Hadamard regularization

Abstract: This paper introduces the inclusion based boundary element method (iBEM) to calculate the elastic fields and effective modulus of a composite containing particles for both three dimensional (3D) and two dimensional (2D) cases. Considering a finite bounded domain containing many inclusions, the isotropic Green’s function has been used to obtain the elastic field caused by source fields on inclusion domains and applied loads on the boundary. Based on Eshelby’s equivalent inclusion method (EIM), the material mismatch between the particle and matrix phases is simulated with a continuously distributed source field, namely eigenstrain, on particles. Because explicit integrals can be obtained for ellipsoidal particles, no mesh is needed for those particles, which enables virtual experiments of a composite containing a large number of particles. The classic Eshelby’s tensor is extended from a constant eigenstrain for the single particle in the infinite domain to a form of a Taylor series for particle-boundary interaction and particle-particle interactions. Using the Hadamard regularization, the 2D formulation is derived from the 3D case by the integral of the elastic solution in the third direction together with an analytical circular harmonic potential integral scheme. The iBEM is particularly suitable to conduct virtual experiments for studying the local elastic field with the integrals of all sources and calculating the effective material properties by the volume average of local fields. A parametric study of accuracy on stress field for uniform, linear, quadratic eigenstrain fields was performed and case studies have been presented to demonstrate the capability of the iBEM for virtual experiments of composites. Some interesting discoveries of microstructure-dependent material behavior are reported with the aid of virtual experiments.