Optimal anisotropic three-phase conducting composites: Plane problem

The paper establishes tight lower bound for effective conductivity tensor K₁ of two-dimensional three-phase conducting anisotropic composites and defines optimal microstructures. It is assumed that three materials are mixed with fixed volume fractions and that the conductivity of one of the materials is infinite. The bound expands the Hashin–Shtrikman and translation bounds to multiphase structures, it is derived using a combination of translation method and additional inequalities on the fields in the materials; similar technique was used by Nesi, 1995, Cherkaev, 2009 for isotropic multiphase composites. This paper expands the bounds to the anisotropic composites with effective conductivity tensor K₁. The lower bound of conductivity (G-closure) is a piece-wise analytic function of eigenvalues of K₁, that depends only on conductivities of components and their volume fractions. Also, we find optimal microstructures that realize the bounds, developing the technique suggested earlier by Albin et al., 2007a, Cherkaev, 2009. The optimal microstructures are laminates of some rank for all regions. The found structures match the bounds in all but one region of parameters; we discuss the reason for the gap and numerically estimate it.     

Sensitivity of the macroscopic elasticity tensor to topological microstructural changes

A remarkably simple analytical expression for the sensitivity of the two-dimensional macroscopic elasticity tensor to topological microstructural changes of the underlying material is proposed. The derivation of the proposed formula relies on the concept of topological derivative, applied within a variational multi-scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. The derived sensitivity—a symmetric fourth order tensor field over the RVE domain—measures how the estimated two-dimensional macroscopic elasticity tensor changes when a small circular hole is introduced at the microscale level. This information has potential use in the design and optimisation of microstructures.     

Eshelby’s problem in an anisotropic multiferroic bimaterial plane

We solve analytically the Eshelby’s problem in an anisotropic multiferroic bimaterial plane. The solution is based on the extended Stroh formalism of complex variables, and is valid for the inclusion of arbitrary shapes, described by a Laurent polynomial, a polygon, or the one bounded by a Jordan curve. Furthermore, the results in the corresponding half plane and full plane can be reduced directly from the bimaterial-plane solution. As such, the solution unifies the complex variable method and the Green’s function method, extending further to the multiferroic bimaterial plane of general anisotropy. The essential eigenfunctions are also identified by which the induced fields can be simply determined. Numerical results are presented to investigate the features of these eigenfunctions as well as the strain, electric and magnetic fields (components of the extended Eshelby tensor). Particularly, we present the values of these fields at the center of the N-side regular polygonal inclusion and also the average values of these fields over the inclusion area. The effect of the half-plane traction-free surface condition as well as the effect of various couplings on the induced fields is discussed in detail. For the N-side regular polygonal inclusion, it is found that, when the inclusion is in the full plane, both the center and average values of the Eshelby tensor are independent of the side number N, except for N = 4. We further show that the piezoelectric and piezomagnetic coupling coefficients could significantly affect the Eshelby tensor. These features should be useful in controlling the Eshelby tensor for the design of better multiferroic composites. Typical contours of the field quantities in and around the inclusion bounded by both straight and curved line segments in a multiferroic bimaterial plane are also presented.     

Strain gradient solution for the Eshelby-type polyhedral inclusion problem

The Eshelby-type problem of an arbitrary-shape polyhedral inclusion embedded in an infinite homogeneous isotropic elastic material is analytically solved using a simplified strain gradient elasticity theory (SSGET) that contains a material length scale parameter. The Eshelby tensor for a polyhedral inclusion of arbitrary shape is obtained in a general analytical form in terms of three potential functions, two of which are the same as the ones involved in the counterpart Eshelby tensor based on classical elasticity. These potential functions, as volume integrals over the polyhedral inclusion, are evaluated by dividing the polyhedral inclusion domain into tetrahedral duplexes, with each duplex and the associated local coordinate system constructed using a procedure similar to that employed by Rodin (1996. J. Mech. Phys. Solids 44, 1977–1995). Each of the three volume integrals is first transformed to a surface integral by applying the divergence theorem, which is then transformed to a contour (line) integral based on Stokes' theorem and using an inverse approach different from those adopted in the existing studies based on classical elasticity. The newly derived SSGET-based Eshelby tensor is separated into a classical part and a gradient part. The former contains Poisson's ratio only, while the latter includes the material length scale parameter additionally, thereby enabling the interpretation of the inclusion size effect. This SSGET-based Eshelby tensor reduces to that based on classical elasticity when the strain gradient effect is not considered. For homogenization applications, the volume average of the new Eshelby tensor over the polyhedral inclusion is also provided in a general form. To illustrate the newly obtained Eshelby tensor and its volume average, three types of polyhedral inclusions – cubic, octahedral and tetrakaidecahedral – are quantitatively studied by directly using the general formulas derived. The numerical results show that the components of the SSGET-based Eshelby tensor for each of the three inclusion shapes vary with both the position and the inclusion size, while their counterparts based on classical elasticity only change with the position. It is found that when the inclusion is small, the contribution of the gradient part is significantly large and should not be neglected. It is also observed that the components of the averaged Eshelby tensor based on the SSGET change with the inclusion size: the smaller the inclusion, the smaller the components. When the inclusion size becomes sufficiently large, these components are seen to approach (from below) the values of their classical elasticity-based counterparts, which are constants independent of the inclusion size.     

弹脆性材料的损伤本构关系及应用

本文根据连续损伤力学方法,对弹脆性材料损伤的力学响应进行一般分析。理论分析中,材料与损伤都是各向异性的。还导出了计算损伤张量、有效弹性张量、真实应力张量以及损伤能耗率张量的实用表达式。     

Some general properties of Eshelby’s tensor fields in transport phenomena and anti-plane elasticity

Consider an infinite thermally conductive medium characterized by Fourier’s law, in which a subdomain, called an inclusion, is subjected to a prescribed uniform heat flux-free temperature gradient. The second-order tensor field relating the gradient of the resulting temperature field over the medium to the uniform heat flux-free temperature gradient is referred to as Eshelby’s tensor field for conduction. The present work aims at deriving the general properties of Eshelby’s tensor field for conduction. It is found that: (i) the trace of Eshelby’s tensor field is equal to the characteristic function of the inclusion, independently of the latter’s shape; (ii) the isotropic part of Eshelby’s tensor field over the inclusion of arbitrary shape is identical to Eshelby’s tensor field over a 2D circular or 3D spherical inclusion; (iii) when the medium is made of an isotropic material and when the inclusion has some specific rotational symmetries, the value of the Eshelby’s tensor field evaluated at the inclusion gravity center and the symmetric average of Eshelby’s tensor fields are both equal to Eshelby’s tensor field for a 2D circular or 3D spherical inclusion. These results are then extended, with the help of a linear transformation, to the general case where the medium consists of an anisotropic conductive material. The method elaborated and results obtained by the present work are directly transposable to the physically analogous transport phenomena of electric conduction, dielectrics, magnetism, diffusion and flow in porous media and to the mathematically identical phenomenon of anti-plane elasticity.     

The non-isothermal elastic dumbbell: A model for the thermal conductivity of a polymer solution

In a flowing polymeric liquid, molecular orientation will give rise to anisotropic conduction of heat. In this paper, a theory is presented relating the thermal conductivity tensor to the deformation history of the fluid. The basis of this theory is formed by the Hookean dumbbell. It is shown that the anisotropy of the thermal conductivity is proportional to the polymer contribution to the extra-stress tensor. This stress-thermal law makes it relatively simple to incorporate anisotropic heat conduction into the numerical simulation of a flowing polymeric liquid.     

Uniform Stress Inside an Anisotropic Elliptic Inclusion with Imperfect Interface Bonding

In this paper we study the two-dimensional deformation of an anisotropic elliptic inclusion embedded in an infinite dissimilar anisotropic matrix subject to a uniform loading at infinity. The interface is assumed to be imperfectly bonded. The surface traction is continuous across the interface while the displacement is discontinuous. The interface function that relates the surface traction and the displacement discontinuity across the interface is a tensor function, not a scalar function as employed by most work in the literature. We choose the interface function such that the stress inside the elliptic inclusion is uniform. Explicit solution for the inclusion and the matrix is presented. The materials in the inclusion and in the matrix are general anisotropic elastic materials so that the antiplane and inplane displacements are coupled regardless of the applied loading at infinity. T.C.T. Ting is Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.     

On micromechanical modeling of particulate composites with inclusions of various shapes

The effective elastic properties of statistically homogeneous two-phase particulate composites are considered. Several first-order micromechanical models are re-written in terms of the inclusion compliance contribution tensor (H-tensor). This tensor is a convenient tool to evaluate contribution of arbitrarily shaped inclusions and cavities to the overall composite properties.For any inclusion shape, the procedure starts with calculation of the H-tensor for a single inclusion. The non-interaction approximation is obtained by direct summation. More advanced micromechanical schemes are derived by substituting the non-interaction inclusion compliance contribution tensor into the formulae provided in the paper. The proposed procedure is illustrated by considering several two-dimensional and three-dimensional examples.     

A new modified couple stress theory for anisotropic elasticity and microscale laminated Kirchhoff plate model

A new modified couple stress theory for anisotropic elasticity is proposed. This theory contains three material length scale parameters. Differing from the modified couple stress theory, the couple stress constitutive relationships are introduced for anisotropic elasticity, in which the curvature (rotation gradient) tensor is asymmetric and the couple stress moment tensor is symmetric. However, under isotropic case, this theory can be identical to modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The differences and relations of standard, modified and new modified couple stress theories are given herein. A detailed variational formulation is provided for this theory by using the principle of minimum total potential energy. Based on the new modified couple stress theory, composite laminated Kirchhoff plate models are developed in which new anisotropic constitutive relationships are defined. The First model contains two material length scale parameters, one related to fiber and the other related to matrix. The curvature tensor in this model is asymmetric; however, the couple stress moment tensor is symmetric. Under isotropic case, this theory can be identical to the modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The present model can be viewed as a simplified couple stress theory in engineering mechanics. Moreover, a more simplified model of couple stress theory including only one material length scale parameter for modeling the cross-ply laminated Kirchhoff plate is suggested. Numerical results show that the proposed laminated Kirchhoff plate model can capture the scale effects of microstructures.     

Homogenization for wave propagation in periodic fibre-reinforced media with complex microstructure. I—Theory

The effective elastic properties of periodic fibre-reinforced media with complex microstructure are determined by the method of asymptotic homogenization via a novel solution to the cell problem. The solution scheme is ideally suited to materials with many fibres in the periodic cell. In this first part of the paper we discuss the theory for the most general situation—N arbitrarily anisotropic fibres within the periodic cell. For ease of exposition we then restrict attention to isotropic phases which results in a monoclinic composite material with 13 effective moduli and expressions for each of these are determined. In the second part of this paper we shall discuss results for a variety of specific microstructures.     

The evolution of Hooke's law due to texture development in FCC polycrystals

Initially isotropic aggregates of crystalline grains show a texture-induced anisotropy of both their inelastic and elastic behavior when submitted to large inelastic deformations. The latter, however, is normally neglected, although experiments as well as numerical simulations clearly show a strong alteration of the elastic properties for certain materials. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of cubic crystal aggregates. The effective elastic properties are determined by orientation averages of the local elasticity tensors. Arithmetic, geometric, and harmonic averages are compared. It can be shown that for cubic crystal aggregates all of these averages depend on the same irreducible fourth-order tensor, which represents the purely anisotropic portion of the effective elasticity tensor. Coupled equations for the flow rule and the evolution of the anisotropic part of the elasticity tensor are formulated. The flow rule is based on an anisotropic norm of the stress deviator defined by means of the elastic anisotropy. In the evolution equation for the anisotropic part of the elasticity tensor the direction of the rate of change depends only on the inelastic rate of deformation. The evolution equation is derived according to the theory of isotropic tensor functions. The transition from an elastically isotropic initial state to a (path-dependent) final anisotropic state is discussed for polycrystalline copper. The predictions of the model are compared with micro–macro simulations based on the Taylor–Lin model and experimental data.     

Eshelby tensors for an ellipsoidal inclusion in a microstretch material

Eshelby tensors for an ellipsoidal inclusion in a microstretch material are derived in analytical form, involving only one-dimensional integral. As micropolar Eshelby tensor, the microstretch Eshelby tensors are not uniform inside of the ellipsoidal inclusion. However, different from micropolar Eshelby tensor, it is found that when the size of inclusion is large compared to the characteristic length of microstretch material, the microstretch Eshelby tensor cannot be reduced to the corresponding classical one. The reason for this is analyzed in details. It is found that under a pure hydrostatic loading, the bulk modulus of a microstretch material is not the same as the one in the corresponding classical material. A modified bulk modulus for the microstretch material is proposed, the microstretch Eshelby tensor is shown to be reduced to the modified classical Eshelby tensor at large size limit of inclusion. The fully analytical expressions of microstretch Eshelby tensors for a cylindrical inclusion are also derived.     

Explicit elasticity-conductivity connections for composites with anisotropic inhomogeneities

Explicit elasticity-conductivity connections for anisotropic heterogeneous materials were recently derived and verified experimentally by the present authors. The constituents were assumed isotropic, so that the overall anisotropy was due solely due to nonrandom orientations of inhomogeneities. Motivated by the materials science applications that deal with strongly anisotropic inclusions, we derive alternative elasticity-conductivity connections that cover these cases. They hold for a broad class of orientation distributions—up to three families of parallel inhomogeneities forming arbitrary angles with each other, with moderate orientation scatter allowed in each of the families. In the case of the isotropic inhomogeneities, this form has substantially better accuracy than the approximate connections derived earlier. The results are compared with experimental data on fiber-reinforced plastics. The agreement is quite good for the entire set of the effective anisotropic constants, and it is achieved without any fitting parameters.     

Bounds and perturbation series for incompressible elastic composites with transverse isotropic symmetry

The effective elastic behavior of a transversely isotropic composite made from two incompressible elastic materials is examined. The set of all effective elasticity tensors for transversely isotropic finite rank laminar microstructures is described. The extremal property of this class of microstructures is used to derive a new more precise characterization of the set of effective shear moduli.The perturbation series for the effective elasticity tensor is considered. An explicit formula for the second order perturbation tensor is derived. We describe precisely the set of tensors that correspond to all second order perturbations consistent with transverse isotropy. We apply analytic methods [cf. 27] to show that all second order perturbation tensors are realized by finite rank laminar microstructures.Supported by NSF through Grant DMS-3907658.     

THEORETICAL MODEL OF EFFECTIVE STRESS COEFFICIENT FOR ROCK/SOIL-LIKE POROUS MATERIALS

Physical mechanisms and influencing factors on the effective stress coefficient for rock/soil-like porous materials are investigated, based on which equivalent connectivity index is proposed. The equivalent connectivity index, relying on the meso-scale structure of porous material and the property of liquid, denotes the connectivity of pores in Representative Element Area （REA）. If the conductivity of the porous material is anisotropic, the equivalent connectivity index is a second order tensor. Based on the basic theories of continuous mechanics and tensor analysis, relationship between area porosity and volumetric porosity of porous materials is deduced. Then a generalized expression, describing the relation between effective stress coefficient tensor and equivalent connectivity tensor of pores, is proposed, and the expression can be applied to isotropic media and also to anisotropic materials. Furthermore, evolution of porosity and equivalent connectivity index of the pore are studied in the strain space, and the method to determine the corresponding functions in expressions above is proposed using genetic algorithm and genetic programming. Two applications show that the results obtained by the method in this paper perfectly agree with the test data. This paper provides an important theoretical support to the coupled hydro-mechanical research.     

Application of results on Eshelby tensor to the determination of effective poroelastic properties of anisotropic rocks-like composites

The present work is devoted to the determination of the macroscopic poroelastic properties of anisotropic elastic porous materials saturated by a fluid under pressure. It makes use of the theoretical results provided by Withers [Withers, P.J., 1989. The determination of the elastic field of an ellipsoidal inclusion in a transversely isotropic medium, and its relevance to composite materials. Philosophical Magazine A 59 (4), 759–781.] for the problem of an ellipsoidal inclusion embedded in a transversely isotropic elastic medium. The particular case of a spherical inclusion is very important for rock-like composites such as argillite and shales. The implementation of these results in a micromechanical theory of poroelasticity allows to quantify the effects of the solid matrix anisotropy and of pore space on the effective poromechanical properties. Closed form expressions of Biot tensor and of Biot modulus are presented as well as numerical applications for anisotropic shales.     

Stress and fabric in granular material

It has been well recognized that, due to anisotropic packing structure of granular material, the true stress in a specimen is different from the applied stress. However, very few research efforts have been focused on quantifying the relationship between the true stress and applied stress. In this paper, we derive an explicit relationship among applied stress tensor, material-fabric tensor, and force-fabric tensor; and we propose a relationship between the true stress tensor and the applied stress tensor. The validity of this derived relationship is examined by using the discrete element simulation results for granular material under biaxial and triaxial loading conditions.     

增量型各向异性损伤理论与数值分析

考虑到目前各向异性损伤理论存在一些不足,该文在增量型各向异性损伤理论的框架下,引入二阶对称张量,构造四阶对称有效损伤张量,建立了有效应力方程．类似于塑性流动分析方法,定义了增量弹性应力．应变关系．利用von Mises塑性屈服准则,并考虑各向异性损伤效应,推导出四阶对称的弹．塑性变形损伤刚度张量,其对称性反映了材料的固有特性．根据物体的变形和现时损伤状态,构造了材料损伤演化方程,方程中各项具有明确的物理意义．通过对A12024-T3金属薄板单向拉伸的有限元分析,确定了损伤演化参数,验证了损伤演化方程的正确性．此外还对含孔口薄板做有限元模拟,讨论了反力—位移曲线的变化规律以及它所揭示变形性质,给出了损伤场的分布规律。     

An approximate solution of the interaction between an edge dislocation and an inclusion of arbitrary shape

An approximate solution of the interaction force between an edge dislocation and an inclusion of arbitrary shape is derived, from which a set of succinct formulas for several special inclusion shapes are obtained. As compared with several classical solutions to special inclusion shapes, the present approximate solution has fairly good accuracy.