Solitary Longitudinal-Bending Waves in Cylindrical Shell Interacting with a Nonlinear Elastic Medium

Nonlinearity and dispersion are the main factors that determine wave processes in thin-walled structures. In contrast to rods and plates, in thin shells the longitudinal and normal components of displacements are connected already in the linear approximation. Therefore, dynamic processes in shells are much more complex and are analyzed after some simplifications. Most often, researchers neglect the inertia of the longitudinal displacements, assuming that the middle surface of the shell is inextensible. In each case, assumptions of this kind must be strictly justified and should match the physics of the phenomenon. This article focuses on the derivation and analysis of nonlinear quashyperbolic equation modeling axisymmetric propagation of longitudinal-bending waves in infinite cylindrical shell interacting with an external nonlinearelastic medium. The shell is studied in the framework of the Timoshenko model that takes into account shear deformation and rotational inertia. Using diagonal Pade approximants for the summation of the perturbation series, the exact solitary-wave solutions of the derived equation in the form of traveling front and traveling pulse are obtained. The study shows that the exact solution in the form of a traveling front exists if the nonlinearity of the elastic medium surrounding the shell is ''soft''. It is established that the derived equation allows an implicit linearization using the Cole --- Hopf transformation. The equation permits a conditional factorization that enables to find solitary-wave solutions using the appropriate Duffing equation. The exact solutions of the derived equation can be used in problems of acoustic diagnostics and nondestructive testing of materials

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