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Monotonic, completely monotonic, and exponential relaxation functions in linear viscoelasticity. (English) Zbl 0855.73026
The 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)

##### MSC:
 74D05 Linear constitutive equations for materials with memory 74D10 Nonlinear constitutive equations for materials with memory
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