A length scale insensitive phase field model based on geometric function for brittle materials

Abstract

The commonly used phase field model has a broad application in static and dynamic scenarios with brittle materials. Following the assumption of irreversibility of fracture, monotonically increasing phase and tensile strain energy are applied in the commonly used phase field model. By doing so, a clear stiffness reduction, however, is observed in stress–strain curve before reaching fracture strength, which conflicts with the linear elastic behavior of brittle materials. The limitation has also been mentioned in publications when length scale is the only parameter to adjust fracture strength. To solve these problems, a generalized quadratic geometric function is proposed to build a length scale insensitive phase field model. Governing equations and one-dimensional analytical solutions (homogeneous and nonhomogeneous solutions) of this length scale insensitive phase field model are presented. By adjusting the extra parameter in the generalized quadratic geometric function and zeroing negative phase, different phase profile (ranged from exponential profile to linear profile) and fracture strength can be obtained, which could be more suitable to cover different types of fractures. Meanwhile, a narrower crack band and linear elastic behavior can be guaranteed. LS-DYNA user-defined element and user-defined material subroutines are used to implement the proposed length scale insensitive phase field model. A tensile test successfully verifies the properties of a narrower crack band and linear elastic behavior. Three representative numerical examples show the feasibility of this model to simulate fracture propagation, including a Mode Ⅰ failure of wedge splitting test, a mixed-mode failure of l-shaped panel test and a mixed-mode failure of notched beam test. From the crack band and force–displacement curve, the proposed model can handle fracture propagation better for brittle materials.