This paper extends our recent work [1] to evaluate the elastic fields and effective modulus of a composite containing arbitrarily shaped inhomogeneities for both two-dimensional (2D) and three-dimensional (3D) problems. Based on Eshelby’s equivalent inclusion method (EIM), the material mismatch between inhomogeneities and matrix phases is represented with a continuously distributed eigenstrain on inclusions. Since there exists singularities at the vertices of polyhedral inhomogeneities, domain discretization of inhomogeneities is used to interpolate the eigenstrain distribution, and high accuracy is obtained by using the closed-form integrals of the source field. The influence of the singularity decays in $\frac{1}{r^2}$ and $\frac{1}{r^3}$ for 2D and 3D problems respectively. Because Eshelby’s tensors depend on the shape instead of the size, the iBEM is particularly suitable for cross scale modeling of composites with a wide range of particle sizes. Although a number of elements are required to provide high fidelity results of the local field around the vertices, in general, very few elements or a single element are enough for each inhomogeneity to obtain the convergent solutions of the effective material properties, which enables virtual experiments of a composite containing many inhomogeneities. This novel numerical method has been verified with the finite element method (FEM) with much more elements. Virtual experiments of composites with many particles demonstrate its versatile capability and great potentials.

本文扩展了我们最近的工作 [1]，以评估包含任意形状不均匀性的复合材料的弹性场和有效模量，用于二维 (2D) 和三维 (3D) 问题。基于 Eshelby 的等效夹杂物方法 (EIM)，不均匀性和基体相之间的材料不匹配用夹杂物上连续分布的本征应变来表示。由于多面体不均匀性的顶点处存在奇点，利用不均匀性的域离散化对特征应变分布进行插值，利用源场的闭式积分获得高精度。奇点的影响衰减$\frac{1}{r^2}$ 和 $\frac{1}{r^3}$分别用于 2D 和 3D 问题。由于 Eshelby 的张量取决于形状而不是尺寸，因此 iBEM 特别适用于具有广泛颗粒尺寸的复合材料的跨尺度建模。虽然需要多个单元来提供顶点周围局部场的高保真结果，但一般来说，很少的单元或单个单元足以让每个不均匀性获得有效材料属性的收敛解，这使得虚拟实验成为可能包含许多不均匀性的复合材料。这种新颖的数值方法已经通过具有更多元素的有限元方法 (FEM) 进行了验证。多颗粒复合材料的虚拟实验证明了其多功能性和巨大潜力。