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.     

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 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.     

How to handle the inelastic collapse of a dissipative hard-sphere gas with the TC model

The inelastic hard sphere model of granular material is simple, easily accessible to theory and simulation, and captures much of the physics of granular media. It has three drawbacks, all related to the approximation that collisions are instantaneous: 1) The number of collisions per unit time can diverge, i.e. the “inelastic collapse” can occur. 2) All interactions are binary; multiparticle contacts cannot occur and 3) no static limit exists. We extend the inelastic hard sphere model by defining a duration of contact t _c such that dissipation is allowed only if the time between contacts is larger than t _c . We name this generalized model the TC model and discuss it using examples of dynamic and static systems. The contact duration used here does not change the instantaneous nature of the hard sphere contacts, but accounts for a reduced dissipation during “multiparticle contacts”. Kinetic and elastic energies are defined as well as forces and stresses in the system. Finally, we present event-driven numerical simulations of situations far beyond the inelastic collapse, possible only with the TC model.     

Edelen’s dissipation potentials and the visco-plasticity of particulate media

Following is an elaboration on D. G. B. Edelen’s (1972–1973) nonlinear generalization of the classical Rayleigh-Onsager dissipation potentials and the implications for the models of viscoplasticity. A brief derivation is given via standard vector calculus of Edelen’s potentials and the associated non-dissipative or “gyroscopic” forces and fluxes. It is also shown that certain extensions of Edelen’s formulae can be obtained by means of a recently proposed source-flux relation or “inverse divergence,” a generalization of the classical Gauss-Maxwell construct. The Legendre–Fenchel duality of Edelen’s potentials is explored, with important consequences for rate-independent friction or plasticity. The use of dissipation potentials serves to facilitate the development of viscoplastic constitutive equations, a point illustrated here by the special cases of Stokesian fluid-particle suspensions and granular media. In particular, we consider inhomogeneous systems with particle migration coupled to gradients in particle concentration, strain rate, and fabric. Employing a mixture-theoretic treatment of Stokesian suspensions, one is able to identify particle stress as the work conjugate of the global deformation of the particle phase. However, in contrast to past treatments, this stress is not assumed to be a privileged driving force for particle migration. A comparison is made with models based on extremal dissipation or entropy production. It is shown that such models yield the correct dissipative components of force or flux but generally fail to capture certain non-dissipative, but mechanically relevant components. The significance of Edelen’s gyroscopic forces and their relation to reactive constraints or other reversible couplings is touched upon. When gyroscopic terms are absent, one obtains a class of strongly dissipative or hyperdissipative materials whose quasi-static mechanics are governed by variational principles based on dissipation potential. This provides an interesting analog to elastostatic variational principles based on strain energy for hyperelastic materials and to the associated material instabilities arising from loss of convexity.     

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.     

Stress distribution in bonded joints: An exploration within a mathematical model

The introduction of “toughened” adhesives with their enhanced capacity to cope with strain has placed increasing emphasis on the importance of mathematical models to be able to cope with the plastic as well as elastic characteristics of adhesives. The changes in stress distribution caused by a variety of modifications to a collar and pin joint have been explored within the framework of an elastic/plastic model. The results indicate quite clearly how important it is to match the adhesive with the adherends and their proposed geometry.     

An additive theory for finite elastic–plastic deformations of the micropolar continuous media

In this paper, the method of additive plasticity at finite deformations is generalized to the micropolar continuous media. It is shown that the non-symmetric rate of deformation tensor and gradient of gyration vector could be decomposed into elastic and plastic parts. For the finite elastic deformation, the micropolar hypo-elastic constitutive equations for isotropic micropolar materials are considered. Concerning the additive decomposition and the micropolar hypo-elasticity as the basic tools, an elastic–plastic formulation consisting of an arbitrary number of internal variables and arbitrary form of plastic flow rule is derived. The localization conditions for the micropolar material obeying the developed elastic–plastic constitutive equations are investigated. It is shown that in the proposed formulation, the rate of skew-symmetric part of the stress tensor does not exhibit any jump across the singular surface. As an example, a generalization of the Drucker–Prager yield criterion to the micropolar continuum through a generalized form of the J ₂-flow theory incorporating isotropic and kinematic hardenings is introduced.     

A macroscopic description of the quasi-static behavior of granular materials based on the theory of porous media

Granular materials fall into the class of porous media. But in contrast to materials like foams and sponges their structure is discontinous on a microscopic level. For this reason the particles may undergo independent displacements and rotations. This is the classical kinematics which may be captured by a micropolar or Cosserat theory on the macroscopic level. The goal of this paper is to combine the theory of porous media as a macroscopic theory dealing with multi-phase systems and the micropolar theory describing extended kinematics and taking care of the discountinous structure of granular media on the micro scale. The resulting micropolar theory of porous media may be used to describe the quasistatic behavior of granular materials. In the present contribution thermodynamically consistent constitutive relations for the elastic response of a dense granular matrix material saturated by a viscous pore fluid are given and applied to some boundary value problems which demonstrate the physical relevance of the proposed model. Received: 9 June 1999     

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.     

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.     

Parkes revisited: On rigid-plastic and elastic-plastic dynamic structural analysis

In this paper we re-examine a classical problem of rigid-plastic structural dynamics solved by Parkes [1], namely that of finding the deformations of a beam carrying a mass at its tip which is subjected to a short pulse loading. A considerable body of “elementary rigid-plastic theory” has been developed based on the neglect of elastic strain components, the idealization of perfectly plastic behavior (absence of strain hardening or strain rate sensitivity), and the assumption of linear field equations ignoring effects of geometry changes. This theory provides a valuable conceptual framework for problems of dynamic plastic structural response, although corrections are required for many practical applications; for example, Parkes introduced a correction for effects of high strain rates shown to be needed by his experiments on steel beams. The neglect of elastic strains — elastic moduli taken infinite — is a crucial assumption, and its validity is the subject of the present paper. We study it by making comparison between “exact” numerical solutions furnished by an advanced finite element computer program for an elastic-plastic beam, and the rigid-plastic solution slightly modified to allow for large deflections. Further insight is gained by comparisons also with a simplified elastic-plastic approach based on treatment of elastic and plastic action in artificially separated stages. The main conclusions are that the initial travelling-hinge phase of the rigid-plastic solution is essentially a fiction, but that the subsequent mode form phase is a prominent and significant feature of the late-time part of the “actual” elastic-plastic response.     

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.     

Damage-induced stress-softening and viscoelasticity of limited elastic materials

The rate-dependent behavior of filled natural rubber (NR) is investigated in tensile regime. In order to describe the viscosity-induced rate-dependent effects, a constitutive model of finite strain viscoelasticity is proposed on the basis of the multiplicative decomposition of the deformation gradient tensor into elastic and viscous parts. The total stress is decomposed into an equilibrium stress and a viscosity-induced overstress by following the rheological models of Poynting–Thomson and Zener types. To incorporate the Mullins stress-softening phenomenon into a viscoelastic material, an invariant-based stress-softening function is also proposed. To identify the constitutive equation for the viscosity from direct experimental observations, an analytical scheme is proposed that ascertains the fundamental relation between the viscous strain rate and the overstress tensor with limited elastic parent material model. Evaluation of the experimental results using the proposed analytical scheme confirms the necessity of considering both the current overstress and the current deformation as variables to describe the evolution of the rate-dependent phenomena. Based on this, an evolution equation is proposed to represent the effects of internal variables on viscosity phenomena. The proposed evolution equation has been incorporated into the finite-strain viscoelasticity model in a thermodynamically consistent way.     

Proportional paths, Barodesy, and granular solid hydrodynamics

Proportional paths as summed up by the Goldscheider Rule (GR)—stating that given a constant strain rate, the evolution of the stress maintains the ratios of its components—is a characteristics of elasto-plastic motion in granular media. Barodesy, a constitutive relation proposed recently by Kolymbas, is a model that, with GR as input, successfully accounts for data from soil mechanical experiments. Granular solid hydrodynamics (GSH), a theory derived from general principles of physics and two assumptions about the basic behavior of granular media, is constructed to qualitatively account for a wide range of observation – from elastic waves over elasto-plastic deformation to rapid dense flow. In this paper, showing the close resemblance of results from Barodesy and GSH, we further validate GSH and provide an understanding for GR.     

A Lagrangian model for hardening behaviour of materials at finite deformation based on the right plastic stretch tensor

In this paper a modified multiplicative decomposition of the right stretch tensor is proposed and used for finite deformation elastoplastic analysis of hardening materials. The total symmetric right stretch tensor is decomposed into a symmetric elastic stretch tensor and a non-symmetric plastic deformation tensor. The plastic deformation tensor is further decomposed into an orthogonal transformation and a symmetric plastic stretch tensor. This plastic stretch tensor and its corresponding Hencky’s plastic strain measure are then used for the evolution of the plastic internal variables. Furthermore, a new evolution equation for the back stress tensor is introduced based on the Hencky plastic strain. The proposed constitutive model is integrated on the Lagrangian axis of the plastic stretch tensor and does not make reference to any objective rate of stress. The classic problem of simple shear is solved using the proposed model. Results obtained for the problem of simple shear are identical to those of the self-consistent Eulerian rate model based on the logarithmic rate of stress. Furthermore, extension of the proposed model to the mixed nonlinear isotropic/kinematic hardening behaviour is presented. The model is used to predict the nonlinear hardening behaviour of SUS 304 stainless steel under fixed end finite torsional loading. Results obtained are in good agreement with the available experimental results reported for this material under fixed end finite torsional loading.     

The plastic potential in the theory of anisotropic elastic-plastic solids

In the paper the yield condition is proposed for the most general anisotropic material. It is one of the possible generalizations of the Huber-Mises-Hencky yield condition for the case of anisotropy. The body considered is anisotropic elastically as well as plastically. It is assumed that the plastic anisotropy tensor is a definite function of the elastic anisotropy tensor. The corresponding flow function is a part of the strain energy and its value remains unchanged when all normal components of stress are increased by the same value. The theory of the eigen states for fourth order tensors is used. The plastic anisotropy tensor proposed has the same deviatoric eigen states as the elastic anisotropy tensor. The proposed yield condition reduces to that of Huber-Mises-Hencky when the anisotropy is vanishingly small. The method presented in this paper can be also applied to describe other types of plastic anisotropy tensor.     

Théorie hydrodynamique des cristaux liquides nématiques biaxes

A hydrodynamic theory of biaxial nematic liquid crystals is presented. The equations of motion are deduced from the principle of virtual power and constitutive equations are proposed by applying the method of local state within the framework of the irreversible thermodynamics. Hence it follows that the dissipative part of the stress tensor contains generally twelve independent viscosity coefficients. The resulting theory is a generalization of the well-known theory of uniaxial nematics.