Discrete mechanics and the Finite Element Method

Summary We explore applications of the Finite Element Method (FEM) to both Veselov and Lee discrete mechanics in this paper. Based on the FEM, disretizations of continuous Lagrangians are developed and corresponding integrators are obtained. Error estimates for variational integrators are also given. This work is supported by the National Natural Science Foundation of China (grant Nos. 90103004, 10171096) and the National Key Project for Basic Research of China (G1998030601).     

基于细观方向平均模型的颗粒材料宏观Cosserat连续体本构关系

提出了基于细观微-方向模型（Micro—Directional Model）的宏观Cosserat连续体本构关系。在细观尺度上考虑颗粒旋转自由度及接触力矩,微结构的影响通过接触分布函数体现。给出均质各向同性Cosserat连续体模型弹性常数的细观参数表达式,并建议了二维情况下内尺度参数的细观力学表达式。对颗粒材料宏观行...     

Trans-scale mechanics： looking for the missing links between continuum and micro/nanoscopic reality

Problems involving coupled multiple space and time scales offer a real challenge for conventional frame-works of either particle or continuum mechanics. In this paper, four cases studies （shear band formation in bulk metallic glasses, spallation resulting from stress wave, interaction between a probe tip and sample, the simulation of nanoindentation with molecular statistical thermodynamics） are provided to illustrate the three levels of trans-scale problems （problems due to various physical mechanisms at macro-level, problems due to micro-structural evolution at macro/micro-level, problems due to the coupling of atoms/ molecules and a finite size body at micro/nano-level） and their formulations. Accordingly, non-equilibrium statistical mechanics, coupled trans-scale equations and simultaneous solutions, and trans-scale algorithms based on atomic/molecular interaction are suggested as the three possible modes of trans-scale mechanics.     

利用连接尺度方法的颗粒材料破碎数值分析

基于已有的颗粒材料连接尺度方法(BSM)[1-2],发展了在细尺度上采用离散颗粒集合体模型与离散单元法(DEM)并引入了颗粒破碎模型,而在粗尺度上采用Cosserat连续体模型与有限单元法(FEM)的BSM。仅在有限局部区域内采用DEM从细观层次关注颗粒材料破碎现象,而在全域采用储存空间和花费时间较少的FEM,同时在粗细两个尺度采用不同的时间步长。讨论了颗粒材料发生破碎时,颗粒材料结构的承载能力与微结构的演变。数值算例结果说明了所提出可模拟破碎的BSM的可用性和优越性,以及颗粒破碎对颗粒材料微观力学行为的影响。     

A discrete element model for simulating saturated granular soil

A numerical model is developed to simulate saturated granular soil, based on the discrete element method. Soil particles are represented by Lagrangian discrete elements, and pore fluid, by appropriate discrete elements which represent alternately Lagrangian mass of water and Eulerian volume of space. Macro-scale behavior of the model is verified by simulating undrained biaxial compression tests. Micro-scale behavior is compared to previous literature through pore pressure pattern visualization during shear tests. It is demonstrated that dynamic pore pressure patterns are generated by superposed stress waves. These pore-pressure patterns travel much faster than average drainage rate of the pore fluid and may initiate soil fabric change, ultimately leading to liquefaction in loose sands. Thus, this work demonstrates a tool to roughly link dynamic stress wave patterns to initiation of liquefaction phenomena.     

A theoretical and computational framework for anisotropic continuum damage mechanics at large strains

The main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of a fictitious, isotropic, undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient. Then, the corresponding Finger tensor – denoted as damage metric – constructs a second order, internal variable. Due to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration, the free energy function can be computed via push-forward operations within the nominal setting. Referring to the framework of standard dissipative materials, associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation. The numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account. Finally, some numerical examples demonstrate the applicability of the proposed framework.     

大规模颗粒系统的精确缩尺和粗粒化离散元方法

计算效率是制约工程尺度大规模颗粒系统离散元计算发展的重要因素,现有的粗粒化处理方法局限于特定应用并且缺少一般的理论依据。本文采用量纲分析方法,描述了在精确缩尺系统中各物理量应当满足的缩放定律;通过在粗粒化系统和原始系统的代表性体积单元之间建立质量、动量和能量的近似守恒关系,采用多尺度的描述方法得到了粗粒化系统与原始系统之间宏观和细观两种不同尺度的缩放关系,即双尺度粗粒化模型;精确缩尺系统中得到的缩放定律及离散元接触模型处理方法,完全适用于粗粒化系统中细观颗粒层面相关物理量的缩放,通过筒仓侧壁压力和休止角两个算例对精确缩尺模型在粗粒化系统中的有效性进行了验证。     

Intermediate asymptotics,scaling laws and renormalization group in continuum mechanics

Scaling laws and self-similar solutions are very popular concepts in modern continuum mechanics. In the present paper these concepts are analyzed both from the viewpoint of intermediate asymptotics, known in classical mathematical physics and fluid mechanics, and from the viewpoint of the renormalization group technique, known in modern theoretical physics. The definition of the renormalization group is proposed, related to the intermediate asymptotics with incomplete similarity. The general presentation is illustrated by examples of essentially non-linear problems where all analytical properties of the solutions and their asymptotics are rigorously proved, as well by an example from turbulence, where the rigorous problem statement is missing. General lecture delivered at the 11th Italian National Congress of Theoretical and Applied Mechanics (AIMETA), Trento, Sept./Oct. 1992.     

离散颗粒集合体-Cosserat连续体模型的连接尺度方法

基于针对分子动力学-Cauchy连续体模型提出的连接尺度方法(BSM)[1,2],发展了耦合细尺度上基于离散颗粒集合体模型的离散单元法(DEM)和粗尺度上基于Cosserat连续体模型的有限元法(FEM)的BSM。仅在有限局部区域内采用DEM以从细观层次模拟非连续破坏现象,而在全域则采用花费计算时间和存储空间较少的FEM。通过连接尺度位移(包括平移和转动)分解,和基于作用于Cosserat连续体有限元节点和颗粒集合体颗粒形心的离散系统虚功原理,得到了具有解耦特征的粗细尺度耦合系统运动方程。讨论和提出了在准静态载荷条件下粗细尺度域的界面条件,以及动态载荷条件下可以有效消除粗细尺度域界面上虚假反射波的非反射界面条件(NRBC)。本文二维数值算例结果说明了所提出的颗粒材料BSM的可应用性和优越性,及所实施界面条件对模拟颗粒材料动力学响应的有效性。     

Investigation of the damage variable basic issues in continuum damage and healing mechanics

Consistent mathematics and mechanics are used here to properly interpret the damage variable within the confines of the concept of reduced area due to damage. In this work basic issues are investigated for the damage variable in conjunction with continuum damage and healing mechanics. First, the issue of the additive decomposition of the damage variable into damage due to voids and damage due to cracks in continuum damage mechanics is discussed. The accurate decomposition is shown to be non-additive and involves a term due to the interaction of cracks and voids. It is shown also that the additive decomposition can only be used for the special case of small damage. Furthermore, a new decomposition is derived for the evolution of the damage variable. The second issue to be discussed is the new concept of independent and dependent damage processes. For this purpose, exact expressions for the two types of damage processes are presented. The third issue addressed is the concept of healing processes occurring in series and in parallel. In this regard, systematically and consistently, the equations of healing processes occurring either consecutively or simultaneously are discussed. This is followed by introducing the new concept of small healing in damaged materials. Simplified equations that apply when healing effects are small are shown. Finally, some interesting and special damage processes using a systematic and original formulation are presented.