A geometrically nonlinear phase field theory of brittle fracture

Phase field theory is developed for solids undergoing potentially large deformation and fracture. The elastic potential depends on a finite measure of elastic strain. Surface energy associated with fracture can be anisotropic, enabling description of preferred cleavage planes in single crystals, or isotropic, applicable to amorphous solids such as glass. Incremental solution of the Euler-Lagrange equations corresponds to local minimization of an energy functional for the solid, enabling prediction of equilibrium crack morphologies. Predictions are in close agreement with analytical solutions for pure mode I or pure mode II loading, including the driving force for a crack to extend from a pre-existing plane onto a misoriented cleavage plane. In an isotropic matrix, the tendency for a crack to penetrate or deflect around an inclusion is shown to depend moderately on the ratio of elastic stiffness in matrix and inclusion and strongly on their ratio of surface energy. Cracks are attracted to (shielded by) inclusions softer (stiffer) than the surrounding matrix. The theory and results apparently report the first fully three-dimensional implementation of phase field theory of fracture accounting for simultaneous geometric nonlinearity, nonlinear elasticity, and surface energy anisotropy.