The results obtained previously for scalar and class P completely monotone relaxation moduli are extended to arbitrary anisotropy. It is shown for general anisotropic viscoelastic media that, if the relaxation modulus is a locally integrable completely monotone function, then the creep compliance is a Bernstein function and conversely. The elastic and equilibrium limits of the two material functions are related to each other. The relaxation modulus or its derivative can be singular at 0. A rigorous general formulation of the relaxation spectrum in an anisotropic viscoelastic medium is given. The effect of Newtonian viscosity on creep compliance is examined. Notation d space dimension; D = d(d + 1)/2; S the linear space of symmetric d × d matrices; G the set of symmetric operators on S; T d the set of linear transformations of a d-dimensional vector space V d ; A. Hanyga (B) 42 A. Hanyga, M. Seredyńska I unit operator on S; A transpose of A; v A w = v k A kl w l ; v, F w := v kl F klpq w pq ; R + =]0, ∞[, R + = [0, ∞[; C + := {z ∈ C | Im z > 0}; C − := C\ ] − ∞, 0]; L p (A; V) the space of measurable functions f : A → V with | f | p ∈ L 1 (A; R + ); M(A; V) the set of positive Radon measures on A with values in V; M + (A; T d ) the set of Radon measures on A with values in T d ; |H|(E) total variation of a Radon measure H ∈ M(A; V); σ, σ1D stress, stress tensor; e, e scalar strain, strain tensor; g, j scalar relaxation modulus and creep compliance; G, J tensorial relaxation modulus and creep compliance.

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Abstract.A method for constructing time-domain asymptotic solutions of hyperbolic partial differential equations with delay, with singular memory kernels, is presented. The asymptotic solutions are expressed in terms of basis functions that are regularizations of a sequence of distributions related by fractional integration.

It is shown that the dispersion and attenuation functions in a linear viscoelastic medium with a positive relaxation spectrum can be expressed in terms of a positive measure. Both functions have a sublinear growth rate at very high frequencies.In the case of power law attenuation positive relaxation spectrum ensures finite propagation speed. For more general attenuation functions the requirement of finite propagation speed imposes a more stringent condition on the high-frequency behavior of attenuation. It is demonstrated that superlinear power law frequency dependence of attenuation is incompatible with finite speed of propagation and with the assumption of positive relaxation spectrum.