Fractional order theory of thermoelasticity

Abstract

a b s t r a c tIn this work, a new theory of thermoelasticity is derived using the methodology of fractional calculus. The theories of coupled thermoelasticity and of generalized thermoelasticity with one relaxation time follow as limit cases. A uniqueness theorem for this model is proved. A variational principle and a reciprocity theorem are derived.

Fractional-order theory of heat transport in rigid bodies

“…[5]). In this section the first-order relaxation of thermal energy transfer by means of the Cattaneo model for the local thermal energy exchange is considered as:…”

“…Such studies have been further developed toward the use of advanced mathematical tools as the fractional-order calculus [4] to capture memory [5,6] as well as non-local effects [7][8][9]. Indeed fractional (real) order integro-differential operators have been introduced more and more often in several contexts of physics and engineering for their capability to interpolate among the well-known integer-order operators of classical differential calculus [10].…”

The finite element method for fractional non-local thermal energy transfer in non-homogeneous rigid conductors

“…Non-local models of thermal energy transport have been used in recent physics and engineering applications to describe "small-scale" and/or high-frequency thermodynamic processes [1][2][3][4][5][6][7][8][9][10].…”

“…In more recent years, fractional models of non-local thermal energy transport have been awarded a growing attention [4][5][6][7][8][9][10]. Being capable of interpolating among the integer-order operators of classical differential calculus, fractional operators have been used indeed in several contexts of physics and engineering .…”

“…Regarding, in particular, heat transfer problems in rigid as well as elastic bodies, fractional operators have been used in ref. [4][5] to model long-memory effects, and in ref. [6][7][8][9][10] to model long-memory with long-range spatial effects, providing fractional time [4][5][6] and fractional time-space [7][8][9] heat conduction equations.…”