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We consider an evolution in phase-field fracture which combines, in a system of PDEs, an irreversible gradient-flow for the phase-field variable with the equilibrium equation for the displacement field. We introduce a discretization in time and define a discrete solution by means of a 1-step alternate minimization scheme, with a quadraticL  2    {L^{2}}   -penalty in the phase-field variable (i.e. an alternate minimizing movement). First, we prove that discrete solutions converge to a solution of our system of PDEs. Then we show that the vanishing viscosity limit is a quasi-static (parametrized) BV-evolution. All these solutions are described both in terms of energy balance and, equivalently, by PDEs within the natural framework ofW   1  ,  2    ⁢   (  0  ,  T  ;   L  2   ){W^{1,2}(0,T;L^{2})}   .

A unilateral L 2 L^{2} -gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement