ANALYSIS OF THE PROPERTIES OF A NONLINEAR SHEAR FLOW MODEL OF THIXOTROPIC MEDIA TAKING INTO ACCOUNT THE MUTUAL INFLUENCE OF STRUCTURAL EVOLUTION AND DEFORMATION PROCESS

Abstract

A systematic analytical study was conducted on the mathematical properties of the previously proposed prototype of a nonlinear Maxwell-type constitutive equation for describing the shear flow of thixotropic media (viscous liquid polymers, viscoelastic melts, concentrated solutions, pastes, emulsions), which takes into account the mutual influence of the deformation process and structural evolution (the kinetics of formation and breaking of intermolecular bonds) on viscosity and shear modulus and the effect of the deformation process 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 system of two nonlinear autonomous differential equations for the stress and the structuredness parameter. It is proved that the equilibrium position of this system is unique. The dependences of the position 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 dependence of the equilibrium stress on the shear rate and to a decreasing apparent viscosity curve, which reflect the typical properties of the experimental flow curves of pseudoplastic media. Using six arbitrary material parameters and a material function that control the model, we analytically study the phase portrait of the nonlinear system of two differential equations, to which the model is reduced, for dimensionless stress and the degree of structuredness in the vicinity of its only equilibrium position. It is proved that the equilibrium position is always stable and can be of three kinds only: a stable node, a degenerate node, or a stable focus. Criteria for each kind are found in the form of explicit constraints on the material function, model parameters, and shear rate. The existence of a stable focus indicates the nonmonotonicity of the system solutions and the existence of deformation modes with (damped) fluctuations of stress and structuredness when stationary values are reached. 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.