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Variational phase-field models for brittle fracture [1] are powerful computational tools for studying Griffith-type crack propagation under complex three-dimensional and multiaxial loading scenarios. However, they struggle to accurately capture fracture nucleation, i.e., the onset of cracks in quasi-brittle materials. While effective for tensile-driven fractures (mode I) [2], they fail under multiaxial loading due to the lack of flexibility in prescribing a material-specific strength surface [3].

Traditional energy decomposition approaches often lead to questionable residual stresses [4], prompting non-variational modifications [5] that sacrifice the physical, mathematical, and numerical advantages of an energy minimization framework. This limitation stems from the fact that classical phase-field models merely regularize the sharp Griffith fracture model [6], which lacks a nucleation concept, unlike sharp cohesive fracture models.

To overcome this, we propose a variational phase-field model that approximates cohesive fracture allowing the inclusion of an arbitrary strength surface as a material property [7]. Additionally, similar to what was observed in [8], we demonstrate that this formulation enables sharp cohesive fractures, providing an approximation that avoids smearing of the displacement field. The model naturally incorporates a sharp non-interpenetration condition, thus eliminating the need for additional energy decompositions.

REFERENCES

 [1] Bourdin, B., Francfort, G.A. and Marigo, J.J., 2000. Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids, 48(4), pp.797-826.

[2] Tanné, E., Li, T., Bourdin, B., Marigo, J.J. and Maurini, C., 2018. Crack nucleation in variational phase-field models of brittle fracture. Journal of the Mechanics and Physics of Solids, 110, pp.80-99.

[3] Lopez-Pamies, O., Dolbow, J.E., Francfort, G.A. and Larsen, C.J., 2025. Classical variational phase-field models cannot predict fracture nucleation. Computer Methods in Applied Mechanics and Engineering, 433, p.117520.

[4] Vicentini, F., Zolesi, C., Carrara, P., Maurini, C. and De Lorenzis, L., 2024. On the energy decomposition in variational phase-field models for brittle fracture under multi-axial stress states. International Journal of Fracture, 247(3), pp.291-317.

[5] Kumar, A., Bourdin, B., Francfort, G.A. and Lopez-Pamies, O., 2020. Revisiting nucleation in the phase-field approach to brittle fracture. Journal of the Mechanics and Physics of Solids, 142, p.104027.

[6] Griffith, A.A., 1921. VI. The phenomena of rupture and flow in solids. Philosophical transactions of the royal society of london. Series A, containing papers of a mathematical or physical character, 221(582-593), pp.163-198.

[7] Vicentini, F., Heinzmann, J., Carrara, P. and De Lorenzis L. Variational phase-field modeling of cohesive fracture with flexibly tunable strength surface. Pre-print.

[8] Alessi, R., Marigo, J.J. and Vidoli, S., 2014. Gradient damage models coupled with plasticity and nucleation of cohesive cracks. Archive for Rational Mechanics and Analysis, 214, pp.575-615.