Response functions in linear viscoelastic constitutive equations and related fractional operators

Abstract

This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators.

Two‐layer flows of generalized immiscible second grade fluids in a rectangular channel

“…Thus, the fractional calculus became an important tool for biology, chemistry, physics and many engineering domains. A good presentation of the theory of fractional operators, together with their properties and applications can be found in Hristov,13,14 Povstenko, 15,16 Zheng et al, 17 Baleanu et al 18,19 and Hilfer. 20 The purpose of this article is to study two-layer unsteady one-dimensional flows of incompressible and immiscible generalized second grade fluids in a rectangular channel.…”

Some New Fractional Trapezium-Type Inequalities for Preinvex Functions

“…Proof. Integrating by parts (12), using (9) and (10) and changing the variables of integration, we have…”

“…This fractional integral operators are a new generalization of fractional integrals such as the Riemann-Liouville fractional integral, the k-Riemann-Liouville fractional integral, Katugampola fractional integrals, the conformable fractional integral, Hadamard fractional integrals, etc. To read more about fractional analysis, see References [10,11,22,27].…”

Linear Viscoelastic Responses: The Prony Decomposition Naturally Leads Into the Caputo-Fabrizio Fractional Operator

“…Last to this point, but not least, as it was commented in Hristov [14], the ratio τ/t 0 is not integer and expressing the memory kernel as exp −β t −s where β = (τ/t 0 ) −1 = α/(1 − α) we get a fractional operator. Therefore, the memory kernel of the Caputo-Fabrizio operator is controlled by a noninteger parameter in the context of what is needed to say that this operator is fractional, despite the fact that it does not repeat exactly the properties of the Classical Riemann-Liouville and Caputo derivatives.…”

“…and relates it to data that can be really recovered from experimental data, such the relaxation and retardation times (see the sequel) The relationship (Equation 14) says that for τ/t 0 = 1 we get α = 1/2. Further, depending on the ratio τ/t 0 we may have fractional orders roughly arranged in two groups [14]: a) when 0 ≤ τ/t 0 ≤ 1 we have fractional orders α ∈ [0.5, 1.0] and the relaxation time τ is shorter than the macroscopic process observation time t 0 , and b) 1 ≤ τ/t 0 < ∞ the fractional orders are α ∈ (0, 0.5] since the relaxation time τ is larger than the macroscopic process observation time t 0 . Qualitatively, for τ/t 0 < 1 the relaxations could be considered as fast (rapid) relaxations, while τ/t 0 > 1 is related to slow relaxations (see in Hristov [14] the comments of numerical values of Prony decomposition and relaxation times)…”