A symmetry invariant formulation of the relationship between the elasticity tensor and the fabric tensor

The fabric tensor is employed as a quantitative stereological measure of the structural anisotropy in the pore architecture of a porous medium. Earlier work showed that the fabric tensor can be used additionally to the porosity to describe the anisotropy in the elastic constants of the porous medium. This contribution presents a reformulation of the relationship between fabric tensor and anisotropic elastic constants that is approximation free and symmetry-invariant. From specific data on the elastic constants and the fabric, the parameters in the reformulated relationship can be evaluated individually and efficiently using a simplified method that works independent of the material symmetry. The well-behavedness of the parameters and the accuracy of the method was analyzed using the Mori–Tanaka model for aligned ellipsoidal inclusions and using Buckminster Fuller’s octet-truss lattice. Application of the method to a cancellous bone data set revealed that employing the fabric tensor allowed explaining 75–90% of the total variance. An implementation of the proposed methods was made publicly available.     

Fabric tensor based boundary element analysis of porous solids

In this paper, the boundary element analysis of porous solids (sintered materials, foams, etc.) is studied utilizing a fabric tensor. The fabric tensor provides a measure of anisotropy in the solid, as well as information concerning the geometry and distribution of the pores. The homogenized, orthotropic elastic properties of a porous solid can then be predicted with the fabric tensor. To illustrate the analysis, the effect of porosity on a trabecular bone-titanium bimaterial is studied. The boundary element analysis uses an anisotropic, bimaterial Green's function so the interface does not require discretization. It is shown that the anisotropic Stroh variables are independent of the structural density and dependent on the eigenvalues of the fabric tensor. An example calculation is presented where the effect of porosity on the in-plane maximum shear stress in a trabecular bone-titanium bimaterial is substantial.     

The generalized curie principle,the hermann theorem,and the symmetry of macroscopic tensor properties of composites

The symmetry of tensor properties of composite materials is discussed in terms of the generalized Curie principle (including the effect of symmetrizing factors), as well as the Hermann theorem concerning the relationship between the rank of a tensor property and the presence or absence of isotropy in a plane normal to a given symmetry axis. A simple configuration for achieving this transverse isotropy for the elastic behaviour of laminated structures is proposed as an illustration of the practical application of the Hermann theorem. A recently published theorem about the point symmetry of composite objects is shown to be invalid.     

Boundary element analysis for effective stiffness tensors: effect of fabric tensor determination method

Second-rank fabric tensors have been extensively used to describe structural anisotropy and to predict orthotropic elastic constants. However, there are many different definitions of, and approaches to, determining the fabric tensor. Most commonly used is a fabric tensor based on mean intercept length measurements, but star volume distribution and star length distribution are commonly used, particularly in studies of trabecular bone. Here, we investigate the effect of the fabric tensor definition on elastic constant predictions using both synthetic, idealized microstructures as well as a micrograph of a porous ceramic. We use an efficient implantation of a symmetric Galerkin boundary element method to model the mechanical response of the various microstructures, and also use a boundary element approach to calculate the necessary volume averages of stress and strain to obtain the effective properties of the media.     

Fabric dependence of an anisotropic strength criterion

A strength criterion with a dependence on fabric is developed. A fabric tensor is a symmetric second rank tensor which characterizes the geometric arrangement of a porous material microstructure, or the local spatial distribution of one phase of a multiphase material relative to the other phases. The development of the strength criterion follows the method of Malmeister and of Tsai and Wu. A formula for the stress interaction coefficients introduced by Tsai and Wu is obtained. This formula relates the stress interaction coefficient to the tensile and compressive strengths in two directions, to the shear strength in the plane of the two directions, and to the fabric tensor. Heretofore there has a theoretical representation of the stress interaction coefficient in terms of other physical parameters.     

Inversions related to the stress-strain-fabric relationship

The stress-strain-fabric relationship is an extension of the anisotropic form of Hooke's law to include a dependence of the elastic coefficients upon a second-rank tensor called the fabric tensor. The fabric tensor represents features of the material microstructure associated with the type and the degree of the anisotropy. The inversion considered first in this work is that in which the stress-strain-fabric relation is constructed from the strain-stress-fabric relation and vice versa. Next, a semi-inversion of the relationship between the fourth-rank tensor of elastic coefficients and the fabric tensor is developed. This latter inversion permits the determination of the fabric tensor from a fourth-rank tensor of elastic constants. Explicit, approximate forms of these results, including a numerical example, are given for the case when the fabric tensor is normalized and terms of order three and higher in the fabric tensor are neglected.     

Dependence of elastic constants of an anisotropic porous material upon porosity and fabric

The elastic properties of an anisotropic porous material can be represented as functions of the material's solid volume fraction (or porosity) and the principal diameters of the material's fabric ellipsoid. The fabric ellipsoid is a measure of the anisotropy of the microstructure of a material. The definitions and measurement techniques for fabric ellipsoids in granular materials, foams, cancellous bone, and rocks are discussed. The principal results presented in this work are algebraic expressions for the dependence of the orthotropic elastic constants upon both solid volume fraction and the fabric ellipsoid.     

An alternative model for anisotropic elasticity based on fabric tensors

Motivated by the mechanical analysis of multiphase or damaged materials, a general approach relating fabric tensors characterizing microstructure to the fourth rank elasticity tensor is proposed. Using a Fourier expansion in spherical harmonics, the orientation distribution function of a positive, radially symmetric microstructural property is approximated by a scalar and a symmetric, traceless second rank tensor. Following this approximation, a general expression of the elastic free energy potential is derived from representation theorems for anisotropic scalar functions. Based on a homogeneity assumption for the elastic constitutive law with respect to the microstructural property, a particular elasticity model is developed that involves three independent constants beside the fabric tensors. Strict positive definiteness of the corresponding elasticity tensor is ensured under explicit conditions on the independent constants for arbitrary fabric tensors.     

On an Arbitrarily Oriented Crack in a Transversely-isotropic Medium

Transversely-isotropic material with an arbitrarily oriented penny-shaped crack is considered. We calculate fourth-rank compliance contribution tensor of the crack and second-rank crack opening displacement tensor and examine their dependence on crack orientation. It is shown that this dependence for the crack opening displacement tensor is negligible if transverse isotropy has elliptic character, i.e. if material symmetry can be described in terms of a certain second rank tensor.     

Apparent stiffness tensors for porous solids using symmetric Galerkin boundary elements

Representative volume elements (RVEs) from porous or cellular solids can often be too large for numerical or experimental determination of effective elastic constants. Volume elements which are smaller than the RVE can be useful in extracting apparent elastic stiffness tensors which provide bounds on the homogenized elastic stiffness tensor. Here, we make efficient use of boundary element analysis to compute the volume averages of stress and strain needed for such an analysis. For boundary conditions which satisfy the Hill criterion, we demonstrate the extraction of apparent elastic stiffness tensors using a symmetric Galerkin boundary element method. We apply the analysis method to two examples of a porous ceramic. Finally, we extract the eigenvalues of the fabric tensor for the example problem and provide predictions on the apparent elastic stiffnesses as a function of solid volume fraction.     

A 3D fourth order fabric tensor approach of anisotropy in granular media

We propose a micromechanical approach for granular media, with a particular account of the texture-induced anisotropy and of the strain localization rule. The approach is mainly based on the consideration of a fourth order fabric tensor able to capture general anisotropy which can be induced by complex distribution of contacts. Incorporation of this fourth order fabric tensor in a suitable homogenization scheme allows to determine the corresponding macroscopic elastic properties of the granular material. For this purpose, in addition to the classical Voigt upper bound, a new kinematics-based localization rule is proposed. It generalizes the one formulated by Cambou et al. [B. Cambou, Ph. Dubujet, F. Emeriault, F. Sidoroff, Eur. J. Mech. A/Solids 14 (1995) 225–276] in the case of an isotropic contact distribution. The results of the complete model compare well to numerical simulations results when available [C.S. Chang, C.L. Liao, Appl. Mech. Rev. 47 (1 Part 2) (1994) 197–207] (case of isotropic distribution of contacts). Finally, the interest of the fourth order fabric tensor based approach combined with the proposed localization rule is shown for different distributions of contacts by comparing its predictions to those given by a second order fabric tensor approach.     

Conservation laws and the material momentum tensor for the elastic dielectric

It is shown that a Langrangian formulation of continuum mechanics can provide not only the equations of motion, but the conservation laws related to the material symmetries in a perfect continuum interacting with an external electric field. These conservation laws in the presence of defects lead to the path-independent integrals widely used in fracture mechanics. They are basically related to the “material force” on a defect in a continuum. The quantity playing the role of the physical stress tensor in this formulation is the material momentum tensor. A material force in the form of a path-independent integral for the elastic dielectric is derived employing Toupin's [1] formulations.     

The energy-momentum tensor and material uniformity in finite elasticity

Summary Eshelby's elastic energy-momentum tensor is shown to satisfy a differential identity which, in the general case of a uniform elastic body with inhomogeneities, is expressible in terms of the torsion of the material connection.     

Evaluation of the Contribution of Irregularly Shaped Holes into the Effective Elastic Moduli by Numerical Conformal Mapping

This paper presents a computational procedure to calculate the contribution of the irregularly shaped defects into the effective moduli of two-dimensional elastic solids. In this procedure, the hole compliance tensor of an individual defect is constructed using the numerical conformal mapping (NCM) technique. The effective elastic properties of a porous solid are predicted in the non-interacting approximation using the elastic potential-based approach.     

On stiffness and strength reduction of solids due to crack development

Modification of the effective elastic and plastic constants of initially homogeneous and isotropic material with regularly distributed cracks is considered in the paper. The stress-strain relation for linearly elastic range is formulated as a tensor function with two independent variables: the stress tensor and damage tensor describing the current state of the cracked solid. This equation made it possible to evaluate all the elastic constants and is a starting point in the analysis of the plastic behavior of the damaged material. The appropriate yield criterion is derived in the form of an isotropic scalar function with the same variables as in the elastic range. To choose the most important terms of the general representation of this function, the energy of the elastic strain was calculated for homogenized equivalent material. This was done employing the stress-strain relation of elasticity for damaged solid proposed in the paper. The theoretical considerations were verified experimentally. To this end the material constants determined theoretically in the elastic and plastic ranges were compared with those measured experimentally for the models simulating the damaged material.     

Characterization of microstructural anisotropy in orthotropic materials using a second rank tensor

A second rank symmetric tensor which describes the degree of orientation in orthotropic materials is presented and shown to reflect accurately patterns of experimental data. The use of this tensor to describe microstructural anisotropy is compared to currently accepted methods and is found to be more useful and accurate in experimental studies. A method for determining the anisotropy tensor in a material is given, based on measurements on any three mutually perpendicular planes, and the fundamental restriction of this method to orthotropic materials is discussed. Experimentally determined anisotropy tensors in five specimens of cancellous bone from five different human bones are given.     

多孔材料代表单元的性质

为了弄清多孔材料代表单元的基本性质,对泡沫镍的力学性能进行了实验研究和计算机模拟,两者结果的变化趋势吻合较好。在此基础上,用离散的弹性梁构成代表单元,结合连续介质力学的方法,建立了多孔材料的理想力学模型,导出了其宏观本构关系,讨论了其代表单元各向异性性质和材料常数之间的关系。结果表明由代表单元周期性排列构成的多孔材料,在宏观上呈各向异性,只有当代表单元无序地随机排列时,多孔材料才在宏观上出现统计各向同性。同时指出了一些文献中存在的错误。     

A method for evaluating the effect of crack geometry on the mechanical behavior of cracked rock masses

Discontinuities like faults and joints (called cracks) are of widespread occurrence in rock masses in situ, with very complicated geological setting. The complexity, especially in their geometry, is no doubt a major obstruction to develop a useful theory for evaluating the mechanical behavior of cracked rock masses. An index measure (called fabric tensor) which has been introduced to show crack geometry is further discussed in this paper to see if it is useful for evaluating the mechanical behavior of rock masses in situ. Based on some acceptable simplification, an overall elastic compliance for cracked materials is successfully formulated in terms of the fabric tensor. Furthermore, with the help of geometrical probability, the fabric tensor is expressed in terms of in situ measurable quantities. These results strongly suggests that the concept of fabric tensor is useful in the analysis of cracked rock masses.     

Limitations of the constitutive equation of a simple elastic solid

Summary The constitutive equation of a simple, isotropic elastic solid can be arranged in such a form as to give rise to a fundamental identity between Lode's stress parameter and a corresponding deformation parameter. Using the concept of a stress intensity function, it is shown that at initial yield the constitutive equation of a simple, isotropic elastic solid satisfies only the von Mises yield criterion. A general form for the deformation response coefficients is obtained by way of the concept of a deformation intensity function. In general, there are two broad classes of deformation intensity function, defined in terms of whether the deformation intensity function is continuously differentiable or whether it is piece-wise linear and continuous. Use of the fundamental identity between Lode's stress parameter and the corresponding deformation parameter leads to the conclusion that the constitutive equation of the simple, isotropic elastic solid is incompatible with any form of piece-wise linear deformation intensity function. The stretching tensor has been expressed in terms of the co-rotational and convected time derivatives of the left Cauchy-Green deformation tensor and its inverse. This form of the stretching tensor is entered into a particular form of constitutive equation of the rate-type for a simple, isotropic elastic solid. By considering infinitesimal deformations from an arbitrary configuration, the constitutive equation of the rate-type is reduced to a constitutive equation of the incremental-type. In a similar way, an incremental-type constitutive equation is obtained from the constitutive equation of a simple, isotropic elastic solid. Comparison of these two incremental-type constitutive equations leads to the identification of a particular form for the material response coefficients associated with the constitutive equation of a simple elastic solid. Further limitations of the constitutive equation of a simple, isotropic elastic solid are considered in the context of two simple modes of deformation.