A Direct Approach to Membrane Reinforced Bodies

Abstract

The paper deals with membrane reinforced bodies with the membrane treated as a two dimensional surface with concentrated material properties. The membrane response is linearized so that it depends linearly on the surface strain tensor. The response of the matrix is treated separately in three cases: (a) as a nonlinear material, (b) as a linear material and finally (c) as a no-tension material. For the general nonlinear material, the principle of minimum energy and complementary energy are proved. For the linearly elastic matrix the surface Korn inequality is used to prove the existence of equilibrium state under general loads. Finally, for the no-tension material a theorem stating that the total energy of the system is bounded from below on the space of admissible displacements if and only if the loads are equilibrated by a statically admissible stress that is negative semidefinite. An example presenting an admissible stress solution is given for a rectangular panel with membrane occupying the main diagonal plane.