Third‐order effects in the propagation of finite amplitude stress waves

Abstract

The propagation of a finite amplitude stress wave through a homogeneous, isotropic elastic solid is analyzed for cases of one‐dimensional dilatation and shear deformation. The strain energy function is expanded in invariant form as a fourth‐order polynomial in displacement gradients. The displacement equations of nonlinear dynamic elasticity are solved by perturbation techniques that yield a uniformly valid approximation. The result for a dilatational wave, which matches previous predictions, is essentially like a planar wave in a fluid, aside from redefinition of the coefficient of nonlinearity. The primary nonlinear effect for a dilatational wave is encountered at the second order. In contrast, the shear wave case, which was not fully analyzed in previous investigations, has no analog in inviscid fluids. The first‐order shear displacement leads to a second‐order dilatational displacement, which then influences the shear displacement at the third order. The overall effect of nonlinearity is to produce amplitude dispersion, as well as a small phase shift in the waveform. [Work supported by the NSF.]