【作者】 李洪涛；

【导师】 李锡夔；

【作者基本信息】 大连理工大学 ， 固体力学， 2010， 硕士

【摘要】 颗粒材料是自然界及工程界存在并应用最为广泛的材料之一,如土壤,沙砾,混凝土等。另外严重危害人类生命财产的地质灾害都与颗粒材料的性质有关,如地震、煤矿倒塌等。因此研究颗粒材料变形行为及其力学性质具有重要的工程实际价值。建立基于细观力学的离散颗粒集合体的等价连续体本构模型对揭示颗粒材料宏观力学行为具有重要的意义。平均场理论框架下的计算均匀化方法是一种直观的细-宏观多尺度本构建模方法。均匀化方法是基于表征元(RVE)概念的,表征元是指在宏观域中以一个典型样本点为中心在其邻域所截取的有限尺度的单元体,以保证其中能够包含足够多的颗粒微结构信息从而达到表征效果。为得到基于细观力学的颗粒材料等价连续体宏观本构关系,需要定义表征元内基于接触颗粒间相对运动的细观局部应变,并由此建立它们与表征元平均应变之间的关系,通常采用Voigt假定,即假定表征元内局部应变均一。然而实际上,颗粒材料表征元内部各处的等价连续体局部应变并不均匀。首先本文基于平均场理论定义了表征元内部由单个颗粒及其周边空隙构成的颗粒材料等价连续体的Voronoi单元,以描述颗粒与其邻近颗粒的接触拓扑关系。假定定义于Voronoi单元等价连续体尺度上的局部应变为常数,摒弃了Voigt假定,由Voronoi单元导出表征元内依赖颗粒材料局部微结构的等价连续体的本构关系。给出了不同类型的表征不同局部微结构的颗粒材料正多边形Voronoi单元等价连续体的本构关系及其弹性模量张量。论证了正多边形Voronoi单元等价连续体本构关系的坐标不变性并分析了Voronoi单元的接触点扰动对宏观本构关系的影响。其次本文基于Voronoi单元应变均一假设得到的位移为真实的颗粒位移的最优拟合推导了颗粒材料等价连续体的本构关系。定义了Voronoi单元等价连续体的平均应变。对比了颗粒材料正多边形Voronoi单元等价连续体的应变均一假设与最优拟合假设的结果,并用实例验证了两种途径的优劣。最后本文基于多项式函数来假定表征元内连续的颗粒材料的位移场,推导了颗粒材料等价Cosserat连续体的本构关系。给出了接触各向同性情况下等价连续体材料弹性常数与颗粒细观参数的关系。更多还原

【Abstract】 Granular material is one of the materials widely existing in nature and used in engineering, such as soil, sands, concrete. On the other hand, the geologic disasters endangering people’s lives and properties, such as earthquake, collapse of coal mine are related to the properties of granular materials. Hence, the study of mechanical properties and responses of granular materials is valuable to engineering practice.It is of great significance to derive the micromechanically informed constitutive model of the continuum equivalent to a discrete particle assembly to explore macro-mechanical behaviors in granular materials. Computational homogenization method is a direct multi-scale micro-macro constitutive modelling based on the conception of representative volume element (RVE) in the framework of average-field theory. RVE is a volume element with finite size centered by a classical sampling point containing enough information of microscopic structures for granular material in order to achieve the purpose of representivity. To derive the micromechanically informed constitutive model of the equivalent continuum, the relation between local strain defined by relative contact displacements of particles and the average strain of RVE should be given. The usual approach is using the Voigt hypothesis assuming that local strain in RVE is uniform. However, local strain, in fact, is not uniform in RVE.Firstly, in light of the average field theory a single Voronoi cell of effective continuum composed of a particle and its neighboring interspaces able to describe different local micro-structures and characterize different contact topologies of a typical particle with its neighboring particles is defined. Assuming the local strain is uniform in the scale of Voronoi cell, however, Voigt hypothesis is not adopted. The constitutive model for the Voronoi cell is formulated by means of the relations of inter-contacting forces and relative displacements between two particles in contact. The constitutive relations and the associated elastic modular tensors for different types of Voronoi cells with regular polygon shapes representing different local micro-structures of granular materials are given. The coordinate invariance of the formulated constitutive relations for the Voronoi cells is demonstrated. Finally the perturbation analysis of contacts to the Voronoi cell is fulfilled. Secondly, overall constitutive relationship for effective continuum of granular material is derived based on the assumption that displacements obtained from uniform assumption of Voronoi cell best fitting the real displacements of particles. After that average strain of equivalent continnum is defined. Uniform local strain hypothesis and best fit hypothesis for Voronoi cell with regular polygon shapes are compared and verified by a numeric example.At last overall constitutive relationship of equivalent Cosserat continuum is derived from expressing the continuum displacement field of granular material by polynomial function in RVE. The relations between the material parameters of equivalent continuum and micro parameters of particles are formed in isotropic contact situation.更多还原