Nonlinear Model of Shear Flow of Thixotropic Viscoelastoplastic Continua Taking into Account the Evolution of the Structure and Its Analysis

We formulate a nonlinear Maxwell-type constitutive equation for shear deformation of polymers in flow state or polymer viscoelastic melts and solutions which takes into account interaction of deformation process and structure evolution, namely, influence of the kinetics formation and breakage of chain cross-links, agglomerations of molecules and crystallites on viscosity and shear modulus and deformation influence on the kinetics. The constitutive equation is governed by an increasing material function and six positive parameters. We reduce it to the set of two nonlinear autonomous differential equations for two unknown functions (namely, stress and relative cross-links density) and prove existence and uniqueness of its equilibrium point and that its coordinates depend monotonically on every material parameter and on shear rate. We derive general equations for model flow curve and viscosity curve and prove that the first one increases and the second one decreases while the shear rate grows. Thus, the model describes basic phenomena observed for simple shear flow of shear thinning fluids.