Multi-scale continuous–discontinuous framework for computational-homogenization–localization

In this paper, a multi-scale technique is proposed for the modeling of microstructured materials up to the point of macroscopic failure. A continuous–discontinuous computational homogenization–localization framework is developed, which involves a discontinuity enhanced macroscale description. The underlying microstructural volume element (MVE) enables the incorporation of a band with high strains, i.e. a localization band. For the multi-scale coupling, special scale transition relations are established to handle the underlying material response of both the bulk and the localization zone. At a macroscopic (integration) point, the macroscale displacement jump and the deformation of the surrounding continuum material are used to formulate kinematical boundary constraints for the microscale MVE problem. Upon proper averaging of the MVE response, the macroscopic generalized stress is obtained. Simultaneously, a microscale effective displacement jump is recovered and returned to the macrolevel, based on the MVE deformation field. The equality of the macro- and effective microscale displacement jumps is enforced at the macroscale by Lagrange multipliers, being cohesive tractions at the interface. The applicability of the developed continuous–discontinuous computational homogenization framework is illustrated on two simple benchmark problems involving evolving macroscale discontinuities as a result of microstructural degradation by void growth.