Modeling the quasi-brittle fracture of structural materials using a mixed stabilized two-field finite element formulation

Abstract

The fracture process zone (FPZ) is typically characterized as a small region around a crack where non-linear phenomena occur, such as plasticity. In brittle materials, this zone is small and can be safely neglected. However, in quasi-brittle materials, which exhibit a combination of brittle and ductile behavior rather than a clear manifestation of either, the material within the FPZ tends to damage and displays a softening curve after reaching peak load. This behavior is frequently observed in structural materials like concrete and timber, and it can be challenging to model. Traditionally, displacement-based irreducible finite element (FE) formulations have been widely used for simulating structural materials. However, this approach comes with significant drawbacks, such as mesh dependence and convergence problems, when applied to certain phenomena like softening, localization, and fracture. To address these challenges, various techniques have been employed, including extended FE methods and phase-field modeling. In this work, the utilization of a mixed FE formulation in which both displacement and strain serve as primary unknowns within the system, is proposed. To ensure satisfaction of the inf-sup condition, which is associated with saddle point stability in mixed formulations, we employ the variational multiscale method to introduce stabilizing terms into the system. The implementation is conducted using FEniCS, an open-source FE software that offers a high-level programming interface written in Python. The implementation is validated by comparing the obtained results with those reported in the literature for bending test in notched specimens. The results demonstrate remarkably good performance in terms of maximum load, softening curve, and structural size effect in various specimens, exhibiting minimal mesh dependence even when using low-order interpolation elements

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