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

Abstract

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 $$\widetilde{W}{\left( {\user2{F}} \right)} = W{\left( \alpha \right)}: = \frac{1}{2}{\sum\limits_{i = 1}^3 {f{\left( {\alpha _{i} } \right)} + \beta } }{\sum\limits_{1 \leqslant i < j \leqslant 3} {f{\left( {\alpha _{i} } \right)}f{\left( {\alpha _{j} } \right)}} },$$ where \(f:{\left( {0,\infty } \right)} \to \mathbb{R}\) is a function satisfying \(f{\left( 1 \right)} = 0,\;f'{\left( 1 \right)} = 1,\;\beta \in \mathbb{R}\), and α 1, α 2, α 3 are the singular values of the deformation gradient \({\user2{F}}\). Two general situations are determined under which \(\widetilde{W}\) is not rank 1 convex: (a) if (simultaneously) the Hessian of W at α is positive definite, \(\beta \ne 0\), and f is strictly monotonic, and/or (b) if f is a Seth strain measure corresponding to any \(m \in \mathbb{R}\). No hypotheses about the range of f are necessary.