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 (W) over tilde (F) = W(alpha) := 1/2 Sigma(3)(i=1) f(alpha(i))(2) + beta Sigma(1 <= i<j <= 3) f(alpha(i))f(alpha(j)), where f : (0, infinity) --> R is a function satisfying f ( 1) = 0, f '( 1) = 1, beta is an element of R, and alpha(1), alpha(2), alpha(3) are the singular values of the deformation gradient F. Two general situations are determined under which (W) over tilde is not rank 1 convex: ( a) if ( simultaneously) the Hessian of W at alpha = (1, 1, 1) is positive definite, beta not equal 0, and f is strictly monotonic, and/or (b) if f is a Seth strain measure corresponding to any m is an element of R. No hypotheses about the range of f are necessary.