A failure of a class of K-BKZ equations based on principal stretches

Abstract

The invariants in the K-BKZ constitutive equation for an incompressible viscoelastic fluid are usually taken to be the trace of the Finger strain tensor and its inverse. The basis for this choice of invariants is not derived from the K-BKZ theory, but rather is due to the perception that this is the most natural choice. Research into using other sets of invariants in the K-BKZ equation, such as the principal stretches or the eigenvalues of the Finger strain tensor (i.e., the squares of the principal stretches) is relatively new. We attempt here to derive a K-BKZ equation based on the squares of the principal stretches that models the behavior of a low-density polyethylene melt in simple shear and uniaxial elongational deformation. In doing so, two assumptions are made as to the form of the strain-dependent energy function: first, that there is a function f(q) such that the energy function can be written as the sum of f(q _i ),i = 1, 2, 3, where the q _i 'sare the squares of the principal stretches, and second that f is a power law. We find that the K-BKZ equation resulting from these two assumptions is inadequate to describe both the shear and elongational behavior of our material and we conclude that the second of the above assumptions is not valid. Further investigation, including predictions of the second normal stress difference and some finite element calculations reveals that the first assumption is also invalid for our material.