Semi-analytical solution for the Lamb’s problem in second gradient elastodynamics

Abstract

In the present paper, we consider the Lamb-type problem of the dynamic response of isotropic half-space subjected to the time-harmonic vertical point load within the simplified strain gradient elasticity theory (SGET). The classical elastodynamics solution of this problem predicts singular displacements and stresses at the point of the load application. In the present study, we use the technique of Lamé potentials and Hankel transform to solve the high-order motion equations of SGET. An explicit solution accounting for traction and double traction boundary conditions is obtained in the transformed domain. The inverse transformation is performed numerically with accurate capturing of the solution behavior around the Rayleigh pole, which position is defined by the modified scale-dependent Rayleigh equation. Based on obtained results, it is shown that SGET solution is bounded at the epicenter of the load, and that it takes into account the spatial dispersion effects for the propagated waves. These effects are important for the analysis of the high-frequency processes in the structured materials. The intensity of non-classical effects in the solution is controlled by the length scale parameters of gradient model. Based on numerical calculations, we derive the universal non-dimensional relations between the amplitudes of the applied load and maximum displacements on the surface, which can be used to identify the length scale parameters of SGET based on the atomistic calculations or experimental data.