Anisotropic viscoelastic models with singular memory

The physical background of singular memory models and in particular the Cole–Cole model is discussed. Three models of anisotropic linear viscoelasticity with frequency-dependent stiffness coefficients are considered. The models are constructed in such a way that anisotropic properties are separated from anelastic effects. Two of these models represent finite-speed wave propagation with singularities at the wavefronts (the exponential relaxation model) and without singularities at the wavefronts (the Cole–Cole model), while a third model called the fractional model is related to the constant Q with unbounded propagation speed. The Cole–Cole and fractional models belong to the class of singular memory models studied earlier because of their applications in polymer rheology, poroelasticity, poroacoustics, seismic wave propagation and other applications. Well-posedness of initial boundary value problems with mixed Dirichlet–Neumann boundary conditions is established for the three models. Regularity properties of the three models are examined.