On Hadamard stability for compressible viscoelastic constitutive equations

Abstract

In this paper, the necessary and sufficient condition of the global Hadamard stabiltiy is formulated for isothermal and isotropic compressible viscoelastic constitutive equations. Two broad classes of quasilinear differential models (especially the Leonov class) and time-strain separable single integral models of either hyper- or nonhyper-viscoelastic type are considered. In order to derive an algebraic form of the stability condition, we modigy and then combine two mathematical procedures employed in formulating the Hadamard stability criteria for incompressible viscoelastic liquids and the strong ellipticity conditions for compressible elastic solids. We also briefly show that the concept of Hadamard stability is equivalent to the strong ellipticity of compressible elastic field equations. After restating inequalities for stability in terms of such separated variables as equivolumetric shear and volumetric components, we conclude that the stability criteria impose much more restrictive constraints on constitutive relations for compressible materials than for incompressible ones.