  In the present work, the evolution of damage in periodic composite materials
is investigated through a novel finite element-based multiscale computational
approach. The methodology is developed by means of the original combination of
homogenization methods with the phase field approach of fracture. This last is
applied at the macroscale level on the equivalent homogeneous continuum, whose
constitutive properties are obtained in closed form via a two-scale asymptotic
homogenization scheme. The formulation allows considering different assumptions
on the evolution of damage at the microscale (e.g., damage in the matrix and
not in the inclusion/fiber), as well as the role played by the microstructural
topology. Numerical results show that the proposed formulation leads to an
apparent tensile strength and a post-peak branch of unnotched and notched
specimens dependent not only on the internal length scale of the phase field
approach, as for homogeneous materials, but also on the inclusion volumetric
content and its shape. Down-scaling relations allow the full reconstruction of
the microscopic fields at any point of the macroscopic model, as a simple
post-processing operation.
