弹性半平面矩形夹杂基本单元解及其应用  

Elementary solution of the elastic half-plane containing a rectangular inclusion:theory and applications

作　　者：谢东东 金晓清[1,2] 蒋志桢 钱厚鹏 李璞 XIE Dongdong;JIN Xiaoqing;JIANG Zhizhen;QIAN Houpeng;LI Pu(College of Aerospace Engineering,Chongqing University,Chongqing 400044,P.R.China;State Key Laboratory of Mechanical Transmissions,Chongqing University,Chongqing 400044,P.R.China;School of Science,Harbin Institute of Technology,Shenzhen 518055,P.R.China)

机构地区：[1]重庆大学航空航天学院,重庆400044 [2]重庆大学机械传动国家重点实验室,重庆400044 [3]哈尔滨工业大学理学院,深圳518055

出　　处：《重庆大学学报》2022年第12期26-35,共10页Journal of Chongqing University

基　　金：国家自然科学基金资助项目(51875059);重庆市科技计划项目(cstc2020jcyj-msxmX0850);重庆市研究生科研创新项目(CYS21011)。

摘　　要：工程材料中存在的夹杂通常会对基体材料的弹性场产生扰动,当夹杂位于表界面附近时,夹杂与表界面的相互作用往往导致问题的解析求解变得复杂、困难。推导了二维半平面基体中矩形夹杂所致位移和应力场的基本单元解,用于通过“离散叠加”来求解半无限平面含任意形状夹杂的弹性场。与之相比,传统的有限元法需要在远大于夹杂尺寸的半无限大基体域上进行网格划分,并且需要在夹杂/基体界面处细化网格以满足计算精度要求。提出了半解析算法,基于矩形夹杂单元封闭解,只需对夹杂区域进行离散,可有效提高计算效率。以半平面基体中含正六边形和圆形夹杂为例,该方法与有限元软件得到的结果进行对比,验证了基于单元解的半解析数值算法的正确性与有效性。Inclusions in engineering materials create a disturbance to the elastic fields.When an inclusion is located in the vicinity of the surface,the interactions between the inclusion and the surface are often difficult to be solved analytically.The elementary solutions of the displacements and stresses caused by a rectangular inclusion contained in an elastic half-plane are derived in closed-form.Accordingly,the resultant elastic fields produced by any arbitrarily shaped inclusion can be solved through a“discretization-superposition”scheme.The traditional Finite Element Method needs to mesh the semi-infinite matrix region which is much larger than the size of the inclusion,while the mesh needs to be sufficiently refined at the inclusion/matrix interface to attain a satisfactory accuracy.By taking advantages of the elementary solutions of rectangular inclusion,the proposed semi-analytical method merely performs the numerical discretization within the inclusion region,leading to remarkable savings on the meshing efforts and memory storage.Benchmark examples are reported for an elastic half-plane containing either regular hexagonal or circular inclusion,and the results are validated with those obtained by the commercial finite element software,demonstrating the correctness and effectiveness of this proposed semi-analytical method.

关 键 词：数值方法 夹杂 基本单元解 半平面问题 

分 类 号：O343.1[理学—固体力学]