On construction of approximate solutions of equations of the shallow spherical shell

Abstract

The new theory of the elastic shells leads to the elliptic system of equations of the 12th order (see I.N. Vekua, The Theory of Thin Shallow Shells with Variable Thickness, Tbilisi, Metsniereba, 1965). In the case of the spherical shell, the solutions of this system of equations may be exactly expressed by six functions wi satisfying the equations of the form ▽2w + ki2w = 0 (i = 1, 2, 3, 4, 5) where k12, k22, k42, k52 are real constants, k32 is a complex constant, w1, w2, w4, w5 are real-valued functions, and w3 is a complex-valued function. These functions are expressed by six arbitrary analytic functions of one complex variable. For the shallow spherical shells the obtained formulae may be essentially simplified. The same method may be also used for the simplification of the equations of the shallow cylindrical shells.