On Hadamard stability and dissipative stability of the molecular stress function model of non-linear viscoelasticity

Abstract

We study the stability characteristics of the molecular stress function (MSF) model, i.e., a molecular constitutive theory for stress that extends the original Doi–Edwards model for linear polymers to the case of branched polymers, by repeating the assumption that the tension in the deformed chain is equal to its equilibrium value. We derive analytical, closed-form conditions for Hadamard stability under general 3-D high-frequency, short-amplitude wave disturbances in bi-quadratic form, which reduce to simple algebraic criteria for the cases of 1-D and 2-D disturbances. Application of the derived conditions in the case of general biaxial extension, which provides a simplified description of many processes encountered in industry and nature, shows that the MSF is Hadamard unstable for strains beyond 2. This casts doubts on its ability in predicting correct elastic response under rapid extensional deformations. The region of instability widens with the strengthening of network connectivity or the alignment strength of the flow. Dissipative stability of the MSF is examined using two necessary criteria: the first and less restrictive criterion requires the stress to be monotonically and unboundedly increasing function of strain in uniaxial elongation and simple shear. The second criterion requires the free energy and the rate of energy dissipation to be bounded functions of deformation. We find that while MSF satisfies the first stability criterion, violates the second, thus revealing thermodynamic inconsistencies in formulating the dissipative terms of the constitutive equation.