Monotonic, completely monotonic, and exponential relaxation functions in linear viscoelasticity.

*(English)*Zbl 0855.73026The authors study restrictions on the relaxation functions for linear viscoelastic materials that are described by the relation \(T(t)= G_0 E(t)+ \int_0^t G' (s) E(t-s) ds\) between strain \(E\) and stress \(T\), assuming essentially only that \(G'\in L^1 (0, \infty)\) and admitting strain histories that have bounded variation. After discussing the regularity of the stress response in such a situation, showing that the work integrals \(\int_p^q T(t) dE(t)\) are well-defined, and introducing accelerated and retarded processes and their strain responses, the authors first generalize results by W. A. Day [Q. J. Mech. Appl. Math. 24, 487-497 (1971; Zbl 0241.73035)] and M. E. Gurtin and I. Herrera [Q. Appl. Math. 23, 235-245 (1965; Zbl 0173.52703)] that characterize relaxation functions that are dissipative or compatible with thermodynamics in the present more general situation. They next show that \(G\) is monotonic (in the sense that \(G'\) is negative semidefinite) if and only if the work done in rectilinear monotonic processes (i.e. \(E(t)= \lambda (t)E\) with monotonic \(\lambda\)) is decreased under retardation. They also generalize a result by W. A. Day [Proc. Camb. Philos. Soc. 67, 503-508 (1970; Zbl 0202.25302)] that characterizes completely monotonic relaxation functions and finally show that the work done in all closed paths in stress-strain space \((E(a)= E(b)\), \(T(a)= T(b))\) is non-negative if and only if \(G\) is a matrix exponential, i.e. the integral model is equivalent to a simple differential model.

Reviewer: H.Engler (Bonn)