Laws of crack motion and phase-field models of fracture

By V. Hakim, Alain Karma. 2008

401 citations recorded by the source corpus.

Recently proposed phase-field models offer self-consistent descriptions of brittle fracture. Here, we analyze these theories in the quasistatic regime of crack propagation. We show how to derive the laws of crack motion either by using solvability conditions in a perturbative treatment for slight departure from the Griffith threshold or by generalizing the Eshelby tensor to phase-field models. The analysis provides a simple physical interpretation of the second component of the classic Eshelby integral in the limit of vanishing crack propagation velocity: it gives the elastic torque on the crack tip that is needed to balance the Herring torque arising from the anisotropic surface energy. This force-balance condition can be interpreted physically based on energetic considerations in the traditional framework of continuum fracture mechanics, in support of its general validity for real systems beyond the scope of phase-field models. The obtained law of crack motion reduces in the quasistatic limit to the principle of local symmetry in isotropic media and to the principle of maximum energy-release-rate for smooth curvilinear cracks in anisotropic media. Analytical predictions of crack paths in anisotropic media are validated by numerical simulations. Interestingly, for kinked cracks in anisotropic media, force-balance gives significantly different predictions from the principle of maximum energy-release-rate and the difference between the two criteria can be numerically tested. Simulations also show that predictions obtained from force-balance hold even if the phase-field dynamics is modified to make the failure process irreversible. Finally, the role of dissipative forces on the process zone scale as well as the extension of the results to motion of planar cracks under pure antiplane shear are discussed.

Subfields: Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/91f008de935d222a256e6c219ba1f0a1110093fd

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In Mechanics of Materials

Citations in the corpus

Phase-field modeling of crack propagation in piezoelectric and ferroelectric materials with different electromechanical crack conditions (2012, methodology): “In fact this equation is a Ginzburg–Landau type evolution equation, where the left-hand side term is the dissipation in the process zone and the mobility constant controls the rate of the energy dissipation (Hakim and Karma, 2009).”
Phase field modelling of anisotropic crack propagation (2017, background): “More recent theoretical investigations proposed by Hakim and Karma (2009) have motivated model experiments on thin films by Takei et al. (2013).”
Phase‐field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy (2015, background): “Therefore, it can describe up to orthorhombic weak anisotropy, as exploited in previous phasefield models of anisotropic fracture [17].”
A phase field model of dynamic fracture: Robust field updates for the analysis of complex crack patterns (2013, methodology): “Conceptually similar are phase field approaches to brittle fracture based on the classical Ginzburg-Landau-type evolution equation as reviewed in [21] (also [22, 23]).”
Phase‐field modeling of ductile fracture at finite strains: A robust variational‐based numerical implementation of a gradient‐extended theory by micromorphic regularization (2017, background): “Karma A, Kessler DA, Levine H. Phase-field model of mode III dynamic fracture.”
A time-discrete model for dynamic fracture based on crack regularization (2011, methodology): “In Aranson et al. (2000); Corson et al. (2009); Eastgate et al. (2002); Hakim and Karma (2009); Karma et al. (2001); Karma and Lobkovsky (2004), the authors introduce a phase function whose evolution is given by some variation of a gradient flow of a Ginsburg-Landau–type of free energy.”
Phase Field Modeling of Fracture in Multi-Physics Problems. Part II. Coupled Brittle-to-Ductile Failure Criteria and Crack Propagation in Thermo-Elastic-Plastic Solids (2015, background): “Regularized Crack Surface Topology Let B0 ⊂ Rδ be the reference configuration of a material body with dimension δ ∈ [2, 3] in space and ∂B0 ⊂ Rδ−1 its surface as depicted in Figure 1.”
A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits (2010, methodology): “Conceptually similar are recently outlined phase field approaches to brittle fracture based on the classical Ginzburg-Landau type evolution equation as reviewed in Hakim & Karma [14], see also Karma, Kessler & Levine [15] and Eastgate et al.”