A micromechanics-based elastoplastic damage model for granular materials at low confining pressure

This paper is devoted to the formulation of a micromechanics-based constitutive model for granular materials under relatively low confining pressure. The constitutive formulation is performed within the general framework of homogenization for granular materials. However, new rigorous stress localization laws are proposed. Some local constitutive relations are established under the consideration of irreversible thermodynamics. Macroscopic plastic deformation is obtained by considering local plastic sliding in a limit number of families of contact planes. The plastic sliding at each contact plane is described by a non-associated plastic flow rule, taking into account pressure sensitivity and normal dilatancy. Nonlinear elastic deformation related to progressive compaction of contacts is also taken into account. Material softening is described by involving damage process related to degradation of microstructure fabric. The proposed model is applied to some typical granular materials (sands). The numerical predictions are compared with experimental data.     

An elasto-plastic model for granular materials with microstructural consideration

In this paper, we have extended the granular mechanics approach to derive an elasto-plastic stress–strain relationship. The deformation of a representative volume of the material is generated by mobilizing particle contacts in all orientations. Thus, the stress–strain relationship can be derived as an average of the mobilization behavior of these local contact planes. The local behavior is assumed to follow a Hertz–Mindlin’s elastic law and a Mohr–Coulomb’s plastic law. Essential features such as continuous displacement field, inter-particle stiffness, and fabric tensor are discussed. The predictions of the derived stress–strain model are compared to experimental results for sand under both drained and undrained triaxial loading conditions. The comparisons demonstrate the ability of this model to reproduce accurately the overall mechanical behavior of granular media and to account for the influence of key parameters such as void ratio and mean stress. A part of this paper is devoted to the study of anisotropic specimens loaded in different directions, which shows the model capability of considering the influence of inherent anisotropy on the stress–strain response under a drained triaxial loading condition.     

An elastoplastic model for granular materials taking into account grain breakage

Granular materials are constituted of an assembly of particles. In spite of the simplicity of this assembly, its mechanical behaviour is complex. In the first stage we propose a framework to establish correlations between parameters of the supposedly continuous medium and grain properties which are assumed to be constant. However, this hypothesis is no longer valid in the case where physical (shape, size…) or mechanical properties (Young modulus E_g, Poisson's ratio ν_g…) of grains evolve during loading, causing the behaviour of the assembly to modify. We study the influence of the physical and mechanical parameters on grain breakage. We subsequently propose a way to model the influence of the grain breakage on granular materials and we introduce this influence in an elastoplastic constitutive model. Validations are made on two kinds of sands under isotropic and triaxial loading. Since the results of numerical computations corresponded well with the experimental data, we believe that the new model is capable of accurately simulating the behaviour of granular materials under a wide range of stresses and of taking into account, through new parameters, the individual strength of grains.     

Prediction on dispersion in elastoplastic unsaturated granular media

This study focuses on the propagation of the plane wave in the elastoplastic unsaturated granular media, and the wave equations and dispersion equations are derived for the media under the framework of Cosserat theory. Due to symmetry, five different wave modes are considered and predicted for the elastoplastic unsaturated granular media based on the Cosserat theory, including two longitudinal waves, one rotational longitudinal wave and two coupled transverse–rotational transverse waves. The correspondence is discussed between these Cosserat wave modes and the classical wave modes. Based on the dispersion equations, the dispersion behaviors are obtained for the five Cosserat wave modes. The results indicated that the different stress-strain stages,including the elastic, hardening and softening stages, have obvious effect on the dispersion behaviors of the Cosserat wave modes.     

A two-phase elastoplastic model for unidirectionally-reinforced materials

A complete analytical formulation for the elastoplastic behaviour of a composite material comprising one single array of reinforcing inclusions perfectly bonded to the matrix is developed in this paper. Fundamental relationships establish the link between the total stress and strain variables, and those pertaining to the individual constituents (matrix and reinforcement) regarded as superposed continuous phases. Assuming that each constituent behaves as an elastic perfectly plastic material, the constitutive equations governing the evolution of the reinforced material as a whole are derived. They reveal a hardening phenomenon arising from the non-compatibility between matrix and reinforcement plastic strains. It is shown in particular that the obtained constitutive law falls within the formalism of generalized standard plasticity: the reinforcement residual stress plays the role of a hardening parameter which controls the evolution of the yield surface, while the associated kinematic variable is the plastic strain discrepancy between matrix and reinforcement phases.Owing to its inherent simplicity, the model is easily amenable to a numerical treatment for structural analysis. It is shown in particular how the classical iterative algorithm can be modified accordingly, and an illustrative application is finally presented in the field of civil engineering.     

An elastoplastic model for unsaturated rocks and concrete

This paper presents an elastoplastic model for unsaturated rocks and concrete. The formulation is based on the poroplastic theory of porous media. The effect of water saturation on plastic deformation is described through a generalised effective stress. Plastic shrinkage and swelling due to suction change are taken into account. All model’s parameters can be determined from specific laboratory tests. Comparisons between numerical simulations and experimental data are provided for a claystone.     

A stress point algorithm for an elastoplastic model in unsaturated soils

Two stress fields, combination of total stresses, liquid pressure and gas pressure have to be considered to explain the deformational behaviour of unsaturated media. Elastoplastic models developed for these materials consider generally two yield surfaces, each one associated to a stress field, and whose intersection produces a corner in the space of generalised stress components. In this paper, a stress point algorithm is proposed to cope with the integration of such constitutive laws, which can be seen as non smooth multisurface plastic models in the space of the two stress fields. The basic model developed by Alonso et al. (Alonso, E.E., Gens, A., 1990. A constitutive model for partially saturated soils. Géotechnique 40 (3), 405–430), which will be used to test the algorithm, is first described. Generalised stress and strain variables are then defined. Implementation of the return mapping algorithm, based on an implicit integration scheme, is presented with special attention devoted to the problem of mixed control imposed by the F.E. formulation generally used to analyse the hydromechanical behaviour of unsaturated media. Validation results on distinct generalised stress paths are given at the end.     

Solution existence conditions for elastoplastic constitutive models of granular materials

In this paper, the conditions of solution existence for stress rates under given strain rates are investigated. The focus of the solution existence investigation is on the non-associated flow rule and elastic stress–strain relationship. Granular materials characterized with strong non-associated plastic flows are used as a particular example for analysis. Various flow rules for granular materials are analyzed, including Rowe’s, Roscoe’s flow rules and their modified versions. In the elastic stress–strain relationships of materials, the effects of Poisson’s ratio on solution existence are investigated. Both isotropic and anisotropic elasticity are considered. Given a granular material and its states, it is found that there exists a critical Poisson’s ratio for a particular non-associated flow rule. When the Poisson’s ratio of a material is above this critical Poisson’s ratio, its constitutive model is susceptible to solution non-existence. It is suggested that special attentions should be paid to the selection of material Poisson’s ratio and non-associated flow rule to ensure the existence of elastoplastic solutions.     

A kinetic model for rapid flows of granular materials

A kinetic model for rapid flows of granular material is considered. An evolution equation for the first order distribution function is developed which includes the effects of intergranular friction force in an average sense. The collision operator is approximated by the BGK relaxation model. The fundamental equations of motion including an equation for dynamics of the fluctuation energy are derived and discussed. The gravity and the plane Couette flows of granular materials are treated as examples of the applications of the present theory.     

Multi-mechanism anisotropic model for granular materials

By representing the assembly by a simplified column model, a constitutive theory, referred to as sliding–rolling theory, was recently developed for a two-dimensional assembly of rods subjected to biaxial loading, and then extended to a three-dimensional assembly of spheres subjected to triaxial (equibiaxial) loading. The sliding–rolling theory provides a framework for developing a phenomenological constitutive law for granular materials, which is the objective of the present work. The sliding–rolling theory provides information concerning yield and flow directions during radial and non-radial loading. In addition, the theory provides information on the role of fabric anisotropy on the stress–strain behavior and critical state shear strength. In the present paper, a multi-axial phenomenological model is developed within the sliding–rolling framework by utilizing the concepts of critical state, classical elasto-plasticity and bounding surface. The resulting theory involves two yield surfaces and falls within the definition of the multi-mechanism models. Computational issues concerning the solution uniqueness for stress states at the corner of yield surfaces are addressed. The effect of initial and induced fabric anisotropy on the constitutive behavior is incorporated. It is shown that the model is capable of simulating the effect of anisotropy, and the behavior of loose and dense sands under drained and undrained loading.     

Micromechanics modeling the solute diffusivity of unsaturated granular materials

This work is devoted to modeling the evolution of the homogenized solute diffusion coefficient within unsaturated granular materials by means of micromechanics approach. On the basis of its distinct role in solute diffusion, the liquid water within unsaturated granular materials is distinguished into four types, namely intergranular layer (interconnected capillary water), isolated capillary water, wetting layer and water film. Application on two sands shows the capability of the model to accurately reproduce the experimental results. When saturation degree is higher than the residual saturation degree Sr^r, the evolution of homogenized solute diffusion coefficient with respect to the saturation degree depends significantly on the connectivity of the capillary water. Below Sr^r, depending on the connectivity of the wetting layer, the homogenized solute diffusion coefficient within unsaturated sands decreases by 2–6 orders of magnitude with respect to that in bulk liquid water. The upper bound of the solute diffusion coefficient contributed by the water films is 4–6 orders of magnitude lower than that in bulk liquid water.     

Mesoscale analysis of the suction stress characteristic curve for unsaturated granular materials

There is still no theoretical framework accounting for linkage between seepage and deformation of unsat-urated granular materials.Using a mesoscale liquid bridg...     

A model for compaction and shear hysteresis in saturated granular materials

Fluid-saturated sands exhibit irreversible compaction and shear hysteresis under cyclic shear loads in both free draining and undrained conditions. Constitutive relations of differential-type are constructed heuristically from typical qualitative response. An influence of pore pressure on compaction is incorporated, and the generation of pore pressure under cyclic shearing is investigated. Parameter variations in the shear relations allow a variety of hysteresis loop behaviours to be described.     

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part II – the formulation of elastoplastic coupling at large strain

The two key phenomena occurring in the process of ceramic powder compaction are the progressive gain in cohesion and the increase of elastic stiffness, both related to the development of plastic deformation. The latter effect is an example of ‘elastoplastic coupling’, in which the plastic flow affects the elastic properties of the material, and has been so far considered only within the framework of small strain assumption (mainly to describe elastic degradation in rock-like materials), so that it remains completely unexplored for large strain. Therefore, a new finite strain generalization of elastoplastic coupling theory is given to describe the mechanical behaviour of materials evolving from a granular to a dense state.The correct account of elastoplastic coupling and of the specific characteristics of materials evolving from a loose to a dense state (for instance, nonlinear – or linear – dependence of the elastic part of the deformation on the forming pressure in the granular – or dense – state) makes the use of existing large strain formulations awkward, if even possible. Therefore, first, we have resorted to a very general setting allowing general transformations between work-conjugate stress and strain measures; second, we have introduced the multiplicative decomposition of the deformation gradient and, third, employing isotropy and hyperelasticity of elastic response, we have obtained a relation between the Biot stress and its ‘total’ and ‘plastic’ work-conjugate strain measure. This is a key result, since it allows an immediate achievement of the rate elastoplastic constitutive equations. Knowing the general form of these equations, all the specific laws governing the behaviour of ceramic powders are finally introduced as generalizations of the small strain counterparts given in Part I of this paper.     

A micro-investigation on water bridge effects for unsaturated granular materials with constant water content by discrete element method

The most common state of surface soil is unsaturated. Changes in water content will substantially impact its strength, leading to geological and engineering catastrophes. This paper used LIGGGHTS software to simulate the water bridge effect of unsaturated granular materials with constant water content and verify the rationality of the simplification of the stress-force-fabric (SFF) relationship. The results showed that the capillary force was not isotropic, which was different from the previous study, thus it cannot be overlooked in the simplification of the SFF relationship. Moreover, the influence of water content on the macroscopic mechanical behavior of unsaturated granular materials was interpreted through the evolutions of coordination number, interparticle force, fabric and force anisotropy, and other microscopic parameters. Compared to the literature, we found that different water bridge models would not change the characteristics of the solid skeleton.     

A generalized continuum theory for granular materials

A generalized continuum theory for granular media is formulated by allowing for the possibility of rotation of granules. The basic balance laws are presented and based on thermodynamical consideration a set of constitutive equations are derived. The theory naturally gives rise to the generation of antisymmetric stress tensor and existence of couple stresses. The basic equations of motion are derived and it is shown that the theory contains Mohr-Coulomb criterion of limiting equilibrium as a special case. The problem of coupled porosity and microrotational wave propagation is investigated and the rectilinear shear flow of granular materials is discussed.     

A poroelastic model for the effective behavior of granular materials with interface effect

Various materials encountered in civil engineering are made up of grains interacting with one another along their interfaces. Their effective mechanical properties are strongly affected by these interfaces, where large local strains are expected. The purpose of the present paper is to derive a linear poroelastic model for fluid-saturated granular media with imperfect interfaces. A new approach based on the definition of an equivalent homogeneous phase is also derived. The latter allows a straightforward analysis of the homogenized behavior of the aforementioned materials.     

Analysis of microstructural strain tensors for granular assemblies

The micromechanical interpretation of strain tensor for granular assemblies has been a subject of considerable scientific interest in recent years. This paper gives an overview on 10 different microstructural strain definitions that can be found in the granular mechanics literature. After a theoretical introduction and comparison, the different versions are compared to each other and to the macro-level strain with the help of discrete element simulations.