Phase‐field model for ductile fracture in the stress resultant geometrically exact shell

Summary We present a phase‐field fracture model for a stress resultant geometrically exact shell in finite deformation regime where the configuration manifold evolves according to deformation and fracture. The Reissner–Mindlin shell problem is first solved via the finite element method, where the independent unknown fields are the displacement and director. The phase‐field ductile fracture model is then coupled with the verified geometrically exact shell by enriching the three‐field elasto‐plastic free energy with a regularized crack surface energy. To capture the geometrical nonlinearity due to the large plastic deformation exhibited in the processing zone, the corresponding finite‐strain stress resultant elasto‐plasticity model is coupled with the phase field model. This coupling between the stress resultant elasto‐plasticity model and the phase‐field fracture model enables us to predict the different amounts of energy dissipation attributed to plastic deformation and fracture and hence simulate the crack propagation in the ductile regime properly. We introduce a mixed finite element model with displacement, director, and phase‐field order parameters as the nodal degree of freedoms formulated on the mid‐surface. An energy‐based arc‐length method is generalized to track the equilibrium path and mitigate instabilities arising from material nonlinearity. Four numerical examples are presented to validate the implementation of the model and demonstrate its capability to simulate ductile fracture in shell structures. These examples include a plane‐stress tension/shear fracture model, the Muscat‐Fenech and Atkins plate, axial tension of a notched cylindrical shell, and ductile fracture of a simply supported plate under uniform pressure.