Standard methods for determining the elastic modulus of polymer materials are based on tension, compression and bending tests of specially prepared specimens. Contrary to steel, the linear elastic area of deformation of polymer materials in standard experiments is rather small. This is the reason for a noticeable measurement error in the results obtained by such methods. The method of determination of the elastic modulus described in this study is based on the solution of the problem of compression of a thin-walled circular cylindrical shell by two parallel planes with regard to the geometrical and physical nonlinearity. The account of nonlinear effects in determination of elastic modulus makes it possible to use a considerably greater range of the loading curve in the elastic region of deformation compared with that in standard methods of testing specimens for tension, compression, and bending. The contact problem is solved by the finite-element method (ANSYS). To solve the nonlinear problem by the finite-element method, a macros-program was elaborated, which specifies the geometry, physical law, and properties of the material, boundary conditions, loading, division into finite elements, and the step scheme of the solution. The deformation of a thin polymer shell is characterized by great displacements and relatively low elastic deformations in a large range of movement of parallel planes. The solution obtained by finite element method allows making the universal loading diagram in dimensionless coordinates in the given range of thicknesses and elastic module. Equation of the universal loading diagram allows us to solve the inverse problem of determination of the elastic modulus according to experimental points on the loading diagram.