Reformulation of strain invariants at incompressibility

Abstract

The description of (quasi-)incompressible materials such as elastomers is well established in modern continuum mechanics for many years now as well as from a theoretical background as in the numerical implementation in commercial software packages in the context of finite elements. Nevertheless, some questions arise in the practical application of that matter describing technical components, e.g., in the discussion What are valid equivalent measures in order to compare different deformations? Here, one could request for expressions that arise in a deformation intensity and an indication of the deformation mode locally at each material point. We propose an extension of the well-known description of incompressible kinematics. We reformulate the strain invariants at incompressibility in terms of (I 1, I 2) leading to an equivalent pair (λ, m) in order to determine a distance of an arbitrary deformation state, e.g., to its equivalent shearing state. Therefore, we postulate and define an associated deformation state and give the mathematical derivation of quite nice relationships, which we demonstrate on a shear example using the finite elements method to visualize the "new" measure quantity. © 2012 The Author(s).