Structure‐preserving space‐time discretization of large‐strain thermo‐viscoelasticity in the framework of GENERIC

Large‐strain thermo‐viscoelasticity is described in the framework of GENERIC. To this end, a new material representation of the inelastic part of the dissipative bracket is proposed. The bracket form of GENERIC generates the governing equations for large‐strain thermo‐viscoelasticity including the nonlinear evolution law for the internal variables associated with inelastic deformations. The GENERIC formalism facilitates the free choice of the thermodynamic variable. In particular, one may choose (i) the internal energy density, (ii) the entropy density, or (iii) the absolute temperature as the thermodynamic variable. A mixed finite element method is proposed for the discretization in space which preserves the GENERIC form of the resulting semi‐discrete evolution equations. The GENERIC‐consistent space discretization makes possible the design of structure‐preserving time‐stepping schemes. The mid‐point type discretization in time yields three alternative schemes. Depending on the specific choice of the thermodynamic variable, these schemes are shown to be partially structure‐preserving. In addition to that, it is shown that a slight modification of the mid‐point type schemes yields fully structure‐preserving schemes. In particular, three alternative energy‐momentum‐entropy consistent schemes are devised associated with the specific choice of the thermodynamic variable. Numerical investigations are presented which confirm the theoretical findings and shed light on the numerical stability of the newly developed schemes.