Variational methods for solving nonlinear boundary problems of statics of hyper-elastic membranes

A number of important results of studying large deformations of hyper-elastic shells are obtained using discrete methods of mathematical physics. In the present paper, using the variational method for solving nonlinear boundary problems of statics of hyper-elastic membranes under the regular hydrostatic load, we investigate peculiarities of deformation of a circular membrane whose mechanical characteristics are described by the Bidermann-type elastic potential. We develop an algorithm for solving a singular perturbation of nonlinear problem for the case of membrane loaded by heavy liquid. This algorithm enables us to obtain approximate solutions both in the presence of boundary layer and without it. The class of admissible functions, on which the variational method is realized, is chosen with account of the structure of formal asymptotic expansion of solutions of the corresponding linearized equations that have singularities in a small parameter at higher derivatives and in the independent variable. We give examples of calculations that illustrate possibilities of the method suggested for solving the problem under consideration.