Micromechanical definition of an entropy for quasi-static deformation of granular materials

A micromechanical theory is formulated for quasi-static deformation of granular materials, which is based on information theory. A reasoning is presented that leads to the definition of an information entropy that is appropriate for quasi-static deformation of granular materials. This definition is based on the hypothesis that relative displacements at contacts with similar orientations are independent realisations of a random variable. This hypothesis is made plausible based on the results of Discrete Element simulations. The developed theory is then used to predict the elastic behaviour of granular materials in terms of micromechanical quantities. The case considered is that of two-dimensional assemblies consisting of non-rotating particles with an elastic contact constitutive relation. Applications of this case are the initial elastic (small-strain) deformation of granular materials. Theoretical results for the elastic moduli, relative displacements, energy distribution and probability density functions are compared with results obtained from the Discrete Element simulations for isotropic assemblies with various average numbers of contacts per particle and various ratios of tangential to normal contact stiffness. This comparison shows that the developed information theory is valid for loose systems, while a theory based on the uniform-strain assumption is appropriate for dense systems.     

Micro–macro analysis of granular material behavior along proportional strain paths

When granular materials are subjected to proportional strain loading paths, they manifest a variety of behaviors depending on the initial void ratio of the specimen as well as the imposed dilatancy/contractancy rate. In some cases, the stress components may vanish over the duration of the test, and the specimen may progressively liquefy. To investigate this behavior, the authors have developed a kinematic approach to be deployed in two parts. First, numerical simulations are performed by means of a discrete element method. Secondly, two micromechanical models have corroborated the DEM results. The performance of these models may explain a number of microstructural mechanisms responsible for the macroscopic constitutive behavior.     

Particle shape consideration in numerical simulation of assemblies of irregularly shaped particles

The mechanical behavior of granular materials depends much on the shape of the constituent particles. Therefore appropriate modeling of particle, or grain, shape is quite important. This study employed the method of direct modeling of grain shape (Matsushima & Saomto, 2002), in which, the real shape of a grain is modeled by combining arbitrary number of overlapping circular elements which are connected to each other in a rigid way. Then, accordingly, a discrete-element program is used to simulate the assembly of grains. In order to measure the effects of grain shape on mechanical properties of assembly of grains, three types of grains—high angular grains, medium angular grains and round grains are considered where several biaxial tests are conducted on assemblies with different grain types. The results show that the angularity of grains greatly affects the behavior of granular soil.     

Analysis of three-dimensional micro-mechanical strain formulations for granular materials: Evaluation of accuracy

An important objective of recent research on micro-mechanics of granular materials is to develop macroscopic constitutive relations in terms of micro-mechanical quantities at inter-particle contacts. Although the micro-mechanical formulation of the stress tensor is well established, the corresponding formulation for the strain tensor has proven to be much more evasive, still being the subject of much discussion. In this paper, we study various micro-mechanical strain formulations for three-dimensional granular assemblies, following the work of Bagi in two dimensions (Bagi, 2006). All of these formulations are either based on an equivalent continuum approach, or follow the best-fit approach. Their accuracy is evaluated by comparing their results, using data from Discrete Element Method simulations on periodic assemblies, to the macroscopic deformation. It is found that Bagi’s formulation (Bagi, 1996), which is based on the Delaunay tessellation of space, is the most accurate. Furthermore, the best-fit formulation based on particle displacements only did unexpectedly well, in contrast to previously reported results for two-dimensional assemblies.     

Micromechanical modelling of the static and cyclic loading of an Al 2124-SiC MMC

The objective of this study was to use micromechanical finite element models to simulate both the static and cyclic mechanical behaviour of a metal matrix composite: a forged Al 2124 alloy reinforced with 17% SiC particles, at two different temperatures: room temperature and 150°C. In the simulations, periodic unit cell models incorporating the explicit representation of the matrix and the reinforcing particles in both 2D and 3D, were used. Micromechanical models with both idealised and realistic reinforcing particle shapes and distributions were generated. The realistic particle shapes and distributions were inferred from experimental SEM micrographs. The pattern and intensity of the plastic deformation within the matrix was studied and the macroscale behaviour of the composite was inferred from average stress and strain values. In order to include the effects of residual stresses due to the processing of the material, a quenching simulation was performed, prior to the mechanical loading, and its effects on the macroscopic tensile behaviour of the MMC was assessed. The effects of removing the periodicity constraint on the models by using a cell embedding technique was investigated. In order to try and model the deformation behaviour of the matrix more accurately, crystal plasticity models, which included the explicit representation of individual grains were examined for different matrix grain morphologies. The results of the simulations were compared with experimental results for the MMC in terms of macroscopic tensile stress–strain curves. Finally, the effects of different matrix strain hardening models were examined in order to investigate the cyclic behaviour of the MMC.     

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.     

Discrete element modeling of inherently anisotropic granular assemblies with polygonal particles

In the present article, we study the effect of inherent anisotropy, i.e., initial bedding angle of particles and associated voids on macroscopic mechanical behavior of granular materials, by numerical simulation of several biaxial compression tests using the discrete element method (DEM). Particle shape is considered to be irregular convex-polygonal. The effect of inherent anisotropy is investigated by following the evolution of mobilized shear strength and volume change during loading. As experimental tests have already shown, numerical simulations also indicate that initial anisotropic condition has a great influence on the strength and deformational behavior of granular assemblies. Comparison of simulations with tests using oval particles, shows that angularity influences both the mobilized shear strength and the volume change regime, which originates from the interlocking resistance between particles.     

Global–local analysis of granular media in quasi-static equilibrium

A method of global–local analysis is developed for quasi-static equilibrium problems for granular media. The two-scale modeling based on mathematical homogenization theory enables us to formulate two separate boundary value problems in terms of macro- and microscales. The macroscale problem governs the equilibrium of a global structure composed of granular assemblies, while the microscale one is posed for the particulate nature of a local structure with the friction-contact mechanism between particles. The local structure is identified with a periodic representative volume element, or equivalently, a unit cell, over which averaging is performed. The mechanical behavior of unit cells is analyzed by a discrete numerical model, in which spring and friction devices connect rigid particles, whereas the continuum-based finite element method is used for the macroscopic one. Representative numerical examples are presented to demonstrate the capability of the proposed two-scale analysis method for granular materials.     

Effect of particle size distribution on micro- and macromechanical response of granular packings under compression

The role of particle size heterogeneity on micro- and macromechanical properties of assemblies of spherical particles was studied using DEM simulations. The response to an imposed load of a granular material composed of non-uniformly sized spheres subjected to uniaxial confined compression was investigated. A range of geometrical and micro-mechanical properties of granular packings (e.g., void fraction, contact force distribution, average coordination number and degree of mobilisation of friction at contacts between particles) were examined, and provided a more accurate interpretation of the macroscopic behaviour of mixtures than has previously been available. The macromechanical study included stress transmission, stiffness and angle of internal friction of the granular assemblies.The degree of polydispersity showed slight effect on both, the void fraction and the elastic properties of the system. The tendency for increase in the lateral-to-vertical pressure ratios was observed with an increasing degree of particle size heterogeneity; however, the different pressure ratios calculated for samples with various degrees of polydispersity lay within the range of data scatter.     

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.     

基于微形态模型的颗粒材料中波的频散现象研究

基于颗粒材料冲击与波动响应特性的调控波传播行为的超材料设计受到广泛关注,设计这类材料需要对颗粒材料的波传播机制及调控机理有深入认识. 波在颗粒材料中传播的频散现象及频率带隙等行为与材料的非均匀性密切相关,通常讨论频散现象是基于弹性理论框架建立微结构连续体或高阶梯度连续体等广义连续体模型来进行. 本研究基于细观力学给出了一个颗粒材料的微形态连续体模型. 在该模型中,考虑了颗粒的平动和转动,且颗粒间的相对运动分解为两部分:即宏观平均运动和细观真实运动. 基于此分解,提出了一个完备的变形模式,得到了对应于不同应变及颗粒间运动的宏细观本构关系. 结合宏观变形能的细观变形能求和表达式,获得了基于细观量表示的宏观本构模量. 应用所建议模型考察了波在弹性颗粒介质的传播行为,给出了不同形式的波的频散曲线,结果显示此模型具有预测频率带隙的能力.      

Quantification of large and localized deformation in granular materials

Quantifying large deformation in granular assemblies using concepts originating from continuum mechanics is a challenging task because of (1) the discontinuous nature of granular displacement, which does not allow the definition of a continuum measure of deformation, and (2) the almost inevitable shear band localization. These problems exist in both real-world granular materials and their numerical idealizations using particle-based simulations. In this work a new method is developed in order to address these issues. Instead of creating a meshed equivalent continuum for quantifying small engineering strains, the new method performs independent random queries on the velocity gradient characteristics of arbitrary sub-domains in the assembly through the novel concept of overlapping reference triangles, thus, enabling rigorous handling of large deformations which are usually associated with localization. The proposed method is illustrated and validated by discrete element method (DEM) simulation of a biaxial compressive test, in which apparent shear banding takes place. The homogenized deformation quantifications based on the new method match the estimations from the imposed boundary conditions. The numerical examples are also applied to (1) quantifying the heterogeneous distribution of deformation over the specimen, (2) visualizing the nucleation process of shear bands, and (3) characterizing shear flow patterns in shear bands. An investigation on the effects of the reference triangle sizes yields some inspiring and practically significant results.     

Three-dimensional micromechanical modeling of voided polymeric materials

A three-dimensional micromechanical unit cell model for particle-filled materials is presented. The cell model is based on a Voronoi tessellation of particles arranged on a body-centered cubic (BCC) array. The three-dimensionality of the present cell model enables the study of several deformation modes, including uniaxial, plane strain and simple shear deformations, as well as arbitrary principal stress states.The unit cell model is applied to studies on the micromechanical and macromechanical behavior of rubber-toughened polycarbonate. Different load cases are examined, including plane strain deformation, simple shear deformation and principal stress states. For a constant macroscopic strain rate, the different load cases show that the macroscopic flow strength of the blend decreases with an increase in void volume fraction, as expected. The main mechanism for plastic deformation is broad shear banding across inter-particle ligaments. The distributed nature of plastic straining acts to reduce the amount of macroscopic strain softening in the blend as the initial void volume fraction is increased. In the case of plane strain deformation, the plastic flow is observed to initiate across inter-particle ligaments in the direction of constraint. This particular mode of deformation could not have been captured using a two-dimensional, plane strain idealization of cylindrical voids in a matrix.The potential for localized crazing and/or cavitation in the matrix is addressed. It is observed that the introduction of voids acts to relieve hydrostatic stress in the matrix material, compared to the homopolymer. It is also seen that the predicted peak hydrostatic stress in the matrix is higher under plane strain deformation than under triaxial tension (with equal lateral stresses), for the same macroscopic stress triaxiality.The effect of void volume fraction on the macroscopic uniaxial tension behavior of the different blends is examined using a Considère construction for dilatant materials. The natural draw ratio was predicted to decrease with an increase in void volume fraction.     

Micromechanical study of plasticity of granular materials

Plastic deformation of granular materials is investigated from the micromechanical viewpoint, in which the assembly of particles and interparticle contacts is considered as a mechanical structure. This is done in three ways. Firstly, by investigating the degree of redundancy of the system by comparing the number of force degrees of freedom at contacts with the number of governing equilibrium equations; Secondly, by determining the spectrum of eigenvalues of the stiffness matrix for the structure that is represented by the particles and their contacts; Thirdly, by investigating the evolution with imposed strain of the continuum elastic stiffness tensor of the system. It is found that, with increasing imposed strain, the degree of redundancy rapidly evolves towards a state with small redundancy, i.e. the system becomes nearly statically determinate. The spectrum of the system shows many singular and near-singular modes at peak shear strength and at large strain. The continuum elastic stiffness tensor becomes strongly anisotropic with increasing imposed strain and shows strong non-affinity of deformation. The assumption of a constant and isotropic elastic stiffness tensor in elasto-plastic continuum constitutive relations for granular materials is generally incorrect. Overall, the plastic continuum behaviour of granular materials originates from the plastic frictional behaviour at contacts and from damage in the form of changes in the contact network.     

Effects of lattice misorientations on strain heterogeneities in FCC polycrystals

It is well documented that the highly heterogeneous deformation behaviour and lattice rotation typically observed within grains in a polycrystal are attributed to microstructural features such as grain structure, topology, size, etc. In this work, the effects of low- and high-angle grain boundaries on the mechanical behaviour of FCC polycrystals are investigated using a micro-mechanical model based on crystal plasticity theory. The constitutive framework relies on dislocation mechanics concepts to describe the plastic deformation behaviour of FCC metallic crystals and is validated by comparing the measured and predicted local and macroscopic deformation behaviour in a thin Al-0.5% Mg polycrystal tensile specimen containing a relatively small number of surface grains. Comparisons at the microscopic (e.g. local slip distribution) and macroscopic (e.g. average stress-strain response) levels elucidate the role of low-angle grain boundaries, which are found to have a profound effect on both the local and average deformation behaviour of FCC polycrystals with a small number of grains. However, this effect diminishes when the number of grains increases and becomes negligible in bulk polycrystals. In light of the widely accepted view that high-angle grain boundaries strongly influence the mechanical behaviour of very fine-grained metals, this work has shown that low-angle grain boundaries can also play an equally important role in the deformation behaviour of polycrystals with a relatively small number of grains.     

The study of relations between loading history and yield surfaces in powder materials using discrete finite element simulations

Only a few studies in the literature have applied the finite-element method to analyse assemblies of meshed particles. These studies illustrated the relevance of this method for granular materials. Here, the compaction of ductile metal powders was studied through a numerical assembly of elastic–plastic and rate-independent spherical particles. The aim of this paper was to understand the evolution of yield surfaces with complex loading paths up to high relative density at the macroscopic scale and at the granular scale. Simulation results revealed that yield surfaces evolved with both isotropic and kinematic hardening mechanisms, depending on the compaction stage. An analysis of the sample microstructure was proposed, and a detailed study of contacts between particles revealed some of the mechanisms that led to the observed evolution of yield surfaces.     

Numerical and experimental direct shear tests for coarse-grained soils

The presence of particles larger than the permissible dimensions of conventional laboratory specimens causes difficulty in the determination of shear strength of coarse-grained soils. In this research, the influence of particle size on shear strength of coarse-grained soils was investigated by resorting to experimental tests in different scale and numerical simulations based on discrete element method (DEM). Experimental tests on such soil specimens were based on using the techniques designated as "parallel" and "scalping" to prepare gradation of samples in view of the limitation of laboratory specimen size. As a second approach, the direct shear test was numerically simulated on assemblies of elliptical particles. The behaviors of samples under experimental and numerical tests are presented and compared, indicating that the modification of sample gradation has a significant influence on the mechanical properties of coarse-grained soils. It is noted that the shear strengths of samples produced by the scalping method are higher than samples by the parallel method. The scalping method for preparing specimens for direct shear test is therefore recommended. The micromechanical behavior of assemblies under direct shear test is also discussed and the effects of stress level on sample behavior are investigated.     

Effect of particle polydispersity on micromechanical properties and energy dissipation in granular mixtures

A series of numerical tests was conducted to study the micromechanical properties and energy dissipation in polydisperse assemblies of spherical particles subjected to uniaxial compression. In general, distributed particle size assemblies with standard deviations ranging from 0% to 80% of the particle mean diameter were examined. The microscale analyses included the trace of the fabric tensor, magnitude and orien- tation of the contact forces, trace of stress, number of contacts and degree of mobilization of friction in contacts between particles. In polydisperse samples, the average coordination numbers were lower than in monodisperse assemblies, and the mobilization of friction was higher than in monodisperse assemblies due to the non-uniform spatial rearrangement of spheres in the samples and the smaller displacements of the particles. The effect of particle size heterogeneity on both the energy density and energy dissipation in systems was also investigated.