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Dynamic finite deformation viscoelasticity and energy‐consistent time integration
Author(s) -
Müller Melanie,
Groß Michael,
Betsch Peter
Publication year - 2009
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200910156
Subject(s) - viscoelasticity , finite strain theory , infinitesimal strain theory , finite element method , midpoint method , mathematical analysis , mathematics , tensor (intrinsic definition) , classical mechanics , isotropy , physics , geometry , quantum mechanics , thermodynamics
In the present work we deal with the conserving integration of viscoelastic bodies undergoing finite deformations. Isotropic viscoelastic materials can be described by using a symmetric viscous internal variable ${\cal C}_i$ for measuring the inelastic strains, and the right Cauchy‐Green tensor $\cal C$ as measure of the total strain (see Reese & Govindjee [1]). Then, by using the unsymmetric product tensor ${\cal C} {\cal C}_i^{-1}$ , the purely elastic strains enter the isotropic free energy function. Alternatively, the i application of the symmetric right stretch tensor ${\cal U}_i$ as internal variable allows to define a symmetric elastic strain tensor ${\cal C}_e$ which enters the free energy (see also Miehe [2]). These two different approaches lead to different evolution equations for the viscous internal variable. In this lecture, both evolution equations are discretised by an ordinary midpoint rule at each Gauss point of a standard nonlinear displacement‐based finite element in space. For discretising the semidiscrete Hamilton's equations of motion in time, we use numerical time integrators which preserve the fundamental conservation laws of the underlying system. In particular, we make use of a modified midpoint rule according to the discrete gradient method, proposed in Gonzalez [3]. Numerical examples demonstrate the difference between both formulations. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)