Development and Application of the Cauchy–Poisson Method to Layer Elastodynamics and the Timoshenko Equation

Abstract

We consider a generalization of the Cauchy–Poisson method to an n-dimensional Euclidean space and its application to the construction of hyperbolic approximations. In Euclidean space, constraints on derivatives are introduced. The principle of hyperbolic degeneracy in terms of parameters is formulated and its implementation in the form of necessary and sufficient conditions is given. As the particular case of a four-dimensional space with preserving operators up to the sixth order a generalized hyperbolic equation is obtained for bending vibrations of plates with coefficients dependent only on the Poisson number. As special cases, this equation includes all the well-known Bernoulli–Euler, Kirchhoff, Rayleigh, and Timoshenko equations. As a development of Maxwell’s and Einstein’s research on the propagation of perturbations with finite velocity in a continuous medium, Tymoshenko’s non-trivial construction of the equation for bending vibrations of a beam is noted.