The phase-field computational framework stemming from the variational approach to fracture of Bourdin et al. (2000) has emerged as a powerful tool for modelling damage and failure of brittle materials. Its most attractive feature is the ability to handle crack topologies of arbitrary complexity in two and three dimensions, with no need for complicated crack tracking procedures. Tann√© et al. (2018) demonstrated that it can also correctly predict the nucleation of cracks dominated by mode-I loading for a wide range of geometry and materials.  Recently, Kumar et al. (2020) emphasized that multiaxial failure surfaces cannot be obtained using the classical isotropic regularizations of the variational models. Similar issues were also pointed out in Amor et al. (2009), when considering energy decompositions to avoid crack lips interpretation under compressive loading. This motivated Kumar et al. (2020) to propose a non-variational model supplementing the equation which governs the evolution of the phase field with an external driving force. 

In this contribution we discuss how energy decompositions, or splits, typically used to avoid crack lips interpenetration and unphysical cracking under compression, also affect the maximum allowable stress under multi-axial loading. Moreover, we propose a generalized energy density decomposition which extends the no-tension model proposed by Freddi and Royer-Carfagni (2010). The flexibility of the newly proposed decomposition enables the straightforward calibration of a multiaxial failure surface of the Drucker-Prager type, recovering in a fully variational framework the features claimed in the non-variational model of Kumar et al. (2020). Moreover, the adopted line of thought may pave the way for even more general energy density decompositions which may fit multiaxial failure surfaces of alternative shapes.

H. Amor, J.-J. Marigo, and C. Maurini. ‚ÄúRegularized formulation of the varia- tional brittle fracture with unilateral contact: Numerical experiments‚Äù. In: Journal of the Mechanics and Physics of Solids 57.8 (2009), pp. 1209‚Äì1229. 

B. Bourdin, G.-A. Francfort, and J.-J. Marigo. ‚ÄúNumerical experiments in re- visited brittle fracture‚Äù. In: Journal of the Mechanics and Physics of Solids 48 (2000), pp. 787‚Äì826. 

F. Freddi and G. Royer-Carfagni. ‚ÄúRegularized variational theories of fracture : a unified approach‚Äù. In: Journal of the Mechanics and Physics of Solids 58 (2010), pp. 1154‚Äì1174. 

E. Tann√©, T. Li, B. Bourdin, J.-J. Marigo, and C. Maurini. ‚ÄúCrack nucleation in variational phase-field models of brittle fracture‚Äù. In: Journal of the Mechanics and Physics of Solids 110 (2018), pp. 80‚Äì99. 


Nucleation under multi-axial loading in variational phase-field models of brittle fracture

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