A simulation based on the cosserat continuum model of the vortex structure in granular materials

Displacement fluctuation is the difference between the real displacement and the affine displacement in deforming granular materials. The discrete element method(DEM) is widely used along with experimental approaches to investigate whether the displacement fluctuation represents the vortex structure. Current research suggests that the vortex structure is caused by the cooperative motion of particle groups on meso-scales, which results in strain localization in granular materials. In this brief article, we investigate the vortex structure using the finite element method(FEM) based on the Cosserat continuum model. The numerical example focuses on the relationship between the vortex structure and the shear bands under two conditions:(a) uniform granular materials;(b) granular materials with inclusions. When compared with distributions of the effective strain and the vortex structure, we find that the vortex structure coexists with the strain localization and originates from the stiffness cooperation of different locations in granular materials at the macro level.     

Modeling shear localization along granular soil–structure interfaces using elasto-plastic Cosserat continuum

The current study presents finite element simulations of shear localization along the interface between cohesionless granular soil and bounding structure under large shearing movement. Micro-polar (Cosserat) continuum approach is applied in the framework of elasto-plasticity in order to overcome the numerical problems of localization modeling seen in the conventional continuum mechanics. The effects of different micro-polar kinematic boundary conditions, along the interface, on the evolution and location of shear band are shown by the numerical results. Furthermore, shear band thickness is also investigated for its dependence on the initial void ratio, vertical pressure and mean grain size. Here, the distribution and evolution of static and kinematic quantities are the main focuses regarding infinite layer of micro-polar material during plane shearing, especially with advanced large movement of bounding structure. The influence of such movement has not been investigated yet in the literature. Based on the results obtained from this study, shear localization appears parallel to the direction of shearing. It occurs either in the middle of granular layer or near boundaries, regarding the assumed micro-polar kinematic boundary conditions at the bottom and top surfaces of granular soil layer. Narrower shear band is observed in lower rotation resistance of soil particles along the interface. It is emphasized that the displacement magnitude of bounding structure has significant effect on the distribution and evolution of state variables and polar quantities in the granular soil layer. However, continuous displacement has no meaningful effect on the thickness of shear band. Here, smooth distributions of void ratio and shear stress components are obtained within the shear band, what the other previous numerical investigations did not receive. Despite indirect linking of Lade’s model to the critical state soil mechanics, state variables tend towards asymptotical stationary condition in large shear deformation.     

Granular vortices: Identification,characterization and conditions for the localization of deformation

We relate the micromechanics of vortex evolution to that of force chain buckling and, on this basis, formulate the conditions for strain localization in a continuum model of dense granular media. Using the traditional bifurcation analysis of shear bands, we show that kinematic vortex fields are in fact solutions to the boundary value problem satisfying null boundary conditions. To establish an empirical basis for our study, we first develop a method to identify the location of the core and boundary of each vortex from a given displacement field in two dimensions. We then employ this method to characterize the residual deformation field (i.e., the deviation of particle motions from the continuum deformation) in a physical experiment and a discrete element simulation of dense granular samples submitted to biaxial compression. Vortices in the failure regime are essentially confined to the shear band. Primary vortices, the clear majority, rotate in the same direction as the shear band; secondary vortices, the so-called wakes, rotate in the opposite direction. Primary vortices align in spatial succession along the central axis of the band; wakes form next to the band boundaries, in between and beside two adjacent primary vortices. Force chain buckling, the governing mechanism for shear bands, is responsible for vortex formation in the failure regime. Vortex dynamics are consistent with stick-slip dynamics. From quiescent conditions of jamming or stick, vortical motions arise from force chain buckling and associated relative particle rotations and sliding; these in turn precipitate intermittent periods of unjamming or slip, evident in the attendant drops in stress ratio and bursts in both kinetic energy and local nonaffine deformation. A kinematic vortex field inside shear bands is proposed that is consistent with the equations of continuum mechanics and the underlying instability of force chain buckling: such a field is periodic with a repeating unit cell comprising a primary vortex at the center of the band, with two trailing wakes close next to the band boundaries.     

Evolving internal length scales in plastic strain localization for granular materials

A numerical model in the Cosserat continuum for strain localization phenomena in granular materials is developed and proposed in this paper. The model assumes a constant internal length scale that is used to describe the shear band thickness. However, it is observed that the internal length scales need to change to accommodate the possible change in the contact surface between the particles, damage of the particles or/and any change in the local void ratio within the domain, which will change the shear band thickness. The mathematical formulations used in the present numerical model were equipped with evolution equations for the length scales through the Micropolar theory, those formulations are proposed and discussed in this paper. The evolution equations of the internal length scales describe any possible change in the contact surface between the particles, damage of the particles if exists and/or any change in the local void ratio within the domain. Hence, the strain localization described by the enhanced model with evolving internal length scales is more accurate and closer to the real solution. The solution for the shear bands thickness shows more accurate correlation with the experimental results and less dependency on the mesh size when such evolution equations are used. Moreover, the shear band thickness and inclination evolve during the deformation process.     

The formation of multiple adiabatic shear bands

In a previous paper, Zhou et al. [2006. A numerical methodology for investigating adiabatic shear band formation. J. Mech. Phys. Solids, 54, 904-926] developed a numerical method for analyzing one-dimensional deformation of thermoviscoplastic materials. The method uses a second order algorithm for integration along characteristic lines, and computes the plastic flow after complete localization with high resolution and efficiency. We apply this numerical scheme to analyze localization in a thermoviscoplastic material where multiple shear bands are allowed to form at random locations in a large specimen. As a shear band develops, it unloads neighboring regions and interacts with other bands. Beginning with a random distribution of imperfections, which might be imagined as arising qualitatively from the microstructure, we obtain the average spacing of shear bands through calculations and compare our results with previously existing theoretical estimates. It is found that the spacing between nucleating shear bands follows the perturbation theory due to Wright and Ockendon [1996. A scaling law for the effect of inertia on the formation of adiabatic shear bands. Int. J. Plasticity 12, 927-934], whereas the spacing between mature shear bands is closer to that predicted by the momentum diffusion theory of Grady and Kipp [1987. The growth of unstable thermoplastic shear with application to steady-wave shock compression in solids. J. Mech. Phys. Solids 35, 95-119]. Scaling laws for the dependence of band spacing on material parameters differ in many respects from either theory.     

金属材料平面应变拉伸变形局部化有限元分析

本文对于涉及韧性金属大变形中颈缩与剪切带断裂一类高度非线性变形局部化问题进行了弹塑性有限元数值模拟。采用改进的Ｊ２形变理论微分形式公式与交叉三角形四边形单元有限元网格，详细研究了应变硬化指数及初始表面不均匀特性的平面应变拉伸颈缩和剪切带形成的综合影响，给出此类问题的断裂机制图。     

粉末烧结钨合金材料的绝热剪切研究

在分离式霍普金森压杆装置上对斜圆柱粉末烧结钨合金试件进行了冲击实验,由于斜圆柱结构中剪切分量在冲击压缩中的持续作用,实验中观察到了宏观破断现象。利用光学显微镜和扫描电子显微镜分别对试件断面和试件的纵截面进行了分析,观察到了贯通钨颗粒的绝热剪切带这一变形局部化现象。对粉末烧结钨合金绝热剪切破坏机制进行了分析。     

Strain localization analysis using a large deformation anisotropic elastic–plastic model coupled with damage

Sheet metal forming processes generally involve large deformations together with complex loading sequences. In order to improve numerical simulation predictions of sheet part forming, physically-based constitutive models are often required. The main objective of this paper is to analyze the strain localization phenomenon during the plastic deformation of sheet metals in the context of such advanced constitutive models. Most often, an accurate prediction of localization requires damage to be considered in the finite element simulation. For this purpose, an advanced, anisotropic elastic–plastic model, formulated within the large strain framework and taking strain-path changes into account, has been coupled with an isotropic damage model. This coupling is carried out within the framework of continuum damage mechanics. In order to detect the strain localization during sheet metal forming, Rice’s localization criterion has been considered, thus predicting the limit strains at the occurrence of shear bands as well as their orientation. The coupled elastic–plastic-damage model has been implemented in Abaqus/implicit. The application of the model to the prediction of Forming Limit Diagrams (FLDs) provided results that are consistent with the literature and emphasized the impact of the hardening model on the strain-path dependency of the FLD. The fully three-dimensional formulation adopted in the numerical development allowed for some new results – e.g. the out-of-plane orientation of the normal to the localization band, as well as more realistic values for its in-plane orientation.     

A numerical methodology for investigating the formation of adiabatic shear bands

In this paper, we present a numerical approach for analyzing thermo-visco-plastic deformation in one dimension. The method, which is accurate to second order, is based on integration along the characteristic lines. It is able to simulate fully localized plastic flow with high resolution and good efficiency. We apply this numerical scheme to the analysis of shear localization, emphasizing the interactions between a single shear band and its surroundings and among the members of a periodic array of shear bands. It is found that a shear band may grow intermittently due to interactions with other bands. The developed method is specifically adequate for analyzing the self-organized multiple adiabatic formation process, which will be discussed in the follow-up paper.     

Localization Phenomena in Liquid-Saturatedand Empty Porous Solids

Localization phenomena occur as a result of local concentrations of plastic deformations in small bands of finite width (shear bands). Porous materials, as, for instance, soil, rock, concrete and sinter materials as well as polymeric and metallic foams exhibit a strong tendency towards shear banding caused by plastic dilatation in the brittle deformation range. This kind of behaviour is of great practical importance in engineering design, for example in the study and computation of failure mechanisms in soil mechanics (base failure, slope failure, etc.). From the mathematical point of view, the computation of localization phenomena, for example within the framework of the finite element method (FEM), yields an ill-posed problem, since each mesh refinement leads to smaller shear bands until one obtains (ideally) a singular surface. Following this, regularization mechanisms should be introduced to obtain reliable and robust results.In the present article, two natural regularization mechanisms for liquid-saturated and empty granular porous materials are discussed. These mechanisms are (1) the inclusion of additional independent degrees of freedom in the sense of the Cosserat brothers for the granular porous solid and (2) the inclusion of pore-fluid viscosity in the saturated case.     

Numerical simulation for the simple shear test in amorphous glassy polymers

In this paper, a driving stress finite element method of elastic-plastic large deformation based on implicit time integrating algorithm and an eight-chain molecular network model is used for the numerical simulation of the simple shear test of polycarbonate (PC) materials. The simulated results are compared with experimental ones. The strain localization propagation for the shear band deformation for simple shear deformation is investigated numerically. The effects of microstructure parameters in the model on strain softening and orientation hardening of the PC are discussed in detail. Supported by the National Natural Science Foundation of China.     

Initiation of localized shear bands in plane strain

This study is concerned with the initiation of localized shear bands in plane-strain tension and compression. A theoretical framework which views the initiation of such shear bands as a bifurcation phenomenon from a homogeneous equilibrium field in an elastic-plastic body is first briefly reviewed, and then the predictions of the theory are compared with some experimental observations on an aged maraging steel. The experiments support the physical relevance of the theory within the framework of continuum mechanics.Specifically, the comparison between theory and experiment is concerned with the critical strains to localization and the orientation of the shear bands relative to the load axis. The theoretical predictions are only in qualitative agreement with the experimental observations. A better agreement is obtained with use of the constitutive equations corresponding to the classical deformation theory (a simple vertex model) than with use of the constitutive equations of the classical flow theory. It is concluded that better constitutive equations for elastic-plastic materials are needed before theoretical predictions can be obtained which might be expected to be in closer quantitative accord with experiment.     

Anisotropic model for granulated sea ice dynamics

A continuum model describing sea ice as a layer of granulated thick ice, consisting of many rigid, brittle floes, intersected by long and narrow regions of thinner ice, known as leads, is developed. We consider the evolution of mesoscale leads, formed under extension, whose lengths span many floes, so that the surrounding ice is treated as a granular plastic. The leads are sufficiently small with respect to basin scales of sea ice deformation that they may be modelled using a continuum approach. The model includes evolution equations for the orientational distribution of leads, their thickness and width expressed through second-rank tensors and terms requiring closures. The closing assumptions are constructed for the case of negligibly small lead ice thickness and the canonical deformation types of pure and simple shear, pure divergence and pure convergence. We present a new continuum-scale sea ice rheology that depends upon the isotropic, material rheology of sea ice, the orientational distribution of lead properties and the thick ice thickness. A new model of lead and thick ice interaction is presented that successfully describes a number of effects: (i) because of its brittle nature, thick ice does not thin under extension and (ii) the consideration of the thick sea ice as a granular material determines finite lead opening under pure shear, when granular dilation is unimportant.     

Constitutive model for plastic deformation of nanocrystalline materials with shear band

A phase mixture model was used to study the plastic deformation behaviors in hardening stage of nanocrystalline materials. The material was considered as a composite of grain interior phase and grain boundary (GB) phase. The constitutive equations of the two phases were determined in term of their main deformation mechanisms. In softening stage, a shear band deformation mechanism was presented and the corresponding constitutive relation was established. Numerical simulations have shown that the predications fit well with experimental data. The investigation using the finite-element method (FEM) provided a direct insight into quantifying shear localization effect in nanocrystalline materials.     

Modeling of textural anisotropy in granular materials with stochastic micro-polar hypoplasticity

The paper deals with a numerical analysis of the effect of textural anisotropy on the behaviour of cohesionless granular materials with consideration of shear localization. For a simulation of the mechanical behaviour of a granular material during a monotonic deformation path, an isotropic micro-polar hypoplastic constitutive model was used. To describe textural effects, spatially correlated random fields of the initial void ratio were subject to rotation against the horizontal axis. The 2D random fields were generated using a conditional rejection method. The results were compared with those obtained with an anisotropic micro-polar constitutive model for a uniform distribution of the initial void ratio. The calculations were carried out with an initially dense granular specimen during plane strain compression under constant lateral pressure.     

Randomness and slip avalanches in gradient plasticity

While localization of deformation at macroscopic scales has been documented and carefully characterized long ago, it is only recently that systematic experimental investigations have demonstrated that plastic flow of crystalline solids on mesoscopic scales proceeds in a strongly heterogenous and intermittent manner. In fact, deformation is characterized by intermittent bursts (‘slip avalanches’) the sizes of which obey power-law statistics. In the spatial domain, these avalanches produce characteristic deformation patterns in the form of slip lines and slip bands. Unlike to the case of macroscopic localization where gradient plasticity can capture the width and spacing of shear bands in the softening regime of the stress–strain graph, this type of mesoscopically jerky like localized plastic flow is observed in spite of a globally convex stress–strain relationship and may not be captured by standard deterministic continuum modelling. We thus propose a generalized constitutive model which includes both second-order strain gradients and randomness in the local stress–strain relationship. These features are related to the internal stresses which govern dislocation motion on microscopic scales. It is shown that the model can successfully describe experimental observations on slip avalanches as well as the associated surface morphology characteristics.     

Dynamics of inelastic deformation of porous rocks and formation of localized compaction zones studied by numerical modeling

The paper presents a numerical analysis of the inelastic deformation process in porous rocks during different stages of its development and under non-equiaxial loading. Although numerous experimental studies have already investigated many aspects of plasticity in porous rocks, numerical modeling gives valuable insight into the dynamics of the process, since experimental methods cannot extract detailed information about the specimen structure during the test and have strong limitations on the number of tests. The numerical simulations have reproduced all different modes of deformation observed in experimental studies: dilatant and compactive shear, compaction without shear, uniform deformation, and deformation with localization. However, the main emphasis is on analysis of the compaction mode of plastic deformation and compaction localization, which is characteristic for many porous rocks and can be observed in other porous materials as well. The study is largely inspired by applications in petroleum industry, i.e. surface subsidence and reservoir compaction caused by extraction of hydrocarbons and decrease of reservoir pressure. Special attention is given to the conditions, evolution, and characteristic patterns of compaction localization, which is often manifested in the form of compaction bands. Results of the study include stress-strain curves, spatial configurations and characteristics of localized zones, analysis of bifurcation of stress paths inside and outside localized zones and analysis of the influence of porous rocks properties on compaction behavior. Among other results are examples of the interplay between compaction and shear modes of deformation.To model the evolution of plastic deformation in porous rocks, a new constitutive model is formulated and implemented, with the emphasis on selection of adequate functions defining evolution of yield surface with deformation. The set of control parameters of the model is kept as short as possible; the parameters are carefully selected to have simple and intuitive physical interpretation whenever possible. Results demonstrate that evolution of the yield surface with deformation has major influence on the resulting characteristics of deformation patterns, which is not sufficiently acknowledged in the literature.     

土的工程力学性质的颗粒流模拟

基于颗粒流理论,引入不同的颗粒接触连接本构模型,分别建立了砂土和粘性土的颗粒流模型．通过颗粒流数值模型试验,对砂土和粘性土的室内平面应变试验及其剪切带形成和发展进行了数值模拟,分别对比了不同围压下颗粒流试样与室内试验的应力应变关系曲线,基本再现了砂土和粘性土试样应力．应变关系．通过砂土和粘性土PFC试样剪切带模拟表明,当围压较小时试样内部颗粒位移量小而且分布范围较广,当围压增大时,试样内部颗粒位移量也增大,而且发生较大位移颗粒的分布范围趋于集中,同时随着围压的增大试样内部形成明显的剪切带．无论砂土还是粘性土的PFC试样,随着围压的增加剪切带的形状趋于集中,而且剪切带宽度在减小．在围压很小时,试样内形成大的破坏区域,在围压较大时出现明显的线破坏区．这些规律基本与室内试验结果相似。     

Simulation and interpretation of diffuse mode bifurcation of elastoplastic solids

The diffuse mode bifurcation of elastoplastic solids at finite strain is investigated. The multiplicative decomposition of deformation gradient and the hyperelasto-plastic constitutive relationship are adapted to the numerical bifurcation analysis of the elastoplastic solids. First, bifurcation analyses of rectangular plane strain specimens subjected to uniaxial compression are conducted. The onset of the diffuse mode bifurcations from a homogeneous state is detected; moreover, the post-bifurcation states for these modes are traced to arrive at localization to narrow band zones, which look like shear bands. The occurrence of diffuse mode bifurcation, followed by localization, is advanced as a possible mechanism to create complex deformation and localization patterns, such as shear bands. These computational diffuse modes and localization zones are shown to be in good agreement with the associated experimental ones observed for sand specimens to ensure the validity of this mechanism. Next, the degradation of horizontal sway stiffness of a rectangular specimen due to plane strain uniaxial compression is pointed out as a cause of the bifurcation of the first antisymmetric diffuse mode, which triggers the tilting of the specimen. Last, circular and punching failures of a footing on a foundation are simulated.