Morteza M. Mehrabadi

Initial planar deformation of dilatant granular materials

Abstract A theory for the initial planar deformation of dilatant granular materials based on a kinematic proposal of R. Butterfield and R.M. Harkness (1972) is presented. The theory introduces an additional parameter called the angle of dilatancy into the traditional structure of plasticity theories for granular materials and soils. When the angle of dilatancy is zero, the present theory reduces to the theory introduced by A.J.M. Spencer in 1964. When the angle of dilatancy is equal to the angle of internal friction, the present theory reduces to the planar form of the theory introduced by D. C. Drucker and W. Prager in 1952. The properties of the theory presented here include coincidence of the stress and velocity characteristics, realistic energy dissipation predictions, and, in general, non-coincidence of the principal axes of stress and strain-rate. However, the angle of dilatancy is assumed to be a constant in this analysis and it does not decrease to zero with increased monotonic shearing deformation as experiment requires that it should, the theory therefore being limited to the initial deformation of dilatant granular materials.

The structure of the linear anisotropic elastic symmetries

An insightful, structurally appealing and potentially utilitarian formulation of the anisotropic form of the linear Hookes law due to Lord Kelvin was independently rediscovered by Rychlewski (1984, Prikl. Mat. Mekh. 48, 303) and Mehrabadi and Cowin (1990, Q. J. Mech. appl. Math. 43, 14). The eigenvectors of the three-dimensional fourth-rank anisotropic elasticity tensor, considered as a second-rank tensor in six-dimensional space, are called eigentensors when projected back into three-dimensional space. The maximum number of eigentensors for any elastic symmetry is therefore six. The concept of an eigentensor was introduced by Kelvin (1856, Phil. Trans. R. Soc. 166, 481) who called eigentensors “the principal types of stress or of strain”. Kelvin determined the eigentensors for many elastic symmetries and gave a concise summary of his results in the 9th edition of the Encyclopaedia Britannica (1878). The eigentensors for a linear isotropic elastic material are familiar. They are the deviatoric second-rank tensor and a tensor proportional to the unit tensor, the spherical, hydrostatic or dilatational part of the tensor. Mehrabadi and Cowin (1990, Q. J. Mech. appl. Math. 43, 14) give explicit forms of the eigentensors for all of the linear elastic symmetries except monoclinic and triclinic symmetry. We discuss two approaches for the determination of eigentensors and illustrate these approaches by partially determining the eigentensors for monoclinic symmetry. With the nature of the eigentensors for monoclinic symmetry known, a rather complete table of the structural properties of all linear elastic symmetries can be constructed. The purpose of this communication is to give the most specifically detailed presentation of the eigenvalues and eigentensors of the Kelvin formulation to date.

Some basic kinematical relations for finite deformations of continua

Abstract This paper summarizes a number of general kinematical identities developed by the authors in early 1981 for finite deformations of continua. Included are explicit relations between various spin tensors and the symmetric and antisymmetric parts of the velocity gradient, various material strain measures, and strain rates. These relations are expressed in the general, coordinate-independent form (which is the main contribution of this work), as well as in terms of their spectral components (due to Hill).

Incremental constitutive relations for granular materials based on micromechanics

Micromechanically based constitutive relations for two-dimensional flow of granular materials are presented. First, overall stresses are related to the interparticle forces and microstructural parameters. Then, the overall velocity gradient is related to measures of relative sliding and rotation of granules. The notion of the class of granules with continuously evolving distribution of contact normals, is introduced. Simple local constitutive relations are considered for the rate of change of the contact forces, the evolution of the contact normals, the mechanism of local failure, and the density of contacts in a particular class. This leads to macroscopic rate constitutive equations through a Taylor averaging method. Due to nonlinearity of the rate constitutive equations, the response is computed by an incremental procedure. As an illustration, the overall response of a two-dimensional assembly of discs subjected to an overall shearing deformation is determined. In addition, explicit results are presented for the evolution of fabric, contact forces, and the history of active and inactive classes of contacts. The stress-strain relations and the evolution of fabric and contact forces are in qualitative agreement with the observed behaviour of granular materials. In light of these results, the mechanisms of failure and inelastic deformation of dense as well as loose granular materials are discussed.

Experimental investigation of fabric-stress relations in granular materials

Abstract A brief summary of some relevant theoretical and experimental results on the microscopic aspects of the response of granular masses is presented. The results of a series of experiments involving simple shearing under a constant confining pressure, performed on photoelastic rod-like granules (plane strain) are reported. In these experiments, the components of various fabric tensors are measured, and their variations over one cycle of shearing are examined and compared. The orientations of the principal axes of all commonly used fabric tensors are observed to change sharply with the reversal of the shearing direction. It is also concluded that, in general, second-order fabric tensors are not adequate to accurately describe the distribution of fabric measures such as the distribution density function of unit contact normals or unit branches which are unit vectors along line segments connecting the centroids of adjacent contacting granules. This is particularly so when the response of the granular mass is highly anisotropic. Finally, the expression for the macroscopic stress in terms of the contact forces and other local quantities, is reviewed and its experimental verification is discussed.

A multidimensional anisotropic strength criterion based on Kelvin modes

A new theory for the prediction of multiaxial strength of anistropic elastoplastic materials is proposed. The resulting failure envelope, in a multidimensional stress space, is piecewise smooth. Each facet of the envelope is expected to represent the locus of failure data by a particular anisotropic elastic deformation mode called a Kelvin mode. It is shown that the Kelvin mode theory alone provides an incomplete description of the failure of some materials, but that this weakness can be addressed by the introduction of a set of complementary modes. A revised theory which combines both Kelvin and complementary modes is developed and illustrated by an applied example.

Six-dimensional orthogonal tensorrepresentation of the rotation about an axis in three dimensions

Abstract The representation of the classical formula that contains Eulers theorem on three-dimensionalrigid body rotations, as an orthogonal tensor in three dimensions, is extended to a six-dimensional representation as a tool for accomplishing coordinate transformations of the anisotropic elasticity tensor.

Stress, dilatancy and fabric in granular materials

Abstract On the basis of a simple micromechanical model, expressions are developed for the overall stress and deformation rate measures in a granular mass which carries the applied loads through contact friction. Various measures of the granular fabric are examined, and explicit connections between a fabric measure and the stress and the deformation rate tensors are obtained. Then, from the balance of energy, a general stress-dilatancy equation is deduced, which includes the effect of fabric. Several other published dilatancy equations are discussed and compared with the new equation.

An energy-based constitutive model for anisotropic solids subject to damage

Abstract This paper presents an energy-based constitutive model for anisotropic solids. The model attempts to characterize initial anisotropic elastic response, elastic degradation due to anisotropic damage microcracking, yield and subsequent plastic flow, and material failure. The theory treats three-dimensional anisotropic materials subject to small strains. The treatment of anisotropic damage microcracking is approached from a macroscopic perspective. The generalized Hookes law can be written in tensorial form in six-dimensional space. The six eigenvalues of the compliance tensor are then constants of proportionality between the stress and strain eigenvectors. Further, the strain energy stored in an anisotropic solid can be additively decomposed into six independent, non-interacting energy modes. Damage, characterized by a symmetrical second rank tensor, is postulated to grow when the energy level in any such mode reaches a critical value. The rate of damage growth is formulated in terms of the energy modes. The yield conditions are formulated in terms of the second invariants of the eigenstresses. An associated flow rule is developed. Material failure is assumed to occur when any one of the energy modes reaches a critical threshold. A numerical algorithm is used to solve the rate constitutive equations. The proposed constitutive model is compared to uniaxial stress-strain data and biaxial failure data for orthotropic paperboard. The model predicts the stress-strain response of this nonlinear material with reasonable accuracy. In the case of biaxial loading, the failure predictions are fairly consistent with the failure data in three of the four quadrants of the stress space.

Some Basic Theoretical and Experimental Results on Micromechanics of Granular Flow

Summary In order to establish guidelines for modeling the macroscopic behavior of granular materials, an experimental study of the evolution of the microstructure of an assembly of granular materials under a uniform confining pressure and subjected to a pure shear was conducted. The granular material used in the study consisted of photoelastically sensitive rod-shaped particles of oval cross-sections. It was found that (i) the distribution of branches and contact normals are almost identical, (ii) the second rank fabric tensor does not adequately describe the microstructure of highly anisotropic samples, (iii) the density of contacts whose normals lie along the major and minor principal stress axes, varies sharply initially and then approaches a constant value in the course of deformation, and (iv) the density of contacts with planes parallel to the maximum shear stress plane remains practically constant throughout the deformation.