A new internal variables homogenization scheme for linear viscoelastic materials based on an exact Eshelby interaction law

• A homogenization scheme for Maxwellian linear viscoelastic heterogeneous materials is proposed.
• An exact solution of the ellipsoidal Eshelby inclusion problem is obtained.
• The results are reported with a Mori–Tanaka homogenization scheme for two-phase composites.
• The effective and phase behaviors are directly solved in the time domain.

A new time-incremental internal variables homogenization scheme for Maxwellian linear viscoelastic heterogeneous materials is proposed. This scheme is based on the exact solution of the ellipsoidal Eshelby inclusion problem obtained in the time domain. In contrast with current existing methods, the effective behavior as well as the evolution laws of the averaged stresses per phase are solved incrementally in the time domain without need to analytical or numerical inverse Laplace–Carson transforms. This is made through a time-differential equation in addition to the more classic strain rate concentration equation. In addition, the new derived interaction law for the Eshelby inclusion problem is provided in a compact matrix form. It is proved that this is an exact formulation for an arbitrary anisotropic ellipsoidal Maxwellian inclusion embedded in an isotropic Maxwellian matrix. In order to show the interest of the present approach, the results are reported and discussed with a Mori–Tanaka homogenization scheme for two-phase composites in comparisons with other exact or approximate methods.