Energy-consistent time integration for nonlinear viscoelasticity

This paper is concerned with the numerical solution of the evolution equations of thermomechanical systems, in such a way that the scheme itself satisfies the laws of thermodynamics. Within this framework, we present a novel integration scheme for the dynamics of viscoelastic continuum bodies in isothermal conditions. This method intrinsically satisfies the laws of thermodynamics arising from the continuum, as well as the possible additional symmetries. The resulting solutions are physically accurate since they preserve the fundamental physical properties of the model. Furthermore, the method gives an excellent performance with respect to robustness and stability. Proof for these claims as well as numerical examples that illustrate the performance of the novel scheme are provided.

[1]  E. Süli,et al.  An introduction to numerical analysis , 2003 .

[2]  T. Itoh,et al.  Hamiltonian-conserving discrete canonical equations based on variational difference quotients , 1988 .

[3]  H. Ch. Öttinger,et al.  Beyond Equilibrium Thermodynamics , 2005 .

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

[5]  P. Betsch,et al.  A comparison of structure-preserving integrators for discrete thermoelastic systems , 2011 .

[6]  Gerhard A. Holzapfel,et al.  Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science , 2000 .

[7]  C. SimoJ.,et al.  The discrete energy-momentum method , 1992 .

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

[9]  Ignacio Romero,et al.  Algorithms for coupled problems that preserve symmetries and the laws of thermodynamics: Part I: Monolithic integrators and their application to finite strain thermoelasticity , 2010 .

[10]  G. Quispel,et al.  Geometric integration using discrete gradients , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

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

[12]  L. E. Malvern.  Introduction to the mechanics of a continuous medium , 1969 .

[13]  Javier Bonet,et al.  Large strain viscoelastic constitutive models , 2001 .

[14]  C. Truesdell,et al.  The Non-Linear Field Theories Of Mechanics , 1992 .

[15]  Laurent Stainier,et al.  A theory of subgrain dislocation structures , 2000 .

[16]  Ignacio Romero,et al.  Thermodynamically consistent time‐stepping algorithms for non‐linear thermomechanical systems , 2009 .

[17]  Gerhard A. Holzapfel,et al.  A new viscoelastic constitutive model for continuous media at finite thermomechanical changes , 1996 .

[18]  Ignacio Romero,et al.  An analysis of the stress formula for energy-momentum methods in nonlinear elastodynamics , 2012 .

[19]  Laura Gotusso,et al.  On the energy theorem for the Lagrange equations in the discrete case , 1985 .

[20]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[21]  K. Hutter.  The foundations of thermodynamics, its basic postulates and implications. A review of modern thermodynamics , 1977 .

[22]  R. D. Wood,et al.  Nonlinear Continuum Mechanics for Finite Element Analysis , 1997 .

[23]  Gerhard A. Holzapfel,et al.  A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computational aspects and applications , 2001 .

[24]  Francisco Armero,et al.  Volume-preserving energy–momentum schemes for isochoric multiplicative plasticity , 2007 .

[25]  B. Leimkuhler,et al.  Simulating Hamiltonian Dynamics , 2005 .

[26]  Tod A. Laursen,et al.  On energy consistency of large deformation plasticity models, with application to the design of unconditionally stable time integrators , 2002 .

[27]  Ignacio Romero,et al.  Algorithms for coupled problems that preserve symmetries and the laws of thermodynamics Part II: fractional step methods , 2010 .

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

[29]  Sanjay Govindjee,et al.  Mullins' effect and the strain amplitude dependence of the storage modulus , 1992 .

[30]  A. R. Humphries,et al.  Dynamical Systems And Numerical Analysis , 1996 .

[31]  Ignacio Romero,et al.  Energy-Entropy-Momentum integration of discrete thermo-visco-elastic dynamics , 2012 .

[32]  Peter Betsch,et al.  Energy–momentum consistent finite element discretization of dynamic finite viscoelasticity , 2010 .

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

[34]  M. Gurtin,et al.  Thermodynamics with Internal State Variables , 1967 .