隔离非线性平面应变单元模型及其在Drucker-Prager模型中的应用  

Inelasticity-separated plane-strain element model and its application to Drucker-Prager model

作　　者：李佳龙 李钢[1] 于龙[1] LI Jia-long;LI Gang;YU Long(State Key Laboratory of Coastal and Offshore Engineering,Dalian University of Technology,Dalian,Liaoning 116024,China)

机构地区：[1]大连理工大学海岸和近海工程国家重点实验室,辽宁大连116024

出　　处：《岩土力学》2020年第5期1492-1501,1509,共11页Rock and Soil Mechanics

基　　金：国家重点研发计划项目资助(No.2018YFD1100404);国家自然科学基金(No.51679038);大连市高层次人才创新支持计划项目(No.2017RD04)。

摘　　要：提出了一种基于隔离非线性理论的平面应变单元分析模型。对于线性的四边形等参单元,采用2×2个高斯积分点作为单元的非线性应变插值点。考虑到平面应变单元面外无应变分量存在,故仅在面内的3个方向建立非线性应变场以及相应的控制方程。采用Woodbury公式与组合近似法联合求解控制方程,使得整个非线性分析过程仅为多次的初始弹性刚度矩阵回代计算以及稀疏矩阵与向量的乘积,极大地提高了控制方程的求解效率。基于时间复杂度的计算效率分析表明:所提算法的计算效率相对传统变刚度法显著提高,且相对于精确的Woodbury公式,则大大提高了非线性自由度的临界比例。将Drucker-Prager准则的转移应力解析解法用于典型平面应变模型的非线性分析,数值算例验证了平面应变单元模型的正确性以及算法的高效性,为平面应变类型的非线性分析提供了一种新思路。In this paper, a plane-strain element analysis model is proposed on the basis of the inelasticity-separated finite element method(IS FEM). The 2×2 Gaussian points as the inelastic strain interpolation points are considered for the linear quadrilateral isoparametric plane-strain element. The inelastic strain field and the corresponding governing equation are established only in the in-plane three directions since there are no strain components out of the plane. In addition, the solving process of the governing equation only contains many times back-substitutions of the initial stiffness matrix and the sparse matrix and vector multiplication by using the Woodbury formula and the combined approximation approach. The computational efficiency of the proposed method based on the time complexity theory indicates that it significantly improves the total efficiency as compared with the conventional FEM with the updated stiffness matrix. The critical ratio of the inelastic degrees of freedom(IDOFs) is also improved by comparing with the exact Woodbury formula. The stress transfer method of the Drucker-Prager criterion is applied for the nonlinear analysis of the classical plane-strain model. Numerical examples verify the accuracy of the proposed model and the efficiency of the proposed algorithm for the nonlinear analysis of the plane-strain model.

关 键 词：隔离非线性法 平面应变单元 Woodbury公式 组合近似法 效率分析 DRUCKER-PRAGER准则 

分 类 号：TU452[建筑科学—岩土工程]