In the present paper, a novel computational model of non-uniform eigenstrain formulation of boundary integral equations (BIEs) with corresponding iterative solution procedures is presented for simulating solids with multiple rectangular inhomogeneities in elastic range. The computational model is developed by introducing the concepts of the equivalent inclusion of Eshelby with eigenstrains into BIEs with the removal of the constant assumption of eigenstrain in previous works that are limited to solve the elliptical or ellipsoidal inhomogeneities. The non-uniform eigenstrains are expressed by Lagrange interpolation polynomials, which are determined in an iterative way for each inclusion embedded in the matrix. Moreover, to deal with the interactions among inclusions, all of the inclusions in the matrix are divided into two groups, namely the near-field group and the far-field group, according to the distance to the current inclusion in consideration. The local Eshelby matrix is constructed over the near-field group to guarantee the convergence of iterative procedure by getting rid of the strong interactions among the inclusions in the near-field group. Due to the unknowns appear only on the boundary of the solution domain in the present model, the solution scale is effectively reduced. The results of the elastic stress distributions across the interface of inclusions are compared with the subdomain BIE method, whereas the overall effective elastic properties of the media are verified by the reference results with doubly periodic square inclusions. In addition, the overall effective elastic properties of a square representative volume element (RVE) with various inclusion distributions are also investigated in considering a variety of factors, including the properties, the aspect ratios, the orientations and the total number of inclusions. Finally, the convergence behaviors and efficiencies of the solution procedure are studied numerically, showing the validity and efficiency of the proposed computational model.

本文提出了一种新的边界积分方程（BIE）非均匀特征应变公式的计算模型，并给出了相应的迭代求解程序，用于模拟弹性范围内具有多个矩形不均匀性的固体。通过将等效应变包含Eshelby与本征应变的概念引入BIE中，并消除了以前只能解决椭圆或椭球不均匀性的本征应变的恒定假设，从而开发了计算模型。非均匀本征应变由拉格朗日插值多项式表示，该多项式以迭代方式确定嵌入在矩阵中的每个内含物。此外，为了处理夹杂物之间的相互作用，将矩阵中的所有夹杂物分为两组，根据到当前包含物的距离，即近场组和远场组。在近场组上构造局部Eshelby矩阵，以通过消除近场组中夹杂物之间的强相互作用来确保迭代过程的收敛。由于在本模型中未知数仅出现在求解域的边界上，因此有效减小了求解规模。将夹杂物界面上的弹性应力分布结果与子域BIE方法进行了比较，而介质的总体有效弹性性能则通过具有双周期正方形夹杂物的参考结果进行了验证。此外，在考虑各种因素的情况下，还研究了具有各种夹杂物分布的正方形代表体积元素（RVE）的总体有效弹性，其中包括性质，纵横比，取向和夹杂物总数。最后，对求解过程的收敛性和效率进行了数值研究，证明了所提计算模型的有效性和有效性。