Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations.

*(English)*Zbl 1202.74014Summary: The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase-field. In this paper, we outline a thermodynamically consistent framework for phase-field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that \(\Gamma \)-converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase-field. Here, we propose alternative rate-independent and viscous over-force models that ensure the local growth of the phase-field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase-field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase-field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase-field formulations of fracture by means of representative numerical examples.

##### MSC:

74A45 | Theories of fracture and damage |

74A15 | Thermodynamics in solid mechanics |

74S05 | Finite element methods applied to problems in solid mechanics |

##### Keywords:

fracture; crack propagation; phase-fields; gradient-type damage; incremental variational principles; finite elements; coupled multi-field problems
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\textit{C. Miehe} et al., Int. J. Numer. Methods Eng. 83, No. 10, 1273--1311 (2010; Zbl 1202.74014)

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