Thermal work and minimum free energy in a heat conductor with memory

Abstract

A rigid linear heat conductor with memory effects is considered. Thus, the behaviour of the material is characterized by a constitutive equation which relates the heat flux to the history of the temperature gradient. A given thermal history is considered and its prolongation via an assigned process is defined. Then, the notion of equivalence is introduced to single out and associate together all those different thermal histories which correspond to the same heat flux. Notably, whenever the heat flux is the same the related thermal work is also the same. An explicit expression of the minimum free energy, related to the maximum recoverable work, is obtained in the frequency domain. † More precisely, it should be required that x ∈ B ⊂ R 3 , where B denotes the bounded closed set in R 3 which represents the configuration domain of the conductor, here not specified since of no interest in the present study. ‡ Throughout the paper we define R + = [0, ∞) while R ++ = (0, ∞).

Some Remarks on Materials with Memory: Heat Conduction and Viscoelasticity

“…In particular, the approach presented by Fabrizio, Gentili and Reynolds in [11], and, subsequently, in [2] is adopted. Accordingly, the key notions comprised in [2] are briefly reviewed.…”

“…The heat flux q ∈ R 3 , when the heat flux relaxation function denoted by k(x, t), is assumed to satisfy the constitutive equation 2) which shows that the heat flux q depends on the time variable not only via the present time t, but also via its past history. In addition, according to [11] and [2], the rigid heat conductor is understood to be an isotropic material, and, hence, focussing the attention on a generic element of the conductor, no dependence on the position in the conductor is considered. Then, the energy e, the heat flux q, the temperature θ and, consequently, all the other quantities are represented by functions of the time variable alone.…”

“…The thermal work associated to a process is considered in the subsequent Section 3 where results obtained in [2] are recalled. In particular, given two different, but equivalent, thermal processes, the thermal work related to them both is the same (and viceversa).…”

“…Accordingly, [15], the same strain history corresponds to equivalent viscoelastic processes. Then, the viscoelastic work is written, under the form obtained by Gentili [15]; it is compared with that one [2], obtained in the case of the thermal work functional. An interesting connection between thermodynamics of rigid bodies and isothermal viscoelasticity appears.…”

A continuum theory for first‐order phase transitions based on the balance of structure order

“…A further free energy, which is the minimum one, m is considered in [19] borrowing from the Golden representation of the free energy for viscoelastic materials [20] (see [21]). …”