On the Rank 1 Convexity of Stored Energy Functions of Physically Linear Stress-Strain Relations

The rank 1 convexity of stored energy functions corresponding to isotropic and physically linear elastic constitutive relations formulated in terms of generalized stress and strain measures [Hill, R.: J. Mech. Phys. Solids 16, 229–242 (1968)] is analyzed. This class of elastic materials contains as special cases the stress-strain relationships based on Seth strain measures [Seth, B.: Generalized strain measure with application to physical problems. In: Reiner, M., Abir, D. (eds.) Second-order Effects in Elasticity, Plasticity, and Fluid Dynamics, pp. 162–172. Pergamon, Oxford, New York (1964)] such as the St.Venant–Kirchhoff law or the Hencky law. The stored energy function of such materials has the form where is a function satisfying , and α ₁, α ₂, α ₃ are the singular values of the deformation gradient . Two general situations are determined under which is not rank 1 convex: (a) if (simultaneously) the Hessian of W at α is positive definite, , and f is strictly monotonic, and/or (b) if f is a Seth strain measure corresponding to any . No hypotheses about the range of f are necessary.