基于SEM的可變形塊體離散元法研究

周東 張青波 李世海

摘 要：針對邊坡工程中巖土體連續(xù)-非連續(xù)漸進(jìn)破壞的特點，提出一種新的變形體離散元方法（DEM）。與傳統(tǒng)有限單元法（FEM）不同，彈簧元法（SEM）通過構(gòu)建一組廣義彈簧系統(tǒng)描述單元的力學(xué)行為。彈簧元法中的一個廣義彈簧可以具有多個方向的剛度系數(shù)，確定廣義彈簧系統(tǒng)的構(gòu)造形式及其各剛度系數(shù)表達(dá)式是彈簧元法的核心。以三角形單元為例，介紹平面彈簧元的基本理論。對任何二維正交廣義彈簧系統(tǒng)，通過定義廣義彈簧變形與單元應(yīng)變之間的關(guān)系，直接對比單元的應(yīng)變能與彈簧系統(tǒng)的彈性勢能即可得到廣義彈簧剛度系數(shù)的表達(dá)形式。定義泊松剛度系數(shù)和純剪剛度系數(shù)兩個系統(tǒng)參數(shù)，描述正交廣義彈簧之間的聯(lián)系。對任意泊松比的材料，該方法都可準(zhǔn)確地描述泊松效應(yīng)的影響，計算結(jié)果與傳統(tǒng)有限元法一致。該方法不需要求得有限元單元剛度矩陣的具體形式，具有直接方便、物理意義明確的優(yōu)點，應(yīng)用該方法給出任意4節(jié)點單元彈簧系統(tǒng)的構(gòu)造形式及其各剛度系數(shù)的表達(dá)式。基于SEM的可變形塊體離散元法，用彈簧元中的廣義彈簧求解塊體變形，用離散元中的接觸彈簧計算塊體間作用力，在單元節(jié)點的控制方程中實現(xiàn)彈簧元-離散元耦合計算，通過接觸彈簧的狀態(tài)實現(xiàn)材料由連續(xù)到非連續(xù)的破壞過程。在基于連續(xù)介質(zhì)離散元法（CDEM）程序的基礎(chǔ)上實現(xiàn)了彈簧元-離散元耦合程序，應(yīng)用耦合程序計算了均質(zhì)土坡在重力作用下的彈塑性變形和基覆邊坡在重力作用下的破壞，初步證明該方法用于邊坡變形漸進(jìn)破壞分析的可行性。

關(guān)鍵詞：彈簧元法 離散元法 彈簧剛度 邊坡工程

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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