Modeling the effective behavior of viscoelastic composites is one of the most challenging of micromechanics. Effective behavior of a linear non ageing composite can be modeled by the classical correspondent principle. However such technique is not applicable for ageing materials such as hardening cement, concrete, clay, etc (Ricaud and Masson, 2009). Recently, a numerical incremental integral technique was bone to deal with such difficulty (Sanahuja, 2013; Lavergne et al., 2016; Barthélémy et al., 2016; Masson et al., 2012). The idea is to approximate the Stieltjes's stress-strain integral relation by an incremental form that is then converted to a linear matrix relation that can be coupled, directly in time domain, with the classical homogenization schemes (Mori-Tanaka, Self-consistent, etc.) to estimate the effective behavior of ageing composites. It is verified that such technique works very well with both ageing and non ageing materials.

Let consider an example of an ageing viscoelastic matrix-inclusion
composite made of an ageing viscoelastic matrix and spherical elastic inclusions
(e.g. ageing concrete made of elastic gravels and ageing cement paste).
Isotropic behavior is assumed for both the matrix and the inclusion phases.
The elastic inclusions are defined by a Young modulus parameter E_{1},
a Poisson ratio ν_{1} and a volume fraction f_{1}.
The ageing viscoelastic matrix is characterized by an ageing uniaxial compliance
(that can be measured by uniaxial creep tests) such as (Sanahuja, 2013):

J(t,t') = 1 / E_{0m} +
e^{-t'/τm} *
[
J_{1} * (1 - e^{-(t - t')/τ1}) +
J_{2} * (1 - e^{-(t - t')/τ2})
]

and a Poisson ratio ν_{m} that is assumed to be a constant.
Below, an online tool is provided to model the effective bulk and shear relaxation moduli of such ageing viscoelastic composite using the Mori-Tanaka scheme.

**An ageing viscoelastic composite made of spherical elastic inclusions and an ageing viscoelastic matrix.**

References

Sanahuja, J. (2013). Effective behaviour of ageing linear viscoelastic composites: Homogenization approach. International Journal of Solids and Structures, 50(19), 2846-2856.

Masson, R., Brenner, R., & Castelnau, O. (2012). Incremental homogenization approach for ageing viscoelastic polycrystals. Comptes Rendus Mécanique, 340(4-5), 378-386.

Ricaud, J. M., & Masson, R. (2009). Effective properties of linear viscoelastic heterogeneous media: Internal variables formulation and extension to ageing behaviours. International Journal of Solids and Structures, 46(7), 1599-1606.

Lavergne, F., Sab, K., Sanahuja, J., Bornert, M., & Toulemonde, C. (2016). Homogenization schemes for aging linear viscoelastic matrix-inclusion composite materials with elongated inclusions. International Journal of Solids and Structures, 80, 545-560.

Barthélémy, J. F., Giraud, A., Lavergne, F., & Sanahuja, J. (2016). The Eshelby inclusion problem in ageing linear viscoelasticity. International Journal of Solids and Structures, 97, 530-542.