  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.
