Viscoelasticity of Fractional Order: New Restrictions on Constitutive Equations with Applications

In this paper, we analyze the restrictions on the coefficients in the constitutive equations of linear Viscoelasticity that follow from the Second Law of Thermodynamics under isothermal conditions. Especially, we analyze the constitutive equations in which fractional derivatives of real and complex order appear. We present the conditions that follow after application of the Bochner-Schwartz theorem. Conditions derived here, representing in certain cases a weak form of the Second law of Thermodynamics, are more general (weaker) than the classical Bagley-Torvik conditions widely used in Viscoelasticity Theory. Several examples that illustrate the theory are presented.