An efficient gradient smoothing meshfree formulation for the fourth-order phase field modeling of brittle fracture

A reproducing kernel gradient smoothing meshfree formulation is proposed for the fourth-order phase field modeling of brittle fracture. In order to circumvent the complexity and lower efficiency of meshfree gradient computation, a reproducing kernel gradient smoothing formulation is presented with particular reference to quadratic basis functions. Both first- and second-order meshfree smoothed gradients are discussed, in which the first-order smoothed gradients are expressed as a reproducing kernel form and meet the quadratic consistency conditions of Galerkin weak form, and the second-order smoothed gradients are then computed through performing a direct differentiation on the first-order smoothed gradients. It turns out that the resulting second-order smoothed gradients satisfy the standard gradient reproducing conditions of meshfree approximations. Subsequently, the smoothed gradients of meshfree shape functions are employed to discretize the Galerkin weak forms of equilibrium and crack phase field equations, where the numerical integration and smoothed gradient construction are specially emphasized. The tensile–compressive strain decomposition is adopted to prevent compressive crack evolution. Numerical results demonstrate the effectiveness of the proposed gradient smoothing meshfree formulation for the fourth-order phase field modeling of brittle fracture.