Failure zone averaging for computational homogenization in phase-field fracture simulations
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Computational homogenization of heterogeneous materials is well established in the linear elastic regime. However, it faces significant limitations when material softening and fracture occur, often resulting in non-physical, size-dependent macroscopic responses [Geers et al. 2010]. These challenges arise from strain localization within narrow failure zones, which violates classical scale-separation and volume-averaging assumptions. This contribution introduces a computational homogenization framework specifically designed to address these issues in the presence of material softening and fracture. The framework combines an extended phase-field fracture model that captures quasi-brittle material behavior with a homogenization strategy based on a failure zone averaging scheme [Nguyen et al. 2010]. The homogenization framework is derived within the context of the phase-field fracture model, and the resulting fracture and fluctuation terms are shown to vanish both theoretically or numerically under periodic boundary conditions. The failure zone is defined as the actively fracturing region within a representative volume, identified by the localized phase-field fracture evolution. At the onset of material softening and fracture, the stress and strain fields are averaged, and an effective fracture property is derived. From the average stress and strain response, the corresponding traction and separation are computed, thus enabling the construction of the traction-separation law. The traction-separation law enables the computation of a macroscopic effective critical fracture energy. The proposed framework is first validated by demonstrating size independence across a range of representative structured volumes, thereby ensuring confidence in its robustness and applicability. The homogenized stress–strain response becomes independent of the representative volume size after the onset of material softening and fracture when using the failure zone averaging approach. Subsequently, the method is applied to random particle-filled microstructures with varying particle sizes, distribution strategies, and volume fractions, highlighting its versatility for complex microstructural analysis. The results demonstrate that the proposed framework extends the applicability of computational homogenization beyond the linear elastic regime, providing a robust tool for multiscale analysis of materials undergoing softening and fracture.
