Surface Green’s functions of a horizontally graded elastic half-plane

Abstract

High-throughput mechanical testing based on functionally graded specimens is very promising for accelerating the development of new materials. However, due to the inhomogeneity-induced complexity, most existing analyses on functionally graded materials have recourse to numerical methods to predict their mechanical responses in reaction to external stimuli. This work investigates the surface Green’s functions for an inhomogeneous half-plane with horizontal exponential material gradient subject to both normal and tangential concentrated forces acting on the surface. The governing equations are first simplified by introducing appropriate potential functions, which facilitates the mathematical derivation of displacements via the Fourier transform technique. In the case of normal force, the vertical surface displacement is derived explicitly under the weak gradient assumption while the horizontal surface displacement is derived directly without the same assumption. In particular, the Meijer G-function and Fox H-function are introduced to express and simplify the vertical displacement. In the case of tangential force, the analytical expressions of surface displacements are also derived similarly. It is noted that the surface Green’s functions not only exhibit singularity and asymmetry properties as expected, but also can be reduced to the classical Boussinesq-Flamant solutions for a homogeneous half-plane. In addition, the analytical results are verified through comparison with the finite element analyses. The surface Green’s functions derived here could be a theoretical basis for developing high-throughput mechanical testing methods which use specimens made of functionally graded materials.