Notes on a duality of stress and deformation fields in plane finite elasticity

Abstract

In [ 1] for a given plane deformation of a homogeneous isotropic compressible elastic material with strain-energy function X, the first Piola-Kirchoff stress potentials are shown to be a possible deformation for another material of the same type having a strain-energy function 22*. A formula for 27* in terms of 27 is given as well as formulae connecting the strain invariants of the two deformations. In these notes we show that the same function 27* arises as a Legendre transformation when inverting Kirchoffs form of the stress-strain relations and that the duality principle established in [11 can be deduced by means of this Legendre transformation. In addition we analyse the formulae given in [1] in terms of principal stretches. We find that if 21 and 22 are the principal stretches of the given deformation then the principal stretches of the "stress 'potential" deformation are essentially the partial derivatives of 2; with respect to 21 and 2z. Although the results of [11 were derived for homogeneous isotropic elastic materials, we emphasize that the method of derivation of the duality principle as given in section 3 also applies to anisotropic homogeneous elastic materials provided they have enough symmetry to allow finite plane strain. The reader is referred to Green and Adkins [2] for details of such materials. These notes are expected to be read in conjunction with [-11. The notation and terminology used here is that given in [-11 except that here for convenience material and spatial rectangular cartesian co-ordinates are denoted by Z K ( K 1, 2, 3) and z i (i = 1, 2, 3) respectively, and the first Piola-Kirchoff stress potentials are denoted by u a (a = 1, 2). We use the convention that equation numbers with a dagger, that is ( )t, refer to equations of [11.