Covariant spacetime formalism for applications to thermo-hyperelasticity

The principle of material objectivity used in classical continuum mechanics states that the description of material behaviors has to be frame-invariant. Moreover, geometric considerations show that objectivity can be extended by use of a covariant principle. Covariance can be naturally expressed using a four-dimensional spacetime. We thus propose to consider a thermodynamic approach within this framework to obtain covariant constitutive models in the case of finite deformation. A four-dimensional form of the Clausius-Duhem inequality is thus derived for continuum thermomechanics applications. In this article, we are able to propose a new thermo-hyperelastic constitutive model from such a generalized thermodynamics approach with the implementation of the spacetime Lie derivative. The resulting model is then illustrated for different mechanical or thermal loadings with numerical simulations.

Automated summary

This article introduces the concepts of material objectivity and covariance used in classical continuum mechanics. The authors propose a new approach to thermodynamics using four-dimensional spacetime in the case of finite deformation, and derive a four-dimensional form of the Clausius-Duhem inequality for continuum thermomechanics applications. The effectiveness of the resulting thermo-hyperelastic constitutive model is then validated through numerical simulations.