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Phase-field approaches to fracture allow for convenient and efficient simulation of complex fracture pattern. In this paper, two variational formulations of phase-field fracture, a common second-order model and a new fourth-order model, are combined with a finite deformation ansatz for general nonlinear materials. The material model is based on a multiplicative decomposition of the principal stretches in a tensile and a compressive part. The excellent performance of the new approach is illustrated in classical numerical examples.