Formulation and implementation of three-dimensional viscoelasticity at small and finite strains

Abstract

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.

Generalized parameter identification for finite viscoelasticity

“…First the Mullins effect, a stress softening occurring in the first load cycle(s) and second a viscoelastic material behavior. Detailed descriptions of the classical theory of rate dependent materials can be found for instance in Findley [17], Eringen [16], Lemaitre and Chaboche [32], Haupt [21], Holzapfel [23], Kaliske and Rothert [27] as well as Simo and Hughes [70]. Constitutive laws which describe the stress softening due to the Mullins effect have been elaborated by, e.g.…”

Finite strain formulation of viscoelastic damage model for simulation of fabric reinforced polymers under dynamic loading

“…This tensor is determined thanks to the Boltzman superposition principle and is expressed in an iterative form [9,10] bỹ…”

“…Accordingly, the spectral viscoelastic model is replaced by a generalized Maxwell model. It has been successfully extended and used in finite strain for anisotropic [9,10] …”

Dynamics and Stability of Rolling Viscoelastic Tires

“…This viscoelastic model is based on a model of Simo [31] which is found in many finite element codes. Similar approaches were used in [17,18]. However, in [28] Govindjee and Reese noted that the model of Simo has not been proven to satisfy the 2nd Law of Thermodynamics and the linearity of the model restricts its validity to strain states near elastic equilibrium.…”

“…The model of Simo is used in many finite element programs and similar approaches are taken [17,18]. That this viscoelastic model can lead to increasing elastic energy is not as surprising if we consider the later work of Govindjee and Reese, which points out some of the drawbacks of the model of Simo.…”