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Zbl 0544.73052
Koeller, R.C.
Applications of fractional calculus to the theory of viscoelasticity. (English)
[J] J. Appl. Mech. 51, 299-307 (1984). ISSN 0021-8936; ISSN 1528-9036/e

Summary: The connection between the fractional calculus and the theory of Abel's integral equation is shown for materials with memory. Expressions for creep and relaxation functions, in terms of the Mittag-Leffler function that depends on the fractional derivative parameter $\beta$, are obtained. These creep and relaxation functions allow for significant creep or relaxation to occur over many decade intervals when the memory parameter, $\beta$, is in the range of 0.05--0.35. It is shown that the fractional calculus constitutive equation allows for a continuous transition from the solid state to the fluid state when the memory parameter varies from zero to one.

MSC 2000:: *74D05 Linear constitutive equations
74D10 Nonlinear constitutive equations
74R05 Brittle damage
74Hxx Dynamical problems
45E10 Integral equations of the convolution type

Keywords: Rabotnov theory; hereditary; spring-pot combined with springs; fractional polynomial operator; connection; fractional calculus; theory of Abel's integral equation; materials with memory; Mittag-Leffler function; continuous transition; solid state; fluid state

Citations: Zbl 0515.73026

Cited in: Zbl 1017.74011 Zbl 0578.73040

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