Variational-based energy–momentum schemes of higher-order for elastic fiber-reinforced continua

Betsch

Numerical Meth Engineering

2019

Summary In this paper, we propose an energy and momentum consistent time‐stepping scheme for nonlinear elastodynamics. The algorithm is based on a mixed Hu‐Washizu–type variational principle that is inspired by the concept of polyconvexity and relies on a tensor cross product of second‐order tensors. In addition, we introduce a new algorithmic stress formula in its eigenvalue representation to model the transient behavior of hyperelastic bodies of Ogden‐type materials. Finally, several numerical examples show the superior performance of the proposed formulation in terms of numerical robustness and stability.

Section: Spectral Decompositionmentioning

confidence: 99%

Structure‐preserving space‐time discretization of a mixed formulation for quasi‐incompressible large strain elasticity in principal stretches

Betsch

Numerical Meth Engineering

2019

“…Very recently, the authors in [34] proposed a new energy-momentum (EM in the sequel) preserving time integrator [6,20,48] for reversible electro-elastodynamics building upon the works [18,22,24,25]. As shown in [34], the new EM time integrator proved to be very robust and accurate for the long-term simulation of EAPs.…”

Section: Introductionmentioning

confidence: 99%

“…In the purely mechanical case [22,24], the thermodynamical consistency of these methods is ensured by virtue of replacing the (exact) derivative of the strain energy with respect to the right Cauchy-Green deformation tensor (i.e. the consistent second Piola-Kirchhoff stress tensor) with its algorithmic counterpart.…”

Section: Introductionmentioning

confidence: 99%

A mixed variational framework for the design of energy–momentum integration schemes based on convex multi-variable electro-elastodynamics

Ortigosa

Computer Methods in Applied Mechanics and Engineering

et al. 2019

“…The EM consistent algorithms are well known for superior numerical robustness in nonlinear elastodynamics and have been successfully applied in the context of mixed variational formulations, anisotropic material behavior, nonlinear visco‐elastodynamics, and nonlinear elasto‐thermodynamics . For more details concerning the evolution of EM schemes from the beginning to recent developments, we refer to the work of Betsch…”

Section: Introductionmentioning

confidence: 99%

An energy momentum consistent integration scheme using a polyconvexity‐based framework for nonlinear thermo‐elastodynamics

Numerical Meth Engineering

Schiebl

et al. 2018

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...