A theory for non-Newtonian viscoelastic polymeric liquids

Abstract

In many existing theories for incompressible polymeric liquids the Cauchy stress is decomposed as T=−p1+Sv+Se, where p is an arbitrary pressure, Sv=2μsD a deviatoric viscous stress with μs a viscosity and D the deviatoric stretching tensor, and Se is a deviatoric elastic stress which is introduced to account for stiffening arising from the alignment of long-chain polymer molecules during flow. A constitutive equation for Se needs to be prescribed and there are a large number of different proposals in the literature, with most proposals involving a hypoelastic rate constitutive equation for Se given in terms of a suitable frame-indifferent rate, which is usually taken as the Oldroyd or upper-convected rate. As is well-known, a hypoelastic equation for the stress is not thermodynamically consistent, in the sense that the constitutive equation for Se is not derived form a free energy function.The purpose of this paper is to present an alternative — thermodynamically-consistent and frame-indifferent — continuum theory for incompressible viscoelastic liquids. The theory is based on a Kröner-type multiplicative decomposition of the deformation gradient F of the form F = FeFp. In this theory the elastic stress Se is derived from a free-energy function which is prescribed in terms of a suitable measure based on the unimodular elastic distortion tensor Fe. This relation is supplemented by an evolution equation for the unimodular plastic distortion tensor Fp — the plastic flow rule.We study the response of the constitutive theory in steady simple shearing and steady extensional flows. We show: (i) that the theory qualitatively reproduces the experimentally-observed transient shear-thinning and normal stress effects during shearing flows of a polymer melt; and (ii) that it also reproduces the transient extensional response of a polymer melt.