Abstract
A phenomenological theory of the non-linear visco-elasticity of three dimensional bodies is presented. We start from the two elementary models of the classical linear theory of visco-elasticity, that is, the so-called Maxwell and Voigt models. These models are extended to three dimensional non-linear case by adopting the energy consideration. The three dimensional Maxwell model is defined by “series connection” of a “non-Hookeian spring”, i.e. the internal elastic mechanism and a “non-Newtonian dashpot” the energy dissipation mechanism, and the three dimensional Voigt model is represented by the “parallel connection” of these mechanisms. Making use of the Maxwell model, we discuss the tensile viscosity for the stationary simple elongation of elasticoviscous liquids. Our models are shown to be successful in analyzing the so-called normal stress effects, i.e., the Weissenberg effect for elastico-viscous liquids and the Poynting effect for visco-elastic solids. The non-linearity in visco-elastic properties is classified into three cases: elastic, viscous or relaxational, and geometric non-linearities. An essential feature of the three dimensional bodies is the geometric non-linearity. The typical examples of this are the said normal stress effects and the velocity gradient dependence of the tensile viscosity coefficient.
