Linear models of dissipation in anelastic solids
01 Apr 1971-Vol. 1, Iss: 2, pp 161-198
About: The article was published on 1971-04-01. It has received 726 citations till now. The article focuses on the topics: Dissipation.
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TL;DR: In this article, the authors introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus, and derive the analytical solutions of the most simple linear integral and differential equations in fractional order.
Abstract: We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.
1,281 citations
Cites background from "Linear models of dissipation in ane..."
...Subsequently, in 1971, Caputo & Mainardi [28] have proved that the Mittag-Leffler function is present whenever derivatives of fractional order are introduced in the constitutive equations of a linear viscoelastic body....
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...We now observe that an alternative definition of fractional derivative, originally introduced by Caputo [19], [27] in the late sixties and adopted by Caputo and Mainardi [28] in the framework of the theory of Linear Viscoelasticity (see a review in [24]), is the so-called Caputo Fractional Derivative of order α > 0 : D ∗ f(t) := J D f(t) with m− 1 < α ≤ m, namely...
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...We now observe that an alternative definition of fractional derivative, originally introduced by Caputo [19], [27] in the late sixties and adopted by Caputo and Mainardi [28] in the framework of the theory of Linear Viscoelasticity (see a review in [24]), is the so-called Caputo Fractional Derivative of order α > 0 : Dα∗ f(t) := J m−αDm f(t) with m− 1 α ≤ m, namely Dα∗ f(t) := 1 Γ(m− α) ∫ t 0 f (m)(τ) (t− τ)α+1−m dτ , m− 1 α m , dm dtm f(t) , α = m. (1.17) This definition is of course more restrictive than (1.13), in that requires the absolute integrability of the derivative of order m....
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...Fractional relaxation is overall a peculiarity of a class of viscoelastic bodies which are extensively treated by Mainardi [24], to which we refer for details and additional bibliography....
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...In this respect A. Carpinteri and F. Mainardi have edited the present book of lecture notes and entitled it as Fractals and Fractional Calculus in Continuum Mechanics....
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TL;DR: In this article, the authors review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics.
Abstract: We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical modelling of viscoelastic bodies (Sect. 1), and unsteady motion of a particle in a viscous fluid, i.e. the Basset problem (Sect. 2). In the former analysis fractional calculus leads us to introduce intermediate models of viscoelasticity which generalize the classical spring-dashpot models. The latter analysis induces us to introduce a hydrodynamic model suitable to revisit in Sect. 3 the classical theory of the Brownian motion, which is a relevant topic in statistical mechanics. By the tools of fractional calculus we explain the long tails in the velocity correlation and in the displacement variance. In Sect. 4 we consider the fractional diffusion-wave equation, which is obtained from the classical diffusion equation by replacing the first-order time derivative by a fractional derivative of order $0< \beta <2$. Led by our analysis we express the fundamental solutions (the Green functions) in terms of two interrelated auxiliary functions in the similarity variable, which turn out to be of Wright type (see Appendix), and to distinguish slow-diffusion processes ($0 < \beta < 1$) from intermediate processes ($1 < \beta < 2$).
1,064 citations
TL;DR: In this article, the authors introduced fractional derivatives of order α in time, with 0 for relaxation, diffusion, oscillations, and wave propagation, and showed that they are governed by simple differential equations of order 1 and 2 in time.
Abstract: The processes involving the basic phenomena of relaxation, diffusion, oscillations and wave propagation are of great relevance in physics; from a mathematical point of view they are known to be governed by simple differential equations of order 1 and 2 in time. The introduction of fractional derivatives of order α in time, with 0
925 citations
01 Jan 2014
TL;DR: In this paper, Dzherbashian [Dzh60] defined a function with positive α 1 > 0, α 2 > 0 and real α 1, β 2, β 3, β 4, β 5, β 6, β 7, β 8, β 9, β 10, β 11, β 12, β 13, β 14, β 15, β 16, β 17, β 18, β 20, β 21, β 22, β 24
Abstract: Consider the function defined for \(\alpha _{1},\ \alpha _{2} \in \mathbb{R}\) (α 1 2 +α 2 2 ≠ 0) and \(\beta _{1},\beta _{2} \in \mathbb{C}\) by the series
$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$
(6.1.1)
Such a function with positive α 1 > 0, α 2 > 0 and real \(\beta _{1},\beta _{2} \in \mathbb{R}\) was introduced by Dzherbashian [Dzh60].
919 citations
References
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TL;DR: In this paper, a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges has been investigated by introducing fractional derivatives in the stressstrain relation, and a rigorous proof of the formulae to be used in obtaining the analytic expression of Q is given.
Abstract: Summary Laboratory experiments and field observations indicate that the Q of many non-ferromagnetic inorganic solids is almost frequency independent in the range 10-2-107 cis, although no single substance has been investigated over the entire frequency spectrum. One of the purposes of this investigation is to find the analytic expression for a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges. This will be obtained by introducing fractional derivatives in the stressstrain relation. Since the aim of this research is also to contribute to elucidating the dissipating mechanism in the Earth free modes, we shall treat the dissipation in the free, purely torsional, modes of a shell. The dissipation in a plane wave will also be treated. The theory is checked with the new values determined for the Q of spheroidal free modes of the Earth in the range between 10 and 5 min integrated with the Q of Rayleigh waves in the range between 5 and 0.6 min. Another check of the theory is made with the experimental values of the Q of the longitudinal waves in an aluminium rod in the range between lo-’ and 10-3s. In both checks the theory represents the observed phenomena very satisfactorily. The time derivative which enters the stress-strain relation in both cases is of order 0.15. The present paper is a generalized version of another (Caputo 1966b) in which an elementary definition of some differential operators was used. In this paper we give also a rigorous proof of the formulae to be used in obtaining the analytic expression of Q; moreover, we present two checks of the theory with experimental data. The present paper is a generalized version of another (Caputo 1966b) in which an elementary definition of some differential operators was used. In this paper we give also a rigorous proof of the formulae to be used in obtaining the analytic expression of Q; moreover, we present two checks of the theory with experimental data. In a homogeneous isotropic elastic field the elastic properties of the substance are specified by a description of the strains and stresses in a limited portion of the field since the strains and stresses are linearly related by two parameters which describe the elastic properties of the field. If the elastic field is not homogeneous nor isotropic the properties of the field are specified in a similar manner by a larger number of parameters which also depend on the position.
3,372 citations
Book•
01 Jan 1963TL;DR: A large and growing number of original papers on both the experimental and the theoretical aspects of stress wave propagation is appearing in the scientific literature, and two international conferences solely concerned with the subject have been held during the last five years as mentioned in this paper.
Abstract: A large and growing number of original papers on both the experimental and the theoretical aspects of stress wave propagation is appearing in the scientific literature, and two international conferences solely concerned with the subject have been held during the last five years. The purpose of this paper is to review recent experimental and theoretical advances in the propagation of deformation waves of arbitrary shape through elastic and anelastic solids, and also to attempt to outline the problems on which present efforts are being directed and to predict probable lines of future development.
2,063 citations