Hydroelastic Solitary Deformation Waves in a Cylindrical Shell with Combined Nonlinearity and with Energy Dissipation

Abstract

We study longitudinal nonlinear solitary deformation waves in an elastic cylindrical shell completely filled with a viscous fluid. In the case under consideration, the state of the shell is assumed to be momentless, and the stress–strain relation for its material has the form of a generalized Hooke’s law which additionally allows for the dependence of the components of the stress tensor on the strain tensor components to the power of 3/2. Moreover, the quadratic nonlinearity of the initial equations of motion of the shell element is also taken into account. The problem of hydroelasticity for the specified shell is formulated, and asymptotic analysis by the method of two-scale expansions yields an evolution equation that generalizes the Schamel–Korteweg–de Vries equation by incorporating the dissipation of the energy of the wave process in the material of the shell and the fluid. The evolution of hydroelastic solitary waves in a shell is studied numerically with a proposed difference scheme similar to the Crank–Nicolson scheme for the heat equation. The scheme is verified with an exact particular solution for the case where the influence of the fluid in the shell and dissipation in its material are excluded from consideration. It is shown that for this case, the speed of deformation waves is supersonic, and they are solitons. On the other hand, it has been established that when the dissipative properties of the shell material and the fluid, as well as the inertia of the latter, are taken into account, the speed of deformation solitons is subsonic, and their destruction is observed.