基于SEM的可变形块体离散元法研究

周东 张青波 李世海

摘 要：针对边坡工程中岩土体连续-非连续渐进破坏的特点，提出一种新的变形体离散元方法（DEM）。与传统有限单元法（FEM）不同，弹簧元法（SEM）通过构建一组广义弹簧系统描述单元的力学行为。弹簧元法中的一个广义弹簧可以具有多个方向的刚度系数，确定广义弹簧系统的构造形式及其各刚度系数表达式是弹簧元法的核心。以三角形单元为例，介绍平面弹簧元的基本理论。对任何二维正交广义弹簧系统，通过定义广义弹簧变形与单元应变之间的关系，直接对比单元的应变能与弹簧系统的弹性势能即可得到广义弹簧刚度系数的表达形式。定义泊松刚度系数和纯剪刚度系数两个系统参数，描述正交广义弹簧之间的联系。对任意泊松比的材料，该方法都可准确地描述泊松效应的影响，计算结果与传统有限元法一致。该方法不需要求得有限元单元刚度矩阵的具体形式，具有直接方便、物理意义明确的优点，应用该方法给出任意4节点单元弹簧系统的构造形式及其各刚度系数的表达式。基于SEM的可变形块体离散元法，用弹簧元中的广义弹簧求解块体变形，用离散元中的接触弹簧计算块体间作用力，在单元节点的控制方程中实现弹簧元-离散元耦合计算，通过接触弹簧的状态实现材料由连续到非连续的破坏过程。在基于连续介质离散元法（CDEM）程序的基础上实现了弹簧元-离散元耦合程序，应用耦合程序计算了均质土坡在重力作用下的弹塑性变形和基覆边坡在重力作用下的破坏，初步证明该方法用于边坡变形渐进破坏分析的可行性。

关键词：弹簧元法 离散元法 弹簧刚度 边坡工程

Abstract：Aiming at the continuous-discontinuous failure process of rock and soil materials in slope engineering，a novel deformable block discrete element method which combined spring element method（SEM） and discrete element method（DEM）together is presented. Compared with the accustomed element in traditional finite element method（FEM），the element in SEM is described as a spring system that contained some orthogonal generalized springs.This generalized springs are defined in 3D space， which means that each spring can has two or three spring stiffness.How to determine the generalized spring stiffness for continuous material is the difficult and most important in SEM.With the triangle element as an example，the basic theory of SEM is introduced in detail.Assuming the relationship between the generalized spring deformation and the element strain，the generalized spring stiffness can be obtained directly by comparing the elastic strain energy of the element and the elastic potential energy of the spring system.The Poisson and shear stiffness coefficients were defined as system parameters to describe the relationship between different generalized springs.The SEM can consider the Poisson effect accurately for any Poissons ratio material；and the result using SEM are the same with using traditional FEM.This method does not need to know the expression of the element-stiffness-matrix. It can be used in 4-node element；and the stiffness expressions of springs are given clearly. With the SEM used to compute the block deformation and the contact-spring used to calculate the interaction between blocks，the combined SEM/DEM program can be used to simulate the failure process of rock and soil material from continuous to discontinuous.The SEM and DEM are combined in the motion equation of each node in each element.The contact-spring in DEM satisfied specific strength criterion.When the contact-spring force exceeded its limit，the material became discontinuous from continuous. The combined SEM/DEM program is implemented easily in the continuum-based discrete element method（CDEM）program.The simulation of a homogeneous soil slope under gravity shows that the SEM is performed as good as FEM when using line elastic constitutive and reasonable when using Mohr-Coulomb strength criterion. The simulation of a bedrock and overburden layer slope shows that the combined program is suitable to simulate the slope failure process.

Key Words：Spring element method（SEM）；Discrete element method（DEM）；Spring stiffness；Slope engineering

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