Bending energy of highly elastic membranes
For a membrane composed of elastic material with strain energy W per unit initial volume, approximations to the energy per unit initial area are obtained by integrating W through the thickness. The usual stretching energy M{r a) is modified by including a bending energy term aB(r a, r ab) that is quadratic in the second derivatives r ab . If W is the strain energy function for a stable material, M need not satisfy the Legendre-Hadamard material stability conditions, but the modified energy M + aB does satisfy these conditions. The special form that B takes when the membrane is isotropic is given.