The influence of non-stationary pressure on a thin spherical shell with an elastic filler

Abstract

This article considers the non-stationary problem of a thin elastic spherical shell filled with an elastic medium under external pressure. The equations of Timoshenko’s model serve as the basis for describing the motion of the spherical shell. The reaction of the elastic filler is described by the equations of the theory of elasticity. The contact between the shell and the filler is assumed frictionless. Based on the principle of superposition, we have obtained an integral relationship between the normal displacements of the shell with filler and the external pressure to solve the problem. The kernel of this integral relationship is an influence function. This function is, in fact, the normal displacements of the shell with filler taken as a solution to the problem of the concentrated instantaneous external pressure effect. It is modeled by the use of Dirac delta functions. To get the influence function, the authors have applied a Fourier series expansion in terms of eigenfunctions of shell and filler sub-problems. A partial separation of variables leads to a system of differential equations for expansion series coefficients. Due to the substitution of the corresponding expansions into the equations of motion of shells, the system contains ordinary differential equations and partial differential equations with respect to the expansions coefficients of elastic displacement potentials in the filler. The relationship between these systems is governed by the contact conditions between the shell and the filler. To determine the series expansion coefficients, we have produced the Laplace transform in the time domain. As a result, the problem is reduced to solving the systems of algebraic and ordinary differential equations. The solution to these equations, taking into account contact conditions, leads to the expressions describing the coefficients for series expansion of the desired influence function. This is achieved through the relation between the modified Bessel functions of the first kind and the elementary functions. We have found the coefficients in the time domain analytically using exponent series expansions. The influence function and the integral relationship are used to solve a set of test problems. Analytical calculations of the corresponding integrals are carried out. The results of calculations are represented graphically with an assessment of their convergence based on the number of coefficients of series expansions of the influence functions.