Steady-state response of a Cosserat medium with a spherical inclusion

Summary Two concepts of asymmetric eigenstrain and eigentorsion are employed to derive a general steady-state theory of inhomogeneous anisotropic micropolar media containing defects with the help of Green's function technique. In particular, a dynamic inclusion problem for homogeneous isotropic centrosymmetric micropolar elasticity is investigated. By means of Green's functions an exact closed-form solution is presented for the case of a spherical inclusion embedded in an infinitely extended Cosserat medium. With this result, the micropolar dynamic Eshelby tensors for the inside and outside elastic fields of the spherical inclusion are defined and determined. It is confirmed that the classical dynamic and static Eshelby tensors are obtained as two special cases of the micropolar dynamic Eshelby tensors, respectively.     

The Composite Eshelby Tensors and their applications to homogenization

Summary In recent studies, the exact solutions of the Eshelby tensors for a spherical inclusion in a finite, spherical domain have been obtained for both the Dirichlet- and Neumann boundary value problems, and they have been further applied to the homogenization of composite materials [15], [16]. The present work is an extension to a more general boundary condition, which allows for the continuity of both the displacement and traction field across the interface between RVE (representative volume element) and surrounding composite. A new class of Eshelby tensors is obtained, which depend explicitly on the material properties of the composite, and are therefore termed “the Composite Eshelby Tensors”. These include the Dirichlet- and the Neumann-Eshelby tensors as special cases. We apply the new Eshelby tensors to the homogenization of composite materials, and it is shown that several classical homogenization methods can be unified under a novel method termed the “Dual Eigenstrain Method”. We further propose a modified Hashin-Shtrikman variational principle, and show that the corresponding modified Hashin-Shtrikman bounds, like the Composite Eshelby Tensors, depend explicitly on the composite properties.     

Eshelby tensors for an ellipsoidal inclusion in a micropolar material

Micropolar Eshelby tensors for an ellipsoidal inclusion are derived in an analytical form, which involves only one-dimensional integral. The numerical evaluation of the Eshelby tensors are also performed, it is found that the micropolar Eshelby tensors are not uniform in the ellipsoidal inclusion, however, their variations over the ellipsoidal domain are not significant. When size of inclusion is large compared to the characteristic length of the micropolar material, the micropolar Eshelby tensor is reduced to the classical one. It is also demonstrated that for a general ellipsoidal inclusion a uniform eigenstrain or eigentorsion produces on average only nonzero strain or torsion, and the average Eshelby relations are uncoupled.     

A new homogenization method based on a simplified strain gradient elasticity theory

Homogenization methods utilizing classical elasticity-based Eshelby tensors cannot capture the particle size effect experimentally observed in particle–matrix composites at the micron and nanometer scales. In this paper, a new homogenization method for predicting effective elastic properties of multiphase composites is developed using Eshelby tensors based on a simplified strain gradient elasticity theory (SSGET), which contains a material length scale parameter and can account for the size effect. Based on the strain energy equivalence, a homogeneous comparison material obeying the SSGET is constructed, and two sets of equations for determining an effective elastic stiffness tensor and an effective material length scale parameter for the composite are derived. By using Eshelby’s eigenstrain method and the Mori–Tanaka averaging scheme, the effective stiffness tensor based on the SSGET is analytically obtained, which depends not only on the volume fractions and shapes of the inhomogeneities (i.e., phases other than the matrix) but also on the inhomogeneity sizes, unlike what is predicted by the existing homogenization methods based on classical elasticity. To illustrate the newly developed homogenization method, sample cases are quantitatively studied for a two-phase composite filled with spherical, cylindrical, or ellipsoidal inhomogeneities (particles) using the averaged Eshelby tensors based on the SSGET that were derived earlier by the authors. Numerical results reveal that the particle size has a large influence on the effective Young’s moduli when the particles are sufficiently small. In addition, the results show that the composite becomes stiffer when the particles get smaller, thereby capturing the particle size effect.     

On the transient elastic field for an expanding spherical inclusion in 3-D solid

Summary. The three-dimensional transient elastic field of an infinite isotropic elastic medium is investigated when a phase transformation is nucleated from a point and proceeds through the crystal dynamically. The phase transformation keeps the spherical shape and expands at a speed of arbitrary time profile. This process is modeled by an expanding spherical inclusion with a spatially uniform eigenstrain. The objective of this paper is to present a general method to determine the transient displacement field for points either covered or not covered by the transformation area. This method can be applied to investigate the nucleation and expanding mechanism of phase transformation. Using a Green's function approach, an explicit procedure is presented to evaluate the 3-D displacement field when the expanding history of the spherical inclusion is given. As numerical examples, the explicit formulations are given for the transient elastic fields, when the spherical inclusion expands at a constant or an exponent damping speed with a pure dilatational eigenstrain or pure shear eigenstrain. It is found that the elastic field inside the expanding inclusion remains constant with respect to time, which is consistent with the well-known Eshelby solution for a static inclusion case.     

Stress concentration due to a hemispherical surface inclusion

Fatigue cracks often initiate at surface inclusions, and sometimes appear at inclusions within the bulk. To clarify the relative efficiency of these crack sources, an approximate solution for the elastic stress of a hemispherical surface inclusion is provided and compared with the existing result of a spherical interior inclusion. The approximate solution is obtained from the Eshelby solution for the elastic field of an ellipsoid inclusion by introducing the Green's function of an elastic half-space. The numerical calculated results indicate that the stress concentration of a surface inclusion is higher when the inclusion is harder than the matrix, while that of an embedded inclusion is higher when the inclusion is softer than the matrix.     

Solution of the Eshelby-type anti-plane strain polygonal inclusion problem based on a simplified strain gradient elasticity theory

The Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion of arbitrary-shape polygonal cross-section is analytically solved using a simplified strain gradient elasticity theory that incorporates one material length scale parameter. The Eshelby tensor (with four nonzero components) is obtained in a general form in terms of two scalar-valued potential functions. These potential functions, as area integrals over the polygonal cross-section, are first converted to two line (contour) integrals using Green’s theorem, which are then evaluated analytically by direct integration. The newly derived Eshelby tensor is separated into a classical part and a gradient part. The former does not contain any elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle (inclusion) size effect. For homogenization applications, the area average of the new position-dependent Eshelby tensor over the polygonal cross-section is also provided in a general form. To illustrate the newly obtained Eshelby tensor, five types of regular polygonal inclusions (i.e., triangular, quadrate, hexagonal, octagonal, and tetrakaidecagonal) are quantitatively studied by directly using the general formulas derived. The components of the induced strain and the averaged Eshelby tensor inside the inclusion are evaluated. Numerical results reveal that the induced strain varies with both the position and the inclusion size. The values of the induced strain components in a polygonal inclusion approach from below those in a corresponding circular inclusion when the inclusion size or the number of sides of the polygonal inclusion increases. The results for the averaged Eshelby tensor components show that the size effect is significant when the inclusion size is small but may be neglected for large inclusions.     

Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem

The solution for the Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter in addition to two classical elastic constants. The Green’s function based on the SSGET for an infinite three-dimensional elastic body undergoing anti-plane strain deformations is first obtained by employing Fourier transforms. The Eshelby tensor is then analytically derived in a general form for an anti-plane strain inclusion of arbitrary cross-sectional shape using the Green’s function method. By applying this general form, the Eshelby tensor for a circular cylindrical inclusion is obtained explicitly, which is separated into a classical part and a gradient part. The former does not contain any classical elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle size effect. The components of the new Eshelby tensor vary with both the position and the inclusion size, unlike their counterparts based on classical elasticity. For homogenization applications, the average of this Eshelby tensor over the circular cross-sectional area of the inclusion is obtained in a closed form. Numerical results reveal that when the inclusion radius is small, the contribution of the gradient part is significantly large and should not be ignored. Also, it is found that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion, the smaller the components. These components approach from below the values of their counterparts based on classical elasticity when the inclusion size becomes sufficiently large.     

Eigenstrain formulation of boundary integral equations for modeling particle-reinforced composites

A novel computational model is presented using the eigenstrain formulation of the boundary integral equations for modeling the particle-reinforced composites. The model and the solution procedure are both resulted intimately from the concepts of the equivalent inclusion of Eshelby with eigenstrains to be determined in an iterative way for each inhomogeneity embedded in the matrix. The eigenstrains of inhomogeneity are determined with the aid of the Eshelby tensors, which can be readily obtained beforehand through either analytical or numerical means. The solution scale of the inhomogeneity problem with the present model is greatly reduced since the unknowns appear only on the boundary of the solution domain. The overall elastic properties are solved using the newly developed boundary point method for particle-reinforced inhomogeneous materials over a representative volume element with the present model. The effects of a variety of factors related to inhomogeneities on the overall properties of composites as well as on the convergence behaviors of the algorithm are studied numerically including the properties and shapes and orientations and distributions and the total number of particles, showing the validity and the effectiveness of the proposed computational model.     

On the capability of micromechanics models to capture the auxetic behavior of fibers/particles reinforced composite materials

This work investigates the possibility to predict the auxetic behavior of composites consisting of non-auxetic phases by means of micromechanical models based on Eshelby’s inclusion concept. Two specific microstructures have been considered: (i) the three-layered hollow-cored fibers-reinforced composite and (ii) a microstructure imitating the re-entrant honeycomb micro-architecture. The micromechanical analysis is based on kinematic integral equations as a formal solution of the inhomogeneous material problem. The interaction tensors between the inhomogeneities are computed thanks to the Fourier’s transform. The material anisotropy due to the morphological and topological textures of the inhomogeneities was taken into account thanks to the multi-site approximation of these tensors. In both cases, the numerical results show that auxetic behavior cannot be captured by such models at least in the case of elastic and isotropic phases. This conclusion is supported by corresponding finite element investigations of the second microstructure that indicate that auxetic behavior can be recovered by introducing joints between inclusions. Otherwise, favorable issues are only expected with auxetic components.     

Modeling of Effective Properties of Multiphase Magnetoelectroelastic Heterogeneous Materials

In this paper an N-phase Incremental Self Consistent model is developed for magnetoelectroelastic composites as well as the N-phase Mori-Tanaka and classical Self Consistent. Our aim here is to circumvent the limitation of the Self Consistent predictions for some coupling effective properties at certain inclusion volume fractions. The anomalies of the SC estimates are more drastic when the void inclusions are considered. The mathematical modeling is based on the heterogeneous inclusion problem of Eshelby which leads to an expression for the strain-electric-magnetic field related by integral equations. The effective N-phase magnetoelectroelastic moduli are expressed as a function of magnetoelectroelastic concentration tensors based on the considered micromechanical models. The effective properties are obtained for various types, shapes and volume fractions of inclusions and compared with the existing results.     

Micromechanics determination of electroelastic properties of piezoelectric materials containing voids

This paper examines the electroelastic properties of piezoelectric materials that contain voids. A unified micromechanics approach is adopted for determining the properties. Voids are treated as spheroidal inclusions with zero elastic moduli. The surrounding material is assumed to be linearly piezoelastic and transversely isotropic. The electroelastic Eshelby tensors for spheroidal inclusions have been evaluated numerically for different aspect ratios. Utilizing these tensors and applying the Mori–Tanaka mean field theory that accounts for the interaction between inclusions and matrix, the effective electroelastic properties of the materials are obtained. Numerical examples are given based on PZT-5H and BaTiO₃. Influences of the volume fraction and aspect ratio of voids on the material properties have been studied. Emphasis has been placed on the piezoelastic coupling effect of the material. For both materials, the piezoelastic coupling provides a stiffening effect on the materials, and the influence is more pronounced when void volume increases and when the aspect ratio of voids becomes shorter.     

考虑体积塑性应变的岩石损伤本构模型研究

采用Eshelby等效夹杂方法建立岩石弹塑性损伤本构模型是一种有效方法,但相关文献目前还极少。在连续介质损伤力学框架内利用细观力学的Eshelby等效夹杂方法建立了考虑损伤相塑性体积变形的岩石的Helmholtz自由比能函数。利用连续介质损伤力学方法推导出了考虑损伤相塑性变形的岩石损伤本构关系,给出了损伤演化方程和塑性应变发展方程。并通过数值模拟证实该模型能够反映岩石体积塑性应变、损伤的变化规律和损伤部分不能承受拉应力等力学特性。     

Stiffness of particulate reinforced metal matrix composites with damaged reinforcements

Abstract

During tensile plastic deformation particulate reinforced metal matrix composites (MMCs) undergo reinforcement damage and a parallel reduction in stiffness. An analytical model is developed to calculate this stiffness reduction using the equivalent inclusion technique proposed by Eshelby. The model considers both damaged and undamaged reinforcement particles as ellipsoidal inclusions but with different stiffness tensors. The effect of the aspect ratio of the reinforcing particles has been accounted for in the model. The model is very flexible and can meet different specific damage situations by designing a suitable stiffness tensor for the damaged reinforcements. Finite element analysis is used to modify a numerical stiffness tensor for cracked reinforcement particles. The model is compared with an earlier model of modulus reduction in MMC materials and with a few experimental measurements made on a 15 vol.-%SiC particulate reinforced aluminium alloy 2618 MMC.     

The elastic strain energy of a coherent inclusion with deviatoric misfit strains

The elastic strain energy of a misfitted coherent inclusion is discussed using the Eshelby method for ellipsoidal inclusions. Deviatoric and tetragonal misfit strains in an elastically inhomogeneous spheroidal inclusion are particularly considered. The variation of the elastic strain energy is evaluated as a function of the shape and orientation of the inclusion. Results obtained are compared with those for a coherent inclusion with purely dilatational misfit strains. Under certain deviatoric misfit strains and elastic moduli of the inclusion, the least elastic strain energy is achieved by a spheroid with an intermediate shape between a plate and a sphere or between a sphere and a needle.     

Eshelby’s tensors for plane strain and cylindrical inclusions based on a simplified strain gradient elasticity theory

The Eshelby tensor for a plane strain inclusion of arbitrary cross-sectional shape is first presented in a general form, which has 15 independent non-zero components (as opposed to 36 such components for a three-dimensional inclusion of arbitrary shape). It is based on a simplified strain gradient elasticity theory that involves one material length scale parameter. The Eshelby tensor for an infinitely long cylindrical inclusion is then derived using the general form, with its components obtained in explicit (closed-form) expressions for the two regions inside and outside the inclusion for the first time based on a higher-order elasticity theory. This Eshelby tensor is separated into a classical part and a gradient part. The latter depends on the position, the inclusion size, the length scale parameter, and Poisson’s ratio. As a result, the new Eshelby tensor is non-uniform even inside the cylindrical inclusion and captures the size effect. When the strain gradient effect is not considered, the gradient part vanishes and the newly obtained Eshelby tensor reduces to its counterpart based on classical elasticity. The numerical results quantitatively show that the components of the new Eshelby tensor vary with the position, the inclusion size, and the material length scale parameter, unlike their classical elasticity-based counterparts. When the inclusion radius is comparable to the material length scale parameter, it is found that the gradient part is too large to be ignored. In view of the need for homogenization analyses of fiber-reinforced composites, the volume average of the newly derived Eshelby tensor over the cylindrical inclusion is obtained in a closed form. The components of the average Eshelby tensor are observed to depend on the inclusion size: the smaller the inclusion radius, the smaller the components. However, as the inclusion size becomes sufficiently large, these components are seen to approach from below the values of their classical elasticity-based counterparts.     

Elastic/plastic indentation hardness and indentation fracture toughness: The inclusion core model

A new model for determining elastic/plastic indentation is presented. This model generalizes Johnson's incompressible core model to a compressible material and allows the indentation pressure to be transmitted via a misfitted inclusion core beneath the indenter which is surrounded by a hemispherical plastic zone. The internal stress field inside the core is obtained by applying Eshelby's spherical inclusion problem together with Hill's spherical-cavity expansion analysis. The plastic deformation considered here exactly ensures compatibility between the volume of a material displaced by the indenter and that accommodated by expansion. The analysis explains the essential relationships between the dimensions of the indentation and plastic zone and the dominant material properties; yield stress, hardness and elastic modulus. The solution is extended to evaluate the indentation fracture toughness by taking into account the reduced half-space constraint by the image force.     

Generalization of the multiparticle effective field method to random structure matrix composites

Summary In this paper linearly thermoelastic composite media are treated, which consist of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated inclusions. Effective properties (such as compliance, thermal expansion, stored energy) as well as the first statistical moments of stresses in the phases are estimated for the general case of nonhomogeneity of the thermoelastic inclusion properties. The micromechanical approach is based on the generalization of the ``multiparticle effective field' method (MEFM, see [7] for references), previously proposed for the estimation of stress field averages in the phases. The refined version of the MEFM takes into account both the variation of the effective fields acting on each pair of fibers and inhomogeneity of statistical average of stresses inside the inclusions. One considers in detail the connection of the method proposed with numerous related methods. The explicit representations of the effective thermoelastic properties and stress concentration factor are expressed through some building blocks described by numerical solutions for both the one and two inclusions inside the infinite medium subjected to the homogeneous loading at infinity. Just with some additional assumptions (such as an effective field hypothesis) the involved tensors can be expressed through the Green's function, Eshelby tensor and external Eshelby tensor. The dependence of effective properties and stress concentrator factors on the radial distribution function of the inclusion locations is analyzed.     

On self-similarly expanding Eshelby inclusions: Spherical inclusion with dilatational eigenstrain

For a subsonically self-similarly expanding spherical inclusion with dilatational transformation strain in a linear elastic solid, the governing system of partial differential equations is shown to be elliptic under scaling of uniform stretching of the variables, and the resulting elliptic equation is solved by satisfying the Hadamard jump conditions on the moving boundary. The solution has the Eshelby constant stress property for the interior domain, and can thus be used for the expanding inhomogeneity with transformation strain according to Eshelby (1957). The driving force on the moving boundary is also obtained.     

The problem of two plastic and heterogeneous inclusions in an anisotropic medium

We demonstrate an integral equation for the total local strain ε^T in an anisotropic heterogeneous medium with incompatible strain ε^p and which is at the same time submitted to an exterior field. The integral equation is solved in the case of an heterogeneous and plastic pair of inclusions, for which we calculate the average fields in each inclusion as well as the different parts of the elastic energy stocked in the medium.The solution is applied to the case of two isotropic and spherical inclusions in an isotropic matrix loaded in shear. The results are compared with those deduced from a more approximate method based on Horn's approximation of the integral equation. In appendix we give a numerical method for calculating the interaction tensors between anisotropic inclusions in an anisotropic medium as well as the analytic solution in the case of two spherical inclusions located in an isotropic medium.