Fundamental solutions of multi-layer elastic media with couple stress theory

Abstract

This thesis presents fundamental solutions of a two-dimensional, elastic, multi-layered medium under surface loadings by taking the influence of material micro-structure into account. An underlying mathematical model for simulating such small-scale influence is established within the continuum-based framework via the well-known couple stress theory. For each material layer, the generalized Navier’s equation governing the displacement field is established and the method of Fourier integral transform is applied to derive its general solution in a transformed space. A set of boundary conditions and the continuity of fields along the material interfaces are enforced to obtain a system of linear algebraic equations governing all unknown degrees of freedom of the whole layered medium in the transformed space. An efficient quadrature is then adopted to carry out all involved integrals arising from Fourier integral transform inversion. A selected set of results is also reported not only to confirm the validity of established solutions but also to demonstrate the capability of the selected mathematical model to simulate the size-dependency when the external and internal length scales are comparable. An approximation of a functionally graded material rested on a rigid base is also investigated using the same model.