Mathematical Vibration Modelling of the Pre-Stressed Viscoelastic Thick-Walled Cylindrical Shell

In the paper the mathematical model of the pre-stressed viscoelastic thick walled cylindrical shell's axisymmetrical vibrations, based on the deduction of the integro-differential vibration equations of such shell, is developed. The deduction of the general and approximate vibrations equations of the thick-walled shell, which in limiting cases can proceed in the vibration equations of column and thin-walled cylindrical shell, is based on the use of exact solutions of the three-dimensional problem of the elasticity theory in transformations concerning potentials of longitudinal and transversal waves. The received equations belong to hyperbolic type and describe the waves distribution concerning dispersion. Herewith proposed approach allows different particular cases and generalizations. Alongside with the vibration equations, the algorithm, allowing by results of the solution of the vibration equations uniquely define the stress—strain state of considered system in its any section at the arbitrary time moment and correctly formulate initial and boundary conditions at the formulation of applied mechanics problems, is developed. Using Laplace transformation and the constants' variation method at the various boundary and initial conditions, a number of applied problems are solved. Reverse transition in the area of originals has been realized by means of the reversion table using shift theorem and theorem of folding function. Plots show how the change in the value of initial displacements affects on the strain—stress distribution.