Energy–momentum consistent finite element discretization of dynamic finite viscoelasticity

This 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 non-linear 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 energy–momentum 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. Copyright © 2009 John Wiley & Sons, Ltd.

[1]  Donald A. French,et al.  Continuous finite element methods which preserve energy properties for nonlinear problems , 1990 .

[2]  F. Armero,et al.  On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part II: second-order methods , 2001 .

[3]  J. C. Simo,et al.  Penalty function formulations for incompressible nonlinear elastostatics , 1982 .

[4]  M. Crisfield,et al.  A co‐rotational element/time‐integration strategy for non‐linear dynamics , 1994 .

[5]  T. Hughes,et al.  Space-time finite element methods for elastodynamics: formulations and error estimates , 1988 .

[6]  Peter Betsch,et al.  Conservation properties of a time FE method. Part IV: Higher order energy and momentum conserving schemes , 2005 .

[7]  Jacob Lubliner,et al.  A model of rubber viscoelasticity , 1985 .

[8]  B. Simeon,et al.  A higher-order time integration method for viscoplasticity , 1999 .

[9]  Laurent Stainier,et al.  A first-order energy-dissipative momentum-conserving scheme for elasto-plasticity using the variational updates formulation , 2008 .

[10]  Michael Groß,et al.  Higher-order accurate and energy-momentum consistent discretisation of dynamic finite deformation thermo-viscoelasticity , 2009 .

[11]  J. C. Simo,et al.  On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects , 1987 .

[12]  Thomas J. R. Hughes,et al.  Improved numerical dissipation for time integration algorithms in structural dynamics , 1977 .

[13]  S. Hartmann.  Computation in finite-strain viscoelasticity: finite elements based on the interpretation as differential–algebraic equations , 2002 .

[14]  Christian Miehe,et al.  Exponential Map Algorithm for Stress Updates in Anisotropic Multiplicative Elastoplasticity for Single Crystals , 1996 .

[15]  Olivier A. Bauchau,et al.  Energy decaying scheme for non-linear beam models , 1996 .

[16]  Christian Miehe,et al.  A formulation of finite elastoplasticity based on dual co- and contra-variant eigenvector triads normalized with respect to a plastic metric☆ , 1998 .

[17]  J. C. Simo,et al.  Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics , 1992 .

[18]  Ted Belytschko,et al.  On the Unconditional Stability of an Implicit Algorithm for Nonlinear Structural Dynamics , 1975 .

[19]  Francisco Armero,et al.  Energy-dissipative momentum-conserving time-stepping algorithms for finite strain multiplicative plasticity , 2006 .

[20]  P. Tallec,et al.  Three-dimensional incompressible viscoelasticity in large strains: Formulation and numerical approximation , 1993 .

[21]  Peter Betsch,et al.  Conservation properties of a time FE method—part II: Time‐stepping schemes for non‐linear elastodynamics , 2001 .

[22]  S. Reese,et al.  A theory of finite viscoelasticity and numerical aspects , 1998 .

[23]  Gerhard A. Holzapfel,et al.  ON LARGE STRAIN VISCOELASTICITY: CONTINUUM FORMULATION AND FINITE ELEMENT APPLICATIONS TO ELASTOMERIC STRUCTURES , 1996 .

[24]  T. Laursen,et al.  Energy consistent algorithms for dynamic finite deformation plasticity , 2002 .

[25]  Peter Betsch,et al.  Numerical implementation of multiplicative elasto-plasticity into assumed strain elements with application to shells at large strains , 1999 .

[26]  Oreste S. Bursi,et al.  Iterative solutions for implicit Time Discontinuous Galerkin methods applied to non-linear elastodynamics , 2003 .

[27]  Albrecht Bertram,et al.  Material systems —A framework for the description of material behavior , 1982 .

[28]  Michael Groß,et al.  Conserving Time Integrators for Nonlinear Elastodynamics , 2004 .

[29]  Laurent Stainier,et al.  An energy momentum conserving algorithm using the variational formulation of visco‐plastic updates , 2006 .

[30]  J. C. Simo,et al.  The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics , 1992 .

[31]  Oscar Gonzalez,et al.  Exact energy and momentum conserving algorithms for general models in nonlinear elasticity , 2000 .

[32]  Gerhard A. Holzapfel,et al.  Entropy elasticity of isotropic rubber-like solids at finite strains , 1996 .

[33]  J. C. Simo,et al.  Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory , 1992 .

[34]  Paul Steinmann,et al.  A consistent time FE-method for large strain elasto-plasto-dynamics , 2008 .

[35]  Higher‐order energy‐momentum consistent time‐stepping schemes for dynamic finite thermo‐viscoelasticity , 2009 .

[36]  C. Bottasso.  A new look at finite elements in time: a variational interpretation of Runge-Kutta methods , 1997 .

[37]  Walter Noll.  A New Mathematical Theory of Simple Materials , 1972 .