Fractional constitutive laws are expressed by differential equations involving non-integer derivatives, raising memory effects that are experimentally observed in materials like rocks, ice and polymers. The present talk adresses the homogeneisation of composite materials taking into account the fractional nature of the local viscoelastic phases, by using an incremental variational approach called "Effective Internal Variable method" (EIV) (Lahellec and Suquet, 2007). The two thermodynamic potentials describing the conservative and dissipative effects, respectively the free-energy and the dissipation potential, are derived into a unique potential to be minimized. The method has been applied to homogeneisation problems with (generalized) Maxwellian constituents phases pointed out to be efficient when extended to phases with several internal variables (Tressou et. al, 2023).

The fractional dashpot behavior (Scott-Blair model) is characterized by a power-law relaxation spectrum (Tshoegl,1989). Its discretization leads to the definition of a generalized Maxwell model which can be used to approximate any type of fractional viscoelastic behaviour. Such behaviours are thus thermodynamically consistent (Lion, 1997) and can be addressed with an incremental variational approach. In this work, we apply the EIV method to a two-phase composite material composed of elastic inclusions and a viscoelastic fractional Zener matrix (Fig. 1). The behaviour of the matrix is approximated by following the discretization procedure proposed by Papoulia et. al (2010). The results are compared with full-field simulations for harmonic loadings.