Effects of pre-stresses on the propagation of nonlinear surface waves in an incompressible elastic half-space

Abstract

The speed υv of Rayleigh waves in a pre-stressed isotropic incompressible elastic half-space is a function of the pre-stress and there exist states of pre-stress which induce zero Rayleigh wave speed. In the special case when the pre-stresses are in the form of an all-round pressure, Rayleigh waves become standing waves (corresponding to υ = 0) when p&¯ = ±2 and degenerate into shear body waves when p&¯ = −1, where p&¯⁠; is the pressure scaled by the shear modulus. In this paper we investigate the effects of pre-stresses on the propagation of nonlinear surface waves in a pre-stressed elastic half-space. The evolution equations are derived using a new approach which we believe to be simpler than any other method which has been proposed before in the literature. When specializing to the case when the pre-stresses are in the form of an all-round pressure with −2 < p&¯ < 2 and p&¯≠ −1, we find that the nonlinear term which involves second-order elastic moduli in the evolution equations is identically zero and so material nonlinearity has no effects on the surface wave at the order considered (the only nonlinear effect is due to interaction between pressure and displacement). We also find that (i) the main effect of varying p&¯ is to change the shock formation time, variation of p&¯ has little effect on the shock amplitude and the manner in which shocks are formed, (ii) as p&¯ tends to the neutral value of 2, the shock formation time decreases like υ, implying that the corresponding near-neutral modes evolve on a shorter time scale, (iii) as p&¯ tends to the other neutral value of −2, the shock formation time increases like υ−1 which is quite unexpected, and (iv) as p&¯ tends to −1 , the surface wave tends to a shear body wave and the shock formation time increases like (⁠p&¯ +1)−4 which is consistent with previous results on the order of shock formation time for shear body waves.