Implicit constitutive models with a thermodynamic basis: a study of stress concentration

Abstract

Motivated by the recent generalization of the class of elastic bodies by Rajagopal (Appl Math 48:279–319, 2003), there have been several recent studies that have been carried out within the context of this new class. Rajagopal and Srinivasa (Proc R Soc Ser A 463:357–367, 2007, Proc R Soc Ser A: Math Phys Eng Sci 465:493–500, 2009) provided a thermodynamic basis for such models and appealing to the idea that rate of entropy production ought to be maximized they developed nonlinear rate equations of the form ${\mathbf{A}\dot{\mathbf{T}}+\mathbf{B}\mathbf{D} = \mathbf{O}}$ where T is the Cauchy stress and D is the stretching tensor as well as ${\mathbf{A}\dot{\mathbf{S}}+\mathbf{B}\dot{\mathbf{E}} = \mathbf{O}}$, where S is the Piola–Kirchhoff stress tensor and E is the Green–St. Venant strain tensor. We follow a similar procedure by utilizing the Gibb’s potential and the left stretch tensor V from the Polar Decomposition of the deformation gradient, and we show that when the displacement gradient is small one arrives at constitutive relations of the form ${\boldsymbol{\varepsilon} = \mathbf{f}(\mathbf{T})}$. This is, of course, in stark contrast to traditional elasticity wherein one obtains a single model, Hooke’s law, when the displacement gradient is small. By solving a classical boundary value problem, with a particular form for f(T), we show that when the stresses are small, the strains are also small which is in agreement with traditional elasticity. However, within the context of our model, when the stress blows up the strains remain small, unlike the implications of Hooke’s law. We use this model to study boundary value problems in annular domains to illustrate its efficacy.