The present contribution provides a new approach to the design of energy momentum consistent integration schemes in the field of nonlinear thermo-elastodynamics. The method is inspired by the structure of polyconvex energy density functions and benefits from a tensor cross product that greatly simplifies the algebra. Furthermore, a temperature-based weak form is used, which facilitates the design of a structure-preserving time-stepping scheme for coupled thermoelastic problems. This approach is motivated by the general equation for nonequilibrium reversible-irreversible coupling (GENERIC) framework for open systems. In contrast to complex projection-based discrete derivatives, a new form of an algorithmic stress formula is proposed. The spatial discretization relies on finite element interpolations for the displacements and the temperature. The superior performance of the proposed formulation is shown within representative quasi-static and fully transient numerical examples. KEYWORDSfinite element method, nonlinear thermo-elastodynamics, polyconvexity-based framework, structure-preserving discretization, tensor cross product INTRODUCTIONThe present contribution aims for the consistent discretization of nonlinear thermo-elastodynamics. The emphasis of this paper is on both theoretical and numerical aspects. In recent decades, thermomechanical constitutive models have been addressed in numerous works (see, eg, the works of Miehe, 1 Holzapfel and Simo, 2 and Reese and Govindjee 3 as well as textbooks by Holzapfel 4 and Gonzalez and Stuart 5 ). Classically, the hyperelastic Helmholtz free-energy density function depends only on the deformation gradient and the temperature. Furthermore, the weak form is deduced from its strong form. Dependent on the chosen material model, eg, for a Mooney-Rivlin model, the consistent linearization may lead to cumbersome expressions. In contrast to the classic approach, the present work is inspired by the concept of polyconvexity (see, eg, the works of Ball 6 and Ciarlet 7 ), where the Helmholtz free energy is a convex function of the deformation gradient, its cofactor, and its determinant and is concave with respect to absolute temperature. In addition to the polyconvexity-based framework, the present work relies on the cross product between second-order tensors, as introduced in the work of de Boer 8 (see also Appendix B 4.9.3 in another work of de Boer 9 ). This tensor cross product has been used in the context of large-strain Int J Numer Methods Eng. 2018;115:549-577. wileyonlinelibrary.com/journal/nme 549 550 FRANKE ET AL.hyperelasticity in the works of Bonet et al 10,11 and remarkably simplifies the algebraic manipulation of the large-strain continuum formulation and is extended herein to the thermomechanical formulation. Furthermore, the polyconvexity-based framework makes possible a wide range of advanced finite element technologies in the context of continuum mechanics. For example, the introduction of mixed finite elements, 10,12 phase-field fracture models, 13 anisotropic...

Variationally consistent phase-field methods have been shown to be able to predict complex three-dimensional crack patterns. However, current computational methodologies in the context of large deformations lack the necessary numerical stability to ensure robustness in different loading scenarios. In this work, we present a novel formulation for finite strain polyconvex elasticity by introducing a new anisotropic split based on the principal invariants of the right Cauchy-Green tensor, which always ensures polyconvexity of the resulting strain energy function. The presented phase-field approach is embedded in a sophisticated isogeometrical framework with hierarchical refinement for three-dimensional problems using a fourth order Cahn-Hilliard crack density functional with higher-order convergence rates for fracture problems. Additionally, we introduce for the first time a Hu-Washizu mixed variational formulation in the context of phase-field problems, which permits the novel introduction of a variationally consistent stress-driven split. The new polyconvex phase-field fracture formulation guarantees numerical stability for the full range of deformations and for arbitrary hyperelastic materials.

Thermomechanical Mortar contact algorithms and their application to NURBS based Isogeometric Analysis are investigated in the context of nonlinear elasticity. Mortar methods are applied to both the mechanical field and the thermal field in order to model frictional contact, the energy transfer between the surfaces as well as the frictional heating. A series of simplifications are considered so that a wide range of established numerical techniques for Mortar methods such as segmentation can be transferred to IGA without modification. The performance of the proposed framework is illustrated with representative numerical examples. © 2014 Elsevier B.V

An energy-momentum time integration scheme based on a convex multi-variable framework for non-linear electro-elastodynamics. Computer Methods in Applied Mechanics and Engineering

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