An exact thermodynamical model of power-law temperature time scaling

In this paper a physical model for the anomalous temperature time evolution (decay) observed in complex thermodynamical system in presence of uniform heat source is provided. Measures involving temperatures T with power-law variation in time as T(t)∝tβ with β∈R shows a different evolution of the temperature time rate T(t) with respect to the temperature time-dependence T(t). Indeed the temperature evolution is a power-law increasing function whereas the temperature time rate is a power-law decreasing function of time.Such a behavior may be captured by a physical model that allows for a fast thermal energy diffusion close to the insulated location but must offer more resistance to the thermal energy flux as soon as the distance increases. In this paper this idea has been exploited showing that such thermodynamical system is represented by an heterogeneous one-dimensional distributed mass one with power-law spatial scaling of its physical properties. The model yields, exactly a power-law evolution (decay) of the temperature field in terms of a real exponent as T∝tβ (or T∝t-β) that is related to the power-law spatial scaling of the thermodynamical property of the system. The obtained relation yields a physical ground to the formulation of fractional-order generalization of the Fourier diffusion equation

A mechanical picture of fractional-order Darcy equation

In this paper the authors show that fractional-order force-flux relations are obtained considering the flux of a viscous fluid across an elastic porous media. Indeed the one-dimensional fluid mass transport in an unbounded porous media with power-law variation of geometrical and physical properties yields a fractional-order relation among the ingoing flux and the applied pressure to the control section. As a power-law decay of the physical properties from the control section is considered, then the flux is related to a Caputo fractional derivative of the pressure of order 0 ≤ β≤1. If, instead, the physical properties of the media show a power-law increase from the control section, then flux is related to a fractional-order integral of order 0 ≤ β≤1. These two different behaviors may be related to different states of the mass flow across the porous media

Laminar flow through fractal porous materials: The fractional-order transport equation

The anomalous transport of a viscous fluid across a porous media with power-law scaling of the geometrical features of the pores is dealt with in the paper. It has been shown that, assuming a linear force-flux relation for the motion in a porous solid, then a generalized version of the Hagen-Poiseuille equation has been obtained with the aid of Riemann-Liouville fractional derivative. The order of the derivative is related to the scaling property of the considered media yielding an appropriate mechanical picture for the use of generalized fractional-order relations, as recently used in scientific literature

The fractal model of non-local elasticity with long-range interactions

The mechanically-based model of non-local elasticity with long-range interactions is framed, in this study, in a fractal mechanics context. Non-local interactions are modelled introducing long-range central body forces between non-adjacent volume elements of the elastic continuum. Such long-range interactions are modelled as proportional to the product of interacting volumes, to the relative displacements of the centroids and to a distance-decaying function that is monotonically-decreasing with the distance. The choice of the decaying function is a key aspect of the model and it has been proved that any continuous function, strictly positive, is thermodynamically consistent and it leads to a material that satisfy the Drucker stability criterion [2]. Such a mathematical model of non-local elasticity has an interesting mechanical counterpart that is described by a point-spring network with multiple springs with distance-decaying stiffness. As the functional class of the distance-decaying function is modelled as a power-law function of the distance of interacting particles, then, in the 1D case, the governing operators are Marchaud-type fractional derivatives as proved by the authors in previous studies [1]. In this study we aim to show that, as we assume that the stiffness associated to long-range interactions is modelled as a self-similar transformation of the Euclidean distance with anomalous and real scaling exponent, the mechanical model of the non-local elasticity is a self-similar fractal object. In more detail, assuming a non-integer power-law decay of the long-range forces between adjacent volumes of an ideal next nearest (NN) model, the scaling law of the stiffness of the long-range bonds is readily obtained. The Hausdorff-Besitckovich (HB) fractal dimension provides the appropriate bounds of the decay coefficient necessary to maintain the self-similarity of the obtained fractal set. The NN model, however leads to mathematically inconsistent governing operator for general class of continuous displacement function and it is proved that in this case only one integer value of the long-range force decay is admissible leading to classical second-order differential operators. A different scenario is involved as we introduce, on mechanical grounds, the long-range interaction concept so that as we refine observation scale, the interactions between particles is still involving the presence of all the new, non-adjacent particles so that the original NN lattice is turned into a more refined and realistic next to the nearest next (NNN) lattice model. Such a model is equivalent to the mechanical model of the long-range interactions introduced by the authors to describe non-local elasticity. The model is constituted of self-similar copies of elastic chains and henceforth it may be considered as a mechanical fractal as we assume an unbounded domain. This fractal set is not coalescing with usual fractals since it retains all the informations of previous observation scales and henceforth it has been dubbed as multiscale fractal. In this context the HB dimension of the mechanical fractal may be obtained as a function of the decaying exponent of the long-range interactions and it may be proved that the governing equation of 1 the problem are Marchaud fractional-type differential operator as already postulated by the authors in a previous study [1]. Some conclusions about the use of fractional operators in the context of multiscale approach to non-local mechanics may be also withdrawn from previous considerations

Exact Mechanical Models of Fractional Hereditary Materials (FHM)

Fractional Viscoelasticity is referred to materials, whose constitutive law involves fractional derivatives of order β R such that 0 β 1. In this paper, two mechanical models with stress-strain relation exactly restituting fractional operators, respectively, in ranges 0 β 1 / 2 and 1 / 2 β 1 are presented. It is shown that, in the former case, the mechanical model is described by an ideal indefinite massless viscous fluid resting on a bed of independent springs (Winkler model), while, in the latter case it is a shear-type indefinite cantilever resting on a bed of independent viscous dashpots. The law of variation of all mechanical characteristics is of power-law type, strictly related to the order of the fractional derivative. Because the critical value 1/2 separates two different behaviors with different mechanical models, we propose to distinguish such different behavior as elasto-viscous case with 0< β <1 / 2 and visco-elastic case for 1 / 2 <β <1. The motivations for this different definitions as well as the comparison with other existing mechanical models available in the literature are presented in the pape

The multiscale stochastic model of fractional hereditary materials (FHM)

In a recent paper the authors proposed a mechanical model corresponding, exactly, to fractional hereditary materials (FHM). Fractional derivation index β; ∈ [0, 1/2] corresponds to a mechanical model composed by a column of massless newtonian fluid resting on a bed of independent linear springs. Fractional derivation index β ∈ [1/2, 1], corresponds, instead, to a mechanical model constituted by massless, shear-type elastic column resting on a bed of linear independent dashpots. The real-order of derivation is related to the exponent of the power-law decay of mechanical characteristics. In this paper the authors aim to introduce a multiscale fractance description of FHM in presence of stochastic fluctuations of model parameters. In this setting the random multiscale fractance may be used to capture the fluctuations of material parameters observed in experimental tests by means of proper analytical evaluation of the model statistic

A non-local model of thermal energy transport: The fractional temperature equation

Non-local models of thermal energy transport have been used in recent physics and engineering applications to describe several "small-scale" and/or high frequency thermodynamic processes as shown in several engineering and physics applications. The aim of this study is to extend a recently proposed fractional-order thermodynamics ([5]), where the thermal energy transfer is due to two phenomena: A short-range heat flux ruled by a local transport equation; a long-range thermal energy transfer that represents a ballistic effects among thermal energy propagators. Long-range thermal energy transfer accounts for small-scale effects that are assumed proportional to the product of the interacting masses, to a distance-decaying function, as well as to their relative temperature. In this paper the thermodynamic consistency of the model is investigated obtaining some restrictions on the functional class of the distance decaying function that rules the strength of the long-range thermal energy transfer. As the distance-decaying function is assumed in the form of a power-law decay a novel temperature equation involving multidimensional spatial Marchaud α-order derivatives (0 ≤ α ≤ 1) of the temperature field in the body is obtained. Some analytical and numerical solutions of the fractional-order temperature equation have been provided in the paper to show the capabilities of the proposed model and the influence of model parameter

Solution strategies for 1D elastic continuum with long-range interactions: Smooth and fractional decay

An elastic continuum model with long-range forces is addressed in this study within the context of approximate analytical methods Such a model stems from a mechanically-based approach to non-local theory where long-range central forces are introduced between non-adjacent volume elements Specifically, long-range forces depend on the relative displacement on the volume product between interacting elements and they are proportional to a proper, material-dependent, distance-decaying function Smooth-decay functions lead to integro-differential governing equations whereas hypersingular, fractional-decay functions lead to a fractional differential governing equation of Marchaud type In this paper the Galerkin and the Rayleigh-Ritz method are used to build approximate solutions to the integro-differential and the fractional differential governing equations Numerical applications show the accuracy of the proposed approximate solutions as compared to the finite difference approximation and to the fractional finite difference approximatio

One-dimensional heterogeneous solids with uncertain elastic modulus in presence of long-range interactions: Interval versus stochastic analysis

The analysis of one-dimensional non-local elastic solids with uncertain Young's modulus is addressed. Non-local effects are represented as long-range central body forces between non-adjacent volume elements. For comparison purpose, the fluctuating elastic modulus of the material is modeled following both a probabilistic and a non-probabilistic approach. To this aim, a novel definition of the interval field concept, able to limit the overestimation affecting ordinary interval analysis, is introduced. Approximate closed-form expressions are derived for the bounds of the interval displacement field as well as for the mean-value and variance of the stochastic respons