Simulation of solids with multiple rectangular inhomogeneities using non-uniform eigenstrain formulation of BIEs

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.