Application of variational principles to the axial extension of a circularcylindical nonlinearly elastic membrane

In this paper stationary potential-energy and complementary-energy principl es are formulated for boundary-value problems for compressible or incompres sible nonlinearly elastic membranes, and full justification for adoption of the complementary principle is provided. The stationary principles are the n extended to extremum principles, which provide upper and lower bounds on the energy functional associated with the solution of a given problem. The principles are then illustrated by their application to the nonlinear probl em of the axially symmetric static deformation of an isotropic elastic memb rane. In its undeformed natural configuration the membrane has the form of a circular cylindrical surface. The cylinder is subject to a prescribed (te nsile) axial force with the ends of the cylinder constrained so that their radii remain constant. The alternative boundary condition in which the axia l displacement of the ends is prescribed instead of the axial force is also considered. The extremum principles are applied first without restriction on the form o f strain-energy function in order to obtain primitive bounds on the energy of Voigt and Reuss type commonly used in composite-material mechanics. Then , for particular forms of strain-energy function, specific bounds are obtai ned by selecting suitable trial deformation and stress fields and the bound s are optimized using a numerical procedure (which is readily adapted for o ther forms of strain-energy function). It is found that these bounds are ve ry close and hence give a good estimate of the actual energy. The associate d deformed geometry of the membrane is described together with the resultin g principal stresses.