-merical approach. The convergence of the algorithm can be achieved within a fewiterations. The validity and the effectiveness of the proposed computational modeland 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 prob-lems. 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 Societyof 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 distri-bution of spheroidal inclusions: a geometrical approach to overall properties. Journal of theMechanics and Physics of Solids 52 (2004) 2309–2327.

5. Cohen I. Simple algebraic approximations for the effective elastic moduli of cubic arrays ofspheres. 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 materialswith 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 materialsreinforced 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: Nu-merical treatment of orientation. Computer Methods in Applied Mechanics and Engineering195 (2006) 1387–1406.


10. Kakavas PA, Kontoni DN. Numerical investigation of the stress field of particulate reinforcedpolymeric composites subjected to tension. International Journal for Numerical Methods inEngineering 65 (2006) 1145–1164.

11. Kanaun SK, Kochekseraii SB. A numerical method for the solution of thermo- and electro-static 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 inclu-sions and voids using a new integral equation technique. International Journal of Solids andStructures 38 (2001) 2789–2802.

13. Dong CY, Cheung YK, Lo SH. A regularized domain integral formulation for inclusion prob-lems of various shapes by equivalent inclusion method. Computer Methods in Applied Me-chanics and Engineering 191 (2002) 3411–3421.

14. Dong CY, Lee KY. Boundary element analysis of infinite anisotropic elastic medium contain-ing 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 variousshapes 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 methodfor the analysis of fiber-reinforced composites based on a rigid-inclusion model. ASME Jour-nal of Applied Mechanics 72 (2005) 115–128.

17. Ma H, Deng HL: Nondestructive determination of welding residual stresses by boundary ele-ment method. Advances in Engineering Software 29 (1998) 89–95.

18. Greengard LF, Rokhlin V. A fast algorithm for particle simulations. Journal of ComputationalPhysics 73 (1987) 325–348.

19. Nakasone Y, Nishiyama H, Nojiri T. Numerical equivalent inclusion method: a new computa-tional method for analyzing stress fields in and around inclusions of various shapes. MaterialsScience and Engineering A285 (2000) 229–238.

20. Ma H, Qin QH: Solving potential problems by a boundary-type meshless method – the bound-ary 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 applicationsin engineering. Berlin: Springer (1984).

22. Ma H, Kamiya N, Xu SQ: Complete polynomial expansion of domain variables at boundaryfor two-dimensional elasto-plastic problems. Engineering Analysis with Boundary Elements21 (1998) 271–275.

23. Ma H, Kamiya N: Boundary-type integral formulation of domain variables for three-dimensional 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 Anal-ysis with Boundary Elements 16 (1995) 205–213.

25. Fairweather G, Karageorghis A. The method of fundamental solutions for elliptic boundaryvalue problems. Advances in Computational Mathematics 9 (1998) 69–95.

26. Wang H, Qin QH, Kang YL. A meshless model for transient heat conduction in functionallygraded materials. Computational Mechanics 38 (2006) 51–60.

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