Abstract
AbstractLarge‐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.
Highlights
Structure-preserving time integration schemes, known as geometric integrators, have been an active topic of research in applied mathematics and computational mechanics for decades
Considering the description of large strain thermoelasticity, a GENERIC-based weak form is derived in Betsch and Schiebl[39] which makes possible the free choice of thermodynamic state variable following the approach presented in Mielke.[25]
Choosing the internal energy density as thermodynamic variable leads to an EM scheme, while the choice of entropy density yields a momentum-entropy method
Summary
Structure-preserving time integration schemes, known as geometric integrators, have been an active topic of research in applied mathematics and computational mechanics for decades (see Leimkuhler and Reich[1] and Hairer et al.[2] for a large number of contributions to the field). Considering the description of large strain thermoelasticity, a GENERIC-based weak form is derived in Betsch and Schiebl[39] which makes possible the free choice of thermodynamic state variable following the approach presented in Mielke.[25] It was shown in Betsch and Schiebl[39] that application of the standard mid-point rule to the GENERIC-based weak form already yields partially structure-preserving schemes. Choosing the internal energy density as thermodynamic variable leads to an EM scheme, while the choice of entropy density yields a momentum-entropy method Despite of their (partially) structure-preserving properties, all of the mid-point type schemes considered in Betsch and Schiebl[39] cannot prevent numerical instabilities.
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