The relationship between the elasticity tensor and the fabric tensor

An algebraic relationship between the fourth rank elasticity tensor of a porous, anisotropic, linear elastic material and the fabric tensor of the material is considered. The fabric tensor is a symmetric second rank tensor which characterizes the geometric arrangement of the porous material microstructure. In developing this result it is assumed that the matrix material of the porous elastic solid is isotropic and, thus, that the anisotropy of the porous elastic solid is determined by the fabric tensor. It is then shown that the material symmetries of orthotropy, transverse isotropy and isotropy correspond to the cases of three, two and one distinct eigenvalues of the fabric tensor, respectively.     

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 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.     

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.     

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.     

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.     

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.     

A note on the anisotropy and fabric of highly porous materials

The dependence of the orthotropic elastic constants of a highly porous material upon the stereological parameters characterizing the anisotropy of the porous microstructure has been considered in two recent papers in this journal. In the first paper [1] dimensional arguments were employed to develop to a relationship between ratios of the orthotropic elastic constants and ratios of the mean intercept lengths for a class of cell wall bending models of highly porous materials. In the second paper [2] the general tensorial form of the relationship between the orthotropic elastic constants and the mean intercept length was described without reference to a specific form or type of porous microstructure. The purpose of this note is to observe that the particular relationships obtained from the class of cell wall bending models used in the first paper are proper special cases of the general relationships given in the second paper.     

Mechanical properties of non-cohesive polygonal particle aggregates

We numerically investigate the effective material properties of aggregates consisting of soft convex polygonal particles, using the discrete element method. First, we construct two types of “sand piles” by two different procedures. Then we measure the averaged stress and strain, the latter via imposing a 10% reduction of gravity, as well as the fabric tensor. Furthermore, we compare the vertical normal strain tensor between sand piles qualitatively and show how the construction history of the piles affects their strain distribution as well as the stress distribution. In the next step, elastic constants are determined, assuming Hooke’s law to be locally valid throughout the sand piles. We determine the relationship between invariants of the stress and strain tensor, observing that the behaviour is nonlinear. While linear elastic behaviour near the centre of the pile is compatible with our data, nonlinearity signals the transition to plastic behaviour near its surface. A similar behaviour was assumed by Cantelaube et al. (Static multiplicity of stress states in granular heaps. Proc R Soc Lond A 456:2569–2588, 2000). We find that the macroscopic stress and fabric tensors are not collinear in the sand pile and that the elastic behaviour is anisotropic in an essential way.     

Stress-induced anisotropy in sand under cyclic loading

The anisotropy of a granular material’s structure will influence its response to applied loads and deformations. Anisotropy can be either inherent (e.g. due to depositional process) or induced as a consequence of the applied stresses or strains. Discrete element simulations allow the interactions between individual particles to be explicitly simulated and the fabric can be quantified using a fabric tensor. The eigenvalues of this fabric tensor then give a measure of the anisotropy of the fabric. The coordination number is a particle scale scalar measure of the packing density of the material. The current study examines the evolution of the fabric of a granular material subject to cyclic loading, using two-dimensional discrete element method (DEM) simulations. Isotropic consolidation modifies and reduces the inherent anisotropy, but anisotropic consolidation can accentuate anisotropy. The ratio of the normal to shear spring stiffness at the particle contacts in the DEM model affects the evolution of anisotropy. Higher ratios reduce the degree of anisotropy induced by anisotropic consolidation. The anisotropy induced by cyclic loading depends on the amplitude of the loading cycles and the initial anisotropy.     

High-pressure physical properties of magnesium silicate post-perovskite from ab initio calculations

The structure, thermodynamic and elastic properties of magnesium silicate (MgSiO₃) post-perovskite at high pressure are investigated with quasi-harmonic Debye model and ab initio method based on the density functional theory (DFT). The calculated structural parameters of MgSiO₃ post-perovskite are consistent with the available experimental results and the recent theoretical results. The Debye temperature, heat capacity and thermal expansion coefficient at high pressures and temperatures are predicted using the quasi-harmonic Debye model. The elastic constants are calculated using stress–strain relations. A complete elastic tensor of MgSiO₃ post-perovskite is determined in the wide pressure range. The calculated elastic anisotropic factors and directional bulk modulus show that MgSiO₃ post-perovskite possesses high elastic anisotropy.     

Anisotropic Carbon Phases Prepared from Fullerite C70 under High Pressure

Abstract

We present the structural and ultrasonic study of carbon phases prepared by quenching after heating of fullerite C₇₀ in the temperature range 300–1100°C at pressures 4 and 7.5 GPa. The main aspect of the work concerns the structural and elastic anisotropy of samples resulted as consequence of quasi hydrostatic pressure with an additional pressure component along the axis of pressure load. Structural anisotropy correlates with anisotropy of the tensor of effective elastic constants taken for the medium with single axis of anisotropy. Comparison of the data obtained with anisotropy effects, earlier observed in the phases synthesized from C₆₀ in the similar quasi hydrostatic conditions, clarifies the role of non‐spherical symmetry of C₇₀ molecule.     

Scan line void fabric anisotropy tensors of granular media

Soil fabric anisotropy tensors are related to the statistical distribution of orientation of different microstructural vector-like entities, most common being the contact normal vectors between particles, which are extremely difficult to determine for real granular materials. On the other hand, void fabric based tensors can be determined by image based quantification methods of voids (graphical approaches), which are well defined and easy to apply to both physical and numerical experiments. A promising void fabric characterization approach is based on the scan line method. Existing scan line based definitions of void fabric anisotropy tensors are shown analytically to inherit a shortcoming, since numerous small void segments in a sample have an inordinate contribution towards unwarranted isotropy. Discrete Element Method (DEM) of analysis subsequently confirms this analytical proof. The fact that such scan line void fabric tensor definitions yield acceptable results when used in conjunction with physical image-based measurements, is shown to be attributed to the natural “cut off” of smaller void segments that occurs during such measurements. This is the motivation to propose using the existing definition of void fabric tensors, with exclusion of void segments less than a “cut off” value associated with an internal length of the granular assembly. In addition, an entirely new void fabric tensor was introduced using the squared length, instead of the length of a void segment, as the weighting factor for the definition of the scan line void fabric tensor. It was found by means of DEM analysis that both alternative definitions are void of the aforementioned shortcoming and compatible with existing image quantification methods of void fabric anisotropy.     

Fabric,force and strength anisotropies in granular materials: a micromechanical insight

In micromechanics, the stress–force–fabric (SFF) relationship is referred to as an analytical expression linking the stress state of a granular material with microparameters on contact forces and material fabric. This paper employs the SFF relationship and discrete element modelling to investigate the micromechanics of fabric, force and strength anisotropies in two-dimensional granular materials. The development of the SFF relationship is briefly summarized while more attention is placed on the strength anisotropy and deformation non-coaxiality. Due to the presence of initial anisotropy, a granular material demonstrates a different behaviour when the loading direction relative to the direction of the material fabric varies. Specimens may go through various paths to reach the same critical state at which the fabric and force anisotropies are coaxial with the loading direction. The critical state of anisotropic granular material has been found to be independent of the initial fabric. The fabric anisotropy and the force anisotropy approach their critical magnitudes at the critical state. The particle-scale data obtained from discrete element simulations of anisotropic materials show that in monotonic loading, the principal force direction quickly becomes coaxial with the loading direction (i.e. the strain increment direction as applied). However, material fabric directions differ from the loading direction and they only tend to be coaxial at a very large shear strain. The degree of force anisotropy is in general larger than that of fabric anisotropy. In comparison with the limited variation in the degree of force anisotropy with varying loading directions, the fabric anisotropy adapts in a much slower pace and demonstrates wider disparity in the evolution in the magnitude of fabric anisotropy. The difference in the fabric anisotropy evolution has a more significant contribution to strength anisotropy than that of force anisotropy. There are two key parameters that control the degree of deformation non-coaxiality in granular materials subjected to monotonic shearing: the ratio between the degrees of fabric anisotropy and that of force anisotropy and the angle between the principal fabric direction and the applied loading direction.     

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.     

Macroscopic model with anisotropy based on micro–macro information

Physical experiments can characterize the elastic response of granular materials in terms of macroscopic state variables, namely volume (packing) fraction and stress, while the microstructure is not accessible and thus neglected. Here, by means of numerical simulations, we analyze dense, frictionless granular assemblies with the final goal to relate the elastic moduli to the fabric state, i.e., to microstructural averaged contact network features as contact number density and anisotropy. The particle samples are first isotropically compressed and then quasi-statically sheared under constant volume (undrained conditions). From various static, relaxed configurations at different shear strains, infinitesimal strain steps are applied to “measure” the effective elastic response; we quantify the strain needed so that no contact and structure rearrangements, i.e. plasticity, happen. Because of the anisotropy induced by shear, volumetric and deviatoric stresses and strains are cross-coupled via a single anisotropy modulus, which is proportional to the product of deviatoric fabric and bulk modulus (i.e., the isotropic fabric). Interestingly, the shear modulus of the material depends also on the actual deviatoric stress state, along with the contact configuration anisotropy. Finally, a constitutive model based on incremental evolution equations for stress and fabric is introduced. By using the previously measured dependence of the stiffness tensor (elastic moduli) on the microstructure, the theory is able to predict with good agreement the evolution of pressure, shear stress and deviatoric fabric (anisotropy) for an independent undrained cyclic shear test, including the response to reversal of strain.     

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.     

Influence of fabrics’ design parameters on the morphology and 3D permeability tensor of quasi-unidirectional non-crimp fabrics

This research addresses the effects of quasi-UD non-crimp fabric (NCF) design parameters on the fabric architecture and on the permeability tensor. These fabrics are designed for the Liquid Resin Infusion (LRI) of large and thick composite parts. Three fabrics’ parameters intended to bring a flow enhancement to the NCF are investigated: the stitch spacing, the stitch pattern and the weft tow lineal weight. Image analysis is undertaken to characterize the morphology of non-crimp fabric composite. A new continuous permeability measurement method based on compressive tests is proposed to relate the permeability of the quasi-UD NCF to the design parameters during the infusion process. The latter are proven to influence significantly both the fabric architecture and the permeability tensor coefficients.     

On approximate symmetries of the elastic properties and elliptic orthotropy

The theory of anisotropic elasticity was originally motivated by applications to crystals, where geometric symmetries hold with high precision. In contrast, symmetries of the effective elastic responses of heterogeneous materials are usually approximate due to various imperfections of microgeometry. A related issue is that available data on the elastic constants may be incomplete or imprecise; it may be appropriate to select the highest possible elastic symmetry that fits the data reasonably well. Some of these problems have been discussed in literature in the context of specific applications, primarily in geomechanics. The present work provides a systematic discussion of the related issues, illustrated by examples on the effective elastic properties of heterogeneous materials. We also discuss a special type of orthotropy typical for a variety of heterogeneous materials - elliptic orthotropy - when the fourth-rank tensor of elastic constants can be represented in terms of a certain symmetric second-rank tensor. This representation leads, in particular, to reduced number of independent elastic constants.