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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, ∞).

SUMMARYA general closed expression is given in the frequency domain for the isothermal minimum free energy of an incompressible viscoelastic fluid, whose constitutive equation is expressed by a linear functional of the history of strain. Another equivalent form of the minimum free energy is also derived and used to study the particular case of a discrete model material response.

In the paper we present an isothermal model for describing damage and fatigue by the use of the Ginzburg-Landau (G-L) equation.Fatigue produces progressive damage, which is related with a variation of the internal structure of the material. The G-L equation studies the evolution of the order parameter, which describes the constitutive arrangement of the system and, in this framework, the evolution of damage. The thermodynamic coherence of the model is proved.In the last part of the work, we extend the results of the paper to a non-isothermal system, where fatigue contains thermal effects, which increase the damage of materials.