Rheological constitutive equation of solids: a link between models based on irreversible thermodynamics and on fractional order derivative equations

Stéphane André (1) , Yves Meshaka (1) , Christian Cunat (1)

1 LEMTA - Laboratoire d'Energétique et Mécanique Théorique et Appliquée

In this paper we focus on the rheological problem of defining a constitutive equation for viscoelastic materials. In this simple case, we show that writing the dissipative component of the observable response to a given excitation as the result of multiple internal processes working for equilibrium recovery (flux of internal hidden variables), can yield a recursive series in time. This can be obtained when use is made of the theorem of created entropy equipartition as a model for fluctuation regression. A distribution (spectrum) for relaxation times naturally follows. The model thus obtained reflects the concept of a hierarchically constrained dynamic behavior. The conclusion is that the operator of non-integer differentiation in time applied to field variables can also be recovered from pure thermodynamic considerations.

Mots clés: Relaxation phenomena; Irreversible thermodynamics; Fractional calculus; Fractal dynamics; Rheology; Viscoelasticity

Domaines: Mécanique des matériaux [physics.class-ph]

Stéphane André, Yves Meshaka, Christian Cunat. Rheological constitutive equation of solids: a link between models based on irreversible thermodynamics and on fractional order derivative equations. Rheologica Acta, 2003, 42, pp.500-515. ⟨10.1007/s00397-003-0305-z⟩. ⟨hal-01462871⟩