A flow rule incorporating the fabric and non-coaxiality in granular materials

Deficiencies of constitutive models in prediction of dilatancy are often attributed to simplifications associated with flow rules such as assumptions of isotropy and coaxiality. It is thus proposed here to develop a comprehensive flow rule for granular materials by including the effect of fabric and without the assumption of coaxiality. A second-order tensor is introduced as a fabric for the distribution of contact normals and contact forces. By using the energy principle in micro-mechanical scale and a suitable dissipation mechanism in granular materials, a stress-dilatancy relation is obtained. Fabric plays a “bridge-like” role in the dilatancy and non-coaxiality. Non-coaxialities between stress-strain-fabric are attributed to the non-coaxiality between stress-fabric and strain-fabric. In this formulation the constants for modeling fabric depend on non-coaxiality of the system rather than the history that determines such a state. Ability of this stress-fabric-dilatancy for modeling the non-coaxiality shows that this relation can predict the behavior of granular materials in the presence of the rotation of principal stress axes.     

A damage model based on Kelvin eigentensors and Curie principle

A general anisotropic damage model is developed that accounts for the thermodynamics of irreversible processes in the framework of generalized standard materials and Kelvin tensor decomposition. Damage is described by fourth-rank tensors, one per eigenspace of the initial stiffness tensor. Their number thus ranges from two for an initially isotropic material to six for an initially triclinic material. The yield criteria are expressed in terms of a limiting energy for each eigenspace. The second-rank eigentensors (at most six) of the fourth-rank damage tensors define the direction of influence of the damage, while the associated eigenvalues characterize its intensity. These eigentensors evolve during loading, inducing an evolution of the symmetry group of the elastic tensor subject to the constraints of the Curie principle.     

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.     

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.     

Spectral decomposition of anisotropic elasticity

Summary A newly developed approach, based on the spectral decomposition principle, which is especially useful in crystallography, is applied in this paper. The compliance fourth-rank tensor of crystalline media belonging to the monoclinic system is spectrally decomposed, its eigenvalues are evaluated, together which its elementary idempotent tensors, which expand uniquely the fourth-rank tensor space into orthogonal subspaces. Next, the compliance tensor is spectrally analysed for anisotropic media of the orthorhombic, tetragonal, hexagonal and cubic crystal systems, by regarding these decompositions as particular cases of the spectral decomposition for monolinic media. Consequently, the characteristic values and the idempotent fourth-rank tensors are derived, as well as the stress and strain second-rank eigentensors for all the above mentioned symmetries.     

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.     

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.     

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.     

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.     

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.     

A view of the relation between the continuum theory of lattice defects and non-euclidean geometry in the linear approximation

A view is presented of the relation between the continuum theory of defects in crystals and the mathematical theory of non-metric, non-Riemannian geometry. Both theories are treated in the linear approximation. The lattice defects consist of disclinations, dislocations, and extra-matter, which are identified with the following three important tensors from non-Euclidean geometry: the Riemann-Christoffel curvature tensor, the Cartan torsion tensor and the non-metric Q-tensor. The correspondence between the two theories is established by finding a relation between the coefficients of linear connection of non-Euclidean geometry and the elastic strain, bend-twist, and quasi-plastic strain of defect theory. The definitions of the important tensors from non-Euclidean geometry then generally correspond to the field equations of defect theory. The identities for the curvature tensor generally correspond to the continuity equations of defect theory. The relation to the conventional formulation of defect theory is pointed out. Two examples are given to illustrate the concepts of the paper. One example is related to the deformations associated with constant dislocation distribution and the other to the deformations of a constant disclination distribution.     

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.     

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.     

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.     

Granular micromechanics model of anisotropic elasticity derived from Gibbs potential

This paper presents a Gibbs potential-based granular micromechanics approach capable of modelingmaterialswith complete anisotropy. The deformation energy of each grain–pair interaction is taken as a functionof the inter-granular forces. The overall classical Gibbs potential of a material point is then defined as thevolume average of the grain–pair deformation energy. As a first-order theory, the inter-granular forces arerelated to the Cauchy stress tensor using a modified static constraint that incorporates directional distributionof the grain–pair interactions. Further considering the conjugate relationship of the macroscale strain tensorand the Cauchy stress, a relationship between inter-granular displacement and the strain tensor is derived.To establish the constitutive relation, the inter-granular stiffness coefficients are introduced considering theconjugate relation of inter-granular displacement and forces. Notably, the inter-granular stiffness introducedin this manner is by definition different from that of the isolated grain–pair interactive. The integral formof the constitutive relation is then obtained by defining two directional density distribution functions; onerelated to the average grain–scale combined mechanical–geometrical properties and the other related to purelygeometrical properties. Finally, as the main contribution of this paper, the distribution density function isparameterized using spherical harmonics expansion with carefully selected terms that has the capability ofmodeling completely anisotropic (triclinic) materials. By systematic modification of this distribution function,different elastic symmetries ranging from isotropic to completely anisotropic (triclinic) materials are modeled.As a comparison, we discuss the results of the present method with those obtained using a kinematic assumptionfor the case of isotropy and transverse isotropy, wherein it is found that the velocity of surface quasi-shearwaves can show different trends for the two methods.     

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.     

Expression of third-order effective nonlinear susceptibility for third-harmonic generation in crystals

Third-harmonic-generation processes in crystals are governed by the fourth-rank tensor ((3))(Xijkl), which reflects the crystal symmetry. In this case, the third-order nonlinear susceptibility tensor can be contracted to the compact matrix form ((3))(Xim). The matrices ((3))(Xim) for isotropic media and all 32 crystallographic point groups are presented. With these matrices, the analytic expressions of third-order effective nonlinear susceptibility can be easily derived.     

Macroscopic material properties from quasi-static, microscopic simulations of a two-dimensional shear-cell

One of the essential questions in the area of granular matter is, how to obtain macroscopic tensorial quantities like stress and strain from “microscopic” quantities like the contact forces in a granular assembly. Different averaging strategies are introduced, tested, and used to obtain volume fractions, coordination numbers, and fabric properties. We derive anew the non-trivial relation for the stress tensor that allows a straightforward calculation of the mean stress from discrete element simulations and comment on the applicability. Furthermore, we derive “elastic” (reversible) mean displacement gradient, based on a best-fit hypothesis. Finally, different combinations of the tensorial quantities are used to compute some material properties. The bulk modulus, i.e. the stiffness of the granulate, is a linear function of the trace of the fabric tensor which itself is proportional to the density and the coordination number. The fabric, the stress and strain tensors are not co-linear so that a more refined analysis than a classical elasticity theory is required. Received: 23 July 1999     

Parametric hypoplasticity as continuum model for granular media: from Stokesium to Mohr-Coulombium and beyond

Fluid-particle systems, in which internal forces arise only from viscosity or intergranular friction, represent an important special case of strictly dissipative materials defined by a history-dependent 4th-rank viscosity tensor. In a recently proposed simplification, this history dependence is represented by a symmetric 2nd-rank fabric tensor with evolution determined by a given homogeneous deformation. That work suggests an essential physical link between idealized suspensions (“Stokesium”) and granular media (“Mohr-Coulombium”) along with possible models for the visco-plasticity of fluid-saturated and dry granular media. The present paper deals with the elastoplasticity of dilatant non-cohesive granular media composed of nearly rigid, frictional particles. Based on the underlying physics and past modeling by others, a continuum model based on parametric hypoplasticity is proposed, which involves a set of rate-independent ODEs in the state-space of stress, void ratio and fabric. As with the standard theory of hypoplasticity, the present model does not rely on plastic potentials but, in contrast to that theory, it is based explicitly on positive-definite elastic and plastic moduli. The present model allows for elastic loading or unloading within a dissipative yield surface and also provides a systematic treatment of Reynolds dilatancy as a kinematic constraint. Some explicit forms are proposed and comparisons are made to previous hypoplastic models of granular media.