Bending energy of highly elastic membranes. II

The strain energy per unit area for a deformed sheet of elastic material is estimated by representing the deformation as a power series in the thickness variable. The membrane energy is the lowest-order approximation obtained in this way. Strain-gradient and bending energies appear in the next order of approximation. Neither the membrane energy nor the higher-order approximation satisfy the Legendre-Hadamard material stability condition if the stress is compressive in some direction, so the theories based on either of these approximations can lead to problems with no stable solution. An energy function that does satisfy the material stability conditions is obtained by omitting the strain-gradient term, provided that certain longitudinal moduli are positive. A modified form for which existence of solutions can be guaranteed is proposed.