Microstructural effects on the response of a multi-layered elastic substrate

Abstract

The size dependent response of a multi-layered elastic substrate under surface loading is investigated in the present work. The effect of material microstructure is modeled using the generalized continuum theory of couple stress elasticity. In the formulation, the displacement field is chosen as a primary unknown and the generalized Navier’s equation is established for a generic material layer. The Fourier integral transform method is directly applied to derive the general solution of the elastic field for the generic layer, and such solution is subsequently utilized to form the layer stiffness equation in the transform space. Boundary conditions together with the field continuity across the material interfaces are enforced, via the standard assembly procedure, to obtain a system of linear algebraic equations governing all unknowns of the multi-layered system. An efficient quadrature is adopted to carry out all involved integrals arising from the integral transform inversion. Detailed results are reported that confirm the validity of our results through the comparison with well-known solutions and demonstrate the capability of the proposed model to simulate various scenarios of layered media including those made of functionally graded materials. The important role of the material length scale effect on the predicted response is also elucidated.