Viscoelastic models with consistent hypoelasticity for fluids undergoing finite deformations

Abstract

Constitutive models of viscoelastic fluids are written with rate-form equations when considering finite deformations. Trying to extend the approach used to model these effects from an infinitesimal deformation to a finite transformation framework, one has to ensure that the tensors and their rates are indifferent with respect to the change of observer and to the superposition with rigid body motions. Frame-indifference problems can be solved with the use of an objective stress transport, but the choice of such an operator is not obvious and the use of certain transports usually leads to physically inconsistent formulation of hypoelasticity. The aim of this paper is to present a consistent formulation of hypoelasticity and to combine it with a viscosity model to construct a consistent viscoelastic model. In particular, the hypoelastic model is reversible.A methodology is proposed in this paper to model hypoelasticity as an equivalent incremental formulation of hyperelasticity. This method is based on the use of Lie derivative to write an equivalent hypoelastic model and simultaneously to respect the general covariance principle. By associating these hypoelastic models with viscous behaviors, a physically re-B E. Rouhaud Mech Time-Depend Mater alistic model is obtained and can be used to represent the behavior of viscoelastic solids or fluids that are submitted to finite transformations or large rates of deformation. The validity of this approach is illustrated with the construction of a model for a Maxwell-like viscoelastic fluid. The model is implemented in the Z-set® numerical package and solved.