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This paper presents a mixed Graph-FEM (Finite Element Method) approach for the phase field modeling of fracture in plates and shells composed of nonlinearly elastic solids. In this approach, the finite element mesh is deemed equivalent to a typical graph in computer science. The equation for phase field evolution is thus discretized by a graph Laplacian on a curved surface. A corresponding solver is herein developed, considering the matrix sparsity of the discretized system, and constraining irreversible phase field variables (to values greater than zero and less than one). An alternating solution strategy between mechanical equilibrium and the phase field equation is proposed. Our method is then applied to the fracture of plates and shells of arbitrary curvature, exhibiting fast convergence and robustness compared with our previous work, as could be concluded from the several numerical benchmarks we …