Advances in applied mechanics. Vol. 42

ADVANCES IN

APPLIED MECHANICS Edited by

Hassan Aref VIRGINIA TECH COLLEGE OF ENGINEERING BLACKSBURG, VA, USA

Erik van der Giessen UNIVERSITY OF GRONINGEN GRONINGEN, THE NETHERLANDS

VOLUME 42

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Preface

Applied mechanics is a classical subject by any standard but is nevertheless vibrant in new areas of science and technology. These days, nanotechnology is one of the emerging fields where mechanics is proving to be a powerful tool. In this and in many other subjects, multiscale modelling is a maturing paradigm, involving solid and fluid mechanics alike. This volume of the series Advances in Applied Mechanics contains three examples of modern directions in mechanics, where homogenization (or coarse graining) is a common notion. The first chapter, co-authored by Duan and Karihaloo from Cardiff University together with Wang from Peking University, deals with a careful consideration of continuum mechanics at very small-length scales. One of the nanoscale properties absent in classical continuum theories is surface stress, an effect becoming more and more important at smaller scales as the surface area increases compared to the volume. While surface stress is an atomistic phenomenon, the authors show that it is possible to extend continuum elasticity with surface elasticity. Subsequently, they demonstrate that these surface effects can have a beneficial influence on the overall stiffness of nano porous materials. The theme is continued in the second chapter where Sevostianov and Kachanov marry the homogenization of elasticity and that of thermal conductivity. While homogenization for each of them is a well established, this chapter details the connection between them, something that is far from trivial since the phenomena are governed by partial differential equations of different order. In particular, the authors derive explicit closed-form relations between effective elastic properties and effective conductive properties for anisotropic materials. The results are applied to various problems involving porous, cracked and/or fiber-reinforced materials. The final chapter, Coarse Graining in Elastoviscoplasticity, is a unique example of how solid and fluid mechanics can cross-fertilize. Here, Hütter and Tervoort from ETH-Zürich review the coarse-graining scheme GENERIC originally developed for fluids and apply it to elastic and viscoelastic deformations of solids. In particular, they show how continuum viscoplasticity can be derived from the motion of the underlying microscopic particles. The theory constitutes a systematic approach, based on techniques from statistical physics, to the averaging over intermediate-length scales as well as time scales. We hope that this volume of Advances in Applied Mechanics not only gives an account of some of the exciting new developments in mechanics but also demonstrates how these tools are applied and needed in modern science and technology. Hassan Aref Erik van der Giessen vii

Theory of Elasticity at the Nanoscale H. L. DUANa,b , J. WANGb and B. L. KARIHALOOa∗ a School of Engineering, Cardiff University, Queen’s Buildings, The Parade, Cardiff CF24 3AA, UK ∗ [email protected] b State Key Laboratory for Turbulence and Complex System and Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing, 100871, P. R. China

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2. Eshelby Formalism for Nano-inhomogeneities . . . . . . . . . . . . . . . . . . . . . 2.1. Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Eshelby Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 6 9

3. Application of Eshelby Formalism for Nano-inhomogeneities . . . . . . . . 3.1. Stress Concentration Factor of a Spherical or Circular Nanovoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Strain Fields in QDs With Multi-shell Structures and in Alloyed QDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Compatible Composition Profiles and Critical Sizes of Alloyed QDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4. Micromechanical Framework for Nano-inhomogeneities with Interface Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Application of Micromechanical Framework for Nano-inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Elastic Moduli of Solids with Spherical Nano-inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Generalized Levin’s Formula and Hill’s Connections . . . . . . . . . . . 5.3. Prediction and Tailoring of Elastic Moduli of Nanochannel-array Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 21 27 30 34 34 40 47

6. Scaling Laws for Properties of Nanostructured Materials . . . . . . . . . . . . 6.1. Elastic Properties with Surface Stress Effect . . . . . . . . . . . . . . . . . . 6.2. Melting Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 58 62

7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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ADVANCES IN APPLIED MECHANICS, VOL. 42 ISSN: 0065-2156 DOI: 10.1016/S0065-2156(08)00001-X

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Copyright © 2008 by Elsevier Inc. All rights reserved.

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H. L. Duan et al.

Abstract We have shown in a series of recent papers that the classical theory of elasticity can be extended to the nanoscale by supplementing the equations of elasticity for the bulk material with the generalized Young–Laplace equations of surface elasticity. This review article shows how this has been done in order to capture the often unusual mechanical and physical properties of nanostructured particulate and porous materials. It begins with a description of the generalized Young–Laplace equations. It then generalizes the classical Eshelby formalism for nano-inhomogeneities; the Eshelby tensor now depends on the size of the inhomogeneity and the location of the material point in it. Then the stress concentration factor of a spherical nanovoid is calculated, as well as the strain fields in quantum dots (QDs) with multi-shell structures and in alloyed QDs induced by the mismatch in the lattice constants of the atomic species. This is followed by a generalization of the micromechanical framework for determining the effective elastic properties and effective coefficients of thermal expansion of heterogeneous solids containing nano-inhomogeneities. It is shown, for example, that the elastic constants of nanochannel-array materials with a large surface area can be made to exceed those of the nonporous matrices through pore surface modification or coating. Finally, the scaling laws governing the properties of nanostructured materials are derived. The underlying cause of the size dependence of these properties at the nanoscale is the competition between surface and bulk energies. These laws provide a yardstick for checking the accuracy of experimentally measured or numerically computed properties of nanostructured materials over a broad size range and can thus help replace repeated and exhaustive testing by one or a few tests.

1. Introduction In any material system, the atoms at a free surface experience a different local environment than the atoms away from the surface because of reduced coordination. As a result, the energy associated with these atoms is generally different from that of the atoms in the bulk. Surface stress relates the variation of the excess free energy to the strain of the surface (e.g., Cammarata, 1997). The same situation arises at the interface between two dissimilar materials; the atoms near the interface experience a different local environment than the atoms farther away from the interface in either of the abutting materials. As these surface/interface stresses act on only a few layers of atoms near the surface/interface, their effect is not felt in the bulk of the material, and it is therefore neglected in the classical theory of elasticity. However, it can no longer be neglected in a nanostructured material whose characteristic length is in the nanometer range because the surface stresses displace atoms from the equilibrium positions which they normally occupy in bulk macroscopic assemblies (Streitz et al., 1994). This change in interatomic distance affects the elastic properties of nanoscale structures. For nanostructured (Gleiter, 2000) and nanochannel-array materials (Masuda and Fukuda, 1995; Martin and Siwy, 2004) with a large surface to the bulk ratio, the surface stress effect can

Theory of Elasticity at the Nanoscale

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be substantial. Thus, materials such as thin films, nanowires, and nanotubes may exhibit unusual properties not noticed at the macroscale. As small devices and nanostructures are all pervasive, and the elastic constants of materials are a fundamental physical property, it is important to understand and predict the size effect in elastic properties of materials at the nanoscale. The concept of surface/interface stress in solids was first introduced by Gibbs (1906) and has been steadily developed since (e.g., Shuttleworth, 1950; Herring, 1953; Orowan, 1970; Murr, 1975; Cahn, 1978; Cammarata, 1994). The surface/ interface stress can be defined in various ways. One definition is the surface excess of bulk stress (Ibach, 1997; Müller and Saúl, 2004). Another definition is given through the Shuttleworth equation (Shuttleworth, 1950). Moreover, Nix and Gao (1998) presented an atomistic interpretation of interface stress. Generally, two types of interface stress in solids have been discussed in the literature. One type is related to a deformation that maintains the interface coherency between the abutting solids (with equal tangential strain). In this kind of deformation, no atomic bonds break in the interface plane. Another type of interface stress is related to the deformation with different tangential strains in the two solids (Weissmüller and Cahn, 1997; Gurtin et al., 1998). In this case, atomic bonds may break in the interface plane, as slipping occurs across the interface. Generally, coherent interfaces are commonly present within materials, and interfaces often remain coherent under a wide range of conditions (Rottman, 1988). Many attempts have been made recently to investigate the elastic properties of nanostructured materials by atomistic simulations. For example, Zhou and Huang (2004) and Villain et al. (2004) have obtained the elastic constants of nanoplates of copper and tungsten; they did not explicitly use the concept of surface stress to explain the variation of the elastic constants with the thickness of the nanoplates, but, obviously, the surface stress effect is implicitly taken into account in some atomistic calculations, for example, ab initio models. Diao et al. (2004) studied the effect of free surfaces on the structure and elastic properties of gold nanowires by atomistic simulations. Although the atomistic simulation is a good way to calculate the elastic constants of nanostructured materials, it is only applicable to homogeneous nanostructured materials (e.g., nanoplates, nanobeams, nanowires, etc.) with limited number of atoms. Moreover, it is difficult to obtain the elastic properties of the heterogeneous nanostructured materials using atomistic simulations. For these and other reasons, it is prudent to seek a more practical approach. One such approach would be to extend the classical theory of elasticity down to the nanoscale by including in it the hitherto neglected surface/interface effect. For this it is necessary first to cast the latter within the framework of continuum elasticity. The mathematical framework incorporating surface stress into continuum mechanics was established by Gurtin and Murdoch (1975, 1978), Murdoch (1976),

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and Gurtin et al. (1998). Miller and Shenoy (2000) compared the results obtained by the continuum model, which explicitly uses surface stress and surface elastic constants, with those obtained by the atomistic simulations for nanobeams and nanowires and found that the results were almost indistinguishable. In essence, the continuum surface stress model assumes that a nanostructure is made of the bulk and surfaces (Shenoy, 2005) with the surface moduli of the nanostructure being different from those of the bulk. Steigmann and Ogden (1999) have generalized the Gurtin–Murdoch theory (Gurtin and Murdoch, 1975) to account for the effect of flexural resistance of elastic films attached to the bounding surfaces of solids. The continuum theory of Gurtin and Murdoch (1975) was further developed by many researchers (Bottomley and Ogino, 2001; Cuenot et al., 2004; Ren and Zhao, 2004; Shenoy, 2005; Duan et al., 2005a,b; Dingreville et al., 2005; Jing et al., 2006) to analyze the elastic properties of homogeneous and heterogeneous nanostructured materials. For example, Cuenot et al. (2004) and Jing et al. (2006) analyzed the elastic properties of silver nanowires with outer diameters ranging from 20 to 140 nm using this theory and compared them with contact atomic force microscopy measurements. They found that the apparent Young modulus of the nanowires decreased with an increase in the diameter. This size dependence of the apparent Young modulus was attributed to the surface stress effect, which includes the effects of the surface stress, the oxidation layer, and the surface roughness. Gao et al. (2006) developed a finite element method taking into account the surface stress effect. The finite element method with surface stress effect can be used to analyze the elastic properties of nanostructured materials with complicated structures. Fang and Liu (2006) studied the size-dependent interaction between an edge dislocation and a circular nano-inhomogeneity. He and Lim (2006) derived the surface Green function for incompressible, elastically isotropic half-space coupled with surface stress by using the double Fourier transform technique. Mi and Kouris (2006) investigated the effect of surface/interface elasticity on the response of nanoparticles embedded in a semi-infinite elastic medium. Huang and Yu (2006) presented a relation between the surface stress and the electric field, that is, surface piezoelectricity, and showed that surface piezoelectricity plays an important role in the electromechanical behavior of piezoelectric nanostructures. Wang and Wang (2006) analyzed the deformation around a nanosized elliptical hole with surface stress effect. Wang et al. (2006) studied the effect of the surface stress on the diffraction of plane waves and the dynamic stress concentration around a nanosized hole. The elegant Eshelby formalism (Eshelby, 1957, 1959) for inclusions/ inhomogeneities is fundamental to the solution of many problems in materials science, solid state physics, and mechanics of heterogeneous materials. Not surprisingly, attempts have been made to extend the Eshelby formalism for

Theory of Elasticity at the Nanoscale

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nano-inclusions/inhomogeneities. For example, Sharma et al. (2003) and Sharma and Ganti (2004) analyzed the elastic field of a spherical nano-inhomogeneity with the interface stress effect subjected to a combination of a uniform dilatational eigenstrain and a hydrostatic remote loading in an infinite medium. Duan et al. (2005a) extended the Eshelby formalism to nano-inhomogeneities. They gave the strain field in a nano-inhomogeneity subjected to an arbitrary uniform eigenstrain in an infinite elastic medium, the elastic field in a nano-inhomogeneity in an infinite medium subjected to an arbitrary uniform remote stress, and the strain energy in solids containing nano-inhomogeneities. Based on the generalized Eshelby formalism, attempts have been made to generalize the micromechanical framework for the prediction of the effective properties (e.g., effective elastic moduli and effective coefficients of thermal expansion (CTE)) of heterogeneous materials containing nano-inhomogeneities. For example, Duan et al. (2005b) gave the general micromechanical framework for the prediction of the effective elastic moduli of the heterogeneous materials containing spherical nano-inhomogeneities. Duan et al. (2006c) predicted the effective elastic moduli of nanochannel-array materials and showed that such materials can be made stiffer than the nonporous parent materials by judicious manipulation of the pore surface elasticity or by coating the pore surface with a nanolayer of a stiffer material. Subsequently, Chen et al. (2007a,b), Chen and Dvorak (2006), He (2006), and Duan and Karihaloo (2007) predicted the effective elastic moduli and effective CTE of the heterogeneous materials containing nano-inhomogeneities and gave some intrinsic relations governing these effective properties, for example, the generalized Levin’s formula and generalized Hill’s connections. Recently, Zhang and Wang (2007) extended a conventional micromechanical method for nonlinear composites (Qiu and Weng, 1992; Hu, 1996) to study the effect of the surface stress on the yield strength of nanoporous materials. As mentioned above, the continuum mechanics framework with surface stress effect has been used extensively to characterize the elastic properties of nanostructured materials. It has been shown that the theoretical predictions are in very good agreement with experiment. The continuum mechanics framework with surface stress effect is thus a very practical approach to characterize the nanostructured materials, as the molecular dynamic and ab initio approaches are limited to very small material samples. Therefore, it seems timely to review the advances made possible by the continuum mechanics framework that takes into account the surface stress effect. This review paper presents a unified framework for several topics at the interface of continuum mechanics and materials science. It provides the methods and fundamental solutions necessary to analyze the elastic and mechanical properties of nanostructured materials in which the surface/interface stress plays an important role. We note in passing that Fried and Gurtin (2004) also

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published a review paper on the interaction of continuum mechanics and materials science. However, they concentrated primarily on the dynamic processes of interface evolution due to the atomic transport at the grain boundaries. This review article is organized as follows. In Section 2, it begins with the description of the generalized Young–Laplace equations for surface elasticity. Then the Eshelby formalism for nano-inhomogeneities is given, which includes the Eshelby tensor the stress concentration tensor, and the Eshelby formula for strain energy. Next, the applications of Eshelby formalism for nano-inhomogeneity, that is, the stress concentration factor of a spherical nanovoid, and the strain fields in quantum dots (QDs) with multi-shell structures and in alloyed QDs induced by the mismatch in the lattice constants of the atomic species, are given in Section 3. This is followed in Section 4 by a generalization of the micromechanical framework for determining the effective elastic properties of heterogeneous solids containing nano-inhomogeneities. In Section 5, applications of micromechanical framework for nano-inhomogeneities are given. It is shown, for example, that the elastic constants of nanochannel-array materials with a large surface area can be made to exceed those of the nonporous matrices through pore surface modification or coating. Finally, the scaling laws governing the properties of nanostructured materials are derived in Section 6. These laws provide a yardstick for checking the accuracy of experimentally measured or numerically computed properties of nanostructured materials over a broad size range and can thus help replace repeated and exhaustive testing by one or a few tests. 2. Eshelby Formalism for Nano-inhomogeneities

2.1. Basic Equations The basic equations for solving boundary value problems of thermoelasticity consist of the following conventional equilibrium equations, constitutive equations, and strain-displacement relations for the constituent materials, that is, the matrix and the inhomogeneity embedded in it:  1 (2.1) ∇ · σ k = 0 , σ k = Ck : εk − T Dk , εk = ∇ ⊗ uk + uk ⊗ ∇ 2 where σ k , uk , and εk denote the stresses, displacements, and strains in k (k = I, m). In subsequent sections, I and m will denote an inhomogeneity and matrix, respectively. Ck are the elastic moduli of I and m . T is the temperature difference, Dk is the stress-temperature tensor in I and m . For the isotropic case, Dk = d k I(2) , where I(2) is the second-order identity tensor in threedimensional space. d k = αk (3λk + 2μk ), αk , λk , and μk are the CTE and elastic

Theory of Elasticity at the Nanoscale

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Lamé moduli of I and m , respectively. In what follows, as we will study both spherical inhomogeneities and cylindrical fibers, we will use the super- and subscript k = p for the former, k = f for the latter, with k = m denoting the matrix. At the nanoscale, Eq. (2.1) have to be supplemented by the surface/interface elasticity equations to complete the mathematical description of the problem. As mentioned in the Section 1, a surface/interface stress can be defined in various ways, for example, the surface/interface excess of bulk stress (Müller and Saúl, 2004). To derive the surface/interface equations of elasticity, we consider a system of two abutting solids I and m with different elastic properties. The equilibrium of a general curved interface  with the unit normal vector n between the two solids I and m requires that (Gurtin and Murdoch, 1975; Povstenko, 1993) [σ] · n = −∇S · τ,

(2.2)

where [σ] = σ I − σ m , σ I , and σ m are the volume stress tensors in I and m , respectively, ∇S · τ denotes the interface divergence of τ at  (Gurtin and Murdoch, 1975). Eq. (2.2) is the generalized Young–Laplace equation for solids. It can be derived in various ways, for example, by the principle of virtual work (cf., the second Chapter of Duan (2005)). Huang and Wang (2006) derived the generalized Young–Laplace equation from the stationary condition of an energy functional for hyperelastic media. For a curved interface  with two orthogonal unit base vectors e1 and e2 in the tangent plane and a unit vector n perpendicular to the interface, ∇S · τ can be expressed as follows (Duan et al., 2005a):     e1 ∂(h2 τ11 ) ∂(h1 τ21 ) ∂h1 τ11 τ22 ∂h2 ∇S · τ = − n+ − + + τ12 − τ22 R1 R2 h1 h2 ∂α1 ∂α2 ∂α2 ∂α1   e2 ∂h1 ∂(h2 τ12 ) ∂h2 ∂(h1 τ22 ) + − . (2.3) τ11 + + τ21 + h1 h2 ∂α2 ∂α1 ∂α1 ∂α2 Here, α1 and α2 are two parameters defining the interface such that α1 = constant and α2 = constant give two sets of mutually orthogonal curves on , and h1 and h2 are the corresponding metric coefficients, R1 and R2 are the radii of the principal curvatures (see, e.g., Appendix A of Benveniste (2006) or the second Chapter of Duan (2005)). τ11 , τ22 , and τ12 are the components of the interface stress tensor τ. As can be seen from Eq. (2.3), the first term on the right-hand side is the classical Young–Laplace equation; the remaining terms signify that a nonuniform distribution of the interface stress or a uniform interface stress on a surface with varying curvature needs to be balanced by bulk shear stresses in the abutting materials. In the case of finite deformation, both the Lagrangian and Eulerian descriptions of the generalized Young–Laplace equations can also

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be obtained from the stationary condition of an energy functional (Huang and Wang, 2006). In addition to the generalized Young–Laplace Eq. (2.2), we also need the interface constitutive equations to solve the boundary value problems with the interface stress effect. Gurtin and Murdoch (1975) have given a linear constitutive equation for surface elasticity based on the principles of constitutive theory. As the creation of a surface or interface will usually induce a residual stress field in a body even in the absence of external loading, and this residual field will change when the body deforms, the determination of the mechanical response of the body in the presence of the surface/interface effect requires the solution of what is essentially a finite deformation problem (Huang and Wang, 2006; Huang and Sun, 2007). Recently, Huang and Wang (2006) derived the constitutive relations for hyperelastic solids with the surface/interface energy effect at finite deformation. Later, Huang and Sun (2007) conducted an infinitesimal analysis based upon this theoretical framework and derived the surface/interface elastic constants in terms of the surface/interface free energy. For a linearly elastic and isotropic surface, the Cauchy stress of the surface is (Gurtin and Murdoch, 1975; Huang and Wang, 2006) τ = γ0∗ 1 + λs (trεs ) 1 + 2μs εs ,

(2.4)

where γ0∗ 1 is the residual surface stress in the reference configuration, λs and μs are the Lamé constants of the surface, εs is the surface strain tensor, and 1 is the second-order unit tensor in a two-dimensional space. From the above equation, a linearized constitutive relation for the surface Piola–Kirchhoff stress of the first kind, which is used in the Lagrangian description, can be written as (Gurtin and Murdoch, 1975; Huang and Wang, 2006)   Ss = γ0∗ 1 + λs + γ0∗ (trεs ) 1 + 2 μs − γ0∗ εs + γ0∗ u∇0s ,

(2.5)

where u∇0s is the displacement gradient of the surface in the reference configuration. It is clear that the Cauchy and the Piola–Kirchhoff stress tensors of the first kind are not the same even under the assumption of infinitesimal deformation. Note that the parameter γ0∗ includes a liquid-like surface tension term. The currently available values of γ0∗ for solids are usually much smaller than those of the surface elastic constants λs and μs (Miller and Shenoy, 2000; Dingreville and Qu, 2007). Therefore, without loss of the essence of surface elasticity, the effect of γ0∗ can be omitted to facilitate theoretical analysis. The omission of this term renders the Cauchy and the Piola–Kirchhoff surface stress tensors of the first kind identical, just as the bulk stress tensors, under the assumption of infinitesimal deformation.

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Thus, when augmented with the thermoelastic coupling term, the thermoelastic constitutive relation of a linearly elastic and isotropic surface/interface can be expressed as (Murdoch, 1976, 2005) τ = 2μs εs + λs (trεs )1 − T D0 ,

(2.6)

where D0 is the stress-temperature tensor of the interface. For the isotropic interface D0 = d0 1, we assume that d0 = αs κs , where αs is the coefficient of thermal expansion (CTE) of the interface and κs = 2(μs + λs ). Note that the elastic moduli of the interface have different units from those of the bulk, whereas the CTEs of the interface and bulk material have the same units. For a coherent interface (Cahn, 1978; Rottman, 1988), the interface strain εs is equal to the tangential strain in the abutting bulk materials. In the rest of this article, the description of surface/interface stresses is based upon the constitutive relation (2.6). Eqs. (2.2) and (2.6) constitute the basic equations to analyze the surface/interface stress effect on the elastic properties of materials, and these equations describe the so-called interface stress model (ISM, Duan et al., 2005a,b).

2.2. Eshelby Formalism The Eshelby formalism (Eshelby, 1956, 1957, 1959) is fundamental to the solution of many problems in materials science, solid state physics, and mechanics of composites. In the terminology of Eshelby (1957) and Mura (1987), an inclusion denotes a subdomain in a homogeneous solid subjected to an eigenstrain; an inhomogeneity is a region with elastic properties distinct from those of the matrix. When an eigenstrain is given to an inhomogeneity, it is called an inhomogeneous inclusion. The Eshelby formalism refers to the following problems (Eshelby, 1956, 1957, 1959): the strain field in an inclusion subjected to an arbitrary uniform eigenstrain in an infinite elastic medium; the elastic field in an inhomogeneity in an infinite medium subjected to a prescribed arbitrary uniform remote stress; and strain energy in solids containing inhomogeneities. We shall present the results for these three problems for a spherical inclusion and inhomogeneity with interface stress effect. 2.2.1. Eshelby Tensors We consider first the spherical inhomogeneity problem with the interface stress effect. For an inhomogeneous inclusion, that is, an inhomogeneity embedded in an alien infinite medium is given a uniform eigenstrain, the Eshelby tensors Sk (x) (k = I, m) relate the total strains εk (x) in the inhomogeneity (k = I), denoted by

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I , and the matrix (k = m), denoted by m , to the prescribed uniform eigenstrain ε∗ in the inhomogeneity εk (x) = Sk (x) : ε∗ ,

(k = I, m), ∀x ∈ I + m ,

(2.7)

where x is the position vector. The displacement field u∗ in the inhomogeneity corresponding to the uniform eigenstrain ε∗ is u∗ = ε∗ · x, ∀x ∈ I .

(2.8)

The interface and boundary conditions for this problem are: uI + u∗ = um , (σ I − σ m ) · n = ∇S · τ um = 0,

at

,

|x| → +∞.

σ m = 0,

(2.9) (2.10)

We now outline the procedure for the solution of the inhomogeneous inclusion problem. Based on the basic equations mentioned above, the elastic solutions of the nanoinhomogeneities with arbitrary shape can be obtained by using the corresponding techniques. Here, we consider a spherical inhomogeneity of radius R embedded in an infinite medium as an example (Fig. 2.1). It is given an arbitrary uniform eigenstrain ε∗ . This elastostatic problem with the interface stress is solved using the principle of superposition, that is, we obtain the complete set of the components of the Eshelby tensors Sk (x) through the consideration of several particular eigenstrains. For this, we first solve the elastic field induced by ε∗zz = 1, and the Z

␪

⌫

r

Y

␸ p X m

Fig. 2.1 A spherical inhomogeneity in an infinite medium.

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solution is given in the spherical coordinate system (r, θ, ϕ) (cf. (Fig. 2.1)). The axisymmetric elasticity problem for spherical domains can be solved in a general fashion in terms of functions of r multiplied by Legendre polynomials of θ. For the present problem, only a solution associated with Legendre polynomials n = 0, 2 is needed (Lur’e, 1964). It is expedient to split the displacement field into its dilatational part, k ukr = Fzz r+

Gkzz , r2

ukθ = ukϕ = 0,

(k = I, m),

(2.11)

and its deviatoric part, ukr = Urk P2 (cos θ), in which

ukθ = Uθk

dP2 (cos θ) , dθ

ukϕ = 0,

(k = I, m),

(2.12)



 k k 2(5 − 4ν D )C k zz zz k r+ −3 4 , Urk (r) = 12νk Akzz r 3 + 2Bzz r2 r

 k k 2(1 − 2ν D )C k zz zz k r+ + 4 . Uθk (r) = (7 − 4νk )Akzz r 3 + Bzz r2 r

(2.13)

(2.14)

k , C k , Dk , Here, P2 (cos θ) is the Legendre polynomial of order two. Akzz , Bzz zz zz k k Fzz , and Gzz are constants to be determined. The subscript zz of these constants indicates that they are solved for the eigenstrain ε∗zz = 0. For the inhomogeneous inclusion problem, these constants are determined from the condition to avoid a singularity at r = 0 inside the inhomogeneity and Eqs. (2.9) and (2.10). Inside the I , DI , and GI vanish; in the matrix (k = m), Am , Bm , inhomogeneity (k = I), Czz zz zz zz zz m and Fzz vanish. Due to the spherical shape of the inhomogeneity and linear property of the problem, the solution due to an arbitrary uniform eigenstrain ε∗ can be obtained from Eqs. (2.11)–(2.14) by superimposing the individual solutions for ε∗xx = 1, ε∗yy = 1, ε∗zz = 1, ε∗xy = 1, ε∗xz = 1, and ε∗yz = 1, respectively. The detailed procedures of the superposition can be found in Duan et al. (2005b). It is found that under ε∗xx = 1, ε∗yy = 1, ε∗zz = 1, ε∗xy = 1, ε∗xz = 1, and ε∗yz = 1, respectively, six constants AIpq (pq = xx, yy, zz, xy, xz, and yz) can be obtained, and they obey the following relation: AIxx = AIyy = AIzz = AIxy = AIxz = AIyz . Moreover, the conI , C m , Dm , F I , and Gm also obey their own respective identities. stants Bpq pq pq pq pq Therefore, for brevity, we introduce constants A, B, C, D, F , and G such that

A ≡ R2 AIpq ,

I B ≡ Bpq ,

C≡

m Cpq

R3

,

D≡

m Dpq

R5

,

I F ≡ Fpp ,

G≡

Gm pp R3

,

(2.15)

12

H. L. Duan et al.

where the subscript pairs pq = xx, yy, zz, xy, xz, and yz denote the eigenstrain cases ε∗xx = 1, ε∗yy = 1, ε∗zz = 1, ε∗xy = 1, ε∗xz = 1, and ε∗yz = 1, respectively. Thus, the repeated subscripts in Eq. (2.15) do not represent summation. Note that the last two expressions in Eq. (2.15) are applicable to pp = xx, yy, and zz only. Therefore, the total strain fields in the inhomogeneity and matrix are expressed in terms of the constants A, B, C, D, F , and G. In view of the geometrical and physical symmetry of the inhomogeneous inclusion problem under consideration, the Eshelby tensors in the two phases are transversely isotropic with any of the radii being an axis of symmetry. However, it should be noted that unlike the classical interior Eshelby tensor for an ellipsoidal inhomogeneity without the interface stress, the interior Eshelby tensor with interface stress is generally position dependent. Using the Walpole notation (Walpole, 1981) for transversely isotropic tensors, a fourth-order tensor Sk (r) with radial symmetry can be expressed as Sk (r) = S1k (r)E1 + S2k (r)E2 + S3k (r)E3 + S4k (r)E4 + S5k (r)E5 + S6k (r)E6 (2.16) or in a concise matrix form T Sk (r) = Sk (r)·E in which



Sk (r) = S1k (r) S2k (r) S3k (r) S4k (r) S5k (r) S6k (r) , 

E = E1 E2 E3 E4 E5 E6 ,

(2.17)

(2.18) (2.19)

where r(r = rn) is the position vector of the material point at which the Eshelby tensor is being calculated. n = ni ei is the unit vector along the radius passing through this point, and r is the distance from this point to the origin (the centre of the spherical inhomogeneity). ni are the direction cosines of r and i = 1, 2, 3 denote x-, y-, and z-directions, respectively. Skq (r) (q = 1, 2, . . . , 6) are functions of r, and Ep (p = 1, 2, . . . , 6) are the six elementary tensors (Walpole, 1981) 1 βij βmn , 2 = αij αmn , 1 = βim βjn + βjm βin − βij βmn , 2 1 βim αjn + βin αjm + βjm αin + βjn αim , = 2 = αij βmn ,

1 = Eijmn 2 Eijmn 3 Eijmn 4 Eijmn 5 Eijmn 6 Eijmn

= βij αmn ,

(2.20)

Theory of Elasticity at the Nanoscale where αij and βij are given by αij = ni nj ,

βij = δij − ni nj ,

13

(2.21)

and δij is the Kronecker delta. The (interior) Eshelby tensor in the inhomogeneous inclusion (I ) is given by Eq. (2.17) with SI (r) being ⎤T ⎡ 1 + B + 2F + 3(7 − 8νI )Ah2 ⎥ ⎢ 1 + 2B + F + 36νI Ah2 ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ 1 + 3B + 3(7 − 4νI )Ah ⎥ (2.22) SI (r) = ⎢ ⎢ 1 + 3B + 3(7 + 2ν )Ah2 ⎥ . I ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ −B + F − 18νI Ah2 −B + F − 3(7 − 8νI )Ah2 In the matrix (m ), the exterior Eshelby tensor is again given by Eq. (2.17) with Sm (r) being ⎡ ⎤T 1 1 ⎢ 6D h5 + 2 [G − 2(1 + νm )C] h3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 12D 1 − 2 [G + 2(5 − 4νm )C] 1 ⎥ ⎢ 5 3 h h ⎥ ⎢ ⎥ ⎢ ⎥ 1 1 ⎢ ⎥ 3D 5 + 6(1 − 2νm )C 3 ⎢ ⎥ h h m ⎢ ⎥ . (2.23) S (r) = ⎢ ⎥ 1 1 ⎢ ⎥ −12D + 6(1 + ν )C m ⎢ ⎥ h5 h3 ⎢ ⎥ ⎢ 1 1 ⎥ ⎢ −6D − 2 [G − (5 − 4νm )C] ⎥ ⎢ h5 h3 ⎥ ⎢ ⎥ ⎣ 1 1 ⎦ −6D 5 + [G + 4(1 + νm )C] 3 h h In Eqs. (2.22) and (2.23), h = r/R, and  2η (4 − 5νm ) κsr + 2μrs η12 A= , B= , 3η11 3η11  (1 − 2νI ) 2 + κsr   , F =− 3 η (1 + νI ) + (1 − 2νI ) 2 + κsr  5η2 (7 + 5νI ) + 4η (7 − 10νI ) 5 + 3κsr + μrs C=− , 6η11 

3η2 (7 + 5νI ) + 4η (7 − 10νI ) 3 + (1 + νm ) κsr − (1 − 2νm ) μrs D=− , 3η11 η (1 + νI )   , G= (2.24) 3 η (1 + νI ) + (1 − 2νI ) 2 + κsr

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H. L. Duan et al.

where

 η11 = −2η2 (7 + 5νI ) (4 − 5νm ) + 7η −39 − 20κsr − 16μrs   + 35νm η 9 + 5κsr + 4μrs + νI η 285 + 188κsr + 16μrs   − 5νI νm η 75 + 47κsr + 4μrs − 4 (7 − 10νI ) 7 + 11μrs   

+ 4 (7 − 10νI ) −κsr 5 + 4μrs + νm 5 + 4κsr + 13μrs + 5κsr μrs , (2.25)

   η12 = 4 (7 − 10νI ) 7 + 11μrs + κsr 5 + 4μrs − νm 5 + 4κsr + 13μrs + 5κsr μrs   + 7η 7 − 5νm + 5νm κsr − 4κsr + νI η 35 + 4κsr + 48μrs  − 5νI νm η 5 + κsr + 12μrs , (2.26) and η = μI /μm with μI and μm being the shear moduli of the inhomogeneity and the matrix. κsr and μrs are two nondimensional parameters, κsr =

lκ , R

lκ =

κs , μm

μrs =

lμ , R

lμ =

μs , μm

(2.27)

lκ and lμ are two intrinsic length scales. It is noted that the Eshelby tensors are functions of lκ and lμ . To reveal the salient features of the Eshelby tensors (2.17) for the inhomogeneous inclusion with the interface stress that are absent in the classical Eshelby tensors, we present below the major components of the interior Eshelby tensor SI (r) in cartesian coordinates (for brevity, we drop the superscript I)  3A  S1111 = F + 2B + 2 7(y2 + z2 ) − 6νI (−2x2 + y2 + z2 ) + 1, (2.28) R S1122 = F − B − S1212 =

 3A  2 2 2 2 7y + 2ν (3x − 4y + z ) , I R2

 1 3 3A  2 2 2 2 2 2 7(x + y + z ) + 2ν (x + y − 2z ) + . B+ I 2 2 2R2

(2.29) (2.30)

Some additional components can be obtained by simultaneous permutation of the subscripts (1, 2, 3) and coordinates (x, y, z); thus, S2222 and S3333 can be obtained from S1111 in (2.28), S2211 , S1133 , S2233 , S3311 , and S3322 from S1122 in (2.29), and S1313 and S2323 from S1212 in (2.30). Other components are as follows: ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ xy ⎬ ⎨ S1222 ⎬ xy ⎬ ⎨ S1211 ⎬ ⎨ ⎨ A A = 6νI 2 −2yz , = 6νI 2 , (2.31) yz S S ⎩ 2311 ⎭ ⎭ ⎩ 2322 ⎭ ⎭ R ⎩ R ⎩ S1311 S1322 xz −2xz

Theory of Elasticity at the Nanoscale ⎫ ⎧ ⎫ ⎧ −2xy ⎬ ⎨ S1233 ⎬ A ⎨ = 6νI 2 , yz S ⎭ ⎩ 2333 ⎭ R ⎩ S1333 xz ⎧ ⎫ ⎫ ⎧ w ⎨ S1123 ⎬ A ⎨ 2⎬ S2223 = 3 2 yz w1 , ⎩ ⎭ ⎭ ⎩ R S3323 w1

⎧ ⎫ ⎧ ⎫ w ⎨ S1112 ⎬ A ⎨ 1⎬ S2212 = 3 2 xy w1 , ⎩ ⎭ ⎩ ⎭ R S3312 w2 ⎧ ⎫ ⎧ ⎫ w ⎨ S1113 ⎬ A ⎨ 1⎬ S2213 = 3 2 xz w2 , ⎩ ⎭ ⎩ ⎭ R S3313 w1

where w1 = −7 + 16νI , w2 = −7 + 10νI ⎧ ⎫ ⎧ ⎫ xy ⎨ S2313 ⎬ A ⎨ ⎬ = 9νI 2 yz , S ⎩ 1213 ⎭ R ⎩ ⎭ xz S1223

15

(2.32)

(2.33)

(2.34)

with S1323 = S2313 , S1312 = S1213 , S2312 = S1223 . In the above expressions (2.28)–(2.34), the constants A, B, and F are given in (2.24), where F is contributed by the dilatational solution (2.11) and A and B by the deviatoric solution (2.12)–(2.14). The Eshelby tensor in the inhomogeneous inclusion (and in the matrix) is size dependent through the two nondimensional parameters κsr and μrs . It is clearly seen from the expressions in Eqs. (2.22) and (2.28)–(2.34) that the interior Eshelby tensor is, in general, not uniform for an inhomogeneous inclusion with the interface stress effect; it is a quadratic function of the position coordinates. The solution without the interface stress effect can be obtained by setting κs = 0 and μs = 0 or letting R → ∞. The interior Eshelby tensor is constant in this case. Under dilatational eigenstrain ε∗ = 0 I(2) , the total strain in the inhomogeneous inclusion is given by εI = 0 SI : I(2) . It can be verified that SI : I(2) is a constant tensor even in the presence of the interface stress effect, and thus the stress field in the inhomogeneous inclusion is uniform, confirming the result of Sharma et al. (2003) for a dilatational eigenstrain. Furthermore, when the elastic constants of the inhomogeneity are the same as those of the matrix, namely η = 1 and νI = νm , the inhomogeneity becomes an inclusion. In this case, for a dilatational eigenstrain ε∗ = 0 I(2) , the strain fields in the inclusion and the matrix reduce to those in the paper of Sharma and Ganti (2004). 2.2.2. Stress Concentration Tensors We shall next consider the problem where a spherical inhomogeneity with the interface stress is embedded in an infinite elastic medium with a prescribed arbitrary uniform remote stress field σ 0 . The interior stress concentration tensor for this problem (i.e., inside the inhomogeneity) has recently been given by Duan et al. (2005b). (We use the terminology stress concentration tensor rather than Eshelby

16

H. L. Duan et al.

tensor because we are not solving the inhomogeneity problem using the Eshelby equivalent inclusion approach. This approach loses its elegance in the presence of a surface stress, and the inhomogeneity problem is better solved directly). Here, we shall present both the interior and exterior stress concentration tensors for completeness. The stress concentration tensors Tk (x) (k = I, m) relate the stresses σ k (x) in the two phases to the prescribed uniform remote stress σ 0 , that is, σ k (x) = Tk (x) : σ 0 ,

(k = I, m), ∀x ∈ I + m .

(2.35)

The interface and boundary conditions for this problem are: uI = um , (σ I − σ m ) · n = ∇S · τ um = u0 ,

σm = σ0,

at

(2.36)

,

|x| → +∞.

(2.37)

As for the inhomogeneous inclusion problem in the previous section, this inhomogeneity problem under remote loading is also solved by the principle of 0 = 1 are superposition (Duan et al., 2005b). For example, the solutions under σzz still given by (2.11)–(2.14) with the constants determined by interface and boundary conditions (2.36), (2.37) and the condition to avoid a singularity at r = 0. I , DI , and GI vanish; in the matrix (k = m), In the inhomogeneity (k = I), Czz zz zz m m Am zz vanishes, and Bzz and Fzz are determined from the corresponding remote 0 = 1, σ 0 = 1, σ 0 = 1, loading conditions in Eq. (2.37). It is found that under σxx yy zz 0 0 0 I I σxy = 1, σxz = 1, and σyz = 1, respectively, Axx = Ayy = AIzz = AIxy = AIxz = I , C m , Dm , F I , and Gm also obey their own respective AIyz . The constants Bpq pq pq pq pq identities. Therefore, we define constants A, B, C, D, F, and G such that A ≡ R2 μI AIpq , D ≡ μm

m Dpq

R5

,

I B ≡ μI Bpq ,

C ≡ μm

2(1 + νI ) I F≡ , μI Fpp (1 − 2νI )

m Cpq

R3

,

G ≡ μm

Gm pp R3

(2.38) .

As before, the subscript pairs pq = xx, yy, zz, xy, xz, and yz denote the remote 0 = 1, σ 0 = 1, σ 0 = 1, σ 0 = 1, σ 0 = 1, and σ 0 = 1, respecstress cases σxx yy zz xy xz yz tively. Again, the last two expressions in Eq. (2.38) are applicable to pp = xx, yy, and zz only. Therefore, the total stress fields in the inhomogeneity and matrix are expressed in terms of the constants A, B, C, D, F, and G. The stress concentration tensors for the spherical inhomogeneity with the interface stress effect can be expressed as (Duan et al., 2005a) T Tk (r)·E Tk (r) =

(k = I, m)

(2.39)

Theory of Elasticity at the Nanoscale

17

in which Tk (r) =

 T1k (r) T2k (r) T3k (r) T4k (r) T5k (r) T6k (r) .

(2.40)

In the inhomogeneity (I ), the stress concentration tensor is given by Eq. (2.39) with TI (r) being ⎤T ⎡ 2B + 2F + 6(7 + 6νI )Ah2 ⎥ ⎢ 4B + F − 12νI Ah2 ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ 6B + 6(7 − 4ν )Ah I ⎥ . (2.41) TI (r) = ⎢ ⎥ ⎢ 2 6B + 6(7 + 2νI )Ah ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ −2B + F + 6νI Ah2 −2B + F − 6(7 + 6νI )Ah2 In the matrix (m ), the stress concentration tensor is again given by Eq. (2.39) with Tm (r) being ⎤T ⎡ 1 1 [G 1 + 12D + 4 − 2(1 − 2ν )C] m ⎢ h5 h3 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 1 + 24D 1 − 4 [G + 2(5 − ν )C] 1 ⎥ m ⎥ ⎢ h5 h3 ⎥ ⎢ ⎥ ⎢ 1 1 ⎥ ⎢ 1 + 6D 5 + 12(1 − 2νm )C 3 ⎥ ⎢ ⎥ ⎢ h h m ⎥ ⎢ (2.42) T (r) = ⎢ ⎥ 1 1 ⎥ ⎢ 1 − 24D + 12(1 + ν )C m ⎥ 5 ⎢ 3 h h ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ −12D − 4 [G − (5 − ν )C] 1 m ⎥ ⎢ h5 h3 ⎥ ⎢ ⎢ 1 1 ⎥ ⎦ ⎣ −12D 5 + 2 [G + 4(1 − 2νm )C] 3 h h in which 5η(1 − νm )(κsr + 2μrs ) 5ηη21 (1 − νm ) , B= , 2η11 2η11 η(1 + νI )(1 − νm ) 5η23 F= , C= , η22 24η11 η23 + 4(7 − 10νI )(1 − νm )(κsr + 2μrs ) D= , 4η11

 −η(1 + νI )(1 − 2νm ) + (1 − 2νI ) 1 − κsr + νm (1 + 2κsr ) G= , 6η22 A=

(2.43)

18

H. L. Duan et al.

where η11 has been given in Eq. (2.25), and

 η21 = − η(7 + 5νI ) − 8νI (5 + 3κsr + μrs ) + 7(4 + 3κsr + 2μrs ) , (2.44)

 η22 = (1 + νm ) η(1 + νI ) + (1 − 2νI )(2 + κsr ) , (2.45) η23 = 2η2 (7 + 5νI ) − 4(7 − 10νI )(2 + κsr )(1 − μrs ) +

 + η 7(6 + 5κsr + 4μrs ) − νI (90 + 47κsr + 4μrs ) .

(2.46)

The stress concentration tensors in the inhomogeneity and matrix exhibit the same features as the interior and exterior Eshelby tensors in an inhomogeneous inclusion, namely they are size- and position dependent. 2.2.3. Eshelby formula The Eshelby formula gives the strain energy in solids containing inhomogeneities (Eshelby, 1956). It is of great value in heterogeneous material analysis because it reduces the usual volume integrations for calculating strain energy to surface integration. Consider an inhomogeneity in an elastic matrix. Such a problem arises, for example, in the prediction of the effective moduli of heterogeneous materials which we shall consider later in this chapter. Here, we give the Eshelby formula for the nano-inhomogeneity. Denote the external surface of the matrix by S and the interface between the inhomogeneity and matrix by . Under uniform strain boundary condition u(S) = ε0 · x, where x is the position vector and ε0 is a constant strain tensor, the Eshelby formula with the interface stress effect can be obtained according to the same procedures as those to obtain the classical Eshelby formula without interface stress effect, that is,    1 n · σ m · u0 − u · σ 0 · n d U = U0 + (2.47) 2  in which 1 U0 = 2

 σ 0 : ε0 dV.

(2.48)

V

Here, U is the elastic strain energy in the heterogeneous material, and σ m and u are the stress and displacement, respectively, at  with the superscript “m” signifying that the stress is to be evaluated on the matrix side. σ 0 and u0 , however, denote the stress and displacement, respectively, in the homogeneous body of volume V made of the matrix only. When σ I = σ m , Eq. (2.47) reduces to the classical Eshelby formula,    1 (2.49) U = U0 + n · σ · u0 − u · σ 0 · n d. 2 

Theory of Elasticity at the Nanoscale

19

Under uniform stress boundary condition (S) = σ 0 · N, where N is the outward unit normal vector to S and σ 0 is a constant stress tensor, the Eshelby formula with the interface stress effect is    1 U = U0 + n · σ 0 · u − u0 · σ m · n d. (2.50) 2  When σ I = σ m , Eq. (2.50) reduces to the classical Eshelby formula,    1 U = U0 + n · σ 0 · u − u0 · σ · n d. 2 

(2.51)

It is noted that the difference between the classical Eshelby formula (Eq. (2.49) or Eq. (2.51)) and the Eshelby formula with interface stress effect (Eq. (2.47) or Eq. (2.50)) is that there is stress discontinuity across the interface due to the interface stress effect. Therefore, for the composites with nano-inhomogeneities, the stress σ m in the Eshelby formula Eq. (2.47) (or Eq. (2.50)) should be evaluated on the matrix side. 3. Application of Eshelby Formalism for Nano-inhomogeneities

3.1. Stress Concentration Factor of a Spherical or Circular Nanovoid Consider a spherical nanovoid in an infinite matrix under an axisymmetric 0 = T and σ 0 = σ 0 = χ T . Without loss of generality, loading at infinity with σzz xx yy the triaxiality parameter χ is allowed to range from zero to one so that the loading conditions change from uniaxial tension (compression) to hydrostatic tension (compression). However, before solving the stress concentration due to a spherical or circular nanovoid, we examine some features of the stress field in a generic spherical inhomogeneity. The stress concentration tensor in the spherical inhomogeneity is not uniform because of the presence of the terms involving AI in Eqs. (2.13) and (2.14). Here, we use AIaxis to denote this constant under axisym0 = T and σ 0 = σ 0 = χ T . Without going into details, it can metric loading σzz xx yy be shown that for the axisymmetric remote loading under consideration, AIaxis is given by 5T(1 − χ)(1 − νm )(κsr + 2μrs ) . (3.1) AIaxis = 2R2 μm η11 It is interesting to note that uniform stress fields appear when AIaxis vanishes. This can happen for a finite size inhomogeneity when: (i) the surface stress effect is absent, κs = μs = 0, (ii) the remote loading is hydrostatic, χ = 1 (confirming

20

H. L. Duan et al.

again the results of Sharma et al. (2003)), and (iii) η → ∞, which is the case if the inhomogeneity is infinitely rigid in comparison with the matrix. In all other cases, the stress field inside the inhomogeneity and outside it is not uniform. For a spherical nanovoid, the normalized hoop stress at the boundary of the 0 = T and σ 0 = σ 0 = χ T is given by void under σzz xx yy σθθm 3 [4 − 5νm + 3χ − 5(1 − χ) cos 2θ] (1 + 2χ)(1 − νm )κsr = − − T 2(7 − 5νm ) 2(1 + νm )(2 + κsr ) −

5(1 − χ)(1 − νm ) (M1 + 3N1 cos 2θ), 8H1 (7 − 5νm )

(3.2)

with

 M1 = 6μrs (−5 + 19νm ) + κsr 1 − 16μrs + νm (7 + 50μrs ) ,

 N1 = 13 + 16μrs − 5νm (1 − 2μrs ) κsr + 2(29 + 5νm )μrs ,

(3.3)

H1 = −7 + 5νm + μrs (13νm − 11) + κsr (5νm μrs − 4μrs + 4νm − 5). The first term on the right-hand side of Eq. (3.2) is the classical solution, and the other terms are due to the surface stress effect. κsr and μrs are given by Eq. (2.27). Following the same procedure, the normalized hoop stress at the boundary of a 0 = T and circular nanovoid in an infinite plate (for the plain stress case) under σxx 0 σyy = χT can be shown to be σφφm (1 + χ)χsr 6χsr (1 − χ) cos 2φ = 1 + χ − 2(1 − χ) cos 2φ − + , T (1 + νm )(4 + χsr ) 4 + 3χsr + νm (4 + χsr ) (3.4) where χsr = κsr + 2μrs . The polar coordinate system (ρ, φ) is used with the ρφplane perpendicular to the voids. Again, the first term on the right-hand side of Eq. (3.4) is the classical solution, and the other terms are due to the surface stress effect. The variation of the maximum normalized hoop stress at the boundary of a spherical nanovoid (at the equator, θ = π/2) is shown in Fig. 3.1 for aluminium (with bulk modulus κm = 75.2 GPa and Poisson ratio νm = 0.3). The free surface properties are taken from Miller and Shenoy (2000) and Sharma et al. (2003). Two sets of surface moduli are used, namely A: κs = −5.457 N/m, μs = −6.2178 N/m for the surface [1 0 0]; B: κs = 12.932 N/m, μs = −0.3755 N/m for the surface [1 1 1]. It is seen from Fig. 3.1 that the stress concentration decreases (increases) with increasing void size due to the surface stress effect until it reaches the classical solution (without the surface stress effect) which is independent of the void size. The effect of the surface stress on the stress concentration also depends on the stress triaxiality. The lower the stress triaxiality, the greater the

Theory of Elasticity at the Nanoscale 2.8

A, x 5 0 B, x 5 0 C, x 5 0

s␪␪m/ T

2.4

A, x 5 0.5 B, x 5 0.5 C, x 5 0.5

21

A, x 5 1 B, x 5 1 C, x 5 1

2.0 1.6 1.3

1

10

20

30 R (nm)

40

50

Fig. 3.1 Stress concentration as a function of surface properties and spherical void radius for different stress triaxialities. In Fig. 2.1, “A” and “B” denote the solutions for the two sets of surface properties, and “C” the classical solution without the surface stress effect. Reprinted from Duan et al. (2005a) with acknowledgment to The Royal Society, 6–9 Carlton House Terrace, London SW1Y 5AG, England.

stress concentration. The surface stress effect on the stress concentration becomes negligible for void radius larger than 50 nm under uniaxial tension (χ = 0), while under hydrostatic loading (χ = 1), the surface stress effect is already negligible for void radius larger than 25 nm.

3.2. Strain Fields in QDs With Multi-shell Structures and in Alloyed QDs The behavior of electronic devices made of alloyed QDs (e.g., Inx Ga1−x As) or QDs with a multi-shell structure (e.g., CdS/HgS/CdS/HgS/CdS, ZnS/CdSe) is strongly affected by their enriched but nonuniform composition. It has been demonstrated that the assessment of the composition profiles and strains is important to both the identification of the dominant growth mechanisms and the modelling of the confining potential of QDs (Rosenauer and Gerthsen, 1999; Malachias et al., 2003; Spencer and Blanariu, 2005). In the following, we assume that the lattice constants or the thermal expansion coefficients of QDs with multi-shell structures or alloyed QDs obey Vegard’s law (Vegard, 1921) and analyze the strain distribution in and around the QDs. Moreover, the critical sizes of dislocation-free QDs will be determined. 3.2.1. Strain Distributions in QDs with Multi-shell Structures First, we consider a spherical QD with a multi-shell structure embedded in an infinite elastic matrix. Let phase 1 denote the innermost core, hereinafter referred to as the particle, and let phase i refer to the shell bounded by the concentric and spherical surfaces with radii ri and ri+1 (i ∈ (1, M)), respectively. Let 1 , k

22

H. L. Duan et al.

(k = 2, . . . i, . . . , M), and M+1 denote the regions occupied by the particle, the multi-shells, and the matrix, respectively. The subscripts k (k = 1, 2, . . . i, . . . , M, M + 1) are used to denote the quantities in the regions k , respectively. The particle, the multi-shells, and the matrix are homogeneous, linearly elastic, and isotropic, characterized by bulk modulus κk , shear modulus μk , and Poisson ratio νk (k = 1, 2, . . . i, . . . , M, M + 1). As stated above, the large ratio of surface/interface atoms to the bulk can have a profound effect on the properties of nanostructures, and this effect can be described by the classical continuum model with consideration of the interface stress effect. Moreover, the strains induced by the mismatch of the lattice constants and CTE in the spherical QD and its multi-shell structure can be treated as eigenstrains (Gosling and Willis, 1995; Rockenberger et al., 1998). Therefore, when uniform eigenstrains ε∗i (i = 1, 2, . . . , M) are prescribed in the particle and the multi-shells, the interface conditions for this problem can be expressed as ui + ε∗i · x = ui+1 + ε∗i+1 · x, (σ i − σ i+1 ) · m = ∇S · σ s , uM+1 = 0, σ M+1 = 0 at |x| → +∞,

(3.5)

where m is the unit normal vector to the interface between the ith and (i + 1)th phases, ∇S · σ s denotes the interface divergence of the interface stress tensor σ s . The interface stress σ s depends on the state of the elastic strain εs (Streitz et al., 1994; Bottomley and Ogino, 2001) via the constitutive relation (2.4) or (2.6). However, the interface elastic moduli are currently not available for QD materials. Thus, we simply consider the effect of a constant interface stress, σ s0 , and/or a constant surface stress, τ s , in this section. When uniform eigenstrains ε∗i (i = 1, 2, . . . , M) are prescribed in the particle and the multi-shells, then according to the Eshelby formalism above, the elastic strains εk (x) in the particle (k = I), the multi-shells (k = 2, 3, . . . , M), and the matrix (k = M + 1) can be expressed as εk = Ski : ε∗i − ε∗k + Hk (σ s0 )

(k = 1, 2, . . . , M + 1).

(3.6)

It is noted that ε∗M+1 = 0 in the above equation. Ski is the Eshelby tensor in the kth phase, which relates the uniform eigenstrain ε∗i prescribed in the ith phase to the strain induced in the kth phase (Eshelby, 1957). Thus, the repeated subscript i in Eq. (3.6) indicates summation from 1 to M. Hk (σ s0 ) is the strain tensor due to residual interface stress σ s0 . Ski satisfying the conditions (3.5) can be determined following the same procedure as that for obtaining Eshelby tensor Sk for spherical nano-inhomogeneity described above. After determining the Eshelby tensor Ski in the core and multi-shells, the elastic fields in them can be determined from Eq. (3.6).

Theory of Elasticity at the Nanoscale

23

In the following, we consider a free-standing spherical QD consisting of a core and concentric multi-shells of nanometer size. Besides the uniform eigenstrains ε∗i (i = 1, 2, . . . , M − 1) prescribed in the particle and the multi-shells, we also assume that an isotropic surface stress vector τ s (τ s = τ s 1, we distinguish it from the interface stress σ s ) acts on the outer surface of this QD. Then the elastic strains in the core and the multi-shells are εk = Ski : ε∗i − ε∗k + Hk (τ s , σ s0 )

(k = 1, 2, . . . , M),

(3.7)

where Hk (τ s , σ s0 ) is the strain tensor due to interface stress σ s0 and isotropic surface stress τ s . Here, the procedures to obtain Ski are similar to those used to obtain the Eshelby tensors for the embedded spherical QDs with multi-shell structures mentioned above. However, the interface and boundary conditions for this case are ui + ε∗i · x = ui+1 + ε∗i+1 · x, (σ i − σ i+1 ) · N = ∇S · σ s , (3.8) σrr = 2τ s /rM at r = rM . Note that ε∗M = 0 in the present case and i ∈ (1, M − 1) in Eqs. (3.7) and (3.8). Moreover, Eqs. (3.6) and (3.7) can apply to arbitrary uniform eigenstrains ε∗i prescribed in the core and multi-shells. For QD structures with a uniform composition, the mismatch of the lattice constants or thermal expansion coefficients of different constituents can induce an initial misfit strain. According to the definitions of an embedded QD (Gosling and Willis, 1995) and a free-standing core-shell particle (Rockenberger et al., 1998), the misfit strains arising from the different lattice constants and the thermal expansion coefficients between different phases are, respectively, ε∗m0 =

ain − aex , aex

ε∗tm0 = (αin − αex )T,

(3.9)

where ain , aex and αin , αex are the lattice constants and the thermal expansion coefficients of the interior and exterior phases, respectively. T is the temperature difference. For example, the misfit strain due to the mismatch of the lattice constants of the CdTe(core)/CdS(shell) structure is 11.6%, and that of ZnS(core)/CdS(shell) is −7.0%. 3.2.2. Strain Distributions in Alloyed QDs The expressions in Eq. (3.9) are the so-called Vegard’s law (Vegard, 1921). Note that Vegard’s law for alloyed materials is a linear function: Ealloy = xEA + (1 − x)EB , where EA , EB , and Ealloy are the respective properties of pure A, pure B, and the alloy Ax B1−x , and x is the fraction of one ingredient in a material point (Bailey and Nie, 2003). For example, experiments have shown that the lattice

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H. L. Duan et al.

constant of the nonuniform QD (Znx Cd1−x S) exhibits a nearly linear relation with the Zn content x, which is consistent with Vegard’s law (Zhong et al., 2003; Li et al., 2005). To reveal the profound effect of a nonuniform composition on the stress state of a QD, consider, for simplicity, a spherical alloyed QD embedded in an infinite matrix. The analytical method is equally applicable to QDs of other shapes, sizes, and composition profiles. We assume that a spherical alloyed QD has a spherically symmetric composition, that is, the nonuniform composition is a function of the radial coordinate r only. Therefore, the misfit eigenstrain ε∗ (r) induced by the mismatch of the lattice constants and the thermal expansion coefficients can be expressed as ε∗ (r) = ε∗rr (r)er ⊗ er + ε∗θθ (r)(eθ ⊗ eθ + eϕ ⊗ eϕ ),

(3.10)

where er , eθ , and eϕ are the local unit base vectors in the spherical coordinate system, and ε∗rr (r) and ε∗θθ (r) are the misfit strains in the radial and tangential directions, respectively. According to Vegard’s law (1921), the misfit strains induced by the mismatch in the lattice constants and those by the mismatch in the thermal expansion coefficients are (Duan et al., 2006a), respectively, ε∗rr (r) = cr (r)ε∗m0 , ε∗θθ (r) = cθ (r)ε∗m0 , ε∗rr (r) = cr (r)ε∗tm0 , ε∗θθ (r) = cθ (r)ε∗tm0 , where ε∗m0 and ε∗tm0 are given by Eq. (3.9). cr and cθ are the concentrations of the ingredient at the location r in the radial and tangential directions. If cr ≡ 1 and cθ ≡ 1, Eq. (3.10) reduces to that for a uniform composition. We assume that the elastic constants of the alloyed QD are uniform. This is a reasonable assumption because the alloyed semiconductor QDs usually contain compounds (e.g., InAs/GaAs, CdTe/CdSe) with nearly identical elastic constants, and it has been validated by comparing isotropic and anisotropic solutions for semiconductor materials (Faux and Pearson, 2000). We begin with the total displacement vector u∗ in the free-standing QD induced by the eigenstrain ε∗ij . According to the theory of infinitesimal elasticity, cf. Eq. (2.1), the governing equation to obtain u∗ is   1 Cijkl u∗k,lj − ε∗kl,j = 0,

(3.11)

1 is the elastic modulus where the eigenstrains ε∗ij (x) are given in Eq. (3.10), and Cijkl tensor of the QD. When the variations of cθ (r) and cr (r) are known, the only nonvanishing component of the displacement vector u∗ , viz. u∗r (r), can be easily determined by Eqs. (3.10) and (3.11). In the following, we will consider the special case of cθ (r) = cr (r) = x(r). Substituting Eq. (3.10) into Eq. (3.11), it follows that the only nonvanishing component

Theory of Elasticity at the Nanoscale

25

of the displacement vector u∗ , viz. u∗r (r), must satisfy the equation r2

∂2 u∗r ∂u∗ (1 + ν1 ) ∗ 2 ∂x + 2r r − 2u∗r − ε r = 0, 2 ∂r (1 − ν1 ) m0 ∂r ∂r

(3.12)

where ν1 is the Poisson ratio of the core. Therefore, Eq. (3.12) and the tractionfree condition at the outer boundary of the free core constitute the basic equations to find u∗r (r). For a known variation of x, for example, it could vary in a linear, logarithmic, or exponential manner with r, u∗r (r) can be easily determined. In particular, if x(r) in Eq. (3.10) is assumed to be a linear function in the radial coordinate r r (3.13) x(r) = k0 + k1 , rco where k0 and k1 are two constants, and rco denotes the radius of the nonuniform core, then the corresponding u∗r (r) is given by Eq. (3.12) and the traction-free condition at the outer boundary of the free core   (1 − 2ν1 )k1 ε∗m0 k1 (1 + ν1 )r 2 ε∗m0 + . (3.14) u∗r (r) = r k0 ε∗m0 + 2(1 − ν1 ) 4rco (1 − ν1 ) When the alloyed QD is embedded in an infinite medium (relative to its size), the constraint imposed by the exterior medium will induce an additional displacement field, identified by superscript 1 in the QD and superscript 2 in the matrix: u1r = F1 r, u2r = G2 /r 2 . The constants F1 and G2 are determined from the following interface conditions: u2r = u1r + u∗r |r=rco ,

2 1 σrr − σrr =

2σ0s . rco

(3.15)

Then the elastic strains in the embedded spherical alloyed QD can be determined. The free-standing QD is assumed to be composed of a nonuniform core and a uniform shell with the nonuniform eigenstrain ε∗ prescribed in the core. Also, an isotropic surface stress τ s = τ s 1 is exerted at the outer surface of the shell. The procedure to obtain the solution due to ε∗ is similar to that for the embedded spherical alloyed QD. Let us compare the elastic strain distribution in the free-standing alloyed QD Znx Cd1−x S with that in the free-standing QD with multishells CdS/ZnS/CdS (M = 3). The elastic constants of bulk ZnS and CdS are as follows: ZnS, bulk modulus 81.6 GPa, Poisson’s ratio 0.4; and CdS, bulk modulus 62.3 GPa, Poisson’s ratio 0.4. The lattice constants of ZnS and CdS are a = 5.409 Å, a = 5.815 Å, respectively. Therefore, the misfit strain due to the mismatch of the lattice constants of ZnS(core)/CdS(shell) is ε∗m0 = −7.0%. We assume that the surface stress is τ s = 1 N/m for the free-standing QD. We consider

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H. L. Duan et al.

two cases for Znx Cd1−x S: Case I: k0 = 0.8, k1 = −0.4; Case II: k0 = 1, k1 = 0, and the radii of the core and shell are rco = 9 nm and rsh = 12 nm, respectively. Case II corresponds to a uniform core. For CdS/ZnS/CdS, the radii of the core and two shells are rco1 = 6 nm, rsh1 = 9 nm, rsh2 = 12 nm, respectively. The normalized elastic strain (εzz /ε∗m0 ) and the normalized biaxial strain (εb /ε∗m0 ) are shown in Figs. 3.2 and 3.3. It can be seen that the distributions of the elastic strains are strongly dependent on the compositions and structures of the QDs, and are different for Znx Cd1−x S and CdS/ZnS/CdS (M = 3). Note that the elastic strain in the core (−1 ≤ z/rco ≤ 1.0) of CdS/ZnS/CdS is uniform, whereas the strain in the core of Znx Cd1−x S is nonuniform. Three factors, namely the particle size, composition, and internal structure, have been used to achieve continuous tuning of the optical properties of QDs (Bailey and Nie, 2003; Liang et al., 2005). From Figs. 3.2 and 3.3 and the above analysis, it can be seen that the strain can be modified by adjusting the composition via the two parameters k0 and k1 in the Inx Ga1−x As/GaAs and CdTex Se1−x QD systems. Likewise, the strain in QDs with multi-shells can be modified by adjusting the thickness and the number (M) of the shells according to Eqs. (3.5)–(3.8) (Figs. 3.2, 3.3). Bailey and Nie (2003) have shown that continuous tuning of the optical properties of the CdTex Se1−x QDs can be achieved by changing the composition (Se:Te molar ratio) and the internal structure without changing the particle size. The composition changes the strain, and the strain affects the optical properties of CdTex Se1−x . Moreover, Bailey and Nie (2003) indicated that the tuning of the optical properties through a change in the composition and the internal structures is more advantageous than through a change in the particle size in some applications such as nanoelectronics, superlattice structures, and biological labeling. 3 Case I, Znx Cd12x S

2 εzz /ε*m 0

Case II, Znx Cd12x S

1

CdS/ZnS/CdS

0 21 22 21.3 21.0

20.5

0.0 z /rco

0.5

1.0 1.3

Fig. 3.2 Distribution of normalized elastic strain εzz /ε∗m0 in the free-standing QDs Znx Cd1−x S and CdS/ZnS/CdS subjected to an eigenstrain ε∗ in Eq. (3.10) and a surface stress τ s . Case I: k0 = 0.8, k1 = −0.4; Case II: k0 = 1, k1 = 0. Reprinted from Duan et al. (2006a) with acknowledgment to IOP Publishing Ltd, Dirac House, Temple Back, Bristol BS1 6BE, England.

Theory of Elasticity at the Nanoscale

27

0.0

εb /ε*m 0

21.5

Case I, Znx Cd12x S Case II, Znx Cd12x S

23.0

CdS/ZnS/CdS

24.5 26.0 21.3 21.0

20.5

0.0 z /rco

0.5

1.0 1.3

Fig. 3.3 Distribution of normalized biaxial strain εb /ε∗m0 in the free-standing QDs Znx Cd1−x S

and CdS/ZnS/CdS subjected to an eigenstrain ε∗ in Eq. (3.10) and a surface stress τ s . Case I: k0 = 0.8, k1 = −0.4; Case II: k0 = 1, k1 = 0. Reprinted from Duan et al. (2006a) with acknowledgment to IOP Publishing Ltd, Dirac House, Temple Back, Bristol BS1 6BE, England.

3.3. Compatible Composition Profiles and Critical Sizes of Alloyed QDs For the considered alloyed spherical QD, the only nonvanishing equation of compatibility of misfit eigenstrains ε∗rr and ε∗θθ represented in Eq. (3.10) reduces to an equation relating the radial and tangential alloy composition profiles r

∂cθ (r) + cθ (r) = cr (r). ∂r

(3.16)

Eq. (3.16) is identically satisfied when the composition is uniform, but it imposes restrictions on cr and cθ when the composition is nonuniform. The strain fields induced by nonuniform composition profiles that meet Eq. (3.16) are vastly different from those induced by profiles that violate this condition, howsoever slightly (e.g., cr (r) = cθ (r) = x(r) in Eq. (3.13)). Thus, the compatibility condition (Eq. (3.16)) provides a theoretical basis for designing the composition profile of an alloyed QD and for estimating its lattice deformation. Without loss of generality, we consider the following two cases of linear composition: Case I: compatible composition profile satisfying Eq. (3.16), for example, cθ (r) = k0 + k1 r/rco , cr (r) = k0 + 2k1 r/rco , where k0 and k1 are two constants and rco is the radius of the QD; Case II: composition profile not satisfying Eq. (3.16), cθ (r) = cr (r) = k0 + k1 r/rco (Eq. (3.13)). The expressions of the elastic strain in the alloyed QD and the uniform matrix due to the misfit eigenstrains for Cases I and II can be found in the papers of Duan et al. (2006a,b). It is found that the strain field in the alloyed QD is uniform irrespective of the composition profile provided the nonuniform misfit eigenstrains satisfy the compatibility

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H. L. Duan et al.

equation (cf., Fig. 3.4 for Case I), but not otherwise (cf., Fig. 3.5 for Case I; Duan et al., 2006a,b). Strains play a very important role in the nanofabrication technology, and strain relaxation through the formation of dislocations is highly undesirable for the εzz /ε*m0

2

z /rco

1 0 21 22 22

21

0

1

2

0.18 0.155 0.13 0.105 0.08 0.055 0.03 0.005 20.02 20.045 20.07 20.095 20.12 20.145 20.17 20.195 20.22 20.245 20.27 20.295 20.32 20.345 20.37 20.395 20.42

y/rco

Fig. 3.4 Distribution of normalized elastic strain εzz /ε∗m0 in the embedded QD (Inx Ga1−x As/GaAs) subjected to nonuniform eigenstrains ε∗ satisfying the compatibility equation (Case I, k0 = 0.8; k1 = −0.4). Reprinted from Duan et al. (2006b) with acknowledgment to American Physical Society. εzz /ε*m0

2

z /rco

1 0 21 22 22

21

0 y/rco

1

2

0.18 0.155 0.13 0.105 0.08 0.055 0.03 0.005 20.02 20.045 20.07 20.095 20.12 20.145 20.17 20.195 20.22 20.245 20.27 20.295 20.32 20.345 20.37 20.395 20.42

Fig. 3.5 Distribution of normalized elastic strain εzz /ε∗m0 in the embedded QD (Inx Ga1−x As/GaAs) subjected to nonuniform eigenstrains ε∗ not satisfying the compatibility equation (Case I, k0 = 0.8; k1 = −0.4). Reprinted from Duan et al. (2006b) with acknowledgment to American Physical Society.

Theory of Elasticity at the Nanoscale

29

performance of semiconductor devices (e.g., Zhang, 1995; Tersoff, 1998; Chen et al., 2003). Kolesnikova and Romanov (2004) obtained the critical radii of a spherical QD and a cylindrical quantum wire (QW) with uniform composition by considering the energy of nucleation of circular prismatic dislocation loops from a spherical QD and a cylindrical QW. We calculate the critical radii of the spherical alloyed QD at which the nucleation of a misfit prismatic dislocation (MD) loop (cf. Fig. 1 in Duan et al. (2006b)) becomes energetically favourable. We will again study the effect of the compatibility of misfit strains. The condition for the nucleation of an MD loop is (Kolesnikova and Romanov, 2004), EL + WIL ≤ 0,

(3.17)

where EL is the elastic energy of the prismatic dislocation loop and WIL is the interaction energy between it and the QD. EL of a dislocation loop of radius rL and Burgers vector of magnitude b in infinite space is (Kolesnikova and Romanov, 2004)   8αrL μb2 rL ln −2 , (3.18) EL ≈ 2(1 − ν) b where α is a parameter that takes into account the energy of the dislocation core. WIL is   1 1 ε∗L σ dV = − (−b)σzz dS, (3.19) WIL = − ij ij VL

SL

where VL and SL are the volume and area of the dislocation loop, respectively. Plastic distortion ε∗L ij of a prismatic dislocation loop located in the xoy plane in the xyz coordinate system with the origin at the loop center is ε∗L zz = ±bH(1 − r/rL )δ(z), H(1 − r/rL ) is the Heaviside function and δ(z) is the Dirac delta function. σij1 are 1 the normal stress in the z-direction. the stresses in the QD, with σzz Substituting Eqs. (3.18) and (3.19) and expressions of σij1 for the cases satisfying or not satisfying the compatibility equation into Eq. (3.17), we can obtain the critical radius RcI for the spherical QD under nonuniform eigenstrains ε∗ satisfying the compatibility equation of strains, RcI =

Rc0 , (k0 + k1 )

(3.20)

and the critical radius RcII for the spherical QD under nonuniform eigenstrains ε∗ not satisfying the compatibility equation, RcII =

Rc0 , (k0 + 0.75k1 )

(3.21)

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2.00 RcI RcII

Rcr /Rc0

1.75 1.50 1.25 1.00 20.5

20.4

20.3

20.2

20.1

0.0

k1

Fig. 3.6 Variation of critical radii for dislocation nucleation in QDs with the composition parameter k1 (k0 = 1). Reprinted from Duan et al. (2006b) with acknowledgment to American Physical Society.

where Rc0 is the critical radius of a spherical QD under uniform hydrostatic eigenstrain ε∗m0 (Kolesnikova and Romanov, 2004)   1.083αRc0 3b Rc0 = ln . (3.22) 8π(1 + ν)ε∗m0 b It can be seen from Eqs. (3.20) and (3.21) that RcI /RcII = 1 − 0.25k1 / (k0 + k1 ). It is evident that k0 > 0, k0 + k1 > 0. Generally, for the “self-capping” alloyed QD (e.g., Inx Ga1−x As on GaAs substrate), the core is enriched in In, whereas the outermost layer becomes progressively depleted in In (Tersoff, 1998). Therefore, in the composition profile chosen here, k1 < 0. Fig. 3.6 shows the variation of the normalized critical radii Rcr /Rc0 with k1 (where Rcr stands for RcI and RcII ). The results show that dislocation nucleation is more difficult in a compositionally nonuniform QD than in a uniform one (cf., Rcr /Rc0 > 1) in both situations. However, it is even more difficult when the compatibility equation is satisfied than when it is not (RcI /Rc0 > RcII /Rc0 for k1 < 0). 4. Micromechanical Framework for Nano-inhomogeneities with Interface Stress The interface stress contributes to the effective moduli of composites in two ways. First, it affects the average stress (strain) in each inhomogeneity, and this effect can be taken into account by the use of the so-called stress (strain) concentration tensor in the inhomogeneity (cf. Section 2.2). Second, the discontinuities in the traction across the interface directly participate in the calculation of the volume average of the strain or stress, and this effect can be taken into account by use of the stress (strain) concentration tensor at the interface. The effective

Theory of Elasticity at the Nanoscale

31

moduli of solids with interface stress effect can be modelled with the well-known micromechanical schemes, provided these are suitably modified to account for the discontinuity in the tractions across the interfaces. For example, Benveniste (1985) has given the general framework with consideration of the displacement discontinuity, and Hashin (1991) has used this framework to predict the effective moduli of composites with linear spring interfaces (displacement discontinuity). In view of the importance of the surface/interface stress effect at the nanoscale, a framework has been proposed by Duan et al. (2005b) to include stress discontinuity in order to take into account the surface/interface stress effect. Many micromechanical schemes have been successfully used for obtaining effective elastic constants of heterogeneous solids. For a comprehensive exposition, one can refer to the monographs of Aboudi (1991), Nemat-Nasser and Hori (1999), Milton (2002), and Torquato (2002). For the prediction of the effective properties of nonlinear composites, one can refer to the works of Ponte Castañeda and Suquet (1998) and Willis (2000). For the sake of simplicity but without loss of the physical essence of the surface/interface stress effect, we shall use three schemes to predict the effective elastic constants of solids containing nanoinhomogeneities with the surface/interface stress effect described by Eqs. (2.2) and (2.6). The three schemes are Hashin’s composite sphere assemblage model (CSA; Hashin, 1962), the Mori–Tanaka method (MTM; Mori and Tanaka, 1973), and the generalized self-consistent method (GSCM; Christensen and Lo, 1979). Consider a representative volume element (RVE) consisting of a two-phase medium occupying a volume V with external boundary S, and let VI and Vm denote the volumes of the two phases I and m , respectively. The interface stress effect is taken into account at the interface  with outward unit normal n between I and m . The heterogeneous material is assumed to be statistically homogeneous with the inhomogeneity moduli CI (compliance tensor DI ) and matrix moduli Cm (compliance tensor Dm ). fI and (1 − fI ) denote the volume fractions of the inhomogeneity and matrix, respectively. To define the effective elastic moduli of a composite, we use the usual concept of homogeneous boundary conditions imposed on a RVE. As in the work of Benveniste and Miloh (2001), we define the average strain ε¯ and average stress σ¯ as follows:  1 (N ⊗ u + u ⊗ N) dS, ε¯ = 2V S  1 (σ · N) ⊗ x dS, σ¯ = V S

(4.1) (4.2)

where N is the outward unit normal vector to S, and x is the position vector. In the presence of interface stress effect (stress discontinuity), the average strain and

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H. L. Duan et al.

average stress are ε¯ = (1 − fI ) ε¯ m + fI ε¯ I , σ¯ = (1 − fI ) σ¯ m + fI σ¯ I +

fI VI

(4.3)

 ([σ] · n) ⊗ x d,

(4.4)



where ε¯ k and σ¯ k (k = I, m) denote volume averages of the strain and stress over the respective phases in the RVE. [σ] = σ I − σ m . As usual, the effective elastic moduli of the composite can then be determined by subjecting the external surface S to homogeneous displacement or traction boundary conditions, defined as u(S) = ε0 · x,

(4.5)

(S) = σ · N,

(4.6)

0

ε0

σ0

where and are constant strain and stress tensors, respectively. In the following, we will first derive formulas relating the average stress (strain) in the inhomogeneities and at the interface to the applied stress (strain) under both types of boundary condition (4.5) and (4.6). These formulas are needed to calculate the effective moduli of the composite according to the dilute concentration approximation and GSCM schemes. It is easy to derive formulas relating the average stress (strain) in the inhomogeneities and at the interface to the average stress (strain) in the matrix, again under both types of boundary condition (4.5) and (4.6). These formulas are needed in MTM. Under homogeneous displacement boundary conditions Eq. (4.5), define a strain concentration tensor R in the inhomogeneity and a strain concentration tensor T at the interface such that  1 I 0 ([σ] · n) ⊗ x d = Cm : T : ε0 ε¯ = R : ε , (4.7) VI  ¯ of the composite From Eqs. (4.3)–(4.5) and (4.7), the effective stiffness tensor C is given by ¯ = Cm + fI (CI − Cm ) : R + fI Cm : T. C

(4.8)

Under homogeneous traction boundary conditions Eq. (4.6), define two stress concentration tensors U (in the inhomogeneity) and W (at the interface) by the relations  1 I 0 (4.9) ([σ] · n) ⊗ x d = W : σ 0 . σ¯ = U : σ , VI  Then the effective compliance tensor S¯ of the composite is given by S¯ = Sm + fI (SI − Sm ) : U − fI Sm : W.

(4.10)

Theory of Elasticity at the Nanoscale

33

Eqs. (4.8) and (4.10) can be used to calculate the effective moduli of composites by using the dilute concentration approximation and GSCM once R, T, U, and W have been obtained. If the inhomogeneity and matrix are both isotropic, and the composite is macroscopically isotropic, then R and T in Eq. (4.7) can be expressed as R = R1 J 1 + R2 J 2 ,

T = T1 J1 + T2 J2

(4.11)

1 J2 = − I(2) ⊗ I(2) + I(4s) 3

(4.12)

in which J1 =

1 (2) I ⊗ I(2) , 3

with I(4s) the fourth-order symmetric identity tensor. R1 , R2 , T1 , and T2 are four scalars to be determined from the adopted micromechanical scheme. Then Eq. (4.8) decouples into κ¯ = κm + fI [(κI − κm )R1 + κm T1 ],

(4.13)

μ ¯ = μm + fI [(μI − μm )R2 + μm T2 ].

(4.14)

Let us now relate the average stress (strain) in the inhomogeneity and the average stress difference at the interface to the average stress (strain) in the matrix, following the Mori–Tanaka procedure (Benveniste, 1987). Under the homogeneous displacement boundary conditions Eq. (4.5), define two strain concentration tensors M (in the inhomogeneity) and H (at the interface) by the relations  1 ([σ] · n) ⊗ x d = Cm : H : ε¯ m , ε¯ I = M : ε¯ m , (4.15) VI 

−1 0 where ε¯ m = I(4s) + f(M − I(4s) ) : ε . Then the effective stiffness tensor can be obtained from −1

   ¯ = Cm + fI CI − Cm : M + Cm : H : I(4s) + fI (M − I(4s) ) . (4.16) C Likewise, under the homogeneous traction boundary conditions Eq. (4.6), the effective compliance tensor can be obtained from −1

   S¯ = Sm + fI SI − Sm : P − Sm : Q : I(4s) + fI (P + Q − I(4s) ) , (4.17) where P and Q are stress concentration tensors in the inhomogeneity and at the interface, respectively. They are defined by  1 ([σ] · n) ⊗ x d = Q : σ¯ m , σ¯ I = P : σ¯ m , (4.18) VI 

34

H. L. Duan et al.

−1 0 where σ¯ m = I(4s) + fI (P + Q − I(4s) ) : σ . Eqs. (4.16) and (4.17) can be used to calculate the effective moduli of composites using MTM. If the inhomogeneity, matrix, and composite are isotropic, then the two strain concentration tensors M and H in Eq. (4.15) can be expressed as M = M1 J1 + M2 J2 ,

H = H1 J1 + H2 J2 ,

(4.19)

where M1 , M2 , H1 , and H2 are four scalars to be determined using MTM. With the use of Eq. (4.19), Eq. (4.16) decouples into κ¯ = κm + fI [(κI − κm )M1 + κm H1 ] [1 + fI (M1 − 1)]−1 , μ ¯ = μm + fI [(μI − μm )M2 + μm H2 ] [1 + fI (M2 − 1)]

−1

.

(4.20) (4.21)

5. Application of Micromechanical Framework for Nano-inhomogeneities

5.1. Elastic Moduli of Solids with Spherical Nano-inhomogeneities For predicting the effective moduli of a composite containing spherical nanoinhomogeneities, it is convenient to split the boundary displacement u0 (S) into its 0 and deviatoric parts u0 , that is, dilatational um e 0 + u0 = ε0 I(2) · x + 2ε0 [e ⊗ e − 1 (e ⊗ e + e ⊗ e )] · x u0 (S) = um z x x y y m e A z 2 (5.1) in which ε0m = (trε0 )/3, ε0 is the imposed strain tensor, ε0A is a constant, and ex , ey , and ez are the unit base vectors of the Cartesian coordinate system. The boundary 0 and u0 will be used to predict the effective bulk and conditions corresponding to um e shear moduli, respectively. In the following, we will predict the effective elastic moduli of a composite with spherical nano-inhomogeneities by three schemes (CSA, MTM, and GSCM). The interface stress effect in each scheme is additionally considered at the interface between the inhomogeneity and matrix.

5.1.1. Prediction of Elastic Moduli by CSA The configuration of the CSA consists of two concentric spheres with radii r = R and r = R0 (fp = (R/R0 )3 ) that correspond to the radius of inhomogeneity and the outer radius of matrix, respectively. The boundary conditions are imposed at the outer boundary of the matrix (r = R0 ). The solutions for finding the effective moduli of the composite with spherical nano-inhomogeneities are given in the spherical coordinate system (r, θ, ϕ). The interface conditions at the interface of the

Theory of Elasticity at the Nanoscale

35

inhomogeneity and matrix (r = R) consist of displacement continuity conditions and Eq. (2.2), and for axisymmetric loading, they can be obtained from Eq. (2.3), that is, ⎧ p p ⎪ ur = um uθ = um r , ⎪ θ , ⎪ ⎪ ⎪ ⎨ (τθθ + τϕϕ ) m p σrr − σrr = , (5.2) R  ⎪ ⎪ ⎪ ⎪ p τθθ − τϕϕ 1 ∂τθθ ⎪ m ⎩ σrθ − σrθ + cot θ. = R ∂θ R The interface stresses ταβ (α, β = θ, ϕ) can be obtained from Eq. (2.6), that is, ταβ = 2μs εsαβ + λs (trεs )δαβ ,

(5.3)

where the interface strain εs is equal to the tangential strain in the abutting bulk materials (inhomogeneity and matrix). The solution for CSA should satisfy the interface conditions Eq. (5.2) and the boundary conditions at the outer surface of the matrix r = R0 . For predicting the effective bulk modulus of the composite with spherical nano-inhomogeneities, assume that the configurations for CSA undergo a hydrostatic deformation with the imposed strain being ε0m = 0. Then the boundary displacements in the spherical coordinate system are u0r = ε0m r,

u0θ = 0,

u0ϕ = 0.

(5.4)

For CSA, the nonvanishing displacement component for finding the bulk modulus of the composite is Gk (5.5) ukr = Fk r + 2 , r where Fk and Gk (k = p, m for CSA) are constants to be determined. It is noted that for CSA, the constants Fk and Gk (k = p, m) need to be determined from the boundary conditions Eq. (5.4) at r = R0 , the interface conditions Eq. (5.2), and the condition to avoid a singularity at r = 0. The bulk modulus is then obtained from Eq. (4.13). It is found that the effective bulk modulus depends on the size of the inhomogeneities κ¯ =

3κm (3κp + 4fp μp ) + 2μp [4fp μp κsr + 3κp (2 − 2fp + κsr )] , 3[3(1 − fp )κm + 3fp κp + 2μp (2 + κsr − fp κsr )]

(5.6)

where κsr is given by Eq. (2.27). When κsr → 0, the result reduces to the classical one (without the interface stress effect). As only bounds can be obtained for the shear modulus of CSA, we do not predict the effective shear modulus of the composites by it.

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5.1.2. Prediction of Elastic Moduli by MTM For a composite with spherical nano-inhomogeneities, the configuration of MTM is a spherical nano-inhomogeneity with radius r = R embedded in an infinite matrix subjected to an imposed remote field equal to the as-yet-unknown average stress (strain) field in the matrix of the composite. Similar to those for predicting the effective bulk modulus by CSA, the boundary displacements and the displacement component for finding the bulk modulus of the composite are still given by Eqs. (5.4) and (5.5). It is noted that for MTM, Fk and Gk (k = p, m) need to be determined from the remote boundary conditions Eq. (5.4), the interface conditions Eq. (5.2), and the condition to avoid a singularity at r = 0. The effective bulk modulus is then obtained from Eq. (4.20), and the expression of effective bulk modulus is identical to Eq. (5.6). In order to predict the effective shear modulus of the composites, the imposed deviatoric strain (see Eq. (5.1)) is chosen as ε0xx = ε0yy = −ε0A , ε0zz = 2ε0A . The corresponding remote displacements in the spherical coordinate system are u0r = 2ε0A rP2 (cos θ),

u0θ = ε0A r

dP2 (cos θ) , dθ

u0ϕ = 0

at

r → +∞. (5.7)

For the present axisymmetric deformation, the corresponding solutions have the same form as Eqs. (2.12)–(2.14) for the MTM configuration. However, the k , C k , and Dk are replaced by A , B , C , and D (k = p, m), constants Akzz , Bzz k k k k zz zz respectively. From the remote displacement conditions Eq. (5.7), it follows that Am = 0 and Bm = ε0A ; to avoid a singularity at r = 0, it is necessary that Cp = Dp = 0. The remaining four constants are determined by the four interface conditions Eq. (5.2). Finally, the effective shear modulus using the MTM can be obtained from Eq. (4.21) and is μ ¯ =

μm [5 − 8fp ξ3 (7 − 5νm )] , 5 − fp (5 − 84ξ1 − 20ξ2 )

(5.8)

where 15(1 − νm )(κsr + 2μrs ) , 4H −15(1 − νm ) ξ2 = [η(7 + 5νp ) − 8νp (5 + 3κsr + μrs ) + 4H + 7(4 + 3κsr + 2μrs )], ξ1 =

ξ3 =

5 {2η2 (7 + 5νp ) − 4(7 − 10νp )(2 + κsr )(1 − μrs ) + 16H + η[7(6 + 5κsr + 4μrs ) − νp (90 + 47κsr + 4μrs )]},

(5.9)

(5.10)

(5.11)

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37

H = −2η2 (7 + 5νp )(4 − 5νm ) + 7η[−39 − 20κsr − 16μrs + 5νm (9 + 5κsr + + 4μrs )] + ηνp [285 + 188κsr + 16μrs − 5νm (75 + 47κsr + 4μrs )] + + 4(7 − 10νp ){−7 − 11μrs − κsr (5 + 4μrs ) + + νm [5 + 13μrs + κsr (4 + 5μrs )]},

(5.12)

with μrs is given by Eq. (2.27), η = μI /μm . The effective shear modulus of the composite depends on the size of the inhomogeneity through two nondimensional parameters κsr and μrs . When κs and μs are equal to zero or κsr → 0, μrs → 0, the shear modulus reduces to the classical result. 5.1.3. Prediction of Elastic Moduli by GSCM The configuration of the GSCM is a three-phase model, that is, a spherical nanoinhomogeneity (r = R) with a matrix shell (outer radius r = R0 ) embedded in an infinite effective medium (i.e., the composite material) and boundary conditions are specified at infinity. In the GSCM scheme, the interface stress effect is additionally considered at the interface between the inhomogeneity and matrix (cf. Eq. (5.2)), and conventional stress and displacement continuity conditions are assumed to prevail at the interface between the matrix shell and effective medium (r = R0 ), that is, e um r = ur ,

e um θ = uθ ,

m = σe , σrr rr

m = σe , σrθ rθ

(5.13)

where the superscript “e” denotes the effective medium. The boundary displacements and the nonvanishing displacement component for finding the bulk modulus of the composite are still given by Eqs. (5.4) and (5.5). For GSCM, the constants Fk and Gk (k = p, m, e) need to be determined from the remote boundary conditions Eq. (5.4), the interface conditions Eqs. (5.2) and (5.13), and the condition to avoid a singularity at r = 0. The bulk modulus is then obtained from Eq. (4.13). It is found that, like the classical case without the interface stress effect, the CSA , MTM, and GSCM give the same prediction of the effective bulk modulus for a given composite, but unlike the classical results, the effective modulus depends on the size of the inhomogeneities In order to predict the effective shear modulus by GSCM, we still apply the boundary condition Eq. (5.7). The corresponding solutions have the same k , C k , and Dk replaced by form as Eqs. (2.12)–(2.14) with the constants Akzz , Bzz zz zz Ak , Bk , Ck , and Dk , respectively. In the GSCM model, the constants Ak , Bk , Ck , and Dk (k = p, m, e) are determined by the interface conditions, Eqs. (5.2) and (5.13), and the remote boundary conditions. To avoid a singularity at r = 0, we must have Cp = Dp = 0; to satisfy the remote displacement conditions Eq. (5.7),

38

H. L. Duan et al.

we must have Ae = 0 and Be = ε0A . The remaining eight constants are determined from the eight interface conditions Eqs. (5.2) and (5.13). Moreover, it can be shown by the energy method proposed by Christensen and Lo (1979) that Ce = 0. This method is based on the argument that the strain energy stored in the embedded composite sphere is given by the product of the average strain energy density of the composite and the volume of the composite sphere. Here, we will prove that Ce = 0 by energy method for nano-inhomogeneity with interface stress effect. For a homogeneous medium containing an nanoinhomogeneity with interface stress effect, the strain energy U, under imposed displacement boundary conditions Eq. (4.5), is given by Eq. (2.47). In Eq. (2.47), U represents the total strain energy stored in the configuration of GSCM (with interface stress effect). Following the procedure of Christensen and Lo (1979), a sphere of radius r = R0 is removed from the effective homogeneous medium and replaced by a composite sphere with interface stress effect (with the outer radius r = R0 and inner radius r = R) such that (R/R0 )3 = fp . The strain energy of the medium as a whole should remain unaffected by this replacement. In terms of the Eshelby formula Eq. (2.47), this implies that if U0 denotes the strain energy of the effective homogeneous medium, and U stands for the strain energy of the same medium but now containing the composite sphere with interface stress effect at r = R, then U = U0 . The integral along  in Eq. (2.47), which is to be evaluated at r = R0 , should then vanish. Substituting the expressions of the displacement and stress of GSCM configuration into the integral in Eq. (2.47) leads to Ce = 0. The effective shear modulus using GSCM can be obtained by Eq. (4.14), in which R1 and T1 are given by p

R1 =

e¯ zz , 2ε0A

T1 =

¯ zz 1 ϒ , 2μm 2ε0A

(5.14)

p

where e¯ zz is the component of volume average deviatoric strain tensor (¯ep ) in ¯ zz is the volume average tensor of stress difference at the the inhomogeneity, ϒ ¯ is defined by interface in the z-direction, and ϒ ¯ = ϒ

1 VI

 ([s] · n) ⊗ x d,

(5.15)



where [s] = sm − sp with sk = 2μk ek (k = p, m) being the deviatoric stress tensor. Then according to Eqs. (5.2), (5.13), (4.14), and (5.14), the effective shear modulus obtained by GSCM is a quadratic equation in η∗ = μ/μ ¯ m, A η∗2 + B η∗ + C = 0,

(5.16)

Theory of Elasticity at the Nanoscale

39

where A , B , and C are functions of the volume fractions and elastic moduli of the inhomogeneity, matrix, and interface. The solution of this quadratic equation gives the normalized effective shear modulus μ/μ ¯ m. Like the Eshelby and the stress concentration tensors in the preceding section, the effective moduli are functions of the two intrinsic length scales lκ and lμ defined in Eq. (2.27). Duan et al. (2005b) calculated the effective moduli of aluminium containing spherical nanovoids using the free-surface properties obtained by molecular dynamic simulations (Miller and Shenoy, 2000). Two sets of surface moduli (Cases A and B mentioned in Section 3.1) are used. The normalized bulk modulus κ¯ /¯κc for different surface properties as a function of the void radius R is plotted in Fig. 5.1, and the variation of μ/ ¯ μ ¯ c (calculated by GSCM) is plotted in Fig. 5.2, where “A” and “B” denote the two sets of surface properties and “C” the

A, [1 0 0] B, [1 1 1] C

1.2

␬/␬c

1.1 1.0 0.9 10

20

30 R(nm)

40

50

Fig. 5.1 Effective bulk modulus as a function of void radius (f = 0.3). A: κs = −5.457 N/m, μs = −6.2178 N/m for the surface [1 0 0]; B: κs = 12.932 N/m, μs = −0.3755 N/m for the surface [1 1 1]; C: the classical results without the surface stress effect. Reprinted from Duan et al. (2005b) with acknowledgment to Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK. 1.05 0.90 ␮/␮c

A, [1 0 0] B, [1 1 1] C

0.75 0.60 10

20

30 R(nm)

40

50

Fig. 5.2 Effective shear modulus as a function of void radius (f = 0.3). A: κs = −5.457 N/m, μs = −6.2178 N/m for the surface [1 0 0]; B: κs = 12.932 N/m, μs = −0.3755 N/m for the surface [1 1 1]; C: the classical results without the surface stress effect. Reprinted from Duan et al. (2005b) with acknowledgment to Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK.

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H. L. Duan et al.

¯ c represent the classical results without the surface stress classical result. κ¯ c and μ effect. It is found that the surface stress effect has a significant effect on the effective bulk and shear moduli, especially when nanovoids are less than 10 nm in radius. The surface stress effect becomes negligible when the radius is larger than 50 nm.

5.2. Generalized Levin’s Formula and Hill’s Connections Levin (1967) derived a simple formula relating the effective CTE to the effective elastic moduli of a two-phase heterogeneous material. Earlier, Hill (1964) had shown that for any two-phase transversely isotropic heterogeneous material containing parallel cylindrical fibers, the five effective elastic constants are interrelated, with only three of them being independent. Both Levin’s formula and Hill’s connections are for a perfect interface in the sense that the field quantities are continuous across it. The question arises, do Levin’s formula and Hill’s connections have counterparts in the presence of surface/interface stress? Here, this question will be answered in the affirmative. We will give the generalized Levin’s formula and Hill’s connections for the heterogeneous materials containing spherical particles or cylindrical fibers, which take into account the surface/interface stress effect between the inhomogeneity and the matrix. In the following, if we assume that there exists a homogeneous deformation field in a RVE of the heterogeneous material, we will prove that, under specific types of external loading and a uniform temperature change, this uniform deformation field can satisfy all basic equations and can be viewed as a characteristic field in various admissible deformation modes (Dvorak, 1990). We shall use it to obtain the generalized Levin’ formula and the generalized Hill’s connections for the heterogeneous media with interface stress effect. 5.2.1. Generalized Levin’s Formula for Spherical Particles Assume that the matrix and the spherical particles are isotropic and the latter are randomly distributed in the former. The effective elastic moduli and the effective CTE of such an heterogeneous medium are also isotropic. Consider a RVE consisting of a volume V of the two-phase medium, and let Vp and Vm denote the volumes of the spherical particles and matrix, respectively. The interface stress effect is taken into account at the interface  between a particle and the matrix. Assume that the RVE is subjected to a hydrostatic deformation together with a uniform temperature change T . We will now prove that, under these deformation and temperature conditions, it is possible to generate the following uniform displacement field in the RVE (in the spherical coordinate system (r, θ, ϕ)): ukr (r, θ, ϕ) = e0 r,

(5.17)

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41

where e0 is a constant. Under this hydrostatic deformation mode, the boundary conditions (Eqs. (2.2) and (2.6)) at the interfaces between the particle and matrix would require that the following equation be satisfied     2κs 2αs κs e0 3κm − 3κp − − T 3αm κm − 3αp κp − = 0. (5.18) R R Thus, if the external deformation e0 and the temperature change T are chosen according to Eq. (5.18), then the uniform deformation field Eq. (5.17) can indeed prevail inside the RVE. In this case, the spherical particles in the matrix can be regarded as neutral ones in the sense that they do not disturb the elastic field Eq. (5.17). The volume average stress σ¯ and strain ε¯ in the RVE with interface stress effects are given by Eqs. (4.3) and (4.4). Under the uniform deformation field in Eq. (5.17), it follows from Eqs. (2.2), (2.6), (4.3), and (4.4) that 

3fk κk (e0 − αk T ) + fp

k=p,m

2κs (e0 − αs T ) = 3¯κ(e0 − αT ¯ ). R

(5.19)

The solution of Eqs. (5.18) and (5.19) gives an exact relationship between the effective bulk modulus κ¯ and the effective CTE α¯ 2fp αs κs 2αs κs fk αk κk − α¯ ¯κ − 3R k=p,m 3R = . 2fp κs 2κs κp − κ m + κ¯ − fk κ k − 3R 3R k=p,m

αp κp − αm κm +

(5.20)

Eq. (5.20) is the sought-after generalized Levin’s formula for a two-phase heterogeneous medium containing spherical particles with interface stress effect. Note that Chen et al. (2007a) also obtained a relation (Eq. (31) in their paper) between κ¯ and α¯ for a heterogeneous medium containing spherical particles with interface stress effect. However, they did not include the thermoelastic term (T D0 ) in the interface constitutive equation Eq. (2.6). If we let αs = 0, Eq. (5.20) reduces to Eq. (31) of Chen et al. (2007a), and it reduces to Levin’s formula (Levin, 1967) when κs = 0 and αs = 0. 5.2.2. Generalized Levin’s Formula and Hill’s Connections for Cylindrical Fibers For a RVE of a heterogeneous medium containing aligned cylindrical fibers of the same radii with interface stress effect, we choose a cylindrical coordinate system (ρ, φ, z) in which the ρ − φ-plane is perpendicular and the z-axis is parallel to the fibers. We assume that the fibers are transversely isotropic so that the

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H. L. Duan et al.

thermoelastic constitutive equations of the cylindrical fibers and matrix in Eq. (2.1) can be expressed in the following form: ⎛ k ⎞ ⎛ ⎞ ⎛ εk − α T ⎞ σρ kT ρ kk + μk kk − μk lk ⎜ k ⎟ ⎜ k ⎟ ⎝ ⎠ kk − μk kk + μk lk ⎝ εφ − αkT T ⎠ , ⎝ σφ ⎠ = (5.21) lk lk nk σk εk − α T z

k = 2p εk , σρz k ρz

z

k = 2p εk , σφz k φz

kL

k = 2μ εk , σρφ k ρφ

where kk , lk , nk , μk , and pk (k = f, m) are the elastic moduli of the fiber and the matrix in Hill’s notation (Hill, 1964), and αfT and αfL are the CTEs of the fibers in the transverse and longitudinal directions, respectively. For an isotropic matrix, αmT = αmL = αm , and km , lm , and nm can be expressed in terms of the bulk and shear moduli κm and μm , namely km = κm + μm /3, nm = κm + 4μm /3, pm = μm , lm = κm − 2μm /3. The heterogeneous medium with aligned cylindrical fibers exhibits transverse isotropy and has five effective elastic constants, that ¯ the longitudinal Young modulus is, the transverse plane-strain bulk modulus k, ¯ ¯ L , and the EL , the longitudinal Poisson ratio ν¯ L , the longitudinal shear modulus μ transverse shear modulus μ ¯ T . The effective CTEs are also transversely isotropic with the transverse and longitudinal components α¯ T and α¯ L . We now prove that there exists a certain axisymmetric loading that includes a uniform temperature change T which creates a uniform deformation field in the RVE, described by the strain field ερρ = e0 ,

εφφ = e0 ,

εzz = ε0 .

(5.22)

For this axisymmetric deformation mode to exist, the interface conditions (Eqs. (2.2) and (2.6)) between the fiber and matrix must satisfy the following equation:     λs λs + 2μs ε0 lm − lf − + e0 2km − 2kf − ρ0 ρ0 (5.23)   αsT κs − lm αmL + 2km αmT − lf αfL − 2kf αfT − T = 0, ρ0 where ρ0 is the radius of the fibers, and αsT is the interface CTE in the transverse direction. Eq. (5.23) can be satisfied by an appropriate choice of the external loading parameters e0 , ε0 , and T . In this case, the cylindrical fibers in the matrix can be regarded as neutral ones in the sense that they do not disturb the elastic fields Eq. (5.22). We now derive the volume average stress and strain in the RVE with aligned cylindrical fibers under the local strain field Eq. (5.22). From Eqs. (4.3) and (4.4),

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43

the volume average strain and stress (¯εij and σ¯ ij , i, j = x, y) in the transverse plane of RVE can be written as f

ε¯ ij = ff ε¯ ij + fm ε¯ m ij , f

σ¯ ij = ff σ¯ ij + fm σ¯ ijm +

ff Sf

 

(5.24)

f

m (σik − σik )nk xj d,

where Sf is the cross-sectional area of a cylindrical fiber. The volume average strain and stress in the axial direction are τzz f m + fm σ¯ zz + 2ff , (5.25) ε¯ zz = ε0 , σ¯ zz = ff σ¯ zz ρ0 where τzz is the interface stress in the z-direction given by Eq. (2.6). From Eqs. (2.2), (5.22), (5.24), and (5.25), we can obtain the following two expressions: 

fk [2kk (e0 − αkT T ) + lk (ε0 − αkL T )]

k=f,m

+

ff [(λs + 2μs )e0 + λs ε0 − αsT κs T ] ρ0

¯ 0 − αeT T ) + le (ε0 − αeL T ), = 2k(e 

(5.26)

fk [2lk (e0 − αkT T ) + nk (ε0 − αkL T)2e0 λm ]

k=f,m

+

2ff [e0 λs + ε0 (λs + 2μs ) − αsL κs T ] ρ0

= 2le (e0 − αeT T ) + ne (ε0 − αeL T ),

(5.27)

where αsL is the CTE of the interface in the longitudinal direction. Solving Eqs. (5.23), (5.26), and (5.27), we obtain two exact expressions. ¯ ¯l, and n) The first expression relates the three effective elastic moduli (k, ¯ ff (λs + 2μs ) λs + 2μs k¯ − fk k k − 2ρ0 k=f,m 2ρ0 = λs f f λs ¯l − lm − l f − fk lk − ρ0 ρ0 k=f,m

k m − kf − l

¯l −

fk l k − k=f,m

= n¯ −

fk n k − k=f,m

ff λs ρ0

2ff (λs + 2μs ) ρ0

,

(5.28)

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H. L. Duan et al.

¯ L + 4k¯ ν¯ 2 , ρ0 is the radius of the fibers, and αsT and αsL where ¯l = 2k¯ ν¯ L , n¯ = E L are the interface CTEs in the transverse direction and in the longitudinal direction, respectively. The two equalities prove that only three of the five effective elastic moduli are independent. Eq. (5.28) is identical to Eq. (13) in the paper of Chen and Dvorak (2006). The two equalities Eq. (5.28) reduce to Hill’s connections (Hill, 1964) when λs = 0 and μs = 0. The second expression relates the effective CTE to the effective elastic moduli 2km αmT + lm αmL − (2kf αfT + lf αfL ) − km − k f − 2k¯ α¯ T + ¯lα¯ L − =

fk (2kk αkT + lk αkL ) − k=f,m

k¯ −

fk k k − k=f,m

2¯lα¯ T + n¯ α¯ L − =

λs + 2μs 2a

αsT κs ρ0

ff (λs + 2μs ) 2ρ0

fk (2lk αkT + nk αkL ) − k=f,m

¯l −

ff αsT κs ρ0

λs fk lk − ff ρ 0 k=f,m

2ff αsL κs ρ0

.

(5.29)

Eq. (5.29) can be regarded as the generalized Levin’s formula for the heterogeneous material containing aligned cylindrical fibers with interface stress discontinuity. The effective CTEs α¯ T and α¯ L can be obtained by solving the two equalities in ¯ ν¯ L , and E ¯ L are given in the paper of Chen et al. Eq. (5.29). The expressions of k, ¯ ν¯ L , (2007b). For the heterogeneous material with aligned cylindrical nanopores, k, ¯ L are given by Duan et al (2006c). Chen and Dvorak (2006) obtained relations and E (Eq. (14) in their paper) between the effective elastic moduli and effective CTEs for a heterogeneous material containing aligned cylindrical fibers with interface stress effect but without taking into account the thermoelastic term (T D0 ) in the interface constitutive equation Eq. (2.6). Eq. (5.29) reduces to their result when αsT = αsL = 0. It reduces to Levin’s formula (Levin, 1967) when κs = 0, αsT = 0, and αsL = 0. It is emphasized that Pathak and Shenoy (2005) have shown that the CTE of the surface of nanostructured materials is different from that of the parent bulk materials. By using the present continuum theory and molecular dynamics simulations, they showed that the CTE of the surface of a nanoslab has a significant effect on its effective CTE, and excellent agreement is found between the continuum theory and molecular dynamics simulations. Therefore, the generalized Levin’s formula with interface stress effect will provide a guideline for tailoring the effective CTE

Theory of Elasticity at the Nanoscale

45

of heterogeneous materials, especially, that of nanostructured materials (e.g., the nanochannel-array materials). 5.2.3. Effective CTE of Composites with Spherical, or Parallel Cylindrical, Nanopores For solids containing spherical nanopores or aligned cylindrical nanopores, the effective CTEs of solids can be obtained from the generalized Levin’s formulas (Eq. (5.20) or (5.29)) by letting the elastic mouli of the spherical particles or cylindrical fibers equal to zero. κ¯ in Eq. (5.20) is given by Eq. (5.6) with κp = ¯ ¯l, and n¯ in Eq. (5.29) for the solids containing aligned cylindrical μp = 0. k, nanopores will be given in the next application example. We will now analyze the effect of the surface elastic and thermoelastic terms on the effective CTEs of solids containing spherical nanopores or aligned cylindrical nanopores. The numerical results are presented for aluminium, and the free-surface elastic moduli are taken from the papers of Miller and Shenoy (2000) and Sharma et al. (2003). These freesurface elastic moduli of Al have been obtained by molecular dynamic simulations (Miller and Shenoy, 2000). Two sets of surface moduli (Cases A and B mentioned in Section 3.1) are used. The CTE of aluminum is αm = 5.01 × 10−5 K −1 (Pathak and Shenoy, 2005), and the volume fraction of spherical or cylindrical nanopores is fp = ff = 0.3. We assume that the surface CTE is αs = 5αm . The normalized effective CTE α/ ¯ α¯ c for spherical nanopores and α¯ T /α¯ cT for aligned cylindrical nanopores for different surface properties as a function of the pore radius are plotted in Figs. 5.3 and 5.4, where “A” and “B” denote the two designated sets of surface properties and “C” the classical result (no surface stress effect). α¯ c and α¯ cT denote the effective CTEs of porous Al with spherical and cylindrical nanopores without the surface stress effect. It is seen from Figs. 5.3 and 5.4 that α/ ¯ α¯ c and α¯ T /α¯ cT decrease (or increase) with an increase in the void size due to the surface stress effect, whereas there is no size effect when the surface stress is absent. The surface stress effect on the effective CTEs becomes negligible for pores larger than 30 nm in radius. It is noted that by using the continuum theory and molecular dynamics simulations, Pathak and Shenoy (2005) showed that the CTE of the surface of a nanoslab has a significant effect on its CTE, and similar curves as Figs. 5.3 and 5.4 for CTEs of Si and Al nanoslabs have been obtained. 5.2.4. Equivalence Between ISM and Interphase Models It can be seen from Eqs. (5.20) and (5.29) that the effective CTE α¯ e for spherical particles and the effective CTEs α¯ T and α¯ L for cylindrical fibers are dependent on the interface elastic moduli and CTEs, which must be experimentally determined. However, as the ISM can be used to simulate the behavior of thin interphases (Wang et al., 2005), these interface parameters can be deduced from the

46

H. L. Duan et al. 1.4 A, [1 0 0] B, [1 1 1] C

␣e / ␣c

1.2

1.0

0.8 1

5

10

15 R(nm)

20

25

30

Fig. 5.3 Effective normalized α/ ¯ α¯ c as a function of the radius of spherical nanopores (fp = 0.3). A: κs = −5.457 N/m, μs = −6.2178 N/m for the surface [1 0 0]; B: κs = 12.932 N/m, μs = −0.3755 N/m for the surface [1 1 1]; C: the classical results without the surface stress effect. Reprinted from Duan and Karihaloo (2007) with acknowledgment to Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK. 1.50 A, [1 0 0] B, [1 0 0] C

␣–eT / ␣–cT

1.25

1.00

0.75 1

5

10

15 20 ␳0(nm)

25

30

Fig. 5.4 Effective normalized α¯ T /α¯ cT as a function of the radius of cylindrical nanopores (fp = 0.3). A: κs = −5.457 N/m, μs = −6.2178 N/m for the surface [1 0 0]; B: κs = 12.932 N/m, μs = −0.3755 N/m for the surface [1 1 1]; C: the classical results without the surface stress effect. Reprinted from Duan and Karihaloo (2007)) with acknowledgment to Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK.

equivalence relationships that exist between the thin interphase and the ISM. Such relationships, however, only exist for elastic moduli (Benveniste and Miloh, 2001; Hashin, 2002; Wang et al., 2005; Benveniste, 2006). The procedures of Hashin (2002) and Wang et al. (2005) can, however, be followed to derive similar relationships for the thermoelastic problem, using appropriate thermoelastic constitutive relations for the inhomogeneity, matrix, and interface (Eqs. (2.1) and (2.6)). We, therefore, derive these relationships giving only the most essential details. For a thin interphase between the inhomogeneity and the matrix, denote the interface between the inhomogeneity and the interphase by 1 , and that between

Theory of Elasticity at the Nanoscale

47

the interphase and the matrix by 2 . For brevity, we illustrate the procedure on a spherical particle; the procedure is the same for a cylindrical fiber. The differences in the displacement and stress (uj and σrj , j = r, θ, ϕ) across the interphase can be approximated with sufficient accuracy (Hashin, 2002) by the relations ' uj = ucj (R + t) − ucj (R) = t ucj, r '1 , (5.30) ' c ' σrj = σrjc (R + t) − σrjc (R) = t σrj, r 1 , where t is the interphase thickness, the subscript 1 means that the corresponding quantities are evaluated at interface 1 , and the superscript “c” denotes the interphase. To obtain the interface conditions equivalent to the thin interphase, the differences in the displacement and stress (uj and σrj ) across the interphase should be equal to the displacement and stress jumps ([uj ] and [σrj ]) across the interface (), [uj ] = uj ,

[σrj ] = σrj .

(5.31)

Here, [uj ] and [σrj ] are defined by p

[uj ] ≡ um j () − uj (),

p

[σrj ] ≡ σrjm () − σrj (),

(5.32)

where  is the interface between the inhomogeneity and matrix. For a thin and stiff interphase, the characteristics of the displacements and stresses across the interphase can be described by  uj = 0, σrj = finite (Wang et al., 2005), which have the same features as those of the ISM (Eq. (2)), p p m namely um j () = uj (), [σrj ] = σrj () − σrj () = σrj . Therefore, the equivalence between a thin and stiff isotropic interphase and an ISM is assured if the elastic moduli and CTE (both for the spherical particle and cylindrical fiber) of the latter meet the following conditions: λs =

2μc νc t , 1 − νc

μs = μc t,

αs = αc ,

(5.33)

where κs = 2(λs + μs ). νc , μc , and αc are the Poisson ratio, shear modulus, and CTE of the interphase, respectively.

5.3. Prediction and Tailoring of Elastic Moduli of Nanochannel-array Materials The fabrication, structure, and properties of nanostructured materials with feature sizes in the tens to several hundreds of nanometer range have attracted considerable recent interest. These materials include nanochannel-array materials,

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nanocrystalline metals, thin films, nanowires, nanobeams, nanobelts, etc. Particularly, the relationship between their structures and functional properties has been the major focus of the studies. As the number of atoms near the surface/interface in these materials is large compared to the total number of atoms, the surface/interface stress effect on their properties can be significant. Gleiter (2000) and Weissmüller et al. (2003) have pointed out that nanostructured materials with a large ratio of the interface/surface to volume open the way to modifying the electronic structure and related properties of solids. Among the above-mentioned nanostructured materials, nanochannel-array materials have been extensively used in nanotechnology. They can be used as filters and in catalytic convertors, as well as templates for nanosized magnetic, electronic, and optoelectronic devices (Masuda and Fukuda, 1995; Martin and Siwy, 2004; Shi et al., 2004). As these materials possess a large surface area, pore surfaces can be modified to create nanoporous materials that are very stiff and light and have very low thermal conductivity. One important immediate application of these materials is as cores in sandwich construction. 5.3.1. Prediction of Elastic Moduli of Nanochannel-array Materials For the nanochannel-array materials with aligned cylindrical nanopores (Fig. 5.5), we choose a Cartesian coordinate system (x, y, z) such that the z-axis is parallel to aligned cylindrical nanopores and the x − y-plane is perpendicular to them. In a cylindrical coordinate system (ρ, φ, z), the ρ − φ-plane is perpendicular to the pores. For generality, we assume that the matrix is linearly elastic

Fig. 5.5 A nanochannel-array material with aligned cylindrical nanopores. Reprinted from Duan et al. (2006c) with acknowledgment to Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK.

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49

and transversely isotropic in the plane perpendicular to the pores, and the surface of the aligned cylindrical nanopores is orthotropic. Then, the surface constitutive equation can be written as follows: s s s s s τzz = C1111 εszz + C1122 εsφφ , τφφ = C2211 εszz + C2222 εsφφ , τφz = 2C1212 εsφz , (5.34) s s s s s , C2222 , C1122 , C2211 , and C1212 are the surface elastic constants. Since where C1111 nanochannel-array materials have aligned and approximately hexagonal closepacked array of cylindrical pores (Masuda and Fukuda, 1995; Miyata et al., 2004), they will exhibit an overall transversely isotropic response. Therefore, we shall ¯ E ¯ L , ν¯ L , μ predict the five effective elastic constants, namely k, ¯ L , and μ ¯ T . The subscript “L” denotes a longitudinal property along the axis of the pores and “T” a transverse property perpendicular to them; here and in the following, the superand subscripts “m,” “e,” and “s” denote the quantities pertaining to the matrix, the effective medium, and the surface, respectively. We shall use two micromechanical models, namely the composite cylinder assemblage model (CCA) (Hashin and Rosen, 1964) and the generalized selfconsistent method (GSCM) (Christensen and Lo, 1979), to predict the effective (overall) elastic constants of the nanochannel-array materials with aligned cylindrical nanopores. As mentioned above, these methods are simple and have been found to give accurate predictions of effective elastic constants of heterogeneous materials. The corresponding micromechanical framework with interface stress effect has been given in Section 4. The configuration of the composite cylinder model for predicting the effective elastic constants consists of two concentric cylinders with radii ρ = ρ0 and ρ = ρ1 , which correspond to the radius of a cylindrical pore and the outer radius of matrix, respectively. The boundary conditions are imposed at the outer boundary of the matrix (ρ = ρ1 ). The configuration of the GSCM is a three-phase model, that is, a cylindrical pore (ρ = ρ0 ) with a matrix shell (outer radius ρ = ρ1 ) embedded in an infinite effective medium, and the boundary conditions are specified at infinity. Unlike the conventional composites containing aligned cylindrical fibers or pores, here the surface stress effect needs to be additionally considered on the surface of the cylindrical pore (ρ = ρ0 ) in each model. The boundary conditions on the surface of the cylindrical pore (ρ = ρ0 ) are given by Eq. (2.3), and in the cylindrical coordinate system, they can be expressed as

m = σρρ

τφφ , ρ0

m σρφ =−

∂τzφ 1 ∂τφφ − , ρ0 ∂φ ∂z

m =− σρz

∂τzz 1 ∂τφz − . ∂z ρ0 ∂φ

(5.35)

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H. L. Duan et al.

The surface stress ταβ (α, β = φ, z) can be obtained from Eq. (5.34). For the GSCM, the interface between the matrix and the effective medium (ρ = ρ1 ) is perfectly bonded, that is, the conventional continuity conditions prevail e um ρ = uρ ,

e um φ = uφ ,

e um z = uz ,

m e σρρ = σρρ ,

m e σρφ = σρφ ,

m e σρz = σρz .

(5.36) The solution for CCA should satisfy the boundary conditions in Eq. (5.35) and the boundary condition at the outer surface of the matrix (ρ = ρ1 ). The solution for GSCM should satisfy the boundary and the interface conditions in Eqs. (5.35) and (5.36), and the remote boundary condition. In the following, the elastic constants ¯ E ¯ L , ν¯ L , μ ¯ L will be predicted by CCA and GSCM, and μ ¯ T by GSCM. k, ¯ To predict k, ¯ we apply nonvanishing strain (a) Transverse Bulk Modulus k. 0 0 0 0 0 components εxx and εyy (εxx = εyy = εm ) to the CCA and GSCM models. Then in the cylindrical coordinate system, the boundary conditions for CCA and the remote boundary conditions for GSCM can be expressed as follows: u0ρ = ε0m ρ,

u0φ = 0,

u0z = 0.

(5.37)

The displacement solutions to determine k¯ are ukρ = ak ρ +

bk , ρ

ukz = 0,

ukφ = 0,

(5.38)

where ak and bk (k = m for matrix in CCA; k = m, e for matrix and effective medium, respectively, in GSCM) are constants to be determined. For CCA, the constants am and bm need to be determined from Eq. (5.37) at ρ = ρ1 and the boundary conditions in Eq. (5.35) at ρ = ρ0 . For GSCM, the constants ak and bk (k = m, e) need to be determined from the remote boundary conditions in Eq. (5.37) and the boundary and the interface conditions in Eqs. (5.35) and (5.36). After the constants are obtained, the effective plane-strain bulk modulus of the nanochannel-array material can be calculated. As for the effective bulk modulus of a medium containing spherical nano-inhomogeneities (Eq. (5.6)), here it is found that both CCA and GSCM give the same prediction of k¯ for the nanochannel-array material k¯ =

∗ ] μTm [2(1 − f )km + (km + fμTm )C2222 , ∗ 2fkm + μTm [2 + (1 − f )C2222 ]

(5.39)

m m )/2 and μ m m + C2233 where km = (C2222 Tm = (C2222 − C2233 )/2, and the nondi∗ s mensional parameters Cklrs = Cklrs /(ρ0 μTm ) have been introduced here and for use below. f denotes the porosity.

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51

¯ L and ¯ L and Poisson’s ratio ν¯ L . To determine E (b) Longitudinal Modulus E 0 ν¯ L , we apply a nonvanishing strain component εzz to the CCA and GSCM models. Then in the cylindrical coordinate system, the displacement field for CCA and the remote boundary conditions for GSCM can be expressed as u0ρ = 0,

u0φ = 0,

u0z = ε0zz z.

(5.40)

¯ L and ν¯ L The general expressions of the displacement components to determine E are bk (5.41) ukρ = ak ρ + , ukφ = 0, ukz = ε0zz z, ρ where again, ak and bk (k = m for CCA; k = m, e for GSCM) are constants to be determined. For CCA, the constants am and bm need to be determined from Eq. (5.40) at ρ = ρ1 and the boundary conditions in Eq. (5.35) at ρ = ρ0 . For GSCM, the constants ak and bk (k = m, e) need to be determined from the remote boundary conditions in Eq. (5.40) and the boundary and the interface conditions in Eqs. (5.35) and (5.36). It is found that both CCA and GSCM give the same ¯ L and ν¯ L predictions of E ¯ L = (1 − f )nm + E

∗ ) 4f(1 − f )km νLm (−2km νLm + μTm C1122 2 , (5.42) − 4k¯ ν¯ L ∗ 2fkm + μTm [2 + (1 − f )C2222 ]

∗ ∗ )νLm + (μTm + km )C1122 (1 − f)(2km − μTm C2222 , ∗ 2(1 − f)km + (km + fμTm )C2222 (5.43) m /(2k ). and νLm = C1122 m

ν¯ L = (1 − f)νLm + f m where nm = C1111

(c) Longitudinal Shear Modulus μ ¯ L . To predict μ ¯ L , the following displacement field in the cylindrical coordinate system is imposed on the outer boundary of the CCA model and at infinity in the GSCM model: u0ρ = ε0xz z cos φ,

u0φ = −ε0xz z sin φ,

u0z = ε0xz ρ cos φ,

(5.44)

where ε0xz is the only nonvanishing strain component. For CCA and GSCM, the general displacement solutions to determine μ ¯ L are ukρ = ε0xz z cos φ,

ukφ = −ε0xz z sin φ,

ukz = ε0xz (ak ρ +

bk ) cos φ, ρ

(5.45)

where ak and bk (k = m for CCA; k = m, e for GSCM) are constants to be determined. For CCA, the constants am and bm need to be determined from Eq. (5.44)

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at ρ = ρ1 and the boundary conditions at ρ = ρ0 in Eq. (5.35). For GSCM, the constants ak and bk (k = m, e) need to be determined from the remote boundary conditions in Eq. (5.44) and the boundary and interface conditions in Eqs. (5.35) and (5.36). Again, it is found that CCA and GSCM give the same predictions of μ ¯L ∗ (1 − f)μLm + (1 + f)μTm C1212 μ ¯ L = μLm , (5.46) ∗ (1 + f)μLm + (1 − f)μTm C1212 m . where μLm = C1212

(d) Transverse Shear Modulus μ ¯ T . As the composite cylinder assemblage model (CCA) only gives bounds on the effective transverse shear modulus, we only use the GSCM to calculate μ ¯ T . For this, as in the work of Christensen and Lo (1979), the imposed remote strain field in this model is chosen as ε0xx = −ε0yy = ε0T , and the corresponding remote displacements (ρ → +∞) in the cylindrical coordinate system are u0ρ = ε0T ρ cos 2φ,

u0φ = −ε0T ρ sin 2φ,

u0z = 0.

(5.47)

The general displacement solutions for the matrix and the effective medium in GSCM are   1 + ςk dk k

2 fk − 4 ρ, (5.48) uρ = −2 cos 2φ (1 − ςk )ak r + bk − r 2 r  ςk dk = 2 sin 2φ (2 + ςk )ak r + bk − 2 fk + 4 ρ, r r 

ukφ

2

(5.49)

where ςk = μTk /kk , (k = m, e), r = ρ/ρ0 . From the remote boundary conditions in Eq. (5.47) at ρ → ∞, we have ae = 0 and be = −ε0T /2. According to Eq. (4.14), μ ¯ T can be obtained from the following expression: μ ¯ T = μTm (1 −

¯ xx f ε¯ xx1 fϒ )+ , 0 εT 2ε0T

(5.50)

¯ in the x-direction, with ε¯ 1 ¯ xx are the components of ε¯ 1 and ϒ where ε¯ xx1 and ϒ ¯ and ϒ defined by    m  m 1 ¯ = 1 u ⊗ n + n ⊗ um d, ϒ σ · n ⊗ x d, (5.51) ε¯ 1 = 2S  S  where  is the surface of ρ = ρ0 and S is the cross-sectional area of a pore. There are six unknown constants in Eqs. (5.48) and (5.49) and two unknown material

Theory of Elasticity at the Nanoscale

53

parameters μ ¯ T and k¯ to be determined from six boundary and interface conditions in Eqs. (5.35), (5.36), and (5.50). Following the procedure for composites with the surface/interface stress effect for the spherical nano-inhomogeneities (Eq. (5.16)), it can be shown that the constant ce related to ςe should vanish, which furnishes an additional condition. As in the conventional composites and in the composites with nano-inhomogeneities, the transverse effective shear modulus of the nanochannelarray material predicted by the GSCM is calculated from a quadratic equation in terms of μ ¯ T /μTm , which is similar to Eq. (5.16), that is,     μ ¯T 2 μ ¯T + c = 0, + b (5.52) a μTm μTm where a , b , and c are functions of the volume fractions and elastic moduli of the inhomogeneity, matrix, and interface. ¯ ν¯ L , E ¯ L, μ ¯ L , and μ ¯ T that the effective It is seen from the expressions of k, elastic constants of nanochannel-array materials with parallel cylindrical pores s /μ . When the effective are governed by the intrinsic length scales lklrs = Cklrs Tm elastic constants are normalized by those of the matrix material, they become functions of the ratios of the intrinsic lengths to the characteristic length of the s material, that is, the radius ρ0 of the nanopores. When Cklrs → 0 or ρ0 → ∞, the results degenerate into the classical ones (without the surface stress effect). 5.3.2. Tailoring of Elastic Moduli of Nanochannel-array Materials When the matrix and the surface of the cylindrical pores are both isotropic, k¯ can be obtained from Eqs. (5.39) and (2.6), (1 − 2νm ) [2(1 − f) + (1 + f − 2fνm )A] k¯ = km , 2(1 + f − 2νm ) + (1 − f)(1 − 2νm )A

(5.53)

where A = (λs + 2μs )/(ρ0 μm ) is a mixed parameter related to the surface elastic properties and the radius ρ0 of the pores. Note that A becomes vanishingly small when the surface stress effect is negligible which is true when the pore radius ρ0 becomes large; Eq. (5.53) then gives the effective bulk modulus of a conventional cellular material that is always smaller than km . However, it is clear from Eq. (5.53) that if A exceeds a critical value Acr , the elastic modulus of a nanocellular material will exceed that of the matrix material. This critical value Acr is independent of the porosity and is simply 2 . (5.54) Acr = (1 − 2νm ) If the Poisson ratio of the matrix material is νm = 0.3, then the critical value Acr is 5 so that the combined surface elastic modulus λs + 2μs = 5ρ0 μm . For aluminium,

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μm = 34.7 GPa so that λs + 2μs should reach 347 N/m for the nanocellular material with ρ0 = 2 nm to be stiffer than the nonporous counterpart. The combined surface elastic modulus λs + 2μs of aluminium obtained by molecular dynamic simulations is in the order 10 N/m (Miller and Shenoy, 2000). This surface elastic modulus is nowhere near enough to make nanocellular aluminium stiffer than nonporous Al. However, the surface elastic constants can be enhanced by, for example, chemical modification of the pore surface, as mentioned above, or a parent material with a smaller Poisson ratio can be chosen to achieve the stiffening effect. For instance, if the Poisson ratio of the parent material is close to zero, then the stiffening effect is possible when Acr is just 2. If the parent material has a negative Poisson ratio, then Acr is further reduced and a much more manageable combined surface elastic modulus λs + 2μs is needed to achieve the stiffening of the parent material. The effective longitudinal shear modulus μ ¯ L of the nanocellular material, which determines its resistance to shearing along the direction of the pores, can be obtained from Eq. (5.46) μ ¯ L = μm

[1 − f + (1 + f)B] . [1 + f + (1 − f)B]

(5.55)

The mixed surface parameter is B = μs /(ρ0 μm ). It is easy to see that there also exists a critical value Bcr = 1, and when B > Bcr , the effective longitudinal shear modulus of the nanocellular material will exceed that of the parent counterpart. The surface elastic constant μs needs to be larger than ρ0 μm to achieve the stiffening effect. In practical applications such as in aerospace engineering, the specific stiffness, that is, stiffness/density ratio, of a material is very important. The ratio of the specific shear stiffness of the nanoporous material (which is critical to the cores of sandwich panels), denoted by μ∗L and calculated from Eq. (5.55), to that of the nonporous solid, denoted by μ∗m , is μ∗L μ ¯L 1 = , ∗ μm (1 − f) μm

(5.56)

where μ ¯ L is given by Eq. (5.55). The variation of μ∗L /μ∗m with the porosity f and the mixed surface parameter B is shown in Fig. 5.6. It is seen that μ∗L /μ∗m > 1 can be obtained for even small values of B. The transverse plane-strain bulk modulus ¯ T are very important for the in-plane stiffness k¯ and the related Young modulus E and thus the stability of a sandwich panel with pores perpendicular to the facets. ∗ and E∗ /E∗ with the porosity f and the mixed surface The variations of k∗ /km m T parameter A are similar to that of μ∗L /μ∗m . Thus, they are not shown here for brevity.

Theory of Elasticity at the Nanoscale

55

2.5

␮*L/ ␮*m

2.0 1.5 1.0 0.0 0.1 0.2 0.3 4 0. 0.5 6 0. f

1.00 0.7 0.50 5 0.25 0.00 B

The ratio μ∗L /μ∗m of the specific effective longitudinal shear modulus of a nanocellular material to that of the nonporous solid versus the porosity f and the surface property B. The light-shaded area represents the region where μ∗L /μ∗m < 1 and the dark-shaded area the region where μ∗L /μ∗m > 1. Reprinted from Duan et al. (2006c) with acknowledgment to Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK.

Fig. 5.6

An alternative route to achieving the stiffening of a transversely isotropic nanocellular material is by coating the cylindrical pore surfaces. It has been be proved (e.g., Wang et al., 2005) that the effect of the surface elasticity is equivalent to that of a thin surface layer on the pore surface. In this case, λs and μs are given by Eq. (5.33). Therefore, by a proper choice of the properties and thickness of a coating layer, materials with cylindrical nanopores can be designed to be stiffer than their nonporous counterparts. For coated cylindrical pores, the effective elastic constants are still given by Eqs. (5.53) and (5.55) but with different expressions for the parameters A and B obtained from Eq. (5.33) A=

t μc 2 , (1 − νc ) ρ0 μm

B=

t μc . ρ0 μm

(5.57)

The critical value Acr for surface stiffening is still given by Eq. (5.54), and Bcr is still equal to 1. If we let νc = 0.3, t = 1 nm, and ρ0 = 10 nm, then when the shear ¯ L > μm . modulus ratio μc /μm exceeds 17.5, k¯ > km ; when μc /μm exceeds 10, μ From the theoretical analysis mentioned above, it can be seen that the stiffening effect can be easily obtained by surface coating. Moreover, the ratios of the specific effective moduli of nanoporous materials to those of the nonporous solids can be written in the general form k∗ k¯ = , ∗ km 1 km

μ∗L μ ¯L = , ∗ μm 1 μm

μ∗T μ ¯T = , ∗ μm 1 μm

(5.58)

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H. L. Duan et al.

where ζI 1 = 1 − f + f ζ

(

) 2t t2 + 2 , ρ0 ρ0

(5.59)

where ζI and ζ are the densities of the coating and the matrix, respectively. As can be seen from Eqs. (5.57)–(5.59), the stiffness of the nanoporous materials is only dependent on the ratio t/ρ0 of the coating thickness to the radius of the pores, and not dependent on the absolute size of the pores. Therefore, the application of the second route is not limited to the nanoporous materials. The same procedures can be used to improve the stiffness of the micro- and macroporous materials. To show the effect of the coating, the variation of μ∗L /μ∗m calculated from Eq. (5.58) versus the porosity f and the mixed surface parameter B is plotted in Fig. 5.7. The parameters used in Fig. 5.7 are t/ρ0 = 0.1 and ζI /ζ = 1. It is seen that μ∗L /μ∗m > 1 can be obtained at a small value of B. If the material of the coating has a smaller density, μ∗L /μ∗m will be larger than that shown in Fig. 5.7. The above procedures to increase the stiffness of the nanoporous materials can find applications in many areas of industry. Sandwich panels used in many industrial applications contain cellular cores, for example, honeycombs, with pores aligned perpendicular to the facets (Gibson and Ashby, 1997). The shear rigidity of such panels and therefore their resistance to shape change are determined by the effective longitudinal shear modulus in Eq. (5.55) of the cellular cores. Thus, a cellular core with a high shear modulus has great potential in the fabrication of lightweight sandwich structures which are vital for aerospace engineering and other transport industries. Although the microstructures of nanoporous materials

␮*L / ␮*m

2

1

0.0 0.1 .2 0 0.3 .4 0 0.5 .6 0 f

1.00 0.75 0.50 0.25 0.00 B

Fig. 5.7 The ratio μ∗L /μ∗m of the specific effective longitudinal shear modulus of a nanocellular material to that of the nonporous solid versus the porosity f and the surface property B (t/ρ0 = 0.1, ζI /ζ = 1). The light-shaded area represents the region where μ∗L /μ∗m < 1 and the dark-shaded area the region where μ∗L /μ∗m > 1. Reprinted from Duan et al. (2006c) with acknowledgment to Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK.

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are not exactly the same as those of the conventional honeycombs, the analysis reported here can be regarded as a first step towards increasing the stiffness of porous materials by surface modification.

6. Scaling Laws for Properties of Nanostructured Materials The study of the variation of the properties of materials with their geometrical feature size has a long history because of its importance in many fields. The interest has been heightened recently at the nanoscale because nanostructures are pervasive in nature (Kamat et al., 2000; Gao et al., 2003) and in modern industry (Goldstein et al., 1992; Streitz et al., 1994; Bertsch, 1997; Miller and Shenoy, 2000; Miyata et al., 2004), and the large ratio of surface atoms to the bulk can have a profound effect on their properties. In physics and chemistry, the effect of particle size on melting and evaporation temperatures has been discussed since 1900s, and this effect is not restricted to any particular material; rather, it is observed in a variety of materials from metals and alloys to semiconductors and semi-crystalline lamellar polymers, and the typical size range over which the melting temperature undergoes a large change is 2–20 nm (Pawlow, 1909; Buffat and Borel, 1976; Couchman and Jesser, 1977; Castro et al., 1990; Peters et al., 1998; Zhao et al., 2001; Nanda et al., 2002b; Dick et al., 2002; Sun et al., 2002; Jesser et al., 2004). Many phenomena in solidstate physics and materials science also exhibit size dependence. For example, reduction in the size of solids also results in a change of their failure mode. When the size of brittle calcium carbonate particles is reduced to a critical value of 850 nm, comminuting becomes impossible (Kendall, 1978; Karihaloo, 1979) and the particles behave as if they were ductile. As shown in the previous sections, surface/interface stress has a profound effect on the properties of nanostructured materials due to the large ratio of surface/interface atoms to the bulk. The properties of the nanostructured materials are affected by the energy competition between the surface and bulk, and a common feature of many physical properties is that when the characteristic size of the object is very large, the physical property under consideration tends to that of the bulk material. By dimensional analysis, the ratio of a physical property at a small size L, denoted by F(L), to that of the bulk, denoted by F (∞), can be expressed as a function of some nondimensional variables Xj (j = 1, . . . , M) and a nondimensional parameter lin /L related to the size F(L) = H(Xj , lin /L), F(∞)

(6.1)

where lin is an intrinsic length scale related to the surface property, and L is the feature size of the object under consideration. We will confirm below that for many

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physical properties the function in Eq. (6.1) can be expanded in powers of lin /L. When the intrinsic length scale lin is small compared with the characteristic size L, we need only retain the linear term. As a result, the size dependence of the properties of materials can be captured accurately by the following simple scaling law: lin F(L) =1+P , F(∞) L

(6.2)

where P is a nondimensional parameter. In what follows, we shall confirm that the scaling law Eq. (6.2) is obeyed by many properties, for example, elastic moduli, thermal conductivity, melting temperature, etc.

6.1. Elastic Properties with Surface Stress Effect For the deformation of homogeneous nanostructured materials, for example, the extension of nanoplates, nanobeams, nanowires, etc., when the surface elasticity, characterized by the surface elastic constant τ, is taken into account, an intrinsic length scale automatically emerges (Miller and Shenoy, 2000; Shenoy, 2002), lin = τ/E,

(6.3)

where E is the Young modulus of the bulk material. Thus, the nondimensional mechanical properties of nanostructures are governed by the scaling law (6.2) with lin given by Eq. (6.3). For metals and some other materials, lin in Eq. (6.3) is typically in the order of 0.01–0.1 nm (Streitz et al., 1994; Miller and Shenoy, 2000). Thus, the scaling law (6.2) can be expected to give good predictions for L >3–5 nm. Ab initio and molecular dynamic simulations and experimental results show that the elastic constants of nanoplates, nanobeams, and nanowires obey the scaling law (6.2) almost exactly in and above the range 1–100 nm (Miller and Shenoy, 2000; Shenoy, 2002; Zhou and Huang, 2004; Cuenot et al., 2004; Villain et al., 2004). In particular, according to Eqs. (6.2) and (6.3), the Young modulus of a nanoplate can be expressed as τ 1 E(L) =1+P , E(∞) a0 E(∞) N

(6.4)

where a0 is the lattice spacing, N is the number of atomic layers along the nanoplate thickness. The elastic constants of nanoplates of copper and tungsten obtained from atomistic calculations by Zhou and Huang (2004) and Villain et al. (2004) are shown in Fig. 6.1. It is seen that the ratio E/E(∞) for these materials almost exactly obeys the scaling law in Eq. (6.2) (or Eq. (6.4)). Zhou and

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1.3 1.2 E/E(`)

1.1 1.0 0.9 0.8 0.7 0.00

{001}/<110>, Cu {111}/<110>, Cu {001}/<100>, Cu {100}/<001>, W

0.05

0.10 1/N

0.15

0.20

Fig. 6.1 Normalized Young modulus versus the inverse of the number of atomic layers (N) along the nanoplate thickness. The data for Cu from Zhou and Huang (2004) and those for W from Villain et al. (2004) are obtained from atomistic calculations. {·} denotes a group of crystallographic planes and · denotes a group of crystallographic directions. Reprinted from Wang et al. (2006) with acknowledgment to The Royal Society, 6-9 Carlton House Terrace, London SW1Y 5AG, England.

Huang (2004) and Villain et al. (2004) did not explicitly use the concept of surface elastic constant to explain the variation of the elastic constants with the thickness of the nanoplates, but, obviously, the surface stress effect is implicitly taken into account in the atomistic calculations. These computations provide a further objective justification of the scaling law (6.2). Moreover, had this scaling law been known, the numerical computations could have been significantly reduced, as only two points are needed to determine a linear function. This is particularly important in the problems of characterization of heterogeneous materials given below for which it may be very difficult and/or time-consuming to perform atomistic simulations. For general heterogeneous nanostructured materials, the elasticity of an isotropic surface is characterized by two surface elastic constants λs and μs , giving rise to two intrinsic length scales lλ =λs /μ and lμ =μs /μ (Duan et al., 2005a,b), where μ is the shear modulus of the bulk material. Thus, the size dependence of nondimensional physical properties associated with the deformation problems of heterogeneous nanosolids can be expected to follow a scaling law similar to Eq. (6.2) but with an intrinsic length scale that is a linear combination of these two scales: F(L) 1 = 1 + (αlλ + βlμ ). F(∞) L

(6.5)

Here, α and β are two nondimensional parameters, F(L) is the property corresponding to a characteristic size L at nanoscale, and F (∞) denotes the same property

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when L→ ∞ or, equivalently, when the surface stress effect is vanishingly small. The scaling law (6.5) is applicable to a wide variety of properties, for example, the maximum stress concentration factor at the boundary of a circular nanopore in a plate under uniaxial tension, the effective elastic moduli of the nanochannel-array material, the Eshelby tensor of a spherical inhomogeneity, and the effective CTE of heterogeneous materials containing aligned cylindrical nanopores, etc. (Wang, et al., 2006; Duan and Karihaloo, 2007). For example, the scaling law of the maximum stress concentration factor k(ρ0 ) at the boundary of a circular nanopore of radius ρ0 in a plate (for the plane stress case) under uniaxial tension (χ = 0) can be determined from Eq. (3.4), that is, 7(lλ + 2lμ ) k(ρ0 ) , =1− k(∞) 3ρ0

(6.6)

where k(∞)=3 is the classical elasticity result. We compare the exact result given by Eq. (3.4) with the scaling law in Eq. (6.6), and the results are shown in Fig. 6.2, where “A” and “B” denote the two sets of surface property parameters given in Section 3.1, and the material of the plate is aluminium. It is seen that the scaling law is very accurate. We have seen above that an isotropic surface/interface is characterized by two surface/interface elastic constants κs and μs , giving rise to two intrinsic length scales lκ = κs /μm and lμ = μs /μm . For many materials, for instance, metals, the intrinsic length scales lκ and lμ are in the order of 0.01–0.1 nm. Thus, the surface/interface stress effect will only become significant at the nanoscale, as evidenced by the elastic moduli of Ag and Pb nanowires (Cuenot et al., 2004)

k (␳0) /k (⬁)

1.3

A, exact A, scaling law B, exact B, scaling law

1.2 1.1 1.0 0.9 0.8

1

5

10

15 20 ␳0 (nm)

25

30

Fig. 6.2 Comparison of the scaling law (Eq. (6.6)) and the exact result of the maximum stress concentration factor k(ρ0 )/k(∞) at the boundary of a circular nanopore of radius ρ0 in a plate (aluminium) under uniaxial tension. A: κs = −5.457 N/m, μs = −6.2178 N/m for the surface [1 0 0]; B: κs = 12.932 N/m, μs = −0.3755 N/m for the surface [1 1 1].

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and the effective elastic constants of heterogeneous solids containing nanovoids (Duan et al., 2005a). It is seen from Section 2 that the Eshelby tensors and the stress concentration tensors are all functions of the ratios between these intrinsic length scales and the size of the inhomogeneity, namely κsr = lκ /R and μrs = lμ /R. Thus, when R is sufficiently large compared with these intrinsic length scales, say R > 3 nm, the nondimensional parameters κsr = lκ /R and μrs = lμ /R will be small. We can then expand the components of the Eshelby and stress concentration tensors in terms of these small nondimensional parameters and retain only the linear terms. In this way, we can obtain a very simple scaling law for the Eshelby and stress concentration tensors. Here, we only show the details of the scaling law for the interior Eshelby tensor for a spherical inhomogeneous inclusion: SI (x) = SI (∞) + SI (R, x)

(6.7)

with SI (R, x) =

α(x)lκ + β(x)lμ . R

(6.8)

Here, SI (∞) is the classical interior Eshelby tensor for a spherical inhomogeneous inclusion, and α(x) and β(x) are two position-dependent tensors. SI (∞) is still given by Eqs. (2.17) and (2.22) but with A, F , and B in Eq. (2.22) replaced with A∞ , F∞ , and B∞ , respectively, A∞ = 0, B∞

F∞ = −

2 (1 − 2νI ) , [η (1 3 + νI ) + 2 (1 − 2νI )]

(7 − 5νm ) = . 3 [−7 + 5ν2 − 2η(4 − 5νm )]

(6.9)

By comparing the values of the components of the Eshelby tensor in Eq. (6.7) with the exact values given in Eq. (2.17) together with the expression in Eq. (2.22), it can be confirmed that the scaling law Eq. (6.7) gives very accurate results for lκ < 0.1 nm, lμ < 0.1 nm, and R > 3 nm. The exterior Eshelby and stress concentration tensors can also be shown to follow scaling laws similar to that in Eq. (6.7). They will not be reproduced here. Many physical properties with the surface/interface stress effect can be shown to obey similar scaling laws, for example, the effective moduli of heterogeneous solids containing nano-inhomogeneities obtained by various micromechanical homogenization schemes (Duan et al., 2005a).

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6.2. Melting Temperature The size dependence of the melting temperature at nanoscale has enormous implications in the production of nanocrystals and thin films and in the thermal stability of QDs (Goldstein et al., 1992). A large body of test data has been accumulated on this size dependence, and a number of theoretical models have been proposed to explain it (Buffat and Borel, 1976; Couchman and Jesser, 1977; Castro et al., 1990; Peters et al., 1998; Zhao et al., 2001; Nanda et al., 2002b; Dick et al., 2002; Sun et al., 2002). It has been demonstrated that the size-dependent melting temperature T(R) of spherical nanoparticles with radius R precisely obeys a scaling law similar to Eq. (6.2), T(R) lin =1−2 , T(∞) R

(6.10)

where lin is given in the paper of Wang et al. (2006). Similar to lin due to the surface elasticity (e.g., Eq. (6.3) or Eq. (6.5)), lin for the melting temperature is also from the contribution of the surface/interface energy. In fact, Eq. (6.10) is actually the Gibbs–Thomson equation following from consideration of the relative thermodynamic contributions of surface and bulk energies. It lends theoretical support to the scaling law Eq. (6.2). For example, the intrinsic length scale lin is in the order of 0.01–0.1 nm for Au and Ag nanoparticles. This value of lin is also true for many other metals. Therefore, the scaling law Eq. (6.10) can be expected to give good predictions for L >3–5 nm. The scaling law not only captures the sizedependent reduction in the melting temperature of unconstrained nanoparticles but also if they are embedded in a matrix. Recently, it has been confirmed that the size-dependent evaporation temperature of nanoparticles (Nanda et al., 2002a) also follows the scaling law (6.2). Moreover, if nanostructured materials are embedded in a matrix and the interfaces between them and the surrounding matrix are coherent or semi-coherent, then the melting temperature actually increases with decreasing nanoparticle size (Zhao et al., 2001).

7. Conclusions This article summarized the recent solutions of some fundamental problems in mechanics of heterogeneous materials where the surface/interface stress is taken into account. These include the Eshelby tensors, stress concentration tensors and their applications, the micromechanical framework, the novel properties of nanochannel-array materials, and the generalized Levin’s formula and Hill’s connections. These solutions show that the surface/interface stress has an important

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effect on the mechanical properties of materials at the nanoscale. When the surface/interface elasticity is taken into account, some length scales emerge automatically. Thus, unlike their classical counterparts, the mechanical properties at the nanoscale become size dependent. Scaling laws governing the properties of nanostructured materials have also been derived. It should be emphasized that this paper has considered the elastic deformation only. For inelastic nanostructured materials, the situation is partly similar in so far as the effect of surface stress is concerned, but there may be additional length scales and size effects. This will need further investigation in the future.

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Connections between Elastic and Conductive Properties of Heterogeneous Materials Igor Sevostianov1 and Mark Kachanov2 1 Department of Mechanical Engineering, New Mexico State University, P.O. Box 30001,

Las Cruces, NM 88003, USA [email protected] 2 Department of Mechanical Engineering, Tufts University, Medford, MA 02155, USA

[email protected]

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2. Overview of Existing Approaches to Cross-property Connections . . . . . 2.1. Bristow’s Elasticity–Conductivity Connection for a Microcracked Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Cross-property Connections Involving the Bulk Modulus . . . . . . . 2.3. Cross-property Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Cross-property Connections for Piezoelectric Fiber-reinforced Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Empirical Observations on Cross-property Relations . . . . . . . . . . .

74 74 76 80 92 94

3. Quantitative Characterization of Microstructures: General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1. Simplest Microstructural Parameters and their Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.2. Microstructural Parameters are Rooted in the Non-interaction Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.3. Cases of Overall Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.4. Benefits of the Proper Microstructural Parameters . . . . . . . . . . . . . 101 4. Materials with Isolated Inhomogeneities: Microstructural Parameters for the Effective Elasticity and Effective Conductivity . . . . . . . . . . . . . . . 4.1. General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Quantitative Characterization of Cracks . . . . . . . . . . . . . . . . . . . . . . 4.3. Example: Two-dimensional Elliptic Holes . . . . . . . . . . . . . . . . . . . . 4.4. Three-dimensional Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Spheroidal Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Microstructural Parameters for Conductive Properties: Comparison with Parameters for Elastic Properties . . . . . . . . . . . . . 4.7. Inhomogeneities of Irregular Shapes . . . . . . . . . . . . . . . . . . . . . . . . . ADVANCES IN APPLIED MECHANICS, VOL. 42 ISSN: 0065-2156 DOI: 10.1016/S0065-2156(08)00002-1

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102 103 106 111 113 115 126 131

Copyright © 2008 by Elsevier Inc. All rights reserved.

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5. Explicit Cross-property Connections for Anisotropic Two-phase Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Elasticity–Conductivity Connections: General Case . . . . . . . . . . . . 5.2. Cases of Overall Isotropy and Transverse Isotropy . . . . . . . . . . . . . 5.3. Materials with Cracks or Rigid Discs . . . . . . . . . . . . . . . . . . . . . . . . 5.4. On the Sensitivity of Elasticity–Conductivity Connection to Shapes of Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Connection between the Degrees of Elastic/Conductive Anisotropies is Insensitive to Inhomogeneity Shapes . . . . . . . . . . . 5.6. On the Effect of Interactions and Nonspheroidal Inhomogeneity Shapes on the Cross-property Connections . . . . . . . . . . . . . . . . . . . . 5.7. On the General Elasticity–Conductivity Constraints . . . . . . . . . . . . 5.8. Connection between the Electric and the Thermal Conductivities . 5.9. Physical Properties that May Not Be Interrelated by Quantitative Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Cross-property Connections for Anisotropic Inhomogeneities . . . . . . . . 6.1. Cross-property Connections for Materials with Parallel Anisotropic Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Moderate Orientation Scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Two or Three Families of Approximately Parallel Inhomogeneities 6.4. Nonlinear Connections for Parallel Isotropic Inhomogeneities of Unknown Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Fiber-reinforced Composites with Transversely Isotropic Piezoelectric Constituents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Applications of Cross-property Connections to Specific Materials Science Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Closed-cell Aluminum Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Microcracked Metal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Short Fiber-reinforced Thermoplastics . . . . . . . . . . . . . . . . . . . . . . . 7.4. Short Fiber-reinforced Plastics: Changes in Properties Due to Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Plasma-sprayed Ceramic Coatings . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6. Piezoelectric Fiber-reinforced Composites . . . . . . . . . . . . . . . . . . . . 8. Contact of Rough Surfaces: The Elasticity–Conductivity Connection . . 8.1. Hertzian Contact of the Circular Shape . . . . . . . . . . . . . . . . . . . . . . . 8.2. General Hertzian Contact of the Elliptic Shape . . . . . . . . . . . . . . . . 8.3. Hertzian Contacts versus “Welded” Areas . . . . . . . . . . . . . . . . . . . . 8.4. Multiple Contacts Between Two Rough Plates: Incremental Compliance, Conductivity, and Cross-property Connection . . . . . .

133 135 141 143 145 146 148 149 152 153 155 156 160 161 164 168 182 183 191 192 195 200 207 210 211 211 214 220

9. Plastic Yield Surfaces of Anisotropic Porous Materials in Terms of Effective Electric Conductivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 9.1. Approximate Constancy of Macroscopic Strain at Yield . . . . . . . . . 227

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9.2. Plastic Yield in Terms of Pore Space Characteristics . . . . . . . . . . . . 229 9.3. Plastic Yield in Terms of Effective Conductivities . . . . . . . . . . . . . . 230 9.4. Cases of Overall Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Appendix A. On Approximate Character of Elastic and Conductive Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Appendix B. Tensor Basis for Transversely Isotropic Fourth-rank Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Appendix C. Series of Associated Elliptic Functions and Relevant Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Abstract We discuss cross-property connections that interrelate effective linear elastic and conductive properties of heterogeneous materials. More precisely, they relate changes in the properties, as compared to the ones of the bulk material, caused by various inhomogeneities (cracks, pores, inclusions). They may also be developed for microstructures formed by multiple contacts, such as rough surfaces pressed against each other. Such connections are especially useful if one property (say, electrical conductivity) is easier to measure than the other (anisotropic elastic constants). For the properties governed by mathematically similar laws (for example, electrical and thermal conductivities), the connections are straightforward. However, for the elasticity–conductivity connections – the main focus of the present work – their very existence is nontrivial: not only the governing equations are different but even the ranks of tensors characterizing the properties are different (fourth-rank tensor of elastic constants versus second-rank conductivity tensor). We overview various approaches to the problem and then advance the approach rooted in similarity of the microstructural parameters that control the given pair of properties. This similarity leads to connections that, albeit approximate, have explicit closed form. They have been experimentally verified on several heterogeneous materials (metal foams, short fiber reinforced composites, metals with fatigue microcracks, sprayed coatings). Moreover, for properties controlled by entirely essentially different parameters (such as permeability or fracture of a microcracked material and its elasticity), the correlations may hold only qualitatively, at best.

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1. Introduction Cross-property connections for heterogeneous materials belong to the fundamental problems of engineering science and physics. They relate changes in different physical properties caused by various inhomogeneities (cracks, pores, inclusions) or, more generally, by the presence of certain microstructure. Their practical usefulness lies in the fact that one physical property (say, electrical conductivity) may be easier to measure than the other (say, anisotropic elastic constants). This allows one to bypass difficulties of expressing the elastic properties in terms of relevant microstructural information (that, in addition, may not be available). Such connections are also helpful in the design of microstructures for the combined conductive/mechanical performance. An example is given by thermal barrier coatings, where demands are made on both thermal conductivity and elastic stiffness. Aside from materials that can be described as continuous matrices containing inhomogeneities – the main focus of the present work – cross-property connections can also be developed for multiple contacts between rough surfaces pressed against each other. They can also be used to relate the yield condition of porous metals to the effective conductivity. Cross-property connections have been discussed in literature for about half a century. Some of them had a character of qualitative observations. For instance, geophysicists noticed that cracks in rocks increase both the elastic compliance and the fluid permeability. In fracture mechanics, attempts have been made to relate lifetime predictions to the loss of elastic stiffness of a material. Some of these connections cannot be upgraded from qualitative statements to quantitative relations that have some generality and are free from fitting parameters, for the reason that the considered pairs of properties may be controlled by entirely different microstructural factors. For example, fracture is quite sensitive to appearance of local clusters of defects, whereas the sensitivity of the effective elastic properties to clustering is, generally, low. The main focus of the present work is the elasticity–conductivity connection. Its very existence is not obvious: besides being governed by different differential equations, they are characterized by tensors of different ranks. Quantitative theoretical results on cross-property connections started to appear in 1960s. In the classical work of Bristow (1960), explicit elasticity–conductivity connection for a microcracked material was derived, in the isotropic case of random crack orientations and low crack density. Levin (1967) interrelated the effective bulk modulus and the effective thermal expansion coefficient of a general two-phase isotropic composite. Rosen and Hashin (1970) extended this result to anisotropic composite. Prager (1969) constructed Hashin–Shtrikman-based bounds for the

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effective magnetic permeability (or electric conductivity) in terms of the effective thermal conductivity of a two-phase isotropic material. Later works can be generally classified into the following groups: A. Qualitative observations that cannot be upgraded to solid theoretical results (free of fitting parameters). B. Empirical relations that fit a particular set of data. C. Universal bounds connecting the two considered properties. D. Explicit cross-property connections for materials with isolated inhomogeneities and for rough surfaces pressed against each other. Works belonging to groups A, B, and C are discussed in Section 2. The present work focuses on the explicit connections D developed by the present authors. They are discussed in Sections 5 and 6, and their applications to materials science problems in Section 7. The connections are derived in the non-interaction approximation. However, this assumption is much less restrictive than may be expected, for the following reasons: • In many cases, predictions of the non-interaction approximation for each of the two properties remain accurate at substantial concentrations of inhomogeneities. For example, 3D analytical results of Sevostianov et al. (2006b) for a matrix with periodic distributions of anisotropic fibers show that for all the effective constants, both elastic and conductive, the non-interaction approximation produces errors less than 10% up to 30% volume fraction of fibers. At higher volume fractions, the results for C1313 (x3 -axis is parallel to the fibers) start losing accuracy, whereas for other effective constants, the non-interaction approximation remains accurate up to 40% volume fraction. We also refer to 3D computational study of the effective elastic properties of solids with multiple cracks (Grechka and Kachanov, 2006) that indicate accuracy of the noninteraction approximation up to crack densities of at least 0.14 (at which the local interaction effects are quite strong, as shown in this study). The underlying reason is that the competing interaction effects, of stress shielding and stress amplification, largely cancel each other. • In cases the interactions do affect the effective properties, it appears that their effect on both effective properties – elasticity and conductivity – is largely similar so that the connection between the two is not affected much. This hypothesis was first suggested by Bristow (1960) who concluded, on the basis of experimental data for microcracked metals, that cross-property connection for a microcracked material derived by him under the non-interaction assumption remains accurate at substantial crack densities. It is further confirmed by experimental data on metal foams (Sevostianov et al., 2006a), plasma-sprayed

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I. Sevostianov and M. Kachanov coatings (Sevostianov et al., 2004), and, in particular, by rather unique experimental data of Choy et al. (1992) on the entire set of orthotropic elastic and conductive constants of composites reinforced with either glass or carbon fibers with volume fraction up to 34% (see Section 7). A related analytical result on fiber-reinforced composites is that the volume fraction of fibers at which the non-interaction approximation remains accurate increases from 30% to at least 40% if this approximation is applied to the cross-property connection rather than to each of the properties separately (Sevostianov et al., 2006b). Moreover, the accuracy range of the non-interaction approximation extends to almost fully dense packing for the connections involving all the elastic constants except C1313 .

The connections D are based on similarity of microstructural parameters that control the given pair of properties. They require, therefore, a careful examination of the parameters that are appropriate for a given physical property. This key issue is discussed in Sections 3 and 4. We note that, for the properties that are controlled by essentially different parameters, the correlations may be only qualitative, at best, as illustrated on two pairs of physical properties – fracture–elasticity and permeability–elasticity – in Section 5.9. 2. Overview of Existing Approaches to Cross-property Connections We overview the history of cross-property connections, with an emphasis on the elasticity – conductivity ones. The first quantitative result of this kind was given, to the best of our knowledge, by Bristow (1960) for solids with microcracks. It was followed by several results on cross-property connections involving the bulk modulus. A different approach to cross-property connections is represented by cross-property bounds, where substantial work involving advanced mathematical methods has been done in the last 20 years. For fiber-reinforced piezoelectric composites, yet another approach has been developed that extends the discussion of Hill (1964) on the number of independent effective elastic constants to materials of this type. We also mention several empirical cross-property connections of data-fitting nature suggested for several specific materials and conditions.

2.1. Bristow’s Elasticity–Conductivity Connection for a Microcracked Material This work (Bristow, 1960) explicitly relates changes in the effective conductivity and in the effective elastic moduli due to randomly oriented microcracks. Although his derivation implicitly assumes low crack density (interactions were

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ignored), experimental data indicate that the derived connections remain relatively accurate at higher-than-expected crack densities. Based on results of Sack (1946) and Segedin (1951) for a single penny-shaped crack, Bristow (1960) derived expressions for changes in effective elastic constants due to multiple randomly oriented penny-shaped cracks as follows:   E0 − E 16 1 − ν02 (10 − 3ν0 ) = ρ, E 45 2 − ν0 K0 − K 16 1 − ν02 = ρ, K 9 1 − 2ν0

(2.1)

G0 − G 32 (1 − ν0 ) (5 − ν0 ) ρ, = G 45 2 − ν0

 where ρ = (1/V ) ai3 is the scalar crack density parameter ( ai is ith crack radius and V is the representative volume). E, K, and G denote the Young’s, bulk, and shear moduli; ν is Poisson’s ratio. Hereafter, superscript “0” indicates properties of the bulk material. Remark In the original equations of Bristow, multiplier 2 − ν0 in denominator of the expression for (G0 − G) /G is missing. Bristow also derived the decrease of electrical conductivity k assuming that cracks are nonconductive: 8 k0 − k = ρ. (2.2) k 9 Observing that changes in both the elastic and the conductive properties are expressed in terms of the same crack density parameter and eliminating it, Bristow (1960) derived explicit elasticity–conductivity connections   2 1 − ν02 (10 − 3ν0 ) k0 − k E0 − E = , E 5 (2 − ν0 ) k   2 1 − ν02 k0 − k K0 − K = , (2.3) K 1 − 2ν0 k 4 (1 − ν0 ) (5 − ν0 ) k0 − k G0 − G = . G 5 (2 − ν0 ) k Comparing these results with experimental data on microcracked metals, he observed that the agreement remained satisfactory when changes in the effective properties due to cracks were substantial, implying non-small crack densities. Based on this observation, he formulated the hypothesis that the connections remain accurate beyond the non-interaction approximation, because crack

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interactions affect the elastic and the conductive properties in a similar manner (so that the connection between the two is not affected). This hypothesis is discussed further in Section 5.6.

2.2. Cross-property Connections Involving the Bulk Modulus A special class of cross-property connections that involve the bulk modulus has been developed starting from 1960s. Some of them were discussed in review of Hashin (1983). Here, we focus on three results of this kind that relate the bulk modulus to • the thermal expansion coefficient (Levin, 1967; Rosen and Hashin, 1970), • the specific heat (Rosen and Hashin, 1970), • the thermal conductivity of composites containing radially inhomogeneous inclusions (Lutz and Zimmerman, 2005). 2.2.1. Levin’s Formula and its Extension to Anisotropic Materials Levin (1967) established the connection between the effective bulk modulus K and the effective thermal expansion coefficient α of a two-phase (phases 0 and 1) isotropic composite with isotropic constituents:  α − α = (α0 − α1 )

1 1 − K0 K1

−1 

  1 1 − , K K

(2.4)

where angle brackets denote volume averaging. It has been generalized to anisotropic two-phase composites by Rosen and Hashin (1970) as follows: 

  (0) (1) (0) (1) −1 Sklrs − Sklrs Srsij − Srsij , αij − αij = αkl − αkl

(2.5)

where Sijkl are elastic compliances and αij are thermal expansion coefficients. We now outline the arguments leading to these connections following a much shorter reasoning privately communicated by Levin (2006). We consider a twophase composite of arbitrary microstructure and start with constitutive equations of thermoelasticity: εij (x) = Sijkl (x)σkl (x) + αij (x)T,

(2.6)

s(x) = αij (x)σij (x) + χ(x)T,

(2.7)

where χ and s are the specific heat and the entropy. Temperature T is assumed to be uniform within the representative volume. We represent the fields as sums of

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their volume averages (denoted by angle brackets) and fluctuations (denoted by prime): S(x) = S + S  (x) ,

α(x) = α + α (x) ,

σ(x) = σ + σ  (x)

so that averaging over the representative volume yields





 (x) σ  (x) + α T, εij = Sijkl σkl  + Sijkl ij kl   s = αij σij + αij (x) σij (x) + χ T.

(2.8)

(2.9)

Due to linearity of the problem, the fluctuations are linear functions of the average stresses and temperature: σij (x) = ijkl (x) σ kl  + λij (x)T,

(2.10)

where (x) and λ(x) are certain tensor coefficients. Substitution into (2.9) yields

 

   λ T + α T, σ  + S εij = Sijkl σkl  + Sijkl klmn mn ij ijkl kl (2.11)

 



s = αij σij + αij ijkl σkl  + αij λij T + χT. Therefore, for the effective compliances Sijkl , the effective thermal expansion coefficients αij , and the effective specific heat χ, defined by

εij = Sijkl σkl  + αij T,



s = αij σij + χT,

(2.12)

we have: 

 Sijkl = Sijkl + Sijmn mnkl , 

 αij − αij = Sijkl λkl = αmn mnij , 

χ = χ + αij λij .

(2.13)

For a two-phase composite, averages over phases are

 0 1 mnkl 0 , = p1 Sijmn − Sijmn 0  

 1 0 mnkl 1 , Sijmn mnkl = p0 Sijmn − Sijmn  Sijmn mnkl



1

(2.14)

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where p0 , p1 are volume fractions of the phases (p0 + p1 = 1) Similar relations  can be written for Sijmn λmn , αmn mnij , and αmn λmn . Then, the following identities hold 





   mnkl = p0 Sijmn mnkl + p1 Sijmn mnkl Sijmn 0 1    0 1 = p0 p1 Sijmn − Sijmn mnkl 0 − mnkl 1 ,





    λmn = p0 Sijmn λmn + p1 Sijmn λmn Sijmn 0 1    0 1 (2.15) = p0 p1 Sijmn − Sijmn λmn 0 − λmn 1 ,





 αmn mnij = p0 αmn mnij 0 + p1 αmn mnij 1  

 = p0 p1 α0mn − α1mn mnij 0 − mnij 1 ,





 αmn λmn = p0 αmn λmn 0 + p1 αmn λmn 1    = p0 p1 α0mn − α1mn λmn 0 − λmn 1 . Substitution of the first and the third of these expressions into Eq. (2.13) yields Eq. (2.5) and, in the case of isotropy, Eq. (2.4).

2.2.2. Rosen–Hashin’s formula For the effective specific heat, substitution of the second and the fourth equations of (2.15) into (2.13) gives    (0) (1) (0) (1) −1  χ − χ = αkl − αkl Sklrs − Sklrs αij − αij .

(2.16)

In the case of the isotropic composite with isotropic constituents, it takes the form  χ − χ = 9 (α0 − α1 )

1 1 − K0 K1

−1

(α − α).

(2.17)

Remark Both Levin’s and Rosen–Hashin’s cross-property connections (2.4), (2.5) and (2.16), (2.17) are obtained without any assumptions on microgeometries. In particular, the inhomogeneities do not have to be isolated and their volume fractions can be arbitrary. Results of Levin (1967) and Rosen and Hashin (1970) were further extended to two-phase electroelastic composites by Dunn (1993). He related the effective thermal expansion and pyroelectric coefficients to the effective elastic, piezoelectric, and dielectric constants.

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79

2.2.3. Lutz–Zimmerman’s result Lutz and Zimmerman (2005) considered an isotropic matrix containing spherical inclusions (marked by subscript “1”) surrounded by an inhomogeneous interphase layer with an arbitrary law of the radial variation in properties. At the outer boundary of the layer, its properties coincide with the ones of the matrix (marked by subscript “0”). The connection discussed below holds exactly if Poisson’s ratio of the interphase layers is zero, otherwise it has approximate character. The connection is based on similarity between governing equations in the conductivity and elasticity problems. In the conductivity problem, the equation has the form   2 d2 f 2 df df − 2 f + k (r) = 0, (2.18) k(r) + r dr dr dr 2 r where k(r) is the thermal conductivity of the inclusion and f(r) is a function characterizing the radial variation of temperature. In the elasticity problem, the radial displacement u(r) obeys the equation: 

   du d2 u 2 du 2 u [λ(r) + 2μ(r)] + − 2 u + λ (r) + 2μ (r) + 2λ (r) = 0, r dr dr r dr 2 r (2.19) where λ(r) and μ(r) are Lamé coefficients. It is seen that, in the case λ = 0 (Poisson’s ratio is zero), Eqs. (2.18) and (2.19) are identical. For a material with multiple inhomogeneities (having identical radial variations of properties), this leads to a direct correspondence between the effective bulk modulus and the effective conductivity: K 1 − 2[(K0 − K1 )/(K1 + 2K0 )]c = , K0 1 + [(K0 − K1 )/(K1 + 2K0 )]c

(2.20)

where volume fraction c is expressed in terms of the effective conductivity by c=

k1 + 2k0 k − k0 . k1 − k0 k + 2k0

(2.21)

The connection (2.20, 2.21) was not explicitly given by Lutz and Zimmerman (2005), but immediately follows from their results. This connection is exact to the first order in volume fraction c. We also observe that it represents relation between Hashin–Shtrikman’s upper bounds for the two properties.

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2.3. Cross-property Bounds Cross-property bounds interrelate, in the form of inequalities, the effective elastic and the effective conductive properties. They are universal, in the sense that they hold for all microgeometries. So far, bounds of this kind have been limited to the isotropic overall properties of two-phase composites, with two trivial exceptions: • elasticity–conductivity connection in the isotropy plane of a transversely isotropic material; • bulk modulus–conductivity connection that holds for the isotropic material can, obviously, be applied to a material with cubic symmetry (since, for both the bulk modulus and the conductivity, the cubic symmetry is indistinguishable from isotropy). Several bounds discussed here have been obtained by different mathematical means. We attempt to present the results and the underlying ideas in a readerfriendly form, simplifying arguments of the original works to the extent possible and refer the reader to the original articles for further details.

2.3.1. Milton’s Inequality Milton’s (1984) inequality is based on the minimum potential energy principle that, for macroscopically isotropic composite with isotropic phases, has the form (Beran, 1968):

 2 

Ktr ε2 + 2Ge : e ≤ K tr εˆ + 2 Gˆe : eˆ ,

(2.22)

where angle brackets denote volume averages, e = ε − (1/3)(tr ε)I is the strain deviator, and I the second-rank unit tensor. Symbol εˆ stands for the “trial” strain field that is kinematically compatible, but the associated stress field is not necessarily divergence-free (and vice versa, the trial stress field is divergence-free, but the associated strain field is not necessarily compatible). It is subject to the restriction ˆε = ε.

Taking εˆ such that εˆ = I/3 brings inequality (2.22) to the form

  2 K ≤ K tr εˆ + 2G eˆ : eˆ

  2 2   2   = λ tr εˆ + 2G εˆ : εˆ − 4G 9 tr εˆ ≤ λ tr εˆ + 2G εˆ : εˆ , (2.23)

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81

where λ = (3K − 2G)/3 is Lamé’s constant. This leads to the following inequality that relates the effective bulk modulus K and the effective conductivity k: k K ≤ , K1 k1

(2.24)

where subscript 1 refers to phase 1 of the composite (the two phases are assigned numbers 0 and 1 in such a way that K0 /K1 ≤ k0 /k1 ). Restricting attention to materials with nonnegative effective Poisson’s ratios, v ≥ 0, this inequality implies the following bound for the effective shear modulus: G 3k ≤ , K1 2k1

(2.25)

as noted by Torquato (1992). We now outline the proof of the above inequalities, following Milton (1984). His essential idea, which is rooted in work of Tartar (1979), is to link the trial displacement field uˆ to the gradient E of the exact electric potential, by choosing uˆ = T such that ˆε =  T = I/3 and relating T to E in such a way that the latter obeys the equation E · E = a2 ∇T : ∇T,

(2.26)

where dimensional constant a is introduced to make the physical dimensions of the left- and right-hand parts compatible. With the usual assumption that the composite is ergodic (the effective properties can be formulated in terms of volume, rather than ensemble, averages), the effective conductivity has the following energy representation: 1 1 kE·E = k E · E 2 2

(2.27)

k = 3k∇T : ∇T .

(2.28)

and

At the same time, the trial strain can be written in terms of T:   εˆ = (1/2) ∇T + (∇T)T .

(2.29)

We assume that Poisson’s ratios of both constituents are nonnegative (i.e., Lamé constant λ is nonnegative) and take into account that 

2

tr εˆ

≤ 3ˆε : εˆ = 3∇T : ∇T.

(2.30)

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Then, inequality (2.23) yields the upper bound for the effective bulk modulus: K ≤ 3 K∇T : ∇T .

(2.31)

With the aid of the characteristic function V (i) (x) that is equal to 1 if x is in ith phase and 0 otherwise, we rewrite (2.28) and (2.31) as

  k k0 (0) = 3 V (1) (x) ∇T : ∇T + 3 V (x) ∇T : ∇T , k1 k1

(2.32)

 

K0 (0) K V (x) ∇T : ∇T , ≤ 3 V (1) (x) ∇T : ∇T + 3 K1 K1

(2.33)

and the bound (2.24) follows. Milton (1984) made the interesting observation that, according to inequality (2.24), a composite with K1  = K0 , k1  = k0 cannot be incompressible and have finite conductivity, but a composite that is compressible and superconducting may exist. 2.3.2. Berryman–Milton’s Bounds Berryman and Milton (1988) constructed elasticity–conductivity bounds for a two-phase isotropic composite that are significantly tighter than the “elementary” bounds discussed above. They utilize bounds derived separately for the effective conductivity (Beran, 1965) and for the effective elastic constants (Beran and Molyneux, 1966; McCoy, 1970). These two sets of bounds take into account not only volume fractions of phases but the information on microgeometry provided by three-point correlation functions. An important observation is that this information is utilized by the two bounds in a similar way, reflecting similarities between effects of microgeometry on the two properties. This leads to cross-property bounds that are independent of microgeometry. Remark This similarity echoes the observation made by Bristow (1960) in a narrower context of cracks, that crack interactions – that are very sensitive to the geometry of crack arrangements – affect the conductivity and the elastic constants in a similar way so that the connection between the two is not affected. The three-point correlation function provides an approximate characterization of microgeometry: it gives the probability that material belonging to a given phase of a composite is located at three given points. For phase 1, this function is defined by

 s3 (r1 , r2 , r3 ) = V (1) (x + r1 ) V (1) (x + r2 ) V (1) (x + r3 ) , (2.34)

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83

where V (1) (x) is the characteristic function of the first phase. Based on the correlation function, the following two dimensionless microstructural parameters are introduced: 9 lim lim ζ= 2c0 c1 →0  →∞

 

  dr



+1 ds −1



150 5ζ1 + lim lim η= 21 7c0 c1 →0  →∞

 

  dr



S3 (r, s, μ) P2 (μ)dμ, rs +1 ds



−1

S3 (r, s, μ) P4 (μ)dμ, rs

(2.35)

where P2 (μ) = (3μ2 − 1)/2 and p4 (μ) = (35μ4 − 30μ2 + 3)/8 are Legendre polynomials of orders 2 and 4 and   S3 r12 , r13 , μ12,13 = s3 (r1 , r2 , r3 ),

(2.36)

  (r2 − r1 ) · (r3 − r1 ) rij = rj − ri  , μ12,13 = . r12 r13

(2.37)

with

This form of the correlation function emphasizes the statistical homogeneity of the microgeometry (invariance with respect to translations in any direction). Parameters ζ and η obey the following inequalities (Milton and Phan-Thien, 1982): 5ζ/21 ≤ η ≤ (16 + 5ζ)/21.

(2.38)

The key observation of Berryman and Milton (1988) is that bounds for different physical properties – magnetic permeability, electrical or thermal conductivity, and elastic constants – can be expressed in terms of the same two parameters, ζ and η. Focusing attention on the elasticity–conductivity pair and excluding the two parameters leads to bounds on the effective bulk and shear moduli in terms of the effective conductivity. The basic arguments are as follows. In addition to the usual volume average   = 0 + ( 1 − 0 )c1 of a certain physical property , Berryman and Milton (1988) introduced averages that involve microstructural parameters (2.35):  ζ = 0 + ( 1 − 0 ) ζ,  η = 0 + ( 1 − 0 )η.

(2.39)

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In terms of these averages, bounds of Beran (1965) for the effective conductivity and bounds of Beran and Molyneux (1966) and McCoy (1970) for the elastic moduli take the form    ≤ k ≤ A1 kζ , (2.40a) A1 1/k−1 ζ    A2 1/G−1 ≤ K ≤ A2 Gζ , (2.40b) ζ    (2.40c) A3 (/6) ≤ G ≤ A3 1 6 , where functions A1−3 (x) are defined as follows −1  1 A1 (x) = − 2x, k + 2x −1  1 − 4x/3, A2 (x) = K + 4x/3 −1  1 − x, A3 (x) = G+x

(2.41)

and the following notations are used =

10 G2 Kζ + 5 G 2K + 3G Gζ + 3K + G2 Gη K + 2G2

,

(2.42) 10 K2 1/Kζ + 5 G 2K + 3G 1/Gζ + 3K + G2 1/Gη = . 9K + 8G2 Note that bounds for the conductivity and for the bulk modulus contain parameter ζ only, whereas bounds for the shear modulus contain both microstructural parameters ζ and η. Substituting ζ from inequality (2.40a) for conductivity, as well as (2.38), into inequalities (2.40b,c) for the bulk and shear moduli would lead to the elasticity–conductivity cross-property bounds. For a general composite material, this procedure is quite cumbersome and, to our knowledge, has not been implemented in explicit form. For a nonconducting porous material filled with conductive fluid, results can be given in the explicit form. In this case, Beran’s bounds degenerate into one inequality:   1 c0 1 c0 k k1 /k ≥ 1 + 1 + , or ζ ≥ , (2.43) 2ζ c1 2 c1 k1 − k

Connections between Elastic and Conductive Properties

85

and Beran–Molyneux’ bounds are ζ ≤1−

3c1 K/4G0 , c0 − K/K0

(2.44)

η≥1−

6R(K0 + 2G0 )2 − 5G0 (1 − ζ)(4K0 + 3G0 ) , (3K0 + G0 )2

(2.45)

where R = (c1 G/G0 )/(c0 − G/G0 ). This leads to cross-property bounds interrelating the effective bulk modulus and the effective conductivity: 3c1 K/4G0 1 c0 k ≤ζ ≤1− . 2 c1 k1 − k c0 − K/K0

(2.46)

To obtain bounds for the effective shear modulus, we introduce dimensionless combinations of elastic constants of phase “0” A0 =

6(K0 + 2G0 )2 5G0 (4K0 + 3G0 ) , B0 = . 2 (3K0 + G0 ) (3K0 + G0 )2

(2.47)

Using (2.38) and rewriting (2.45) in the form 5 c1 G/G0 ζ ≤ η ≤ 1 + B0 − B0 ζ − A0 21 c0 − G/G0 implies the following cross-property bound   21A0 6c1 G/G0 1 c0 k 1 + B0 . − ≤ ζ ≤ 21 2 c1 k1 − k 5 − 21B0 5 − 21B0 c0 − G/G0

(2.48)

(2.49)

Remark 1. Bounds (2.46) and (2.49) involve volume fractions c1 and c0 ; therefore, they become trivial for inhomogeneities of vanishing volume, in particular for cracks. Remark 2. Berryman and Milton (1988) noted that, since inequalities (2.46), (2.51) for parameters ζ and η are independent of the properties of phase “1,” they can be specialized, for example, for a porous material and then used for generating bounds for material of the same microgeometry but with pores replaced by a constituent with different properties. Similar ideas were developed by Bergman (1978) and Kantor and Bergman (1984). 2.3.3. Gibiansky–Torquato’s Translation-based Cross-property Bounds These bounds are based on the translation method outlined below. Its application to cross-property bounds was pioneered by Cherkaev and Gibiansky (1992) in the context of conductive-magnetic bounds for two-dimensional composites; for a

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review of the translation method, see the book of Cherkaev (2000). Gibiansky and Torquato (1993, 1995, 1996a) applied this method to the elasticity–conductivity bounds for 2D isotropic composites and to the bulk modulus–conductivity bounds for 3D composites. They specified these bounds for several microstructures: microcracked material (Gibiansky and Torquato, 1996b), dry and fluid-saturated porous rocks (Gibiansky and Torquato, 1998a), porous or cracked nonlinear conductors assuming the elastic properties to be linear (Gibiansky and Torquato, 1998b) and 2D cellular solids (Torquato et al., 1998). They also applied these results to a design of porous materials for a combined dielectric-stiffness performance (Torquato et al., 2005). The methodology utilizes two concepts – of the translation tensor and of the Y-tensor. The translation tensor T and its application to bounding the effective properties was introduced by Tartar (1979), Murat and Tartar (1985), Lurie and Cherkaev (1982, 1984), and Milton (1990). We consider some physical property of a heterogeneous material, say elastic compliance tensor, its local value S(x), and its effective value S. Translation tensor T is a special tensor that possesses the following property: if S(x) is shifted, at each point x, by a multiple of T: S (x) = S (x) − κT,

(2.50)

then S of this new medium is translated by κT as well: S = S − κT.

(2.51)

The multiplier κ chosen in such a way that S is semi-positive definite. The translation tensor can be introduced for other physical properties as well, including coupled properties (in which case the translation tensor has to be replaced by several ones arranged in a matrix). For a discussion of the method in detail, see books of Cherkaev (2000) and Milton (2002). The Y-tensor interrelates field fluctuations averaged over a given phase. It was introduced by Milton (1997) based, to some extent, on earlier works of Berryman (1980a,b, 1982). For elastic fields in phase 1 of a two-phase composite, the fourthrank Y-tensor is defined by



 V (1) (x) (ε − ε) = −Y : V (1) (x) (σ − σ) . (2.52) Since characteristic functions are interrelated by V (1) (x) = 1 − V (0) (x), the Ytensor for phase 0 is the same:



 V (0) (x) (ε − ε) = −Y : V (0) (x) (σ − σ) . (2.53)

Connections between Elastic and Conductive Properties

87

For the conductivity problem, the second-rank Y-tensor can be introduced in a similar way. Construction of the Cross-property Bounds We now consider a 3D composite that consists of two isotropic phases and is isotropic, overall, and derive the conductivity–elasticity bounds following Gibiansky and Torquato (1996a). Their derivation requires specifying the sign of the product (k1 − k0 )(G1 − G0 ). We consider the case (k1 − k0 ) (G1 − G0 ) ≤ 0.

(2.54)

We write pointwise Hooke’s law in the following form σ (x) = [3K (x) 1 + 2G (x) 2 ] : ε (x),

(2.55)

where two tensors (1 )ijkl =

1 δij δkl , 3

(2 )ijkl =

 1 1 δik δjl + δil δjk − δij δkl , 2 3

(2.56)

supplemented by (3 )ijkl =

 1 δik δjl − δil δjk , 2

(2.57)

constitute an orthogonal basis for fourth-rank isotropic tensors (any such tensor can be expressed as their linear combination). For the conductive properties, the following procedure suggested by Milton (1984) is used. Three electric fields e(1) (x), e(2) (x) and e(2) (x) are applied to the composite. They form columns of 3 × 3 matrix-valued field E(x). The three resultant fields of currents j (1) (x), j(2) (x), and j(3) (x) form columns of 3 × 3 matrix-valued field J(x). The conductivity law then takes the form:   J(x) = k(x)I 4 : E(x), (2.58) 4 = δ δ is the fourth-rank identity tensor. Now, the constitutive relation where Iijkl ik jl for the elastic and conductive properties can be written in the combined form:    ε σ (0) (1) = V L0 + V L1 , (2.59) J E

where

 3Ki 1 + 2Gi 2 Li = 0

 0 , ki 1 + ki 2 + ki 3

i = 0, 1.

(2.60)

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The combined effective elastic and conductive constants have a similar form:  L=

3K1 + 2G2 0

0 k1 + k2 + k3

 (2.61)

and the associated Y-tensors can be arranged in a matrix as follows  Y=



3yK 1 + 2yG 2

0

0

yK 1 + yK 2 + yK 3

,

(2.62)

where the y-parameters, yK = −c0 K1 − c1 K0 −

c0 c1 (K0 − K1 )2 , K − c 0 K0 − c 1 K1

yG = −c0 G1 − c1 G0 −

c0 c1 (G0 − G1 )2 , G − c 0 G0 − c 1 G1

yk = −c0 k1 − c1 k1 −

(2.63)

c0 c1 (k0 − k1 )2 , k − c0 k0 − c1 k1

are y-transformations of the effective bulk and shear moduli and of the conductivity introduced by Berryman (1982). For the translation, Gibiansky and Torquato (1996a) use the following tensor   −t1 (21 − 2 + 3 ) −t3 (21 − 2 + 3 ) T = . (2.64) −t3 (21 − 2 + 3 ) −t2 (21 − 2 + 3 ) The requirement that Li − T is semi-positive definite for both phases, that is, ⎛ ⎜ ⎝

(3Ki + 2t1 ) 1 + (2Gi − t1 ) 2 + t1 3 t3 (21 − 2 + 3 )

t3 (21 − 2 + 3 )

⎞

⎟ (ki + 2t2 ) 1 + (ki − t2 ) 2 + (ki + t2 ) 3 ⎠ ≥ 0 (2.65)

decouples, due to orthogonality of 1 , 2 , 3 , into three inequalities     2Gi − t1 3Ki + 2t1 2t3 −t3 ≥ 0, ≥ 0, 2t3 ki + 2t2 −t3 ki − t 2  t1 t3

t3 ki + t 2

 ≥ 0,

(2.66)

Connections between Elastic and Conductive Properties

89

where index i = 0, 1 indicates the phase number. The necessary and sufficient condition for these inequalities to hold is that each of the six matrices is semipositive definite, so that the following two conditions have to be met: 0 ≤ t1 ≤ min (2G0 , 2G1 ) ,

(2.67)

and determinant of each matrix has to be nonnegative, that is, t1 , t2 , t3 must satisfy the inequality (3y1 /2 − t1 ) (y2 /2 − t2 ) − t33 ≥ 0

(2.68)

for each of the following six pairs of (y1 , y2 ): (k0 , K0 ), (k1 , K1 ), (−2k0 , −4G0 /3), (−2k1 , −4G1 /3), (2k0 , 0), (2k1 , 0). (2.69) It is possible now to construct bounds that interrelate the effective conductivity k and the effective bulk modulus K (but not the effective shear modulus!) in the plane of parameters (yk , yK ). These bounds are formed by the outermost of five curves: four segments of hyperbolae passing through the points indicated in brackets: Hyp [(−2k0 , −4G0 /3) , (−2k1 , −4G1 /3) , (k0 , K0 )] , Hyp [(−2k0 , −4G0 /3) , (−2k1 , −4G1 /3) , (k1 , K1 )] ,

(2.70)

Hyp [(−2k0 , −4G0 /3) , (−2k1 , −4G1 /3) , (2k0 , 0)] , Hyp [(−2k0 , −4G0 /3) , (−2k1 , −4G1 /3) , (2k1 , 0)] , and the straight line connecting points (−2k0 , −4G0 /3) and (−2k1 , −4G1 /3). It is further stated (Gibiansky and Torquato, 1996a) that bounds in the (k, K) plane are formed by two hyperbolae that pass through points (k0 , K0 ) and (k1 , K1 ) and touch the set given by the five mentioned lines More explicit bounds can be obtained in the case when the volume fractions ci of constituents are known. Then the outermost pair of following five hyperbolae gives the bounds: Hyp [(k0∗ , K0∗ ) , (k1∗ , K1∗ ) , (k0 , K0 )] , Hyp [(k0∗ , K0∗ ) , (k1∗ , K1∗ ) , (k1 , K1 )] , Hyp [(k0∗ , K0∗ ) , (k1∗ , K1∗ ) , (k0# , Kh )] , Hyp [(k0∗ , K0∗ ) , (k1∗ , K1∗ ) , (k1# , Kh )] , Hyp [(k0∗ , K0∗ ) , (k1∗ , K1∗ ) , (ka , Ka )] ,

(2.71)

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where the following notations are used k0∗ = c0 k0 + c1 k1 −

c0 c1 (k0 − k1 )2 , c1 k0 + c0 k1 + 2k0

k1∗ = c0 k0 + c1 k1 − k1# = c0 k0 + c1 k1 −

c0 c1 (k0 − k1 )2 , c1 k0 + c0 k1 + 2k1

c0 c1 (k0 − k1 )2 , c1 k0 + c0 k1 − 2k0

k2# = c0 k0 + c1 k1 −

c0 c1 (k0 − k1 )2 , c1 k0 + c0 k1 − 2k1

ka = c0 k0 + c1 k1 , K0∗ = c0 K0 + c1 K1 −

c0 c1 (K0 − K1 )2 , c1 K0 + c0 K1 + 4G0 /3

K1∗ = c0 K0 + c1 K1 −

c0 c1 (K0 − K1 )2 , c1 K0 + c0 K1 + 4G1 /3

Ka = c0 K0 + c1 K1 , Kh = (c0 /K0 + c1 /K1 )−1 . Note that Gibiansky–Torquato’s bounds are realizable and therefore cannot be improved without additional information on microstructure. Fig. 2.1 shows that these bounds are substantially narrower than Berryman–Milton’s bounds. In contrast with Berryman–Milton’s bounds that are formulated in volume fractions and, therefore, become trivial for a cracked material, Gibiansky–Torquato’s bounds produce an inequality in this case, which, assuming v0 ≥ 0, has the following form (Gibiansky and Torquato, 1996b): 

1 1 − K K0

 ≥

3k0 1 − ν0 2G0 1 + ν0



 1 1 . − k k0

(2.72)

This inequality requires, however, an additional assumption: k0 1 − ν0 6k1 ≤ G0 1 + ν 0 3K1 + 4G1

(2.73)

that is somewhat unclear since k1 , K1 , and G1 are zeros for cracks. It is interesting to compare inequality (2.72) for cracks with the explicit elasticity–conductivity connection (2.3) of Bristow (1960). Fig. 2.2 compares Gibiansky–Torquato’s bound (2.72), Milton’s inequality (2.24), and Bristow’s cross-property connection (2.3) for the effective bulk modulus of a microcracked material.

Connections between Elastic and Conductive Properties

91

20.0 15.0

1

Milton

2

Berryman–Milton

3

Gibiansky–Torquato

K /K1

1 10.0

K2 / K1 5 20

2

k2 / k1 5 20

5.0

n0 5 n1 5 0.3

3 0.0 0.0

5.0

10.0

15.0

20.0

k/k1 Fig. 2.1 Comparison of different cross-property bounds. The inner region corresponds to Berryman–Milton bounds for fixed volume fraction c1 = 0.2; its shaded part to Gibiansky–Torquato bounds. 1.00 n0 5 0.3

K /K1

0.75

1

Bristow

2

Milton

3

Gibiansky–Torquato

0.50 3

2 0.25

1

0.00 0.00

0.25

0.50

0.75

1.00

k/k1 Fig. 2.2 Comparison of the Bristow cross-property connection (2.3) for bulk modulus of a microcracked material (randomly oriented cracks) with Milton inequality (2.24) and Gibiansky–Torquato bound (2.72).

2.3.4. Cross-property Bounds Implied by Hashin–Shtrikman’s Bounds Hashin–Shtrikman’s (HS) bounds formulated separately for the effective elastic moduli (Hashin and Shtrikman, 1963), and the effective conductivity (Hashin and Shtrikman, 1962) can be used to produce cross-property bounds. Indeed, rewriting the HS bounds for conductivity as inequalities for volume fractions (assuming for certainty that k0 > k1 ) (k − k1 ) (2k1 + k0 ) 3k0 (k − k1 ) ≤ c0 ≤ ≤ 1, (2k0 + k) (k0 − k1 ) (2k1 + k) (k0 − k1 ) (k0 − k) (2k0 + k1 ) 3k1 (k0 − k) 0≤ ≤ c1 ≤ ≤1 (2k1 + k) (k0 − k1 ) (2k0 + k) (k0 − k1 )

0≤

(2.74)

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and substituting them into similar inequalities for the elastic constants, we find cross-property bounds that relate the bulk and shear moduli to conductivity: K1 +

f(k) (K0 − K1 ) (3K1 + 4G1 ) ≤K (3K1 + 4G1 ) + 3 (K0 − K1 ) [1 − f(k)]

(2.75a) [1 − f(k)] (K1 − K0 ) (3K0 + 4G0 ) , ≤ K0 + (3K0 + 4G0 ) + 3 (K1 − K0 ) f(k) 5f(k)G1 (G0 − G1 ) (3K1 + 4G1 ) G1 + ≤G 5G1 (3K1 + 4G1 ) + 6 (G0 − G1 ) (K1 + 2G1 ) [1 − f(k)] (2.75b) 5 [1 − f(k)] G0 (G1 − G0 ) (3K0 + 4G0 ) , ≤ G0 + 5G0 (3K0 + 4G0 ) + 6 (G1 − G0 ) (K0 + 2G0 ) f(k) where f(k) =

3k0 (k − k1 ) (2k0 + k) (k0 − k1 )

(2.76)

and where it is assumed that K0 > K1 , G0 > G1 . However, cross-property bounds obtained this way are quite wide – they are much wider than the HS bounds themselves. This is particularly clear in the case of a porous material, when thus obtained bounds reduce to trivial statements. Note that, in the 2D case, cross-property bounds of this kind were given by Zhao et al. (2004). Remark In terms of the numerical implementation, the HS-based cross-property bounds are probably the simplest, but they are the least tight (becoming useless in the case of a porous material). The Gibiansky–Torquato’s translation-based bounds are the most tight ones, but they are, probably, the hardest to implement. The Berryman–Milton’s bounds are intermediate from the point of view of tightness and difficulties of the numerical implementation.

2.4. Cross-property Connections for Piezoelectric Fiber-reinforced Composites Cross-property connections for piezoelectric composites reinforced by infinite fibers were derived by Benveniste and Dvorak (1992) and Schulgasser (1992). The linear constitutive equations for a transversely isotropic piezoelectric material have the form: σ11 = C11 ε11 + C12 ε22 + C13 ε33 − e31 E3 , σ22 = C12 ε11 + C11 ε22 + C13 ε33 − e31 E3 ,

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σ33 = C13 ε11 + C13 ε22 + C33 ε33 − e33 E3 , σ12 = C66 ε12 , σ13 = C44 ε13 − e15 E1 ,

(2.77)

σ23 = C44 ε23 − e15 E2 , D1 = e15 ε13 + ζ11 E1 , D2 = e15 ε23 + ζ11 E2 , D3 = e31 ε11 + e31 ε22 + e33 ε33 + ζ33 E3 , where εij and σij are strains and stresses, Di are the electric displacements (components of the electric induction vector), and Ei are components of the electric field vector. The equations contain five elastic stiffnesses Cij , three piezoelectric constants eij , and two dielectric permeabilities ηij . In the text to follow, the matrix and fiber constants will be denoted by superscripts “0” and “1,” correspondingly, the effective constants will be marked by an asterisk, and quantities marked by superscript “d” will stand for the differences between corresponding constants of d = C1 − C0 . the end of the matrix, for instance C11 11 11 Extending well-known Hill’s (1964) universal relations for a fiber-reinforced linear elastic composite d + Cd C11 12 d 2C13

=

∗ + C∗ − Cv − Cv ∗ − Cv C11 C13 11  12 13  12∗ = ∗ − Cv v C33 2 C13 − C13 33

(2.78)

to the piezoelectric one. Benveniste and Dvorak (1992) derived the following two groups of cross-property relations reducing the number of independent effective constants from ten to five: d + Cd C11 12 d 2C13 d + Cd C11 12

2ed31

=

=

∗ + C∗ − Cv − Cv ∗ − Cv C13 C11 e∗31 − ev31 11  12 13  12∗ = = , ∗ − Cv v C33 e∗33 − ev33 2 C13 − C13 33

(2.79)

∗ + C∗ − Cv − Cv v C11 e∗ − ev31 C∗ − C13 11  12  12∗ = − 31 = 13 ∗ ∗ − ζv , v v e33 − e33 ζ33 2 e31 − e31 33

(2.80)

v = where superscript v denotes the Voigt arithmetical average: for instance, C13 0 1 (1 − f ) C13 + fC13 . Benveniste and Dvorak (1992, pp. 1305–1306) also showed that the conditions of the theorem of Milgrom and Shtrikman (1989) are satisfied for uniaxial fiber-reinforced composites with transversely isotropic phases (orientation of fibers coincides with the symmetry axes of phases). This theorem yields the following

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universal relation for transversely isotropic piezoelectric composites that involves constants C44 , e15 , and ζ11 ⎡  ∗ 2 ⎤ e15 e∗15 1 ∗ + ζ ⎢ ∗ 11 ∗ ∗ ⎥ C44 ⎥ ⎢ C44 C44 ⎥ ⎢   2 ⎢ 1 e015 ⎥ e015 ⎥ ⎢ 0 (2.81) det ⎢ 0 ζ11 + ⎥ = 0. 0 0 ⎥ ⎢ C44 C44 C 44 ⎥ ⎢  1 2 ⎥ ⎢ e e115 ⎦ ⎣ 1 1 ζ11 + 151 1 1 C44 C44 C44 This relation further reduces the number of independent effective piezoelectric constants to only four. Remark Schulgasser (1992) gave a connection between C44 , e15 , and ζ11 that is different from (2.81): ⎡ ∗ ⎤ ∗ C44 e∗15 ζ11 ⎢ ⎥ 0 0 ⎥ ⎢ 0 det ⎢C44 e15 ζ11 ⎥ = 0. ⎣ 1 ⎦ 1 C44 e115 ζ11 It is based on application of Milgrom and Shtrikman (1989) relation for magnetoelectric effect to the piezoelectric problem. Such an application is not fully justified, however, for the reason that, in the magnetoelectric case, the matrix is symmetric and positive definite, whereas, in the piezoelectric case, it is not.

2.5. Empirical Observations on Cross-property Relations Aside from theoretical developments, cross-property correlations of the datafitting nature have been suggested in literature for several materials. In the work of Zamora et al. (1993), thermal conductivity of approximately isotropic rock specimens (Fontainebleau sandstone, 99.8% quartz) at three different porosities (4.3%, 9.7%, and 16.2%) was related to wavespeeds VP = √ √ (K + 4μ) /3ρ and VS = μ/ρ (where ρ is material density). The three available data points were approximated by the following best-fit linear relations WS VP = αWS P k − βP ,

WS VS = αWS S k − βS

(water saturated rock),

where fitting parameters are chosen as follows 3 2 −1 αWS W−1 , βPWS = 1.96 · 103 ms−1 , P = 1.05 · 10 m Ks 3 2 −1 αWS W−1 , βSWS = 2.71 · 103 ms−1 , S = 0.88 · 10 m Ks

(2.82)

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and D DS D VP = αD P k − βP , VS = αS k − βS

(dry rock),

(2.83)

where 3 2 −1 αD W−1 , βPD = 2.36 · 103 ms−1 , P = 1.12 · 10 m Ks 3 2 −1 αDS W−1 , βSD = 1.68 · 103 ms−1 . S = 0.77 · 10 m Ks

The reliability of these correlations is somewhat reduced by the fact that the conductivity was measured under uniaxial stress of 500 bars, whereas the wavespeeds were measured in a stress-free state. Besides, these relations make no reference to the properties of the bulk material and, therefore, do not explicitly interrelate changes due to porosity. In the work of Kim et al. (2005), several physical properties of aluminum foams (Al-3Si-2Cu-2Mg) were experimentally measured, such as electrical conductivity, Young’s modulus, and macroscopic yield stress σ y . The authors found that the following elasticity-conductivity correlation gives the best fit of the data:  4/3  2/3 k k E = 1.23 − 0.015 . (2.84) E0 k0 k0 However, the accuracy of this connection is somewhat reduced by the fact that it involves an intermediate interpolation, separately for E and k. The authors also give empirical connection between the macroscopic yield stress and the effective conductivity:  2/3   k k σy − 0.30 . (2.85) y = 1.15 k k σ0 0 0 3. Quantitative Characterization of Microstructures: General Considerations Quantitative characterization of microstructures means identification of the proper microstructural parameters, in whose terms the considered physical property (such as effective elasticity or conductivity) is to be expressed. The key requirement is that they should represent individual inhomogeneities – or, more generally, microstructural features – in accordance with their actual contributions to the property considered; otherwise, the property may not be a unique function of these parameters. For example, the crack density parameters for the

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effective elasticity and effective conductivity take contributions of individual cracks proportionally to their sizes cubed. The proper parameters are, generally, different for different physical properties. For example, tensor crack density parameters for nonrandom crack orientations that have to be used in the effective elasticity and in the effective conductivity problems do not fully coincide. For materials that can be described as continuous matrices with inhomogeneities (cracks, pores, inclusions), such parameters should characterize the distribution of inhomogeneities over shapes, sizes, and orientations. They contain more information than just volume fractions and may be nontrivial, particularly in anisotropic cases of nonrandom orientations. The key step in their identification is finding individual inhomogeneity contributions to the property considered, followed by averaging over all the inhomogeneities in the representative volume V . For granular materials, or two rough surfaces pressed against each other, both the effective elasticity and the effective conductivity are controlled (aside from the bulk material properties) by contact geometries. In such cases, the microstructural parameters should characterize the distribution of contacts (their geometries, orientations, and spatial distribution). The key challenge is finding individual contact contributions to the property. The identification of the mentioned individual contributions and reducing them to simple forms is a challenging task; it requires distinguishing the microstructural features that have a dominant effect on the given property from the ones of minor importance and developing a sufficiently simple characterization of the dominant ones. The progress made in this direction is discussed in Section 4. The micromechanical approach, based on individual inhomogeneity contributions to the overall property, is rooted in a number of classical works. Mackenzie (1950) considered effective elastic properties of a solid with spherical pores on the basis of individual pore contributions to the overall compliance. Kröner (1958) analyzed the isotropic matrix with spherical anisotropic inhomogeneities and operated with a contribution of an individual sphere to the overall property. Hill (1965) considered ellipsoidal inhomogeneities of identical aspect ratios and expressed their contributions to the overall elasticity using Eshelby’s results (1957, 1961); dependencies on the ellipsoids’ aspect ratios were worked out in detail by Wu (1966). Walpole (1969) gave the mentioned dependencies in a more transparent form and, in the case of spheroids, reduced them to elementary functions. In the important case of cracks, their contributions to the overall elasticity and conductivity were given by Bristow (1960). A thorough review of the micromechanical approach was given by Markov (2000). If, for a given pair of physical properties, the individual inhomogeneity contributions – and, therefore, the microstructrural parameters – are sufficiently similar,

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this leads to cross-property connections, elasticity–conductivity connection being a primary example. If, however, they are principally different, such connections cannot be established (at least, in a quantitative way). For example, effective elastic properties of a cracked material have relatively low sensitivity to appearance of clusters of several closely spaced cracks, whereas clustering is of key importance for the fracture-related properties.

3.1. Simplest Microstructural Parameters and their Limitations We start with overviewing several frequently used microstructural parameters. • For a material with ellipsoidal inhomogeneities of identical shapes (Hill, 1965) that is isotropic, overall, the volume fraction of inhomogeneities c=

1 $ (i) V V

(3.1)

i

is the adequate microstructural parameter (here, V (i) stands for the volume of ith inhomogeneity). Indeed, the overall elastic and conductive properties can be expressed in terms of c and the shape factor common for all inhomogeneities. • For cracks, their concentration is characterized by crack density parameters. In the isotropic case of randomly oriented circular cracks (of radii ai ), the scalar crack density parameter introduced by Bristow (1960) is ρ=

1 $ (i)3 a . V

(3.2a)

1 $ (i)2 a , A

(3.2b)

i

In the two-dimensional case, ρ=

i

where 2a(i) are crack lengths and A is the representative area. These definitions are rooted in the fact that the individual crack contributions to the overall compliance are proportional to their sizes cubed (squared, in 2D). For nonrandom crack orientations, the scalar crack density was generalized to second-rank crack density tensor in the work of Kachanov (1980) 1 $  3 (i) 1 $  2 (i) α= a nn a nn ). (3.3) (in 2D case, α = V A i

i

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I. Sevostianov and M. Kachanov In this work, the fourth-rank tensor β=

i 1 $ 3 a nnnn V

(3.4)

i

was identified as a second crack density parameter but shown to play a relatively minor role (provided crack faces are traction free). Here, n is a unit normal to a crack, and nn and nnnn denote dyadic products. • For a granular material consisting of spheres of radius R and nonconducting intergranular space, the effective conductivity (in the isotropic case) was expressed √ by Batchelor and O’Brien (1977) in terms of the parameter φN R−1 S, where φ is the volume fraction of spheres, N is the average number of contacts per sphere, and S is the contact area (assumed to be √ the same for all contacts). This parameter reflects the fact that product R−1 S represents the contribution of an individual contact to the overall conductivity. These microstructural parameters have certain limitations. The volume fraction (3.1) may be insufficient in the following cases: • Mixtures of inhomogeneities of diverse shapes. The proper parameters become nontrivial, as illustrated by a simple 2D example of randomly oriented elliptical holes (Section 4). Shape “irregularities” further complicate the matter. • Nonrandomly oriented inhomogeneities. The proper parameters are tensorial, and their rank is not immediately obvious. Crack density parameters (3.2–3.4) may be inadequate in the following cases: • Nonplanar cracks. • Cracks that are not traction free, such as sliding cracks under compression (in this case, the fourth-rank tensor (3.4) starts to play a major role, Kachanov, 1982, 1993) or fluid-filled crack-like pores of diverse aspect ratios (for which tensor (3.4) has to be modified, Kachanov and Shafiro, 1997). √ The parameter φN R−1 S for a granular material may become inadequate in cases of noncircular or non-Hertzian contacts. Remark For certain microstructures with known microgeometries, the simplest scalar microstructural parameters (volume fractions, crack density) may still be sufficient – provided they are supplemented by characteristics of this specific microgeometry. For example, for two families of parallel spheroids with different properties, the effective properties can be expressed in terms of their partial volume fractions (Taya and Chow, 1981). As another example, several families of parallel

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cracks can be characterized, for the purpose of effective properties, by partial scalar crack densities, plus angles between the families (Piau, 1980). Such expressions are restricted to the particular geometries considered. If, for instance, an orientation scatter is introduced into examples above, the concentration parameters should be revised. Materials science applications increasingly address more complex microstructures – both man-made and naturally occurring. Examples are cortical bone (several systems of pores of diverse geometries and orientations); various sprayed materials (oblate, crack-like pores, mixed with round ones); various reinforced composites (that, in addition to embedded inhomogeneities, may develop microcracks or micropores); geological materials (intersecting fractures at large scale, nonflat microcracks mixed with intergranular pores of complex geometries at microscale). In such cases, the proper microstructural parameters may be entirely nontrivial. In summary, much attention has been paid in literature to constructing dependencies   (3.5) effective property = f microstructural parameter . &' ( % ?

The argument of this function has received much less attention. This issue is discussed in Section 4.

3.2. Microstructural Parameters are Rooted in the Non-interaction Approximation Contributions of individual inhomogeneities to the overall property are affected by interactions between them. Consider, for example, a solid with parallel circular cracks of radius a that are either (A) coplanar or (B) stacked. In case (A), crack contributions to the effective compliances depend on a stronger than a3 (amplifying interactions); in case (B), weaker than a3 (shielding interactions). Strictly speaking, interactions should be reflected in the crack compliance contributions. The corresponding microstructural parameter would correctly reflect “relative weights” of individual cracks and would depend on locations of crack centers. The effective constants would then be linear functions of this parameter. However, incorporating interactions into the microstructural parameter may not be a practical approach – it amounts to solving the interaction problem. Therefore, microstructural parameters usually ignore the interactions and take contributions of individual inhomogeneities by treating them as isolated ones (in particular, they do not reflect the mutual positions of inhomogeneities). The effect of interactions

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on the overall properties is then reflected in a nonlinear dependence of the property on the parameter that is defined in the non-interaction approximation. Although this approach is a practical and is taken in the present work, one should be aware of its limitations: • Beyond the non-interaction approximation, the effective property is, generally, a nonunique function of thus defined microstructural parameter. This may be acceptable if sufficiently narrow bounds can be constructed for this function, which is not always the case (pores and cracks are examples). Besides, constructing bounds in anisotropic cases is a difficult mathematical problem. • An attempt to account for the mutual positions of inhomogeneities, while retaining the microstructural parameter that does not reflect them, may or may not work. Consider, for example, a periodic arrangement of diverse inhomogeneities (diverse shapes or orientations). It is not clear, a priori, whether the effective properties can be represented in terms of the product of a usual microstructural parameter, rooted in the non-interaction approximation and certain periodicity parameter. Thus, the microstructural parameters obtained by summing the contributions of individual inhomogeneities – treated as isolated ones – to overall property are rigorously proper in the non-interaction approximation. It seems logical to use them in various approximate schemes that use the non-interaction approximation as the basic building block, by placing inhomogeneities treated as isolated ones into some “effective” environment-effective homogeneous matrix or effective uniform field (self-consistent, differential, Mori–Tanaka’s schemes). The accuracy of such schemes, however, has to be judged on the case-by-case basis. Having accepted the parameters defined in the noninteraction approximation, the main challenge is to incorporate shapes and orientations of inhomogeneities into these parameters.

3.3. Cases of Overall Isotropy We now consider materials that are isotropic, overall, and consist of an isotropic matrix and, generally, nonspherical inhomogeneities. For the overall isotropy, the inhomogeneities must have random distributions over orientations and over sizes and, in addition, uncorrelated mutual positions. The requirement of randomness of orientations and mutual positions can actually be somewhat relaxed, without affecting the overall isotropy. Since the elastic properties are characterized by a fourth-rank tensor, these distributions may have the hexagonal symmetry; for conductive properties characterized by a second-rank tensor, this requirement can be further relaxed by allowing the cubic symmetries as well.

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Equivalent for conductivity

Not equivalent for elasticity

Fig. 3.1 For the conductive properties, any isotropic mixture of diverse inhomogeneities is equivalent to a certain volume fraction of spheres. This equivalence does not hold for the elastic properties.

Isotropic conductivity is characterized by one effective constant. Therefore, any isotropic mixture of diverse inhomogeneities is equivalent, in its effect on conductivity, to an appropriate volume fraction of spheres, that reflects both the volume fraction and the average shape of inhomogeneities. Isotropic elasticity is characterized by two independent effective constants. Therefore, there is, generally, no equivalence to spheres (Fig. 3.1).

3.4. Benefits of the Proper Microstructural Parameters Identification of the proper microstructural parameters yields the following benefits: • Mixtures of inhomogeneities of diverse shapes and orientations (that are typical for materials science applications) are covered in a unified way. • The overall anisotropy is identified: it is determined by the rank and symmetry of the tensor microstructural parameter. This issue is most relevant for the elastic properties since the conductive ones, being characterized by secondrank symmetric tensor, always have the orthotropic symmetry. Thus identified elastic anisotropy may possess symmetries that the geometrical patterns of microstructure may not have. • Guidance in quantitative modeling of various microstructural features. The proper parameters reflect those microstructural elements that have a dominant effect on the property considered and ignore less important “details.” For example, parameters for the elastic/conductive properties take the individual inhomogeneities’ contributions proportionally to their sizes cubed. This implies that small inhomogeneities can be ignored, unless they vastly outnumber the larger ones. Yet another example concerns pores of strongly oblate shapes (aspect ratios < 0.10–0.15): their aspect ratios – and,

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therefore, porosity – are irrelevant for the elastic/conductive properties, and the concentration of pores should be characterized by the crack density parameter. • Design of microstructures for the prescribed effective properties. An example is given by plasma-sprayed thermal barrier coatings that must have low thermal conductivity in the direction normal to the coating and high elastic compliance in the direction parallel to the coating. This requires identification of the microstructural parameters that actually control the mentioned properties. • Recovery of information on microstructure from the effective properties. Such information recovery is, obviously, nonunique. Nevertheless, certain information can be extracted. Identification of the proper microstructural parameters is crucial; it is these very parameters that can be extracted. • It leads to explicit cross-property connections between different physical properties, if the controlling microstructural parameters for them are sufficiently similar. This is the case for the elasticity–conductivity connections that constitute the focus of the present work as well as the electric – thermal conductivity connection discussed in Section 5.8.

4. Materials with Isolated Inhomogeneities: Microstructural Parameters for the Effective Elasticity and Effective Conductivity We consider materials that can be described as a continuous matrix containing isolated inhomogeneities (cracks, pores, foreign particles). Motivated by materials science applications, we assume that the inhomogeneities have generally diverse shapes and orientations. The first problem is to identify the proper microstructural parameters – in other words, the proper parameters of concentration of inhomogeneities – in whose terms the property of interest (effective elasticity or effective conductivity) is to be expressed. As discussed in Section 3, such parameters must represent individual inhomogeneities according to their actual contributions to the considered property. We first discuss effective elasticity, where this problem is most challenging, and then proceed to effective conductivity. We focus on similarities between microstructural parameters for the two properties since the explicit elasticity– conductivity connections developed in Chapters 5 and 6 hinge on them. The similarities are not immediately obvious – the two physical properties obey different field equations and are characterized by tensors of different ranks (second-rank conductivity tensor versus fourth-rank tensor of elastic constants). One of the key

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challenges is to represent the elasticity tensor (more precisely, its change due to inhomogeneities) in terms of certain second-rank tensor that can be related to the conductivity changes. The background material will be assumed isotropic, k0 is its conductivity and E0 and v0 are Young’s modulus and Poisson’s ratio. Although we focus on 3D microstructures that are of main interest for applications, several 2D models will be used as illustrations. In 2D cases, E0 and v0 will denote 2D elastic constants that coincide with 3D ones for plane stress; for plane strain, they are obtained from 3D ones by dividing them over 1 − vˆ 20 and 1 − vˆ 0 , respectively, where vˆ 0 is 3D Poisson’s ratio. In both cases, 2D bulk modulus is related to 2D constants E0 and v0 by K0 = E0 / (2 − 2v0 ). The elastic behavior is assumed to be linear. This may be a limitation if compressive loads are applied to a solid with narrow, crack-like pores. For the linear elastic modeling to be adequate in such cases, compressive loads should be sufficiently low, as not to cause noticeable pore closures. Results of this Section have been obtained mostly by Sevostianov and Kachanov (2002a) and Kachanov and Sevostianov (2005). Sections 4.1 to 4.6 focus on the elasticity-related microstructural parameters, and Section 4.7 on the conductivityrelated ones.

4.1. General Considerations We assume that the solid is subjected to “remotely applied” stress σ that, in absence of inhomogeneities, would have been uniform within representative volume V (“homogeneous boundary conditions,” Hashin, 1983). The problem of finding the effective elastic properties is best formulated in terms of the elastic potential in stress f(σ) (or in strain, g (ε)) since the structure of the potential aids in identification of the proper microstructural parameters. The effective compliances S ijkl are obtained by differentiation: εij = Sijkl σkl = ∂f/∂σij . We represent f(σ) as a sum f = f0 + f,

(4.1)

where f0 = [(1 + v0 )/2E0 ] σij σji − (v0 /2E0 ) (σkk )2 is the potential in the absence of inhomogeneities. This implies a similar sum for the effective com0 + S . As shown in the text to follow, the structure of f pliances: Sijkl = Sijkl ijkl implies the proper parameters of concentration of inhomogeneities.

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We represent f as a sum of terms corresponding to inhomogeneities contained in V ⎛ ⎞ $ $ (p) $ 1 1 H(p) : σ ⎝≡ H σkl σmn ⎠, (4.2)

f =

f (p) = σ : 2 2 p klmn p where the fourth-rank compliance contribution tensor H(p) gives the extra strain, per reference volume V, due to the presence of the pth inhomogeneity:

ε(p) = H(p) : σ.

(4.3)

The H-tensors reflect shapes and orientations of inhomogeneities, and they scaled as characteristic sizes of inhomogeneities cubed. Treating the inhomogeneities as isolated ones, H-tensors were calculated for a number of 2D and 3D shapes (Kachanov et al., 1994; Sevostianov and Kachanov, 2002a). Thus, the structure of f identifies the general proper microstructural parameter, namely the sum $ H(p) (4.4) subject to symmetrization (ij ↔ kl for the ijkl components) imposed by (4.2). Summation over inhomogeneities can be replaced by integration over their orientations, if computationally convenient. Since the H-tensors represent individual inhomogeneities in accordance with their actual contributions to the effective elastic properties, using parameter (4.4) insures that the effective elastic constants are unique functions of this parameter, at least in the non-interaction approximation. This general parameter (4.4) covers, in a unified way, mixtures of inhomogeneities of diverse shapes and orientations. The challenges are: • to calculate this sum for the classes of inhomogeneity shapes that are of interest; • to explore whether it can be replaced by simpler parameters, most notably, by a second-rank tensor or by scalars, in cases of isotropy. A dual formulation is in terms of stiffness contribution tensors N(p) . Tensor N of an inhomogeneity gives the extra average stress in V due to its presence,

σ = N : ε, under the assumption that, in its absence, the strain field within the site of the inhomogeneity would have been uniform. The elastic potential in strains g(ε) is given by 1 $ (p) g = g0 + g = g0 + ε : N : ε, 2

(4.5)

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105

where g0 = [E0 /2 (1 + v0 )] εij εji + [v0 E0 /2 (1 + v0 ) (1 − 2v0 )] (εkk )2 is the potential in the absence of inhomogeneities, thus identifying the parameter dual to (4.4) as $

N (p)

(4.6)

subject to the same symmetrization as (4.4). Similarly to H-tensors, N-tensors are calculated by treating the inhomogeneities as isolated ones. The H- and N-tensors are interrelated as follows. The overall compliance tensor of the representative volume containing one inhomogeneity, S0 + H, is the inverse of its stiffness tensor C 0 + N. Linearizing with respect to a small change due to  −1 = S0 , we have the inhomogeneity and taking into account that C 0 N = −C 0 : H : C 0

(4.7)

or, in the case of isotropy,   −Nijkl = λ20 Hmmnn δij δkl + μ20 Hijkl + λ0 μ0 δij Hmmkl + δkl Hmmij ,

(4.8)

where λ0 and μ0 are Lame constants of the background material. Remark That it is H (or N) tensors to which the summation is to be applied, has not always been clear in literature. For example, Johannesson and Pedersen (1998) suggested to average Eshelby’s tensor over orientations. The simplest way to see that this is not the quantity to be averaged is to observe that, in the isotropic case of random orientations of nonspherical inhomogeneities, the averaged Eshelby’s tensor coincides with the one for a sphere. However, it is generally impossible to match two effective isotropic constants by one parameter, the volume fraction of spheres. In other words, averaged Eshelby’s tensor is not a proper microstructural parameter. The dual formulations (4.2) and (4.5) would yield the same results if exact expressions for f and g were known, as in the limit of small concentration of ellipsoidal inhomogeneities. The choice between the two formulations is important in the context of the accuracy of various approximate schemes – formulated in terms of either H- or N-tensors. Of particular importance is application of the non-interaction results to finite concentration of inhomogeneities. For example, the potential in stresses (4.1) would yield the effective Young’s modulus (or other effective stiffnesses) in the form E/E0 = (1 + Kc)−1 ,

(4.9)

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where c is the inhomogeneities’ concentration and K is some constant, whereas potential (4.5) leads to E/E0 = 1 − Kc.

(4.10)

The question arises which of the two results (4.9) or (4.10) is more accurate at nonsmall values of c. We cannot formulate a fully rigorous criterion of choice, and it seems that this problem has not received sufficient attention. The following criterion appears reasonable: the change due to inhomogeneities should not lead to the effective constants becoming negative when the concentration parameter reaches a certain value. Based on this criterion, the choice is clear in the cases of cracks and rigid discs: • For cracks, the potential in stresses f(σ) that implies summation of the compliance contributions of cracks is the preferred choice. This is in agreement with the fact that cracks are sources of extra strains, see Eq. (4.11) below. • In the opposite limit of rigid discs, similar considerations imply that g(ε) and N-tensors should be used. In-between these two extremes, the choice is less clear. For pores or inclusions that are softer than the matrix, the potential f(σ) appears preferable; for inclusions that are stiffer than the matrix, the opposite is true. We note that this recommendation may not be generally valid: inclusions may have, for example, low shear modulus and high bulk modulus. We now discuss various classes of inhomogeneities and explore the possibility to replace the general parameters (4.4) and (4.6) by simpler ones.

4.2. Quantitative Characterization of Cracks We first consider 2D rectilinear cracks when results are particularly transparent and then discuss 3D geometries that are of main interest for applications. 4.2.1. Two-dimensional Solid with Rectilinear Cracks Denoting crack lengths by 2a(p) and unit normals to cracks by n(p), the strain per representative area A has the form ε = S0 : σ +

1 $1 (bn + nb)(p) 2a(p) A p 2 , % &' (

ε

(4.11)

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107

where the summation is over all the cracks in A (it may be replaced by integration over orientations, if computationally convenient) and vector b = < u+ − u− > is the average displacement discontinuity on a crack. Representation)(4.11), as well as its extension to pores in terms of integrals over pore boundaries (un + nu)ds, is an immediate consequence of a footnote remark of Hill (1963); in the explicit form, it was given, for example, by Vavakin and Salganik (1975). The following fact is of importance: for each crack, the vector b is parallel to the traction vector n · σ induced at the crack site (in a continuous material) by the applied loading σ: b=

␲a n · σ, E0

(4.12)

where E0 is the 2D Young’s modulus. This follows from equality of crack compliances in the normal and shear modes: if p and τ are uniform normal and shear tractions applied to crack faces, then the corresponding average displacement discontinuities are * + ␲a bn p = , (4.13) τ bτ E0 with the same proportionality coefficient ␲a/E0 . This implies that the potential change f = (1/2) σ : S : σ = (1/2) σ : ε due to multiple cracks of diverse orientations and sizes is 1 $1 1 1 $ (p) (bn + nb)(p) 2a(p) =

f = σ : n · σ · b(p) a(p) 2 A p 2 A p =

␲ ␲ 1 $  2 (p) (σ · σ) : (σ · σ) : α, = a nn E0 A p E0

thus identifying the 2D crack density tensor 1 $  2 (p) a nn , α= A

(4.14)

(4.15)

as the proper crack density parameter. We emphasize that it is not introduced a priori but is implied by the structure of f. Being a sum of symmetric dyads, it is a symmetric second-rank tensor. It generalizes the 2D scalar crack density  ρ = (1/A) a(p)2 : its linear invariant tr α = ρ. For randomly oriented cracks, α = (1/2) ρ I, where I is 2D unit tensor; for parallel cracks, α = ρ nn. Using tensor α as the crack density parameter yields the following benefits: • The effective constants are obtained in a unified form that covers all orientation distributions.

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• It shows that all the details of cracks distribution over orientations and sizes (for example, orientation scatter about certain preferential orientation) that are relevant for the effective elastic properties enter through two parameters only – the principal values of α. • It identifies anisotropy due to cracks as orthotropy with principal axes coaxial to the ones of α (since α is a symmetric second-rank tensor). This result may not be intuitively obvious; it applies to any orientation distribution of cracks (for example, to several families of cracks forming arbitrary angles with each other). Moreover, the orthotropy is of a rather special type, characterized in the 2D case by only three independent constants (Kachanov, 1980, 1993). The possibility to characterize an arbitrary field of cracks by α, without involving tensors of higher ranks, is due to equality of the normal and shear crack compliances (4.13). 4.2.2. Three-dimensional Solid with Circular Cracks For a circular (penny shaped) crack of radius a, the normal and the shear compliances are not equal. However, they are relatively close, differing by a factor of 1 − v0 /2. For multiple cracks, this leads to the following expression for the potential change (Kachanov, 1980):   , 16 1 − v20 V0 (σ · σ) : α −

f = σ:β:σ , (4.16) 3 (2 − v0 ) E0 2 where α=

1 $  3 (p) a nn V

is the 3D crack density tensor and fourth-rank tensor (p) 1 $ 3 β= a nnnn V

(4.17)

(4.18)

is the second crack density parameter that emerges due to the difference between the normal and shear crack compliances. The impact of the β-term is, typically, relatively small, due to the relatively small factor v0 /2. Direct computational studies show that it is even smaller than one may expect (Grechka and Kachanov, 2006a). Thus, retaining α as the sole crack density parameter constitutes a good approximation. This implies approximate orthotropy coaxial with the principal axes of α. Moreover, the orthotropy due to cracks is of a rather special “elliptic” type, characterized by only four independent constants (see Kachanov, 1980, 1993 for details). The concept of approximate symmetries of the elastic and conductive properties, which is used throughout the present work, is discussed in Appendix A.

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109

Expression (4.16) and its approximate simplified form (without β-term) cover all orientation distributions of cracks in a unified way. We mention one orientation distribution that is relevant for many materials science applications – a preferential orientation with certain scatter. We describe this distribution by the following function, containing the scatter parameter λ (Sevostianov and Kachanov, 2000):  1  2 λ + 1 e−λϕ + λe−λ␲/2 (4.19) Pλ (ϕ) = 2␲ (0 ≤ φ ≤ ␲/2 is the angle between normal n to a given crack and the preferential orientation of n). The extreme cases of fully random and ideally parallel orientations correspond to λ = 0 and λ = ∞, respectively. Fig. 4.1 shows orientation patterns that correspond to several values of λ. This orientation distribution is transversely isotropic. Therefore, the crack density tensor has two components: α11 = α22 = f1 (λ) ρ;

α33 = f2 (λ) ρ,

(4.20)

where ρ = tr α a is the overall crack density and:      2 18 − λ λ2 + 3 e−λ␲/2 λ + 3 3 + λe−λ␲/2     f1 = , f2 = . 6 λ2 + 9 3 λ2 + 9

(4.21)

20 1

2

3

15 l 5 2.5

l5 0 4

10

l 5 5.0

5

5 4

l5 7.5

l 5 10

5 3 2 0

0

1 p/4 w

p/2

Fig. 4.1 Orientation distribution function Pλ (ϕ) at several values of scatter parameter λ and the corresponding orientation patterns.

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4.2.3. Cracks of Irregular Shapes The question arises, whether the findings above can be extended to noncircular cracks that are typical in material science applications; in other words, whether certain equivalent distribution of circular ones exists. For flat elliptical cracks, the answer is positive provided deviations from circles are random; moreover, the density of the equivalent distribution of circular cracks can be explicitly expressed in terms of the ellipses’ distribution (Kachanov, 1993). For a distribution of flat cracks of irregular shapes, computational studies of Grechka and Kachanov (2006b,c) show that the replacement by an equivalent distribution of circular cracks is possible as well, provided the shape irregularities are random. Therefore, the characterization by crack density tensor α can be retained, implying overall orthotropy. We mention one irregularity factor of particular importance – “islands” of cohesion between crack faces. These features are common in various microstructures (for example, in sprayed coatings). Such islands – even small ones – produce a strong stiffening effect, substantially reducing the density of equivalent circular cracks. The normal compliance of a circular crack with a concentric island is given by (Sevostianov and Kachanov, 2002b):   1 16a3 1 − v20 Hnnnn = V 3␲E0 % &' ( circular crack ⎡  √  √ 2 ⎤ λ2 2 2 3␲2 ⎣ 3 2 2 (2 − λ) + 2λ3 −λ − 1 ln (1 − λ) + − 1 ⎦, × 8 ␲ 2 ␲ % &' ( correction for island (4.22) where λ = c/a (ratio of internal and external radii of the annular crack). Fig. 4.2 shows radius of the equivalent circular crack in terms of the island size; an almost vertical drop indicates a strong stiffening effect of even a small island (the effect of an island on conductivity, relevant for cross-property connections, is also shown). Remark In the limit of λ  1 (narrow  ring),  we have approximately plane strain conditions and Hnnnn = 2␲ac2 1 − v2 /E, as expected (multiplier 2␲a represents the length of the  ring). In the limit of λ → 1, Hnnnn , formally speaking, should approach 16 1 − v2 a3 / (3E) – its value for the circular crack of radius a. Eq. (4.22) does not yield this limiting value; moreover, the logarithmic term tends to infinity in this limit. This is due to the fact that the value of K I used

Connections between Elastic and Conductive Properties

111

1.00

s

c r r 5a 2 c

Reff /a

0.75 a

0.50 n0 5 0.10 n0 5 0.25 n0 5 0.50

0.25

r5a s

0.00 0.00

0.25

0.50 l 5 a/c

0.75

1.00

Fig. 4.2 Annular crack and the radius of the equivalent circular crack as function of λ = a/c at various values of the Poisson s ratio of the matrix.

in the derivation of (4.22) is an approximation of numerical results that may have up to 3% error. However, even at 1 − λ = 10−20 , Hnnnn , as given by (4.22), still remains lower than Hnnnn for the circular crack. For irregular crack geometries with an island, the possibility to replace a random (in the sense of shapes) distribution of such cracks by an equivalent set of circular ones hinges on the equality of the normal crack compliances Hnnnn to the average, over the in-plane directions, shear crack compliances. Computational studies of Grechka and Kachanov (2006a) show that this equality holds with good accuracy so that characterization by crack density tensor can be retained, with reduced effective value of α. Remark Results for cracks apply, with good accuracy, to strongly oblate pores (aspect ratios smaller than 0.10–0.15). Indeed, as shown by Kachanov et al. (1994), the correction to the compliance contribution due to nonzero opening is of the order of the aspect ratio. Therefore, the effect of such pores on the elastic properties is close to the one of cracks, and they can be characterized by crack the density parameters. As a consequence, porosity is an irrelevant parameter for microstructures containing strongly oblate pores. We now discuss general inhomogeneities. We first illustrate the issue of proper parameters on a simple 2D example of elliptic holes and then proceed to 3D geometries.

4.3. Example: Two-dimensional Elliptic Holes This example demonstrates the guidance provided by the elastic potential in identification of the proper microstructural parameters. The compliance

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contribution tensor of an elliptical hole has the form (Kachanov et al., 1994) . (a + b)2 1 ␲ (mn + nm) a(2a + b)nnnn + b(2b + a)mmmm + H= E0 A 2 (mn + nm) − ab (mmnn + nnmm)] ,

(4.23)

where a and b are semi-axes and m, n are unit vectors along them. It is not obvious  at all that the sum H (p) can be expressed in terms of a symmetric second-rank tensor. However, the potential change due to the hole, after some algebra, takes the form: 0   1 ␲/ 

f = ab 4σ : σ − (trσ)2 + 2 (σ · σ) : a2 nn + b2 mm − abI . 2E0 A (4.24a) For a mixture of elliptic holes of diverse eccentricities and orientations, we have, therefore,

f = where

 0 1 / 2σ : σ − (trσ)2 p + 2 (σ · σ) : ω , 2E

(4.24b)

p=

1 $ (ab)(r) ␲ A r

ω=

(r) 1 $ 2 ␲ a nn + b2 mm (second-rank hole concentration tensor). A r



2D porosity



The structure of f identifies the porosity p and the second-rank symmetric tensor ω as the proper microstructural parameters. In the limit of cracks, p = 0 and ω/␲ reduces to the crack density tensor α. Note that no degeneracy occurs in this limit if parameters p and ω are used. In the case of circles, ω = pI(I is the 2D unit tensor) so that porosity p is the sole microstructural parameter. Representation (4.24b) covers all mixtures of diverse ellipses in a unified way (for example, a mixture of circular holes and cracks). In the isotropic case of randomly oriented ellipses, the tensor ω = (r) (r)  2  2 (␲/2A) a + b2 or, a + b2 I can be replaced by the scalar (1/A) equivalently, by the average eccentricity parameter 1 $  (r) q= ␲ a − b(r)2 . (4.25) A r

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113

and the effective Young’s and bulk moduli in this case are: 1 1 E G K = = , , K0 1 + (2p + q) / (1 − ν0 ) E0 1 + 3p + q G0 1 = . 1 + (4p + q) / (1 + ν0 )

(4.26)

Thus, in the isotropic case, p and q constitute the proper microstructural parameters. In the limit of cracks, p = 0 and q/␲ reduces to the usual 2D scalar crack density. Remark If one wishes to plot the effective moduli in terms of one parameter only, Eq. (4.26) shows that this parameter should be different for the three moduli: (2p + q) for K, (3p + q) for E, and (4p + q) for G. We emphasize that microstructural parameters – p and ω in the general anisotropic case, and p and q in the isotropic case – are not introduced a priori but are implied by the structure of f and that they are nontrivial. They are necessary; expressing the effective constants in terms of some other parameters would be nonunique, even in the low concentration limit (as seen from (4.26), the effective moduli cannot be expressed in terms of porosity p alone).

4.4. Three-dimensional Inhomogeneities For the ellipsoidal inhomogeneities, the H- and N-tensors are expressed in terms of Eshelby’s tensor sijkl that is a function of ellipsoid’s geometry and Poisson’s ratio of the matrix (Eshelby, 1957, 1961; see, also, the book of Mura, 1987). Utilizing the solution of Eshelby’s problem in the form given by Kunin and Sosnina (1971), we relate uniform strains and stresses inside the ellipsoidal inclusion to the uniform field at infinity (either σij or εij ): (int)

εij

= ijkl εkl ,

(int)

σij

= ijkl σkl .

(4.27)

The strain and stress concentration tensors introduced by Wu (1966) are given by   −1 1 0 − Cmnkl , ijkl = Jijkl + Pijmn Cmnkl  −1  1 0 − Smnkl , ijkl = Jijkl + Qijmn Smnkl

(4.28)

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−1 where the inverse of symmetric fourth-rank tensor Xijkl is defined by the rela    −1 −1 tion Xijmn Xmnkl = Xijmn Xmnkl = Jijkl , and Jijkl = δik δlj + δil δkj /2 is the symmetrized fourth-rank unit tensor. Tensors P and Q can be expressed in terms of Eshelby’s tensor s: 0 Pijkl = sijmn Smnkl ;

0 (Jmnkl − smnkl ), Qijkl = Cijmn

(4.29)

and tensors P and Q , and are interrelated as follows:   0 0 0 0 Qijkl = Cijmn Jmnkl − Pmnrs Crskl , Pijkl = Sijmn Jmnkl − Qmnrs Sirskl , ∗ 0 ijkl = Sijmn mnrs Crskl ,

∗ 0 ijkl = Cijmn mnrs Srskl .

(4.30)

Utilizing these results, the compliance and stiffness contribution tensors of the inclusion are given by the equations -−1 , −1 V1  1 0 Sijkl − sijkl + Qijkl , Hijkl = V Nijkl =

-−1 , −1 V1  1 0 Cijkl − Cijkl + Pijkl , V

(4.31)

where V1 is the volume of the inhomogeneity. In particular, H = (V1 /V ) Q−1 for a pore and N = (V1 /V ) P−1 for a rigid inclusion. Remark It is seen that H- and N-tensors – quantities relevant for the effective properties – utilize an incomplete set of Eshelby’s tensor components since expressions in the brackets of (4.31) are symmetric with respect to (ij) ↔ (kl). This imposes three constraints on 12 components of Eshelby’s tensor for an ellipsoid (one constraint on six components, in the case of a spheroid). An important observation is that, since Eshelby’s tensor is independent of the elastic constants of the inhomogeneity, the H- and N-tensors for an inhomogeneity of arbitrary elastic properties can be explicitly expressed in terms of H- and Ntensors for a pore, or for a rigid inclusion, respectively:  −1  V1  1 0 1 0 pore ˜ pore + Sklmn ˜ mnrs Sijkl − Sijkl H H − S , (4.32) Hijrs = klmn klmn V Nijkl =

 −1  V1  1 rigid 0 1 0 ˜ rigid + Cklmn ˜ mnrs N N − C , Cijkl − Cijkl klmn klmn V

˜ = (V/V1 )H and N ˜ = (V/V1 )N. where H

(4.33)

Connections between Elastic and Conductive Properties

115

Remark Chen and Young (1977) showed that Eq. (4.31) holds for an inhomogeneity of general shape as the zeroth approximation with tensor sijkl in (4.29) understood in terms of average over the inhomogeneity strain (discussed in detail by Rodin, 1996). If all inhomogeneities have identical shapes and their orientations are either  (p)  (p) random or strictly parallel, the general parameter H (or N ) can be replaced by volume fraction c plus the shape factor (that, for ellipsoids, is known). For some simple mixtures of diverse ellipsoids, partial volume fractions may still be adequate as microstructural parameters. An example is given by two families of spheroids of two different aspect ratios, all of them strictly parallel (Taya and Chow, 1981). In the case of overall isotropy (random orientations of inhomogeneities), the general tensor parameter can be replaced by two scalars that reflect, in an integral way, the volume fraction of inhomogeneities and their distribution over shapes.

4.5. Spheroidal Inhomogeneities A surprising simplification of key importance is possible for a mixture of spheroids of diverse aspect ratios and orientations: the fourth-rank general parameter, (4.4) or (4.6), can be expressed, with good accuracy, in terms of certain symmetric second-rank tensor (and unit tensors). The possibility of such a representation was demonstrated in Section 4.3 for 2-D elliptic holes (in which case the representation is exact). We first consider a single spheroidal inhomogeneity (semi-axes are a1 = a2 and a3 and unit vector of the spheroid’s axis is n). In the text to follow, the largest semi-axis of the spheroid will be denoted by a (a = a1 for the oblate spheroids and a = a3 for the prolate spheroids). This definition of a leads to equations that cover, without degeneracy, the entire range of spheroid shapes, from a thin disc to a needle. Our analysis requires explicit analytic inversions and contractions of fourthrank tensors. They can be done by representing the tensors in terms of a “standard” tensor basis (Kunin, 1983; Walpole, 1984; see Appendix A). In the cases of the transversely isotropic symmetry, this basis consists of six tensors T(1) , . . . , T(6) , so that 6 6   pk T(k) , Q = qk T(k) , P= H=

k=1 a3 4␲γ t

V

3

k=1 6  k=1

(k)

hk T ,

6 a3 4␲γ t  N= nk T(k) , V 3 k=1

thus reducing the problem to calculation of factors pk , qk , hk , and nk .

(4.34)

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Representations for the H- and N-tensors involve power exponent t that is equal to 1 for the oblate shapes (γ < 1) and to −2 for the prolate shapes (γ > 1). The fact that aspect ratio γ enters (4.34) as either γ or γ −2 is rooted in the representation of the volume of a spheroid in terms of its largest semi-axis a. Using the representations for the stiffness and compliance tensors, Eshelby’s tensor and unit tensor in terms of the mentioned basis yields the following relations for coefficients pi , qi : 1 1 [(1 − κ) f0 + κf1 ], p2 = [(2 − κ) f0 + κf1 ], 2G0 2G0 κ 1 [1 − f0 − 4κf1 ], p3 = p4 = − f1 , p5 = G0 G0 1 [(1 − κ) (1 − 2f0 ) + 2κf1 ] p6 = G0

p1 =

(4.35)

and q1 = G0 [4κ − 1 − 2 (3κ − 1) f0 − 2κf1 ], q3 = q4 = 2G0 [(2κ − 1) f0 + 2κf1 ],

q2 = 2G0 [1 − (2 − κ) f0 − κf1 ] ,

q5 = 4G0 [f0 + 4κf1 ],

q6 = 8G0 κ [f0 − f1 ] ,

(4.36)

where κ−1 = 2 (1 − v0 ). The following functions of the aspect ratio γ are used hereafter:    γ 2 2γ 2 + 1 g − 3 γ 2 (1 − g)  , f1 (γ) = f0 (γ) =  2 , (4.37)  2 2 γ −1 4 γ2 − 1 where g (γ) =

⎧ ⎪ ⎨ √1 ⎪ ⎩

1−γ 2 √1 γ γ 2 −1

γ

√

1−γ 2

arctan γ , oblate shape (γ < 1)  5 ln γ + γ 2 − 1 , prolate shape (γ > 1).

(4.38)

If the material of an inhomogeneity is transversely isotropic, with the axis of elastic symmetry coinciding with the spheroid’s axis, the hk factors entering H-tensor in (4.34) are given by  d  d d ψ S1111 + S1122 ψ6 − 2S1133 4 d /ψ2 , , h2 = 2S1212 h1 = 4 (ψ1 ψ6 − ψ3 ψ4 ) d ψ d ψ S d ψ6 − S3333 S d ψ1 S1133 3 4 d h3 = h4 = 1133 /ψ5 , h6 = 3333 , , h5 = 8S1313 2 (ψ1 ψ6 − ψ3 ψ4 ) ψ1 ψ6 − ψ 3 ψ4 (4.39)

Connections between Elastic and Conductive Properties where ψ1 =

117

 1 d d d + q1 S1111 − q3 S1133 + S1122 , 2 

d ψ2 = 1 + 2q2 S1212 , d d d S1111 + q6 S1133 + S1122 ,

d d + q3 S3333 ψ3 = −2q1 S1133 , ψ4 = q3  d d d + 2q3 S1133 . , ψ6 = 1 + q6 S3333 ψ5 = 2 1 + q5 S1313

(4.40)

Factors nk entering N-tensor in (4.34) are  d  d d ϕ C1111 + C1122 ϕ6 − 2C1133 4 d /ϕ2 , , n2 = 2C1212 n1 = 4 (ϕ1 ϕ6 − ϕ3 ϕ4 ) n3 = n4 =

d ϕ − Cd C1133 6 3333 ϕ3 , 2 (ϕ1 ϕ6 − ϕ3 ϕ4 )

d /ϕ5 , n6 = n5 = 8C1313

d d ϕ ϕ1 − C1133 C3333 4 , ϕ1 ϕ6 − ϕ 3 ϕ4 (4.41)

where  1 d d d d + p1 C1111 − p3 C1133 + C1122 , ϕ2 = 1 + 2p2 C1212 , 2  d d d d d + p3 C3333 , ϕ4 = p3 C1111 + C1122 , (4.42) + p6 C1133 ϕ3 = −2p1 C1133  d d d ϕ5 = 2 1 + p5 C1313 + 2p3 C1133 . , ϕ6 = 1 + p6 C3333

ϕ1 =

We now write the general representation for the H- and N-tensor based on their transversely isotropic symmetry: H= ⎡

a3 1 V E0

⎤

⎢ ⎥ ⎢ ⎥ × ⎢ W1 II + W2 J + W3 (Inn + nnI) + W4 (J · nn + nn · J) + W5 nnnn⎥, &' ( ⎣% ⎦ isotropic terms (4.43a) a3 N = G0 V ⎡

⎤

⎢ ⎥ ⎢ ⎥ × ⎢ U1 II + U2 J + U3 (Inn + nnI) + U4 (J · nn + nn · J) + U5 nnnn⎥, ⎣ % &' ( ⎦ isotropic terms (4.43b)

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where n is a unit vector of the spheroid’s axis and I and J are unit tensors of the second and fourth ranks. Coefficients Wi and Ui are functions of the aspect ratio γ as well as elastic constants of the matrix and the inhomogeneities. In the text to follow, they are referred to as “shape factors.” Using (4.34), we express them in terms of hi and ni : W1 =

4␲γ t E0 (h1 − h2 /2) , 3

W3 =

4␲γ t E0 (2h3 + h2 − 2h1 ) , 3

W5 =

4␲γ t E0 (h6 + h1 + h2 /2 − 2h3 − h5 ) , 3

U1 =

4␲γ t 4␲γ t 4␲γ t (n1 − n2 /2) , U2 = (2n3 + n2 − 2n1 ), n2 , U3 = 3G0 3G0 3G0

U4 =

4␲γ t 4␲γ t (n5 − 2n2 ) , U5 = (n6 + n1 + n2 /2 − 2n3 − n5 ) . (4.44b) 3G0 3G0

W2 =

4␲γ t E0 h2 , 3

W4 =

4␲γ t E0 (h5 − 2h2 ) , 3 (4.44a)

We now consider multiple inhomogeneities. We assume that the distribution over aspect ratios is statistically independent of the distributions over sizes and over orientations of inhomogeneities. This assumption allows one to repre(p)   3 (p)   3 (p)   3 (p)   3 sent sums Wa nn , Wa nnnn , Ua nn , Ua nnnn as products of the average shape factors ∞ wi =

∞ Wi (γ)F(γ)dγ,

ui =

0

Ui (γ)F(γ)dγ,

(4.45)

0

where F(γ) is the aspect ratio distribution density and second- and fourth-rank tensors ω=

1 $  3 (k) a nn , V k

=

(k) 1 $ 3 a nnnn V

(4.46)

k

leading to the following representation  E0 S − S0 = ρ(w1 II + w2 J) + [w3 (ωI + Iω) + w4 (ω·J + J·ω)] + w5 , (4.47a)

Connections between Elastic and Conductive Properties

119

 0 = ρ (u1 II + u2 J) + [u3 (ωI + Iω) + u4 (ω · J + J · ω)] + u5 , G−1 0 C−C (4.47b)   3 (k) where ρ = trω = (1/V ) a (in the case of cracks, it coincides with crack density); wi and ui are the average shape factors related to the individual shape factors Wi and Ui by (4.45) Remark Equations above assume that the distribution over aspect ratios is statistically independent of the distributions over sizes and over orientations of inhomogeneities (for a discussion in more detail, see Section 5.1). Explicit cross-property connections for a solid with multiple inhomogeneities  hinge on the possibility to get rid of terms w5 in the sum H (p) by appropriately adjusting the coefficients at the other terms. More precisely, we seek to  approximate terms σ : H (p) : σ in f by a linear combination of terms σ · σ : ω and (trσ)σ : ω where ω is a certain second-rank symmetric tensor. Alternatively,  such approximations are sought for N (p) . In order to derive the mentioned approximation for multiple inhomogeneities, we start with approximating relations (4.43) for a single inhomogeneity by expressions in terms of the second-rank tensor nn, with terms containing nnnn omitted and coefficients at the remaining terms appropriately adjusted. Thus, we seek to approximate at least one of the H- and N-tensors by the expressions ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢B1 II + B2 J +B3 (nnI + Inn) + B4 (nn · J + J · nn)⎥, (4.48a) &' ( % ⎣ ⎦ isotropic terms ⎡ ⎤

H=

a3 1 V E0

N=

⎢ ⎥ a3 ⎢ ⎥ G0 ⎢D1 II + D2 J +D3 (nnI + Inn) + D4 (nn · J + J · nn)⎥, (4.48b) &' ( % ⎣ ⎦ V isotropic terms

where a is the largest semi-axis whether the spheroid is oblate or prolate; factors Bi , Di are to be found. Representation (4.48a) requires that hi -factors obey the relation h6 + h1 + h2 /2 − 2h3 − h5 = 0

(4.49)

that, with the exception of a sphere, does not hold exactly. We now replace tensor H by a fictitious tensor Hˆ , with coefficients hˆ i obtained from hi by multiplication of hi by either (1 + δ) or (1 − δ) and choose δ in such a way that condition (4.49)

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is satisfied exactly for hˆ i :   hˆ 1 = h1 1 − δ sign h1   hˆ 2 = h2 1 − δ sign h2

  hˆ 3 = h3 1 + δ sign h3 ,   hˆ 5 = h5 1 + δ sign h5 ,   hˆ 6 = h6 1 − δ sign h6 ,

(4.50)

where δ=

h6 + h1 + h2 /2 − 2h3 − h5 . |h6 | + |h1 | + |h2 |/2 + 2|h3 | + |h|5

(4.51)

The error of replacing H by Hˆ , estimated by the norm max ijkl, Hijkl =0   ˆ ijkl /Hijkl |, is equal to |δ|. Smallness of this norm guarantees that strain | Hijkl − H responses to all stress states of the actual and of the fictitious inclusions are close. Substitution of (4.50) into (4.44a) yields the factors Bi entering (4.48a):  4␲ 4␲ B1 = E0 γ t hˆ 1 − hˆ 2 /2 , B2 = E0 γ t hˆ 2 , 3 3   4␲ 4␲ B3 = E0 γ t 2hˆ 3 + hˆ 2 − 2hˆ 1 , B4 = E0 γ t hˆ 5 − 2hˆ 2 , (4.52) 3 3 where power exponent t = 1 for oblate shapes and t = −2 for prolate shapes. Similarly, the factors Di for the N-tensor are obtained as  4␲ t  4␲ t γ nˆ 1 − nˆ 2 /2 , D2 = γ nˆ 2 , 3G0 3G0   4␲ t  4␲ t  D3 = γ 2nˆ 3 + nˆ 2 − 2nˆ 1 , D4 = γ nˆ 5 − 2nˆ 2 , 3G0 3G0 D1 =

(4.53)

where nˆ i are given in terms of ni by relations identical to (4.50). Figs. 4.3 and 4.4 show Bi and Di as functions of the aspect ratio, for several combinations of elastic constants. Note that nonsmooth behavior of B1 , B2 and D1 , D2 at γ = 1 is related to the definition of a in (4.52) and (4.53) as the largest semi-axis. Remark Figs. 4.3 and 4.4 show that in the limit γ → 0, the compliance and stiffness contributions of very “soft” inclusions (low, but finite ratios E1 /E0 ) tend to zero and thus do not converge to the ones of cracks. Indeed, cracks provide nonzero contributions due to displacement discontinuities across their faces. In contrast, for a soft thin inclusion (of a small but finite compliance), the difference between displacements of its two faces vanishes as the aspect ratio γ → 0. The case of a crack corresponds to setting v1 = 0, E1 = 0 and going to the limit γ → 0.

Connections between Elastic and Conductive Properties

121

4

B1

5

2

6

4

0 3

2

22

1

24 0.01

0.1

1.0 

10

100

10

100

10

100

15 1 2

B2 5

3

25

5

4

6

215 0.01

0.1

1.0 

2 1

B3 1

2

0 3 21 22 0.01

4

5

6 0.1

1.0 

5 1 3

B4

2

1

3 4

21 23 0.01

5

6 0.1

1.0 

10

100

Fig. 4.3 Factors entering approximate expression (4.48a) for the compliance contribution tensor H as functions of inclusions aspect ratio γ. Poisson’s ratios v0 = v1 = 0.3 Curves 1 to 6 correspond to ratios E1 /E0 = 0.01, 0.1, 0.33, 3.0, 10, and 100.

122

I. Sevostianov and M. Kachanov 5 6 5 0 4 1

25

D1 210 0.01

3 2

0.1

1.0 

10

100

10

100

10

100

10

D2

5 4

5

6

0 3

25 1 210 0.01

0.1

2

1.0 

1 0

5

21

3

22

4

2

23 24

6

D3

25 0.01

1 0.1

1.0 

2 6

1 0

4

21 22

3

5 2 1

23 0.01

D4 0.1

1.0 

10

100

Fig. 4.4 Factors entering approximate expression (4.48b) for the stiffness contribution tensor N as functions of inclusion’s aspect ratio γ. Poisson’s ratios v0 = v1 = 0.3. Curves 1 to 6 correspond to ratios E1 /E0 = 0.01, 0.1, 0.33, 3.0, 10, and 100.

Connections between Elastic and Conductive Properties 15

0

10 4 5

25 0.01

3

0.1

26 29

0 1

2

23

2 Di

Bi

123

1

1 3

4

212 0.01

0.1

1

1

B1, D1

3

B3, D3

2

B2, D2

4

B4, D4

Fig. 4.5 Factors entering expressions (4.48a) and (4.48b) in the case of pores (oblate shapes). The curves are almost flat at γ < 0.15: strongly oblate pores produce the same effect as cracks.

Fig. 4.5 covers the case of oblate pores (for prolate shapes, the curves for very soft inclusions provide a very accurate approximation of pores). In particular, it provides the correct limit of a crack. In this limit, the curves become practically flat at aspect ratios smaller than 0.15, indicating that strongly oblate pores can be identified with cracks, as far as their effect on the linear effective elastic properties is concerned. Accuracy of the approximations for H- and N-tensors is illustrated in Figs. 4.6 and 4.7. These figures show that the accuracy is, generally, better for N- tensors. The accuracy of both approximations is generally good, particularly for pores (Fig. 4.8) but worsens considerably if the inclusion – matrix contrast in the bulk moduli is very different from the one in the shear moduli (such as fluid-like inclusions with negligible shear modulus). We note, in addition, that both approximations, for H- and N-tensors, become exact when Poisson’s ratio of the matrix v0 = 0; the approximation for N-tensor also becomes exact when v0 = 0.5. Remark The fact that the approximations in terms of a second-rank tensor lose accuracy when the mentioned two contrasts are very different is best explained on the example of cracks. For a dry crack, its compliances in the normal and shear modes are close, and this makes it possible to characterize a cracks solely by the second-rank crack density tensor α defined by (3.3). However, filling a crack with fluid (of negligible shear modulus) reduces its normal compliance but does not affect the shear one. The difference between the two compliances gives rise to fourth-rank tensor β defined by (3.4). Remark Since representations (4.48) are approximate, they are not interrelated by Eq. (4.7) that holds for the exact expressions of H and N (although it may hold approximately).

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I. Sevostianov and M. Kachanov Rigid inclusions 100 v0 5 0.25 E0

0.01

g

1

100

v1 5 0.25

needle-like

crack-like

E1 v1 5 0.33 v1 5 0.45

0.01 Soft inclusions

E1

E0

E0

1

g

0.01 Soft inclusions

100

0.01

1

g

100

needle-like

E1 crack-like

0.01

Rigid inclusions 100 v0 5 0.45

needle-like

crack-like

Rigid inclusions 100 v0 5 0.33

0.01 Soft inclusions

Fig. 4.6 Accuracy maps for the approximate representation (4.48a) of H tensor in terms of a second-rank tensor. The combinations of parameters (elastic contrast E1 /E0 and aspect ratio γ) corresponding to accuracy better than 10% lie in the region centered at point 1 and bounded by the curves shown.

For materials with multiple inhomogeneities, the approximation in terms of the  second rank tensor ω = (nn)(p) has the form: ⎤ ⎡  3 $ a 1 ⎢ ⎥ + b2 J + b3 (ωI + Iω) + b4 (ω · J + J · ω)⎦, H (p) = ⎣ %b1 II &' ( V E0 isotropic terms (4.54a) ⎡ ⎤  3 $ a ⎢ ⎥ G0 ⎣ d1 II + d2 J + d3 (ωI + Iω) + d4 (ω · J + J · ω)⎦, N (p) = % &' ( V isotropic terms (4.54b)

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125

Rigid inclusions 100 v0 5 0.25 E1

0.01

1

g

100

v1 5 0.25

needle-like

crack-like

E0

v1 5 0.33 v1 5 0.45

0.01 Soft inclusions Rigid inclusions 100 v0 5 0.45 E1

0.01

1

g

100

0.01 Soft inclusions

crack-like

E0

needle-like

crack-like

E1

E0 0.01

1

g

100

needle-like

Rigid inclusions 100 v0 5 0.33

0.01 Soft inclusions

Fig. 4.7 Same as Fig. 4.6 for the N tensor.

where factors bi , di depend on the average, in a certain sense, aspect ratio of inclusions and on the elastic constants. These representations utilize, again, the statistical independence of the distributions over orientations and over aspect ratios. Remark Aside from being a key point for cross-property connections, representations (4.54) – if they are possible – have important implications for the overall elastic anisotropy: a solid with an arbitrary mixture of spheroidal inhomogeneities is approximately orthotropic (orthotropy axes are coaxial with the principal axes of ω). This result may seem counterintuitive since it covers cases when, geometrically, the patterns of inclusions’ distribution do not have the orthotropic symmetry (such as several families of parallel inclusions at arbitrary angles to each other). Moreover, the orthotropy is of a simplified (“elliptic”) type: the elasticity tensor is expressed in terms of symmetric second-rank tensor ω and is characterized by five independent elastic constants instead of nine. This generalizes a

126

I. Sevostianov and M. Kachanov 0.15

(a) 1

1

H

2

d

2

0.10

3

3 4

0.05

4

5

5 0.01

0.1

1.0 g

10

100

0.04

(b)

N

0.03

4

d

3

4 3 5

0.02

2

0.01

5 2

1

1

0.01

0.1

crack

1.0 g sphere

10

100 needle

1

v0 5 0.1

4

v0 5 0.4

2

v0 5 0.2

5

v0 5 0.5

3

v0 5 0.3

Fig. 4.8 Accuracy of the approximate representation of the pore compliance contribution tensor H (a) and pore stiffness contribution tensor N (b) as a function of aspect ratio γ for several values of the Poisson’s ratio v0 . Note a much higher accuracy (better than 4% in all cases) for tensor N.

similar finding for cracks when the number of independent constants is further reduced to four (Kachanov, 1980, 1993).

4.6. Microstructural Parameters for Conductive Properties: Comparison with Parameters for Elastic Properties Deep similarities exist between microstructural parameters for effective conductivity and effective elasticity. In particular, contributions of individual inhomogeneities to these properties depend on their sizes in the same way, and their orientation dependence is similar (although not identical). At the same time, there are differences in shape dependence of the mentioned contributions. The similarities lead to cross-property connections; the differences result in approximate

Connections between Elastic and Conductive Properties

127

character of the connections and in their sensitivity – albeit mild – to the average shapes of inhomogeneities. We consider the thermal conductivity problem for a background material with isotropic conductivity k0 containing an inhomogeneity with the isotropic thermal conductivity k1 . Limiting cases k1 = 0 and k1 = ∞ correspond to an insulator and a superconductor. Assuming a linear conduction law (linear relation between farfield temperature gradient G and the heat flux vector U per representative volume V), the change in G required to maintain the same heat flux if the inhomogeneity is introduced is:

G = R · U,

(4.55a)

where the symmetric second-rank tensor R can be called the resistivity contribution tensor of an inhomogeneity. Treating the inhomogeneity as an isolated one allows one to calculate R-tensors for certain shapes, such as an ellipsoid or a crack with a cohesion island. Alternatively, this relation can be written in a dual form

U = K · G,

(4.55b)

where K is the conductivity contribution tensor of an inhomogeneity. Remark The problem of electric conductivity is mathematically identical to the thermal conductivity problem with the role of G played by the gradient of electric potential, and the role of U by the vector of electric current. In contrast with H ↔ N relation (4.7) in the elasticity problem, tensors R and K are simply proportional to each other. Indeed, the conductivity of the representative volume containing one inhomogeneity, k0 I + K, is inverse of the resistivity k0−1 I + R; linearization with respect to a small change due to the inhomogeneity implies R = −k0−2 K.

(4.56)

The two dual tensors yield dual formulations for the effective conductive properties of a material with multiple inhomogeneities in the non-interaction approximation: (A) The effective conductivity k is a sum k = k0 I +

$

K (p) .

(4.57a)

(B) The effective resistivity r is a sum r = k0−1 I +

$

R(p) .

(4.57b)

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I. Sevostianov and M. Kachanov

In the asymptotics of small concentration of inhomogeneities, Eqs. (4.57a) and (4.57b) coincide. An important practical question affecting the choice between the two is the accuracy of the non-interaction results in cases of finite concentration. From this point of view, the formulation remaining accurate at larger concentrations is preferable. This problem parallels the one encountered in the context of elastic properties, where the choice is between formulations where either compliance- or stiffness contribution tensors are summed up. It seems that there is no universal rigorous guidance for choosing between the two formulations. However, the following criterion – that is similar to the one taken in the context of elasticity – appears reasonable. If inhomogeneities are more conductive than the matrix, they are treated as sources of extra conductivity, leading to summation of the conductivity contributions (4.57a). If inhomogeneities are less conductive than the matrix, the formulation in resistivities (4.57b) is the one of choice. This criterion prevents the overall conductivity/resistivity from becoming negative at certain concentration of inhomogeneities. Thus, we identify the proper microstructural parameter for the conductivity problem as the sum over all inhomogeneities $ (4.58a) R(p) if k1 > k0 or, alternatively,

$

K (p) if k1 < k0 .

(4.58b)

Tensor R is expressed in terms of second-rank Eshelby’s tensor for conductivity sC :  −1 V1 1 k0 C R= I−s , (4.59) V k0 k1 − k 0 where t = 1 for oblate shapes (γ < 1) and t = −2 for prolate shapes (γ > 1). For an ellipsoidal inhomogeneity, explicit expressions for sC are known. For spheroidal inhomogeneity of aspect ratio γ, relevant results were given by Carslaw and Jaeger (1959). They lead to the following expression for the Eshelby’s tensor: sC = f0 (I − nn) + (1 − 2f0 ) nn,

(4.60)

where n is the unit vector along the spheroid’s axis of symmetry and function f0 (γ) is defined by (4.37). Substitution into (4.59) yields K = −k02 R =

a3 k0 (A1 I + A2 nn), V

(4.61)

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129

where dimensionless factors A1 and A2 (plotted in Fig. 4.9) are as follows: 1 k0 − k11 4␲γ t   1 , 3 k0 + k11 − k0 f0 (γ) 2  1 k33 − k0 (1 − 3f0 (γ)) 4␲γ t  1    1  1  . A2 = 3 k33 − 2 k33 − k0 f0 (γ) k0 + k33 − k0 f0 (γ)

A1 =

(4.62)

In particular, for a sphere, γ = 1 and f0 = 1/3; for a cylinder, γ → ∞ and f0 = 1/2. For a crack, γ → 0 and 1 8a3 1 R= nn. (4.63) V 3 k0 Remark Similarly to the elasticity problem, R for a crack can be used, with good accuracy, for strongly oblate pores (aspect ratio γ smaller than 0.10–0.15). Indeed, Fig. 4.9 shows that A1  A2 for the strongly oblate shapes, whereas A2 is almost flat at small γ. One has to be careful, however, with applying these results to overly 10 1 2 3

5 0

4 5

25 210 215 0.01

6

A1 0.1

6

1.0 

10

100

10

100

6

4 1

5

2

3

0 22 24 0.01

2

4

A2 0.1

1.0 

Fig. 4.9 Factors Ai entering expressions for the conductivity and resistivity contribution tensors as functions of inclusion’s aspect ratio γ. Curves 1 to 6 correspond to ratios k1 /k0 = 0, 0.1, 0.33, 3.0, 10, and 100.

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narrow cracks. As noted by Zimmerman (1996), cracks with aspect ratios smaller than 0.01 may not be accurately modeled as insulators since the contribution from conductivity through air becomes significant. For a thin superconducting disk (γ → 0, k1 = ∞), formulation in terms of the conductivity contribution tensor is more appropriate: K=

16a3 k0 (I − nn). 3

(4.64)

For spheroids, the general microstructural parameter for the effective conduc  tivity R( p) (or the dual parameter K ( p) that, in view of (4.56), is proportional to it) is specified as follows: (p) $ 1 1 $ 3 (p) 1 1 $ 3 R( p) = a A1 a A2 nn , I+ (4.65) V k0 V k0 implying that the proper microstructural parameters in this case are  3 (p) • Scalar V1 a A1 (in the case of identical aspect ratios γ, it reduces to the volume fraction times factor A1 ); (p)  3 • Second-rank symmetric tensor V1 a A2 nn . Assuming that the distribution over aspect ratios is statistically independent of the distributions over sizes and over orientations of inhomogeneities k0 r − I = I − k/k0 = a1 ρI + a2 ω,

(4.66)

where ω is given by (4.46) and ∞ ai =

Ai (γ)F(γ)dγ.

(4.67)

0

In the case of overall isotropy (random orientations of spheroids), the secondrank tensor in the last term in (4.65) is proportional to unit tensor I and adds up to the first term. The overall scalar coefficient at I is the proper microstructural parameter in this case. Generally, it does not reduce to the volume fraction but reflects the average – in the sense implied by (4.65) – aspect ratio. Comparing R- and K-tensors with H- and N-tensors of the elasticity problem, the following observations can be made. • In their size dependence, all these tensors are proportional to the inhomogeneity size cubed. • In their orientation dependence, the difference is that H- and N-tensors contain fourth-rank nnnn-terms (although, in many cases, they can be eliminated with satisfactory approximation, see Section 4.5).

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131

• The main difference between (H, N)- and (R, K)-tensors is in their shape dependence. In the case of spheroidal shapes, this means differences between the aspect ratio dependencies of A1,2 -factors (in the conductivity problem) and of B1−4 - or D1−4 -factors (in the elasticity problem).

4.7. Inhomogeneities of Irregular Shapes In materials science applications, inhomogeneities often have “irregular” shapes. Their analysis requires both theoretical guidance and numerical studies. The main (and, perhaps, the only) general theoretical tool is provided by the so-called comparison (or “auxiliary”) theorem of Hill (1963). 4.7.1. Hill’s Theorem and its Implications This theorem focuses on changes in the effective properties caused by replacing inhomogeneities by inscribed/circumscribed shapes. Hill formulated it in energy terms; we rephrase it here in terms of compliances/stiffnesses. Let C 0 , S0 and C 1 , S1 be the stiffness/compliance tensors of the matrix and of the inhomogeneities, correspondingly (both, generally, anisotropic), and  be the space occupied by all inhomogeneities. To be specific, we assume that the material of inhomogeneities is “softer” than the one of the matrix. More precisely, this means that the eigenvalues 1 − S 0 are nonnegative or, equivalently, that the eigenof the 6 × 6 matrix Sijkl ijkl 1 − C 0 are nonpositive. Let us enlarge  (some of the values of the matrix Cijkl ijkl inhomogeneities are replaced by circumscribed ones, or new ones are introduced). Then the effective properties become “softer”: eigenvalues of the change in the effective compliance tensor Sijkl are nonnegative. The opposite inequality signs will hold, of course, if  is shrunk. If the matrix and the inhomogeneities have the same type of elastic anisotropy and their anisotropy axes coincide, nonnegative eigenvalues of the difference S1 − S0 imply that each of the eigenvalues of S1 is not smaller than the corresponding eigenvalue of S0 ; the opposite inequality will hold for the stiffness tensor. In the fully isotropic case when (1) both the matrix and the inhomogeneities are isotropic and (2) inhomogeneity orientations are random so that the overall properties are isotropic, Hills’s theorem reduces to statements on the bulk and shear moduli, K and G. Assuming, for example, that G1 < G0 , K1 > K0 , Hill’s theorem reduces to an intuitively obvious statement that enlargement of inhomogeneities results in decrease of the effective shear modulus G and increase in the effective bulk modulus K. Remark The statement on K and G does not necessarily imply similar statements on Young’s modulus or Poisson’s ratio.

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I. Sevostianov and M. Kachanov

Taking the circumscribed/inscribed shapes as ellipsoids generates bounds that can be explicitly calculated (at least, in the case when the matrix is isotropic). The bounds are tight in the important case of strongly oblate inhomogeneities (that may be irregularly shaped) since both the circumscribed and the inscribed shapes can be taken as strongly oblate ellipsoids for which the compliance/stiffness contribution tensors are only weakly dependent on the aspect ratio. In general, however, the bounds formed by ellipsoids may be wide and hence less useful. Hill’s bounds can be somewhat narrowed, as follows. The space in-between the original shape and the circumscribed/inscribed ellipsoids is filled with smaller ellipsoids, with their contributions subtracted/added. However, as demonstrated for cracks of irregular shapes, the narrowing may be only moderate (Sevostianov and Kachanov, 2002b), due to a strong effect of the remaining ligaments. Hill’s theorem applied to the conductivity problem has the same implications as in the elasticity problem. Namely assuming, to be specific, that eigenvalues of kij1 − kij0 are nonnegative, the theorem implies that enlargement of inhomogeneities leads to nonnegative eigenvalues of the change kij of the effective conductivity (nonnegative change in the effective conductivity in any direction). Hill’s theorem implies that small scale shape details are unimportant as long as they do not change the connectivity of the region. In particular, it leads to the following observations: • Slight “jaggedness” of inhomogeneity boundaries can be ignored, as far as the effective properties are concerned. (In the case of crack-like pores, “jaggedness” is unimportant provided it does not produce contacts between crack faces). • It is unimportant whether various corner points of inhomogeneities are sharp or blunted since the difference between the two can be tightly bounded. • As discussed in Section 4.2, small partial contacts between crack faces produce strong effect on the compliance and resistivity of a crack. This does not contradict Hill’s bounds since such contacts change the connectivity of the region. Remark In the context of elasticity–conductivity connections, the irregularity of shapes may be less important, for the following reasons. If certain irregularity factor produces substantial but similar effects on the elastic and on the conductive properties, it is of minor importance for the connections. For example, small partial contacts between crack faces produce strong but almost identical effects on the elastic and the conductive properties, Fig. 4.10 (Sevostianov, 2003). Therefore, in the case of multiple cracks, reductions in the “adjusted” crack densities are very close for the two properties.

Connections between Elastic and Conductive Properties

133

1.00

Reff /a

0.75

Elasticity problem

0.50

Conductivity problem 0.25

0.00 0.00

0.25

0.50

0.75

1.00

l Fig. 4.10 Reduction of the crack compliance (for different values of Poisson’s ratio) and resistivity contribution in the normal to the crack direction due to a cohesion island. Reff is the radius of an equivalent circular crack (producing the same effect). An almost vertical drop indicates that even a very small island produces a strong effect.

The relative importance of various microstructural features for the effective elastic and conductive properties is illustrated in Fig. 4.11. 5. Explicit Cross-property Connections for Anisotropic Two-phase Composites We now derive explicit conductivity–elasticity connections for materials that are anisotropic, overall, and can be described as isotropic matrices containing inhomogeneities of generally nonrandom orientations and diverse shapes. They have been developed recently by the present authors (Kachanov et al., 2001; Sevostianov and Kachanov, 2002a; Sevostianov, 2003). In the present section, we assume that the material of inhomogeneities is isotropic. Section 4 extends the connections to anisotropic inhomogeneities, for a broad class of composite microgeometries. The connections explicitly relate the entire set of anisotropic elastic constants to conductivities. This makes them well suited for various materials science applications, as demonstrated in Section 7. The connections are approximate, their accuracy dependent on the inhomogeneity shapes and on the elastic (but not the conductive) properties of the constituents. The connections are based on similarities between the microstructural parameters that control the conductive and the elastic properties (parameters of inhomogeneities’ concentration in whose terms the said properties are to be expressed).

134

I. Sevostianov and M. Kachanov FACTORS THAT HAVE STRONG EFFECT ON OVERALL ELASTIC AND CONDUCTIVE PROPERTIES

Cohesion Islands between crack faces

Orientational scatter about a preferential orientation

Convexity/concavity of inhomogeneity shapes

vs

(same area) Strong (several times) reduction of crack contribution to overall constants

Effect on overall constants: of the first order in scatter parameter

Concave shapes have stronger effect on the overall properties

FACTORS THAT HAVE MINOR EFFECT ON OVERALL ELASTIC AND CONDUCTIVE PROPERTIES Moderate jaggedness of inhomogeneity boundaries

»

»

Sharpness of corner points

Non-circularity of multiple planar cracks (if uncorrelated with crack sizes and orientations)

»

»

»

Fig. 4.11 Influence of various “irregularity factors” on the elastic and conductive properties.

The connections are derived under two main assumptions: • The inhomogeneities are spheroidal (so that the diversity of shapes means diversity of the aspect ratios). • The non-interaction approximation (NIA). As discussed in the Introduction, these assumptions are actually much less restrictive than they may seem – the connections continue to hold under much broader conditions – at finite, or even large, concentrations of inhomogeneities

Connections between Elastic and Conductive Properties

135

that, in addition, may have nonspheroidal, “irregular” shapes (see Section 5.6 and experimental data of Section 7). This is explained by the fact that both complicating factors – the interactions and the “irregular” shapes – affect the elastic and the conductive properties in a similar way, so that the connection between the two is not affected much.

5.1. Elasticity–Conductivity Connections: General Case The proper microstructural parameters for the elastic and conductive properties reflect the individual inhomogeneity contributions to the considered property. These contributions have dual forms: compliance (H ) ↔ stiffness (N ) contribution tensors (in the context of elasticity) and resistivity (R ) ↔ conductivity (K) contribution tensors (in the context of conductivity) leading to dual microstruc (i)   (i)  tural parameters, H ↔ N (i) and R ↔ K (i) . The cross-property connections relate one of the parameters of the first pair to one of the second pair, resulting in four different forms of the connections that have different accuracies. Changes in the effective resistivity tensor r and in the effective conductivity tensor k due to spheroidal inhomogeneities have the form (see Section 4.6): (i) 1 $ 3 a (A1 I + A2 nn) , k0 r − I = I − k/k0 = (5.1) V i

where n(i) is a unit vector along ith spheroid’s axis, k0 is the conductivity of the bulk material, and aspect ratio dependent factors A1,2 are given by (4.62). The changes in the effective compliance and the effective stiffness tensors are, correspondingly, .  (i) (i) $ $ 1 E0 S − S 0 = II + a 3 B1 a 3 B2 J + V i i (5.2) 6 (i) (i) $ $ 3 3 + + , a B3 (Inn + nnI) a B4 (J · nn + nn · J) i





G−1 C − C0 = 0

1 V

i

.

$

a3 D1

i

(i)

II +

$

a3 D2

(i)

J+

i

(5.3) 6 (i) $  (i) $ 3 3 + , a D3 (Inn + nnI) a D4 (J · nn + nn · J) + i

i

where coefficients Bi and Di are given by (4.52) and (4.53).

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I. Sevostianov and M. Kachanov

These formulas apply to an arbitrary mixture of spheroidal inhomogeneities of diverse aspect ratios and orientations. They contain A-, B-, and D-factors that depend on the inhomogeneity aspect ratios and on material constants of the constituents. Since the conductivity factors A1,2 are different from the elasticity factors Bi and Di , changes in the conductive properties due to inhomogeneities (5.1) cannot, generally, be expressed in terms of changes in the elastic properties (4.47); these tensors may not even be coaxial. However, this difficulty is overcome if the aspect ratios of inclusions are not correlated with either their orientations n(i) or sizes a(i) . This statistical independence implies that coefficients Ai , Bi , and Di can be replaced by their mean values ∞ ai =

∞ Ai (γ)F(γ)dγ,

0

bi =

∞ Bi (γ)F(γ)dγ,

0

di =

(5.4)

Di (γ)F(γ)dγ, 0

(where F(γ) is the aspect ratio distribution density and functions Bi (γ), Di (γ), and Ai (γ) are given by Eqs. (4.52), (4.53), and (4.