A multiscale hybrid method for materials containing defects and inhomogeneities

The Finite Element Method (FEM) requires full-domain discretization and is generally more suitable for problems with comparable characteristic length scales. When applied to multiscale problems, FEM becomes inefficient or even practically impossible, as the number of discretized elements can be prohibitively large. In this work, a Multiscale Hybrid Method (MHM) is developed for the micromechanics of materials containing defects and inhomogeneities of different length scales. MHM is based on a hybrid boundary integral and volume integral equation approach. It is capable of providing the accurate solution of elastostatic 2-D and 3-D multiscale problems in bounded and unbounded solids containing multiple interacting inclusions of essentially different scales. In MHM, the interaction between the inhomogeneities and the domain boundary is explicit in the governing equations. Furthermore, there is no discretization of the transmitting medium (matrix), and the discretization at the inhomogeneity length scale is fully independent of that at the domain boundary scale. As a result, the need for mesh transition, which is the primary means that conventional methods rely on in handling interaction between the various scales, is eliminated in MHM. The accuracy and numerical efficiency are enhanced by the use of fundamental solution in the governing equation and the Eshelby's inclusion solutions. Different from Eshelby's approach in which strains and stress are the basic variables, the displacements are taken as the basic variables in the present method. The displacement-based approach lowers the order of singularity in the governing integral equations and offers high accuracy at reduced computational cost. A coordinate origin shift scheme is used to remove the weak singularity in the integro-differential equation, while the subtraction technique and the Eshelby's solution are employed to remove the strong singularity in the calculation of stresses inside the inhomogeneities. Moving Least Squares Approximation is used to construct a smooth solution field over the inclusions, A special numerical integration scheme with symmetric quadrature points for spherical inhomogeneities is devised to elevate the accuracy of numerical integration. The subtraction' method for the source points outside inhomogeneities is developed to further enhance accuracy. The method can handle arbitrary geometry and general loading conditions at the macro scale and the interaction of microstructural features at the micro scale, and offers orders of magnitude of increased efficiency over FEM. (Abstract shortened by UMI.)...