Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations

<jats:title>Abstract</jats:title><jats:p>The computational modeling of failure mechanisms in solids due to fracture based on <jats:italic>sharp</jats:italic> crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a <jats:italic>diffusive</jats:italic> 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 <jats:italic>regularized crack surface functional</jats:italic> that Γ‐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 <jats:italic>growth</jats:italic> of the phase‐field. Next, we define an energy storage function whose <jats:italic>positive</jats:italic> 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 <jats:italic>two‐field setting</jats:italic> for rate‐independent response and a time‐regularized <jats:italic>three‐field formulation</jats:italic> with viscous over‐force response. It is then shown that these balances follow as the Euler equations of incremental <jats:italic>variational principles</jats:italic> 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. Copyright © 2010 John Wiley & Sons, Ltd.</jats:p>