Analysis of the Properties of a Nonlinear Model for Shear Flow of Thixotropic Media Taking into Account the Mutual Influence of Structural Evolution and Deformation

A systematic analytical study was conducted on the mathematical properties of a previously proposed prototype of a nonlinear Maxwell-type constitutive equation for describing the shear flow of thixotropic substances (viscous liquid polymers, viscoelastic melts, concentrated solutions, pastes, emulsions). The equation takes into account the mutual influence of deformation and structural evolution (the kinetics of intermolecular bond formation and breaking) on viscosity and shear modulus and the effect of deformation on this kinetics. In the uniaxial case, the constitutive equation is governed by a nondecreasing material function and six positive parameters. The equation is reduced to a set of two nonlinear autonomous differential equations for the stress and the crosslinking degree. It is proved that the equilibrium point of this set is unique. The dependences of the point coordinates on all material parameters and on the shear rate for an arbitrary nondecreasing material function are investigated in general form, and all the dependences are proved to be monotonic. Equations for the flow and viscosity curves are derived and investigated. It is proved that the model leads to an increasing shear rate dependence of the equilibrium stress and to a decreasing apparent viscosity curve, which reflect the typical properties of the experimental flow curves of pseudoplastic materials. Using six arbitrary governing material parameters and the governing material function, we analytically study the phase portrait of the nonlinear set of two differential equations, to which the model is reduced, for dimensionless stress and crosslinking degree near its only equilibrium point. It is proved that the equilibrium point is always stable and can be of three types only: a stable node, a degenerate node, or a stable focus. The existence criteria for each type are found in the form of explicit constraints on the material function, model parameters, and shear rate. A stable focus indicates the nonmonotonicity of the set solutions and the existence of deformation modes with damped oscillations of stress and crosslinking degree upon reaching steady-state values. The influence of the material parameters and material function on the type of equilibrium point and on the behavior of the model integral curves is analyzed.