The technique suggested in the present study regarded as an alternative for determining the elastic modulus of polymeric materials. It is based on the solution of a geometrically nonlinear problem of transverse deformation of thin-walled circular cylindrical shells. The account of nonlinear effects makes it possible to use a considerably wider range of the loading curve in the elastic area of deformation, thus increasing the accuracy of results. In addition, we employ the “principle of a unified specimen” i.e., the possibility of measuring the elastic modulus as a function of the processes proceeding in a specimen and affecting the structure of a polymeric material (aging, sorption, and desorption of liquid media and vapors by the material, temperature variations, etc.).