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 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-mechanical investigation of material instability in granular assemblies

It is now well established that for non-associated materials such as geomaterials, there exists a wide domain strictly within the plastic limit where different failure modes can coexist. In particular, material instability in the sense given by Hill, related to the vanishing of the second-order work, can potentially occur. In this paper we examine the occurrence of such instabilities from the simulation of drained triaxial paths, followed by the computation of two-dimensional, then fully three-dimensional Gudehus response-envelopes using the micro-directional model. This model can be seen in the continuity of Hill’s multi-slip theory, because it accounts for the association of a large number of elementary elasto-plastic bodies. Each body is linked to a contact direction in physical space and therefore takes into account the behavior of the contacts oriented along that direction. Simulations confirmed that some loading directions led to the vanishing of the second-order work. In line with the research initiated by Mandel, a micro-mechanical analysis of the origin of these potential instabilities revealed that this macro-scale phenomenon could be directly related to the constitutive nature of the local contact model between neighboring particles. Finally, this investigation provides a clear physical understanding of Hill’s material stability condition for frictional materials.     

Statics and kinematics of discrete Cosserat-type granular materials

A theoretical framework is presented for the statics and kinematics of discrete Cosserat-type granular materials. In analogy to the force and moment equilibrium equations for particles, compatibility equations for closed loops are formulated in the two-dimensional case for relative displacements and relative rotations at contacts. By taking moments of the equilibrium equations, micromechanical expressions are obtained for the static quantities average Cauchy stress tensor and average couple stress tensor. In analogy, by taking moments of the compatibility equations, micromechanical expressions are obtained for the (infinitesimal) kinematic quantities average rotation gradient tensor and average Cosserat strain tensor in the two-dimensional case. Alternatively, these expressions for the average Cauchy stress tensor and the average couple stress tensor are obtained from considerations of the equivalence of the continuum force and couple traction vectors acting on a plane and the resultant of the discrete forces and couples acting on this plane. In analogy, the expressions for the average rotation gradient tensor and the average Cosserat strain tensor are obtained from considerations of the change of length and change of rotation of a line element in the two-dimensional case. It is shown that the average particle stress tensor is always symmetrical, contrary to the average stress tensor of an equivalent homogenized continuum. Finally, discrete analogues of the virtual work and complementary virtual work principles from continuum mechanics are derived.     

On the elastic moduli of three-dimensional assemblies of spheres: Characterization and modeling of fluctuations in the particle displacement and rotation

The elastic moduli of four numerical random isotropic packings of Hertzian spheres are studied. The four samples are assembled with different preparation procedures, two of which aim to reproduce experimental compaction by vibration and lubrication. The mechanical properties of the samples are found to change with the preparation history, and to depend much more on coordination number than on density.Secondly, the fluctuations in the particle displacements from the average strain are analyzed, and the way they affect the macroscopic behavior analyzed. It is found that only the average over equally oriented contacts of the relative displacement these fluctuations induce is relevant at the macroscopic scale. This average depends on coordination number, average geometry of the contact network and average contact stiffness. As far as the separate contributions from particle displacements and rotations are concerned, the former is found to counteract the average strain along the contact normal, while the latter do in the tangential plane. Conversely, the tangential components of the center displacements mainly arise to enforce local equilibrium, and have a small and generally stiffening effect at the macro-scale.Finally, the fluctuations and the shear modulus that result from two approaches available in the literature are estimated numerically. These approaches are both based on the equilibrium of a small-sized representative assembly. The improvement of these estimate with respect to the average strain assumption indicates that the fluctuations relevant to the macroscopic behavior occur with short correlation length.     

DISCRETE GRADIENT METHOD FOR IMAGE-BASED BIOMECHANICAL STRESS ANALYSIS 1)

介绍一种可用于计算生物力学的离散梯度方法,此方法可利用离散的点云模型直接进行数值模拟分析而不需要传统的几何模型。将离散梯度法应用于点云模型需要首先确定模型中点之间的相邻关系和每个点所分配的材料体积,然后通过用广义的有限差分的形式定义了梯度插值向量,并以此向量来近似函数在每个离散点上的梯度。从弱形式出发,推导建立了适用于弹性固体大变形问题的求解器,并具有和有限元法中双线性四边形单元一致的准确性和收敛性。着重描述了一种可以从医学图像中快速提取材料点并建立点云模型的方法,以及利用三角划分和重心划分确定材料点之间的相邻关系和每个材料点体积的具体过程。通过腹主动脉瘤膨胀的静力学模拟分析,展示了离散梯度法的实用性和准确性。该算法实现了基于医学图像进行生物力学分析的过程自动化,为病体特异性的研究和治疗提供便利和实用的工具。     

Interior formulation of axisymmetric Levinson plate theory

In this study, we show that the axisymmetric Levinson plate theory is exclusively an interior theory and we provide a consistent variational formulation for it. First, we discuss an annular Levinson plate according to a vectorial formulation. The boundary layer of the plate is not modeled and, thus, the interior stresses acting as surface tractions do work on the lateral edges of the plate. This feature is confirmed energetically by the Clapeyron's theorem. The variational formulation is carried out for the annular Levinson plate by employing the principle of virtual displacements. As a novel contribution, the formulation includes the external virtual work done by the tractions based on the interior stresses on the inner and outer lateral edges of the Levinson plate. The obtained plate equations are consistent with the vectorially derived Levinson equations. Finally, we develop an exact plate finite element both by a force-based method and from the total potential energy of the Levinson plate.     

Computation of Full-field Strains Using Principal Component Analysis

The primary output from several full-field deformation measurement techniques, e.g., Digital Image Correlation (DIC), is the displacement vector at a dense grid of points covering the area of interest. Since such displacement data inherently contain noise, they are usually smoothed first and then differentiated to obtain strains. Another common approach is to use finite-element shape functions for the strains and compute them by treating the measured displacements as nodal displacements. In this paper, we propose a novel method for strain calculation from full-field data, based on the multivariate analysis technique of Principal Component Analysis (PCA) using which we first obtain the singular values and singular vectors for each component of the displacement field. By choosing only the dominant singular values and their corresponding singular vectors, we show that the dimensionality of the displacement data is sharply reduced and a significant portion of the noise is eliminated. Moreover, the shapes of the dominant singular vectors offer physical insight into dominant deformation patterns. We demonstrate the accuracy of the proposed technique by applying it to two cases each of homogeneous and inhomogeneous strain fields and show that in all cases the proposed method yields excellent results.     

A moiré microscope for finite deformation micro-mechanical studies

A new version of a moiré microscope is presented that embodies the theory of optical moiré interferometry. To interrogate the deformed specimen grating, the device uses a transmission diffraction grating that allows for a simple and quick change of the virtual reference grating vector without disturbing the optical alignment of the other components in the optical train. To analyze deformation from the acquired moiré interference fringe patterns, the displacement light-itensity moiré optical law introduced by Sciammarella is revisited. The analysis of deformation is consistent with the continuum principles of finite deformation and can readily be used to obtain micro-mechanical quantities of interest such as the local strains, stretches and rotations.     

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.     

Grain- and macro-scale kinematics for granular micromechanics based small deformation micromorphic continuum model

Macro-scale deformation of granular solids comprising large number of grains (>10⁶) are most efficiently described within the framework of continuum mechanics. It is notable, however that the micro-scale deformations in these materials are concentrated at the grain-boundaries or grain-contacts. Thus, the deformation energies in these systems must be modeled by considering the deformations concentrated in the neighborhood of the grain-boundaries or grain-contacts. To address this issue, grain-interactions has been widely described in the Hertzian sense by considering the relative movement of points on either side of a grain boundary or contact treated as an imperfect interface. This communication introduces the relevant kinematic variables given in the terms of the grain displacements, spins and size that can be used to estimate the relative movement of a grain boundary or contact. The macro-scale kinematic variables useful for continuum modeling are then identified with the grain-scale kinematic variables. The deformation energy density of the granular solid can thus be expressed both in terms of the grain-scale as well as the macro-scale kinematic variables providing the necessary pathway for micro-macro identification which can lead to non-classical micromorphic continuum models that incorporate grain-scale representation.     

Finite element formulation of slender structures with shear deformation based on the Cosserat theory

This paper addresses the derivation of finite element modelling for nonlinear dynamics of Cosserat rods with general deformation of flexure, extension, torsion, and shear. A deformed configuration of the Cosserat rod is described by the displacement vector of the deformed centroid curve and an orthogonal moving frame, rigidly attached to the cross-section of the rod. The position of the moving frame relative to the inertial frame is specified by the rotation matrix, parameterised by a rotational vector. The shape functions with up to third order nonlinear terms of generic nodal displacements are obtained by solving the nonlinear partial differential equations of motion in a quasi-static sense. Based on the Lagrangian constructed by the Cosserat kinetic energy and strain energy expressions, the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements. A cantilever is presented as a simple example to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure. The corresponding nonlinear dynamical responses of the structures are presented through numerical simulations using the MATLAB software. In addition, a MicroElectroMechanical System (MEMS) device is presented. The developed equations of motion have furthermore been implemented in a VHDL-AMS beam model. Together with available models of the other components, a netlist of the device is formed and simulated within an electrical circuit simulator. Simulation results are verified against Finite Element Analysis (FEA) results for this device.     

A new tracking algorithm of PIV and removal of spurious vectors using Delaunay tessellation

 A new algorithm of Delaunay Tessellation Particle Tracking Velocimetry (DT-PTV in abbreviation) is proposed for tracking particles in images of a PIV system by making use of the Delaunay tessellation (DT). The algorithm is tested by using numerically simulated particle images. The calculation results based on DT are compared with those obtained by a conventional algorithm of Binary Image Cross-correlation method (BICC). The new algorithm shows higher performance of obtaining more identical particles in two consecutive images correctly with shorter computation time even if the images contain many particles. A further application of DT to elimination of spurious vectors is also discussed. Received: 24 November 1997/Accepted: 7 August 1998     

用双视场电子散斑干涉实现检测表面的变尺度同时测量

结构或材料中所含的微小缺陷对其宏观力学性能具有重要的影响，探讨这些微小缺陷引起的微变形及其在结构整体变形中所占的份额，对于有效地评估结构和材料的整体性能及局部演化过程都具有重要的理论和工程价值，本文应用双视场电子散斑干涉技术，实现了对被测对象表面的全场变形和局部微观场变形的同时实时检测，给出了宏观及局部位移场分布，并结合有限元数值分析与实验结果进行了比较。     

Modified virtual internal bond model based on deformable Voronoi particles

In last time, the series of virtual internal bond model was proposed for solving rock mechanics problems. In these models, the rock continuum is considered as a structure of discrete particles connected by normal and shear springs(bonds). It is well announced that the normal springs structure corresponds to a linear elastic solid with a fixed Poisson ratio, namely, 0.25 for threedimensional cases. So the shear springs used to represent the diversity of the Poisson ratio.However, the shearing force calculation is not rotationally invariant and it produce difficulties in application of these models for rock mechanics problems with sufficient displacements. In this letter, we proposed the approach to support the diversity of the Poisson ratio that based on usage of deformable Voronoi cells as set of particles. The edges of dual Delaunay tetrahedralization are considered as structure of normal springs(bonds). The movements of particle's centers lead to deformation of tetrahedrals and as result to deformation of Voronoi cells. For each bond, there are the corresponded dual face of some Voronoi cell. We can consider the normal bond as some beam and in this case, the appropriate face of Voronoi cell will be a cross section of this beam. If during deformation the Voronoi face was expand, then, according Poisson effect, the length of bond should be decrees. The above mechanism was numerically investigated and we shown that it is acceptable for simulation of elastic behavior in 0.1–0.3 interval of Poisson ratio. Unexpected surprise is that proposed approach give possibility to simulate auxetic materials with negative Poisson's ratio in interval from –0.5 to –0.1.     

Plastic deformation and fracture of polycrystalline Ni–Ti with stress concentrators of different scales

Stress concentrators of different scales inherent in polycrystals tend to form mesosubstructures under load. Television-optical images of displacements are obtained for the Ni–Ti alloy to study the deformation and fracture behavior. Discussed are similarity laws in connection with the formation of dissipative structures at the micro- and macro-scale level.     

The influence of particle fluctuations on the average rotation in an idealized granular material

Granular materials are a simple example of a Cosserat continuum in that the average particle rotations may differ from the rotation of the average deformation. In the absence of couple stress, this difference insures that the stress is symmetric. This has been shown in theories that assume that the displacement at particle contacts is given by the average deformation and spin. Here, we indicate how the difference between the average rotation of the particles and the average rotation of the deformation can be determined when fluctuations in particle displacements and rotations satisfy local force and moment equilibria in a random aggregate of identical spheres. The predictions based on this model are in better agreement with numerical simulation than that given by the simple average strain theory.     

The role of texture in tension–compression asymmetry in polycrystalline NiTi

The purpose of the present study is to thoroughly understand the stress–strain behavior of polycrystalline NiTi deformed under tension versus compression. To do this, a micro-mechanical model is used which incorporates single crystal constitutive relationships and experimentally measured polycrystalline texture into the self-consistent formulation. For the first time it is quantitatively demonstrated that texture measurements coupled with a micro-mechanical model can accurately predict tension/compression asymmetry in NiTi shape memory alloys. The predicted critical transformation stress levels and transformation stress–strain slopes under both tensile and compressive loading are consistent with experimental results. For textured polycrystalline NiTi deformed under tension it is demonstrated that the martensite evolution is very abrupt, consistent with the Luders type deformation experimentally observed. The abrupt transformation under tension is attributed to the fact that the majority of the grains are oriented along the [111] crystallographic direction, which is soft under tensile loading. Since single crystals of the [111] orientation are hard under compression it is also demonstrated that under compression the martensite in textured polycrystalline NiTi evolves relatively slower.     

A coupled micro-mechanical based model for saturated soils

It is generally accepted that the apparent behavior of geo-materials is the representation of the average micro-mechanical behavior of its constituents. Constitutive models that do not incorporate these micro-mechanical features in calibrating the material parameters cannot address various material localization features in large strain problems such as shear bands and slope failures, etc. In the absence of such micro-mechanical features the calibration of such models may be incorrect.A rigorous formulation that incorporates these micro-mechanical based mechanisms into the general behavior of the saturated soils is presented here. The plastic rotation of particles, the interaction of particles, the rate dependency, the damage, and the coupling of particles with pore fluid pressure are incorporated through the plastic spin, the gradient theory, the visco-plasticity, the damage theory, and the coupled theory of mixtures, respectively. The link between the micro-mechanical mechanisms and the macro-mechanical behavior is made through the use of RVE (representative volume element). As a result, a full formulation for the micro-mechanics implemented continuum plasticity for saturated soils is presented here.     

一种大变形曲壳单元

本文用壳中面节点的位移矢量和节点处壳中面单位法线矢量的矢端位移矢量构造了一种大变形曲壳单元,它是大变形曲壳单元的一般形式,能包括已有的大变形曲壳单元,且公式最简单,计算中采用的载荷增量或位移增量可以很大。