Micromechanical study of elastic moduli of loose granular materials

In micromechanics of the elastic behaviour of granular materials, the macro-scale continuum elastic moduli are expressed in terms of micro-scale parameters, such as coordination number (the average number of contacts per particle) and interparticle contact stiffnesses in normal and tangential directions. It is well-known that mean-field theory gives inaccurate micromechanical predictions of the elastic moduli, especially for loose systems with low coordination number. Improved predictions of the moduli are obtained here for loose two-dimensional, isotropic assemblies. This is achieved by determining approximate displacement and rotation fields from the force and moment equilibrium conditions for small sub-assemblies of various sizes. It is assumed that the outer particles of these sub-assemblies move according to the mean field. From the particle displacement and rotation fields thus obtained, approximate elastic moduli are determined. The resulting predictions are compared with the true moduli, as determined from the discrete element method simulations for low coordination numbers and for various values of the tangential stiffness (at fixed value of the normal stiffness). Using this approach, accurate predictions of the moduli are obtained, especially when larger sub-assemblies are considered. As a step towards an analytical formulation of the present approach, it is investigated whether it is possible to replace the local contact stiffness matrices by a suitable average stiffness matrix. It is found that this generally leads to a deterioration of the accuracy of the predictions. Many micromechanical studies predict that the macroscopic bulk modulus is hardly influenced by the value of the tangential stiffness. It is shown here from the discrete element method simulations of hydrostatic compression that for loose systems, the bulk modulus strongly depends on the stiffness ratio for small stiffness ratios.     

Micromechanical study of elastic moduli of three-dimensional granular assemblies

In micromechanics of granular materials, relationships are investigated between micro-scale characteristics of particles and contacts and macro-scale continuum characteristics. For three-dimensional isotropic assemblies the macro-scale elastic characteristics are described by the bulk and the shear modulus, which depend on the micro-scale characteristics of the coordination number (i.e. the average number of contacts per particle) and the interparticle contact stiffnesses in directions normal and tangential to the contact.     

Analysis of local behaviour in granular materials

A local scale, called the meso-scale, has recently been introduced to the multi-scale approach for 2D granular materials. This local scale is defined at the level of meso-domains enclosed by particles in contact. Stress and strain have been defined at this local scale, and their relation with the local structure has been studied. The purpose of this paper is to analyse the behaviour of granular materials at the meso-scale, i.e. the stress–strain–structure relationship at this scale. Analyses are performed on a 2D numerical granular sample subjected to a biaxial compression test and simulated with the Discrete Element Method (DEM). The sample is quite dense and it is loaded at a relatively low strain rate so that the state of the sample can be considered as being quasi-static. The size of sub-domains in the sample varies largely from 3 to 12 particles. It is shown that the evolution of the internal state of the sample corresponds, at the meso-scale, to a clear evolution of the quantity of meso-domains oriented in different directions. In addition, the behaviour of meso-domains is highly governed by their orientation rather than their density, especially for the strongly elongated meso-domains: the meso-domains oriented in the compression (resp. extension) direction behave like a dense (resp. loose) granular material.     

On comminution and yield in brittle granular mixtures

The mechanics of granular mixtures are pivotal in many industrial applications. Unravelling the relation between yielding and comminution, the action of mechanically induced grain size reduction, in confined mixture systems is a common and open challenge. This paper attacks this problem by adopting the breakage mechanics theory, which was originally proposed for single mineral materials. We present an extension to the theory that allows predicting: (1) the yielding pressure in granular mixtures, (2) the yield pressure increase/hardening with increasing breakage, and (3) the evolution of the grain size distributions of the separate species—all of these novel capabilities are tested and validated with experiments. Of particular appeal is the finding that the average yielding pressure is a simple generalized mean with an exponent −3/2 of the yielding pressures of the homogeneous components.     

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.     

Exploring the relationship between critical state and particle shape for granular materials

The relationship between critical state and particle shape corresponds to the most fundamental aspect of the mechanics of granular materials. This paper presents an investigation into this relationship through macro-scale and micro-scale laboratory experiments in conjunction with interpretation and analysis in the framework of critical state soil mechanics. Spherical glass beads and crushed angular glass beads of different percentages were mixed with a uniform quartz sand (Fujian sand) to create a sequence of mixtures with varying particle shape. On the micro-scale, particle shape was accurately measured using a laser scanning technique, and was characterized by aspect ratio, sphericity and convexity; a new shape index, taken as the average of the three shape measures and referred to as overall regularity, was proposed to provide a collective characterization of particle shape. On the macro-scale, both undrained and drained triaxial tests were carried out to provide evidence that varying particle shape can alter the overall response as well as the critical states in both stress space and volumetric compression space. The mixtures of Fujian sand and spherical glass beads were found to be markedly more susceptible to liquefaction than the mixtures of Fujian sand and crushed angular glass beads. The change in liquefaction susceptibility was shown to be consistent with the change in the position of the critical state locus (CSL) in the compression space, manifested by a decrease in the intercept and gradient of the CSL due to the presence of spherical glass beads. Quantitative relationships have been established between each of the critical state parameters and each of the shape parameters, thereby providing a way to construct macro-scale constitutive models with intrinsic micro-scale properties built in.     

Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers

A micromechanically based constitutive model for the elasto-viscoplastic deformation and texture evolution of semi-crystalline polymers is developed. The model idealizes the microstructure to consist of an aggregate of two-phase layered composite inclusions. A new framework for the composite inclusion model is formulated to facilitate the use of finite deformation elasto-viscoplastic constitutive models for each constituent phase. The crystalline lamellae are modeled as anisotropic elastic with plastic flow occurring via crystallographic slip. The amorphous phase is modeled as isotropic elastic with plastic flow being a rate-dependent process with strain hardening resulting from molecular orientation. The volume-averaged deformation and stress within the inclusions are related to the macroscopic fields by a hybrid interaction model. The uniaxial compression of initially isotropic high density polyethylene (HDPE) is taken as a case study. The ability of the model to capture the elasto-plastic stress-strain behavior of HDPE during monotonic and cyclic loading, the evolution of anisotropy, and the effect of crystallinity on initial modulus, yield stress, post-yield behavior and unloading-reloading cycles are presented.     

Nonlinear elasto-plastic model for dense granular flow

This work proposes a model for granular deformation that predicts the stress and velocity profiles in well-developed dense granular flows. Recent models for granular elasticity [Jiang, Y., Liu, M., 2003. Granular elasticity without the Coulomb condition. Phys. Rev. Lett. 91, 144301] and rate-sensitive fluid-like flow [Jop, P., Forterre, Y., Pouliquen, O., 2006. A constitutive law for dense granular flows. Nature 441, 727] are reformulated and combined into one universal elasto-plastic law, capable of predicting flowing regions and stagnant zones simultaneously in any arbitrary 3D flow geometry. The unification is performed by justifying and implementing a Kröner–Lee decomposition, with care taken to ensure certain continuum physical principles are necessarily upheld. The model is then numerically implemented in multiple geometries and results are compared to experiments and discrete simulations.     

Boundary-layer corrections for stress and strain fields in randomly heterogeneous materials

Boundary-layer effects on the effective response of fibre-reinforced media are analysed. The distribution of the fibres is assumed random. A methodology is presented for obtaining non-local effective constitutive operators in the vicinity of a boundary. These relate ensemble averaged stress to ensemble averaged strain. Operators are also developed which re-construct the local fields from their ensemble averages. These require information on the local configuration of the medium. Complete information is likely not to be available, but averages of these operators conditional upon any given local information generate corresponding conditional averages of the fields. Explicit implementation is performed within the framework of an approximation of Hashin-Shtrikman type. Two types of geometry are considered in examples: a half-space and a crack in an infinite heterogeneous medium. These are representative, asymptotically, of the field in the vicinity of any smooth boundary, and in the vicinity of a crack tip, respectively. Results have been obtained for the case of anti-plane deformation, realized by the imposition of either Dirichlet or Neumann conditions on the boundary; those for the Neumann condition are presented and discussed explicitly. The stresses in both fibre and matrix adjacent to a crack tip are shown to differ substantially from the values that would be predicted by ordinary homogenization.     

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.     

On the effective stress in unsaturated porous continua with double porosity

Using mixture theory we formulate the balance laws for unsaturated porous media composed of a double-porosity solid matrix infiltrated by liquid and gas. In this context, the term ‘double porosity’ pertains to the microstructural characteristic that allows the pore spaces in a continuum to be classified into two pore subspaces. We use the first law of thermodynamics to identify energy-conjugate variables and derive an expression for the ‘effective’, or constitutive, stress that is energy-conjugate to the rate of deformation of the solid matrix. The effective stress has the form , where σ is the total Cauchy stress tensor, B is the Biot coefficient, and is the mean fluid pressure weighted according to the local degrees of saturation and pore fractions. We identify other emerging energy-conjugate pairs relevant for constitutive modeling of double-porosity unsaturated continua, including the local suction versus degree of saturation pair and the pore volume fraction versus weighted pore pressure difference pair. Finally, we use the second law of thermodynamics to determine conditions for maximum plastic dissipation in the regime of inelastic deformation for the unsaturated two-porosity mixture.     

Micromechanical analysis of kinematic hardening in natural clay

This paper presents a micromechanical analysis of the macroscopic behaviour of natural clay. A microstructural stress–strain model for clayey material has been developed which considers clay as a collection of clusters. The deformation of a representative volume of the material is generated by mobilizing and compressing all the clusters along their contact planes. Numerical simulations of multistage drained triaxial stress paths on Otaniemi clay have been performed and compared the numerical results to the experimental ones in order to validate the modelling approach. Then, the numerical results obtained at the microscopic level were analysed in order to explain the induced anisotropy observed in the clay behaviour at the macroscopic level. The evolution of the state variables at each contact plane during loading can explain the changes in shape and position in the stress space of the yield surface at the macroscopic level, as well as the rotation of the axes of anisotropy of the material.     

A strong contrast homogenization formulation for multi-phase anisotropic materials

Homogenization relations, linking a material's properties at the mesoscale to those at the macroscale, are fundamental tools for design and analysis of microstructure. Recent advances in this field have successfully applied spectral techniques to Kroner-type perturbation expansions for polycrystalline and composite materials to provide efficient inverse relations for materials design. These expansions have been termed ‘weak-contrast’ expansions due to the conditionally convergent integrals, and the reliance upon only small perturbations from the reference property. In 1955, Brown suggested a different expansion for electrical conductivity that resulted in absolutely convergent integrals. Torquato subsequently applied the method to elasticity, with good results even for high-contrast materials; thus it is commonly referred to as a ‘strong contrast’ expansion. The methodology has been applied to elasticity for two phases of isotropic material, generally assuming macroscopic isotropy (with noted exceptions), thus resulting in a rather elegant form of the solution.

More recently, a multi-phase form of the solution was developed for conductivity. This paper builds upon this result to apply the method to elasticity of polycrystalline materials with both local and global anisotropy. New spectral formulations are subsequently developed for both the weak and strong contrast solutions. These form the basis for efficient microstructure analysis using these frameworks, and subsequently for inverse design applications. The process is taken through to demonstration of a property closure, which acts as the basis for materials design; the closure delineates the envelope of all physically realizable property combinations for the chosen properties, based on the particular homogenization relation being used.     

Numerical solution to the shearing flow of granular materials between two plates

We study the shearing flow of granular materials between two horizontal flat plates where the top plate is moving with a constant speed. The constitutive relation used for the stress is based on the continuum model proposed by Rajagopal and Massoudi (DOE Report, DOE/PETC/TR-90/3, 1990). The material coefficients such as viscosity and normal stress coefficients are based on the model of Boyle and Massoudi (Int. J. Eng. Sci 28 (1990) 1261). The governing equations are non-dimensionalized and the resulting system of non-linear differential equations is solved numerically using finite difference technique.     

SPH-based simulation of granular collapse on an inclined bed

The paper presents a numerical model for simulating a granular flow and its deposition on an inclined bed. A granular material is described as an elastic–plastic continuum and its constitutive law, namely Hooke's law, is discretized on the basis of the Smoothed Particle Hydrodynamics (SPH) method. In the equation of motion, however, the artificial viscosity, which is widely used in SPH, is not applied. The diffusive term derived from Hooke's law is introduced with a diffusion coefficient that varies depending on the stress and strain rate based on the Drucker–Prager yield function. The model is verified and validated through two numerical tests. It is shown that the basic elastic–perfectly plastic characteristics are reproduced with a simple shearing test. The effects of the diffusion coefficient and spatial resolution are investigated to show the validity of the model. In the simulation of the gravitational collapse of a granular column on an inclined bed, the performance of the model from the final deposition profile, the time history of the front position of the granular flow, the maximum runout distance, and the velocity profile are investigated for several cases of basal inclinations. The calculated results show good agreement with the experimental results.     

The role of a realistic volume constraint in modelling a two dimensional granular assembly

This paper presents a deceptively simple mathematical model for the deformation of granular materials composed of rigid particles. The model captures many of the diverse features of the behaviour of such a material and emphasises the importance of volume constraints in situations where the deformation is mainly by particle rearrangement. It is constructed using a simple dissipation function and a rather more complicated dilatancy rule containing an updateable reference strain. This allows the solid-like and fluid-like properties of granular materials to be reconciled in a single model.The model has been used to simulate experiments that use an analogue of an ideal granular material [Joer, H.A., Lanier, J., Fahey, M., 1998. Deformation of granular materials due to rotation of principal axes. Geotechnique 48 (5), 605-619] consisting of a two dimensional assembly of thin PVC rods. These experiments clearly illustrate: partially reversible dilatancy in direct shear tests; cyclic shearing leading to liquefaction in constant volume shear tests; and non-coaxiality of the principal axes of stress and strain increment in circular loading tests. These radically different modes of deformation provide a challenging data set that allows the model's potential to be clearly demonstrated.The authors believe that the comparison of these experimental results and our simulations give strong support to the assertion that volume changes associated with shear deformation are responsible for the rotational kinematic hardening seen in granular materials, and hence, the non-coaxiality of the stress and strain-rate tensors.     

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.     

Micromechanics-based thermoelastic model for functionally graded particulate materials with particle interactions

Thermoelastic behavior of functionally graded particulate materials is investigated with a micromechanical approach. Based on a special representative volume element constructed to represent the graded microstructure of a macroscopic material point, the relation between the averaged strains of the particle and matrix phases is derived with pair-wise particle interactions, and a set of governing equations for the thermoelastic behavior of functionally graded materials is presented. The effective coefficient of thermal expansion at a material point is solved through the overall averaged strain of two phases induced by temperature change under the stress-free condition, and is shown to exhibit a weak anisotropy due to the particle interactions within the graded microstructure. When the material gradient is eliminated, the proposed model predicts the effective coefficient of thermal expansion for uniform composites as expected. If the particle interactions are disregarded, the proposed model recovers the Kerner model. The proposed semi-analytical scheme is consistent and general, and can handle any thermal loading variation. As examples, the thermal stress distributions of graded thermal barrier coatings are solved for two types of thermal loading: uniform temperature change and steady-state heat conduction in the gradation direction.     

Statistics of particle interactions in dense granular material under uniaxial compression

Stress evolution in a dense granular material is closely related to interactions of contacting particles. We investigate statistics related to particle interactions and the relationship between the averaged local relative motion and the macroscopic motion. The validity of the Voigt and Reuss assumptions is examined, and extensions to these assumptions are proposed. Effects of history in the dense granular material are investigated. Statistical samples used in this paper are obtained using three-dimensional numerical simulations of dense granular media under uniaxial cyclical compression. The results show that stresses arise mostly from normal forces between particles, and direct contributions from frictional tangential forces between particles are small. Tangential friction, however, significantly increases the particle contact time, and thus reduces the rate of contact breakage. The contact breakage rate is demonstrated to be a stress relaxation rate. Therefore, stress increases significantly with friction between particles as a result of prolonged relaxation time.