A para-universal relation for orthotropic materials

Abstract

Abstract A universal relation is an algebraic relation between stress and strain that holds for any material within a certain class, irrespective of the exact form of the material response function and parameter values. Classical universal relations, such as Rivlin’s famous relation for simple shear, apply to stress components produced by one and the same deformation. We present a family of relations that connect stress components under different deformations, which we call para-universal relations to highlight this difference. The proposed para-universal relations hold for any orthotropic material whose response function is additively decomposed into terms, each of which possesses a symmetry with respect to one of the axes of orthotropy. Using basic properties of the permutation group S3, we demonstrate that such an additive decomposition implies the proposed identities. The established para-universal relations hold for an arbitrary local deformation and, like classical universal relations, are linked to material symmetry and apply to a wide class of materials. Since the proposed para-universal relations do not hold for all orthotropic material models, they present a convenient way to test for the suitability of additively split strain-energy functions, which are often used to model the nonlinearly elastic response of soft tissues. Such a test can be performed on collected experimental data prior to choosing an exact form of the response function and fitting its parameters. We use published experimental data for human myocardium and also synthetic data to illustrate this.