Gibbs free energy based representation formula within the context of implicit constitutive relations for elastic solids. (May 2020) - Průša, Vít; Rajagopal, K.R.; Tůma, Karel
Abstract: We derive a representation formula for a class of solids described by implicit constitutive relations between the Cauchy stress tensor and the Hencky strain tensor. Using a thermodynamic framework, we show that the Hencky strain tensor can be obtained as the derivative of the specific Gibbs free energy with respect to a stress tensor related to the Cauchy stress tensor. Unlike previous studies that have considered implicit relations between the Cauchy stress tensor and the Hencky strain we work with quantities that allow us to split the deformation into two parts. One part is connected to deformations that change the volume and the other to deformations where volume is preserved. Such a decomposition allows us to clearly characterise the interplay between the corresponding parts of the stress tensor, and to identify additional restrictions regarding the admissible formulae for the Gibbs free energy. We also show that if the constitutive relations of this type are linearised under the small strain assumption, then one can transparently obtain linearised models with density/pressure/stress dependent elastic moduli in a natural manner. Highlights: Elastic solids described by implicit constitutive relations are considered. Thermodynamic basis for the given class of materials is developed. Representation formula relating the Hencky strain and the Cauchy stress is derived. Representation formula is given in terms of the Gibbs free energy. Deformation is split into the volume-changing and volume-preserving part.