Analysis of simple constitutive equations for viscoelastic liquids

Abstract

The behavior of general Maxwell-like constitutive equations is analyzed. The total set of the constitutive equations includes the evolution equations for the “configuration tensor” c and an expression for the stress tensor through the tensor c. Several problems are considered. (i) It is shown that the general form for these equations derived recently from the Poissonbracket formalism, is more easily obtained by using a local approach of irreversible thermodynamics for both compressible and incompressible cases. Also, general Maxwell-like constitutive equations, with mixed upper and lower convected time derivatives, are derived using this method, (ii) The problem of Hadamard stability for these constitutive equations is then studied by using the method of “frozen coefficients”. The necessary and sufficient conditions for evolution are established. These conditions demonstrated the common results of 3D instability for the Johnson-Segalman and Phan-Thien-Tanner generalized constitutive equations, with exceptions only for equations with upper and lower convected derivatives. Therefore only Maxwell liquids with a “perfect elastic limit” are further studied. For these liquids, the evolution conditions are expressed explicitly as some constraints imposed on possible forms of free energy (elastic potential). It is also demonstrated that, for the latter liquids, the evolution conditions are the same as those for thermodynamic stability. For the incompressible case, with det c = 1, simple sufficient conditions for global evolutionarity are also found. (iii) It is shown that the configuration tensor c is positive definite for any piecewise smooth strain history. Also, simple sufficient conditions are found for the boundedness of the stress tensor at a given strain history. (iv) The results are applied to some viscoelastic constitutive equations proposed in the literature. (v) The importance of det c in the formulations of the constitutive equations and the relations between the instabilities in rheological equations and instabilities observed in flows of polymer liquids are then discussed.