Fractional integral operators and Fox functions in the theory of viscoelasticity

Abstract

This investigation presents an exactly solvable fractional model of linear viscoelastic behavior. In recent years both phenomenological-and molecular-based theories for the study of polymers and other viscoelastic materials came up with integral or differential equations of fractional order. Exact (analytical) solutions of such equations can be obtained by making use of the fractional calculus. Fox functions play a dominant part: they offer a wide spectrum of applications; however, they are applied little and they have not been used up to now within the context of viscoelasticity. A well-defined fractional initial value problem is derived by starting out with the Zener model. When the initial value problem is solved, a Fox function representation of the stress relaxation function is obtained. Further viscoelastic functions will be calculated analytically. Data sets of stress-strain experiments carried out on polyisobutylene and natural rubber are analyzed and compared with the predictions of the fractional theory. The agreement covers more than 10 orders of magnitude.

Microscopic Formulation of Fractional Theory of Viscoelasticity

“…More recently methods of fractional calculus have been used [5][6][7][8] to generalize these models leading to the so-called Scott-Blair model characterized by the operator equation…”

“…Mechanical models involving a spring-mass connected to a dashpot have been used to explain the elastic and viscous behavior.The mathematical structure of the theory and the spring-dash-pot type of mechanical models used and the so called Standard Linear Solid have all been only too well known [1][2][3][4]. In recent years methods of fractional calculus have been applied to develop viscoelastic models especially by Caputo and Mainardi [5,6] , Glockle and Nonenmacher [7], and Gorenflo and Mainardi [8]. A recent monograph by Mainardi [9] gives extensive list of references to the literature connecting fractional calculus, linear viscoelasticity and wave motion.All these works treat the phenomenon of viscoelasticity as a macroscopic phenomenon exhibited by matter treated as an elastic continuum albeit including a viscous aspect as well.…”

Correlations in a generalized elastic model: Fractional Langevin equation approach

“…[35]. As interesting applications of H−functions we could mention an exactly solvable model of linear viscoelastic behavior [36], the H−function representation of non-Debye relaxation [37,38] and of the solution of the space-time fractional diffusion equations [39,40]. We present the defintion and the basic properties of the H−functions in Appendices B and C, respectively.…”

Indentation analysis of fractional viscoelastic solids

“…Galanov [1982] established 1 There is a special case in which (25), with different fractional derivatives on stress and strain, can be thermodynamically stable. That is, when the fractional derivative acting on strain is greater than that acting on stress, and only below a certain limiting frequency [Glöckle and Nonnenmacher 1991]. Figure 4.…”