Purely elastic material models have a limited validity. Generally, a certain amount of energy absorbing behaviour can be observed experimentally for nearly any material. A large class of dissipative materials is described by a time-and frequency-dependent viscoelastic constitutive model. Typical representatives of this type are polymeric rubber materials. A linear viscoelastic approach at small and large strains is described in detail and this makes a very efficient numerical formulation possible. The underlying constitutive structure is the generalized Maxwell-element. The derivation of the numerical model is given. It will be shown that the developed isotropic algorithmic material tensor is even valid for the current configuration in the case of large strains. Aspects of evaluating experimental investigations as well as parameter identification are considered. Finally, finite element simulations of time-dependent deformations of rubber structures using mixed elements are presented.

IntroductionResearch on realistic and reliable descriptions for rubber material has been carried out over the last 50 years and is still under way. Parallel to the development of fast computer facilities, the efficient numerical realization of these material formulations in the context of a finite element method was and is still of great importance. The two aspects mentioned -material formulations and their numerical implementation -are main objectives of this paper. A generally applicable, explicit representation for a certain class of hyperelastic materials at finite strains is presented in a form easy to implement and to run efficiently on a computer. The formulation is derived with respect to the current configuration and, therefore, restricted to isotropy. This Eulerian description is of certain interest because a spatial finite element approach has the same simple structure as the linear formulation. Moreover, true stresses are defined in the current configuration. Subsequently, this framework is applied to several constitutive models. Large-scale three-dimensional numerical investigations illustrate computations with standard elements and mixed elements utilizing the material models shown. Some recent papers also draw attention to the field of computational modelling of rubber-like materials. Miehe (1994) derives the theoretical background for finite elasticity in an Eulerian setting, i.e. with respect to the current configuration. Main emphasis is put on the comparison of different mixed finite element implementations, especially for plane problems. Compressible and incompressible materials are considered. Van den Bogert and de Borst (1994) mainly investigated a non-homogeneous shear experiment numerically, and this will be described later in this paper. The applied material constants are determined only by uniaxial elongation tests. A similar formulation as that given in our paper is presented by Liu et al. (1994). They point out the performance of different element formulations. Moreover, a huge number of papers deal with constitutive modelling (e.g.

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Based on the generalized Maxwell-model, a viscoelastic material approach for steady-state rolling structures has been developed. Unlike a transient ®nite element formulation the ®nal state is attained in a few load increments and just one time step. The consistent linearization of the steady-state viscoelastic constitutive formulation leads to additional coupling matrices so that the number of non-zero entries in the global stiffness matrix is increased. The steady-state formulation of the viscoelastic material approach as well as the transient formulation allow the addition of so-called Prandtl-elements to consider elastoplastic effects, too. Numerical results con®rm the robustness, reliability and capability of the steady-state viscoelastic material formulation.