Fractional Thermoelasticity

Abstract

The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids.The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design.The median level of presentation is to the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.More information about this series at

Propagation of nonlinear thermoelastic waves in porous media within the theory of heat conduction with memory: physical derivation and exact solutions

“…Our aim is to discuss the role of memory effects by using modified non‐local in time Fourier and Darcy laws, through a fractional calculus approach. These generalizations have been recently studied and discussed in the fractional theory of thermo‐elasticity and also experimentally validated . From the physical point of view, we would like to stress with this paper that both memory and nonlinear effects should be considered in problems of thermoelasticity arising in physics of solid earth.…”

“…In this paper, the relation between the flux and the gradient of the temperature field is given by This equation describes a general heat flux history model depending by the particular choice of the relaxation kernel a ( t ). In recent years, different authors (e.g., and the references therein) have considered the case in which the kernel in is a power law function, so that the integral operator appearing in the memory flux relation is a Riemann–Liouville fractional integral. In particular, recalling that the Riemann–Liouville fractional integral is defined by (e.g., ) taking the relaxation function a ( t ) = t α − 1 /Γ( α ), we have the following fractional integral memory flux law This equation is the starting point for the investigations about fractional thermoelasticity, started with the seminal paper of Povstenko and widely studied in different papers.…”

“…In particular, recalling that the Riemann–Liouville fractional integral is defined by (e.g., ) taking the relaxation function a ( t ) = t α − 1 /Γ( α ), we have the following fractional integral memory flux law This equation is the starting point for the investigations about fractional thermoelasticity, started with the seminal paper of Povstenko and widely studied in different papers. We refer for example to the recent paper for the fractional advection–diffusion equation and to the monography for a complete review. We underline that, taking together Equation and the energy balance Equation , we obtain a physical derivation of the time‐fractional heat–wave equation, that is where D = k / ρ c , ∂ α + 1 / ∂ t α + 1 is the Caputo fractional derivative and we have used the fact that, for α > 0 and f ( t )∈ C [0, t ], (see , pag.95, Lemma 2.21]) …”

The method of approximate particular solutions to simulate an anomalous mobile‐immobile transport process

“…The main advantage of the fractional model is that they are excellent instruments for the description of the hereditary and memory properties of various phenomena and materials. The interested readers can easily find additional information about theory of fractional calculus and their applications in previous studies . Although, mathematical models of many natural phenomena have been developed by using constant order fractional operators, moreover, new findings indicate that several physical processes exhibit fractional order behaviors that vary with time or space.…”

“…The interested readers can easily find additional information about theory of fractional calculus and their applications in previous studies. [1][2][3][4] Although, mathematical models of many natural phenomena have been developed by using constant order fractional operators, moreover, new findings indicate that several physical processes exhibit fractional order behaviors that vary with time or space. Fortunately, the theory of fractional calculus allows us to introduce and develop fractional mathematical models with variable-order fractional derivatives.…”