Energy–momentum consistent finite element discretization of dynamic finite viscoelasticity

Abstract

SUMMARYThis paper is concerned with energy-momentum consistent time discretizations of dynamic finite viscoelasticity. Energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics. The discretization is energy-momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations. The performed approximation is based on a three-field formulation, in which the deformation field, the velocity field and a strain-like viscous internal variable field are treated as independent quantities. The new nonlinear viscous evolution equation satisfies a non-negative viscous dissipation not only in the continuous case, but also in the fully discretized system. The initial boundary value problem is discretized by using finite elements in space and time. Thereby, the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments (incremental objectivity). Although the present approach makes possible to design schemes of arbitrary order, the focus is on finite elements relying on linear Lagrange polynomials for the sake of clearness. The discrete energymomentum consistency is based on the collocation property and an enhanced second Piola-Kirchhoff stress tensor. The obtained coupled non-linear algebraic equations are consistently linearized. The corresponding iterative solution procedure is associated with newly proposed convergence criteria, which take the discrete energy consistency into account. The iterative solution procedure is therefore not complicated by different scalings in the independent variables, since the motion of the element is taken into account for solving the viscous evolution equation. Representative numerical simulations with various boundary conditions show the superior stability of the new time-integration algorithm in comparison with the ordinary midpoint rule. Both the quasi-rigid deformations during a free flight, and large deformations arising in a dynamic tensile test are considered.

On the derivation of energy consistent time stepping schemes for friction afflicted multibody systems

“…Generalized forms of these models can be found in many textbooks about constitutive modeling [17,30,43]. A theoretical framework which has been investigating viscoelastic materials in the context of energy consistent time integration was studied by Gross et al [24][25][26][27]. In this contribution, we will focus on the one dimensional case, since the dissipation will always be acting on augmented coordinates.…”

“…Since for this purpose we need to apply thermodynamically consistent models, we rely in this contribution on rheological models originating from constitutive modeling in continuum mechanics. The ideas to this approach are similar to those developed for modeling viscoelastic material behavior in dynamics, applying an energy consistent integration scheme by Gross et al [24][25][26][27]. For finite elasto-plasto-dynamics energy consistent schemes have been developed by Noels et al [42,41], Meng and Laursen [34], Mohr et al [35][36][37][38] or Armero [1,2].…”

A comparison of structure-preserving integrators for discrete thermoelastic systems

“…While both the TC scheme [30] and the energy-momentum consistent method [21] (leading to the MPD integrator) are second-order accurate finite difference formulations, the Galerkin-based ehG method makes possible the design of structure-preserving integrators of arbitrary order. Though the present work is confined to the second-order ehG method, Groß [16] has implemented the ehG method up to order six. It is further worth noting that the Galerkin-based approach comes along with a constructive procedure for the design of discrete derivatives (see also [19] for purely elastic systems).…”

“…We next apply the structure-preserving integrator developed by Groß [16] for general continuum thermoviscoelastodynamics. This scheme emanates from a new hybrid (continuous/discontinuous) Galerkin method in time and can be regarded as extension of the previously developed energy momentum (EM) schemes of Betsch and Steinmann [7] and Groß et al [19] to coupled thermoelastic problems.…”

“…An alternative approach to the design of structurepreserving integrators for coupled thermomechanical problems is based on the so-called enhanced hybrid Galerkin (ehG) method developed by Groß [16,17] in the realm of nonlinear thermoviscoelastodynamics (see also [18] for an application of the ehG method to dynamic finite viscoelasticity).…”