Weakly nonlocal thermoelasticity of type II: Rayleigh waves in coated half-spaces

Abstract

It is well known that Eringen’s nonlocal thermoelasticity theory using the nonlocal Fourier law of heat conduction and its modifications has been widely applied to harmonic plane wave problems. However, it has recently been proven that a significant class of these problems such as Rayleigh wave propagation in layered nonlocal thermoelastic half-spaces is ill-posed under this theory in the sense of yielding no solutions because the number of required boundary conditions exceeds the number of constants to be determined. Therefore, it is necessary to develop nonlocal thermoelasticity theories under which the number of required boundary conditions and the number of constants to be determined are the same for all harmonic plane wave problems. In this paper, we introduce such a theory, called the weakly nonlocal thermoelasticity theory of Green–Naghdi type II (without energy dissipation). Employing this theory, we investigate Rayleigh wave propagation in a nonlocal type II thermoelastic half-space coated with a nonlocal type II thermoelastic layer. Departing from previous methodologies, we establish the explicit transfer matrix for a layer and the explicit surface impedance matrix of a half-space in the framework of this new theory. These results allow us to obtain directly the explicit secular equation of Rayleigh waves, which is a 3 × 3 determinant equation, along with the explicit H/V ratio formula. Based on these analytical results, we numerically examine the effects of nonlocality and material parameters on Rayleigh wave characteristics, focusing on both phase velocity and displacement polarization.