Asymptotic Equations for Weakly Nonlinear Elastic Waves in a Cubic Crystal

The complex Burgers equation as a model for collinear interactions of weakly nonlinear shear plane waves in anisotropic elastic materials

“…The possibility of the appearance of quadratically nonlinear coupling in the evolution equations for shear elastic waves propagating along the threefold symmetry acoustic axis direction in a cubic crystal was shown for the first time in our paper [3]; see also [4,5]. Experimental confirmation of this fact was announced in [20].…”

“…(1) by the relation between the time derivative of the deformation gradient and the gradient of velocity that follows from (3) and (4):…”

“…Our aim in this work is to derive equations similar to those in [2] but by a different method. Here, unlike the presentation in [2] (see also [3][4][5][6][7][8]), where the method of weakly nonlinear geometric optics was used, we will apply a double-scale expansion. Instead of introducing a new fast variable and applying geometric optics-type asymptotics, which leads to evolution equations with three independent variables, we introduce here just two new independent variables: a slow time variable τ and a characteristic variable θ .…”

Riemann waves in an elastic medium with small cubic anisotropy

“…For further analysis, one must transform its expression to new axes, rotated through a suitable angle, so that the x3 axis points along the direction of propagation of the plane wave, and the strains sii themselves are expressed in terms of the components ui. Such a representation has been accomplished [5] for three special directions of motion of the wave:…”

Shock waves in an elastic medium with cubic anisotropy

“…4. When using the components of the gradients of the displacement u i instead of ij for waves propagating along the principal diagonal of the cube, the elastic potential of an incompressible medium must be taken in the form 5 (1.1)…”