Generalized model of nonlinear elastic foundation and longitudinal waves in cylindrical shells

A non-integrable quasi-hyperbolic sixth-order equation is derived that simulates the axisymmetric propagation of longitudinal waves along the generatrix of a cylindrical Kirchhoff – Love shell interacting with a nonlinear elastic medium. A six-parameter generalized model of a nonlinear elastic medium, which is reduced in particular cases to the models of Winkler, Pasternak, and Hetenyi, is introduced into consideration. The equation was derived by the asymptotic multiscale expansions method under the assumption that the dimensionless parameters of nonlinearity, dispersion, and thinness have the same order of smallness. The use of the introduced model made it possible to reveal additional high-frequency and low-frequency dispersions characterizing the response of the external environment to bending and shear. It is shown that non-classical shell theories should be used to reveal nonlinear effects that compensate for dispersion. It was found that the Pasternak model admits a “dispersionless” state when the dispersion due to the inertia of normal displacement is compensated by the dispersion generated by the reaction of the nonlinear elastic foundation to shear.