Effect of compressibility on the mechanics of hyperelastic membranes

Elastic membranes are usually studied assuming material incompressibility. However, in several applications they are made of compressible materials such as polymeric foams, hydrogels, and certain kinds of elastomers. Only a few works attempted to incorporate volume changes into membrane problems, but with significant limitations. The models proposed were designed for nearly incompressible materials and lacked a foundation in experimental data, leading to results of limited value. In this work, we investigate the effect of compressibility in membrane problems adopting a consistent model based on the real response of materials to large volume changes. We consider three benchmark problems of nonlinear elasticity: (i) inflation of a circular flat membrane; (ii) inflation of a thin-walled cylindrical tube; (iii) inflation of a thin-walled spherical balloon. Four types of materials divided by increasing degree of compressibility are studied. The results indicate that volumetric deformations have a significant impact on both the limit pressure and the deformed shape. The proposed solutions represent benchmarks for developing new applications of compressible membranes made of polymeric foams and hydrogels, playing an increasingly important role in engineering technologies.