Convexity conditions for strain-dependent energy functions for membranes

The forms that the convexity, polyconvexity, and rank-one convexity inequalities take when the strain energy is required to be a function of the strain G are studied. It is shown in particular that W(G) must be an increasing function of G, in the sense that W(G)W(G) if G – G is non-negative definite. Relatively simple sufficient conditions in terms of G alone are given. Necessary and sufficient conditions in terms of G alone are found to be rather complex.