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-迭加”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 will 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 may be solved through a "discretization-superposition" scheme. In contrast, 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 to show the correctness and effectiveness of this proposed semi-analytical method.

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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)。