Shear waves in a nonlinear elastic cylindrical shell

Asymptotic integration methods have been used to model the propagation of a shear wave beam along a nonlinear-elastic cylindrical shell of the Sanders – Koiter model. The shell is assumed to be made of a material characterized by a cubic dependence between stress and strain intensities, and the dimensionless parameters of thinness and physical nonlinearity are considered to have the same order of smallness. The multiscale expansion method is used, which makes it possible to determine the wave propagation speed from the equations of the linear approximation, and in the first essentially nonlinear approximation, to obtain a nonlinear quasi-hyperbolic equation for the main term of the expansion of the shear displacement component. The derived equation is a cubically nonlinear modification of the Lin – Reisner – Tsien equation modeling unsteady near-sonic gas flow and can be transformed into the modified Khokhlov – Zabolotskaya equation used to describe narrow beams in acoustics. The solution of the derived equation is found in the form of a single harmonic with slowly changing complex amplitude, since in deformable media with cubic nonlinearity the effect of self-induced wave essentially prevails over the effect of generation of higher harmonics. As a result, a perturbed nonlinear Schrödinger equation of defocusing type is obtained for the complex amplitude, for which there is no possibility of modulation instability development. In terms of the elliptic Jacobi function, an exact physically consistent solution, periodic along the dimensionless circumferential coordinate, is constructed.