On the use of multidimensional differential geometry to model covariant behaviors of viscoelastic or hyperelastic structures, illustrated with numerical simulations using spacetime finite element analysis

Abstract

In the present article, a covariant spacetime formalism is used to model the behavior of viscoelastic and hyperelastic solids, within a thermodynamical framework. The latter aims to ensure the validity of thermodynamics second principle and to derive reversible or irreversible models for thermomechanics. The use of the Lie derivative is of particular interest to achieve such goals. Covariance enables to address finite deformation. Coupled to a covariant finite element analysis, it allows numerical simulations that simultaneously ensure the physical balance of energy and momentum for thermomechanical applications. Different mechanical loadings are considered, bending or uniaxial extension ones, with quasi-static or time exponential or time cyclic evolution. We also provide quantification of the different performances of the numerical simulations, and show the advantages and drawbacks of the spacetime approach.