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In the present work, the evolution of damage in periodic composite materialsis investigated through a novel finite element-based multiscale computationalapproach. The methodology is developed by means of the original combination ofhomogenization methods with the phase field approach of fracture. This last isapplied at the macroscale level on the equivalent homogeneous continuum, whoseconstitutive properties are obtained in closed form via a two-scale asymptotichomogenization scheme. The formulation allows considering different assumptionson the evolution of damage at the microscale (e.g., damage in the matrix andnot in the inclusion/fiber), as well as the role played by the microstructuraltopology. Numerical results show that the proposed formulation leads to anapparent tensile strength and a post-peak branch of unnotched and notchedspecimens dependent not only on the internal length scale of the phase fieldapproach, as for homogeneous materials, but also on the inclusion volumetriccontent and its shape. Down-scaling relations allow the full reconstruction ofthe microscopic fields at any point of the macroscopic model, as a simplepost-processing operation.

A phase field approach for damage propagation in periodic microstructured materials