A Finite-Difference Time-Domain Technique for Nonlinear Elastic Media and Its Application to Nonlinear Lamb Wave Propagation

Abstract

A finite-difference time-domain technique for nonlinear elastic media is proposed, which can be applied to analyze finite amplitude elastic waves in solids. The kinematic and the material nonlinearities are considered, employing a general expression for the strain energy of an isotropic solid containing the second- and third-order terms of the strain components. The accuracy of the proposed technique is demonstrated by comparison with the analytical solution for the plane longitudinal wave propagation with finite amplitude. Two-dimensional simulations are performed to demonstrate the effectiveness of this formulation for Lamb waves. First, numerical simulations without the nonlinear effects are carried out, and the spectral peaks obtained from the calculated waveforms are shown to agree well with the theoretical dispersion curves of Lamb waves. As an example with the nonlinear effects, the harmonic generation in Lamb wave propagation is also demonstrated. The results show that the growth of the second-harmonic mode occurs for an incident-wave frequency selected in accordance with the analytical phase matching condition.