Cones of material response functions in 1D and anisotropic linear viscoelasticity

Viscoelastic materials have non-negative relaxation spectra. This property implies that viscoelastic response functions satisfy certain necessary and sufficient conditions. It is shown that these conditions can be expressed in terms of each viscoelastic response function ranging over a cone. The elements of each cone are completely characterized by an integral representation. The 1:1 correspondences between the viscoelastic response functions are expressed in terms of cone-preserving mappings and their inverses. The theory covers scalar and tensor-valued viscoelastic response functions. Notation. ]a, b] – the set of real numbers x satisfying a < x ≤ b; R,R± – the set of real, positive/negative numbers; R± – topological closure of R±; C – the set of complex nubers; C− := C \ R−; z∗ – complex conjugate of z ∈ C; L(V ) – the set of linear mappings of a linear space V into itself; S – the set of symmetric tensors w of rank 2; T – the set of symmetric linear operators A on S; 1 ar X iv :0 90 6. 18 93 v1 [ co nd -m at .m tr lsc i] 1 0 Ju n 20 09 A ≥ 0 if 〈w,A w〉 ≥ 0 for all w ∈ S; A > 0 if 〈w,A w〉 > 0 for all w 6= 0 in S; T+ – the set of positive semi-definite symmetric linear operators (A ≥ 0) on S; SC, TC – complexifications of S, T ; L(f) = f̃ – Laplace transform; C – the set of CM functions; CT – the set of T -valued CM functions; L – the set of locally integrable completely monotone functions; LT – the set of T -valued locally integrable completely monotone functions; B – the set of Bernstein functions; BT – the set of T -valued Bernstein functions; F – the set of complete Bernstein functions; FT – the set of T -valued complete Bernstein functions; S – the set of Stieltjes functions; ST – the set of T -valued Stieltjes functions.