The Isogeometric Boundary Element Method Analysis For 2D Inhomogeneities Problems

Recently, composite materials have received a lot of attentions in industry and scientific research area since these materials have excellent mechanical properties. And the research of the problems in composite materials are mainly focus on the inhomogeneity problems. In this paper, we are mainly focus on the inclusion problems of inmohogeneity for both the macro scale and nano scale. For the macro scale, we are mainly focus on the effective mechanical properties of the composites and its effective thermal conductivity under steady steate heat loading. And for the nano-inhomogeneities, due to its own volume effect, quantum size effect, and surface effect, it presents the different properties from macro composite materials such as mechanical, thermal and the special performance properties of the original component. The development of nanocomposites prompted scientists to study inclusion theory in the nanometer scale. It is very meaningful to study stress field around inclusions or that the inhomogeneous composite overall mechanical properties. Nano inclusions in the material are widespread, so it has important practical value to study the local mechanical properties.This paper studies the problem of the effective elastic modulus of the composite materials which contain complex inclusion under doubly periodic loading of macro scale or the stress field around the inclusions, which are embedded in the infinite elastic isotropic domain under remote axis loading of nano scale, also the effective thermal conductivity of composite materials which contain complex inclusion under steady steat heat flow of macro scale is taken into consideration. The first part of our work is mainly consider the macro scale inclusion problems, where we taken doubly periodic loading into consideration and use the IGBEM to build complex models and do the analysis work. Then based on the first part`s fine result, we will consider the nano scale problem. Since the problem is under the nanometer level, the interface/surface effect should be considered. In numerical implementation process, we choose Gurtin-Murdoch model to simulate the inclusion-matrix-interface, where the stress field of nano inclusion composite is studied. And then, we will consider the effective thermal conductivity of the infinite composite materials contain complex inclusions under the steady steat heat flow. In this paper, we take the IGBEM framework into consideration and use the NURBS tools to build the isogeometric model and use the quadratic isogeometric element to discretize the boundary.Some classical examples are taken into consideration to test the accurate and effective of the IGBEM and the IGBEM`s results are also compared with tranditional BEM, and they are in good agreement. The results in this paper are also contrasted with other people`s results, and we find that our results is accurate and effective.