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• An adaptive consistent element-free Galerkin method (CEFG) for four-order phase-field model is proposed. • Two different consistent integration schemes, namely quadratically consistent one-point integration scheme and three-point integration scheme, are employed to evaluate the fourth-order phase-field model. • The adaptive criterion of local mesh refinement for four-order phase-field model is proposed. An adaptive fourth-order phase-field model is proposed using consistent element-free Galerkin method (EFG). Convenient construction of high-order interpolation functions in EFG method is fully utilized. Requirement of C 1 continuity of shape functions is satisfied in fourth-order phase-field model. Two different consistent integration schemes with high efficiency and high accuracy, namely quadratically consistent one-point integration scheme and three-point integration scheme, are employed to evaluate the fourth-order phase-field equation and displacement equation, respectively. To reduce computational cost, the adaptive strategy of local background mesh refinement is established on basis of strain energy history and phase field. Background mesh is refined via the insertion of nodes at the midpoints of each side of the integration element. Comparing with EFG method, computational accuracy and efficiency are enhanced by the proposed method. Load-displacement response, the smoothness of the resulting stress and crack paths are predicted well. [ABSTRACT FROM AUTHOR]

• An efficiently adaptive EFG for phase-field of brittle fracture of FGMs is developed. • The adaptive criterion for phase-field model of brittle fracture of FGMs is proposed. • Fracture mechanism of FGMs is revealed using phase-field model. Gradient distributions of material properties are taken into account in phase-field model of fracture and the adaptive consistent element-free Galerkin method (EFG) for the fracture analysis of functionally graded materials (FGMs) is presented. Considering the fact that both the moving least-square approximation and the consistent integration scheme contribute to producing the accurate stress fields, the phase-field model of fracture for FGMs is numerically solved by the consistent EFG method in this paper. The corresponding adaptive criterion is established and 2D and 3D crack propagation of FGMs is simulated. Numerical results indicate that the proposed method can accurately reflect the influence of different distributions of material parameters on the crack propagation and has high efficiency compared with the standard EFG method and linear FEM. At some level, the fracture mechanism that crack propagation is driven by both the strain energy history and the critical energy release rate is revealed using phase-field model. [ABSTRACT FROM AUTHOR]

Consistent element-free Galerkin (CEFG) method for phase-field model of 3D cracks with adaptive nodal refinement. Element-free Galerkin (EFG) method for three-dimensional phase-field model of cracks is described. Quadratic moving least-squares (MLS) approximation is constructed for both the phase field and the displacement. The quadratically consistent 4-point (QC4) integration scheme using background tetrahedral elements (cells) is employed to efficiently compute the domain integrals in weak forms. As cracks propagate, adaptive refinement of the approximation nodes around crack fronts is implemented with the use of background integration cells. The refinement is triggered by the maximum residual strain energy history. New approximation nodes are inserted at the midpoints of the element edges. Several benchmark examples are investigated. Numerical results show that three-dimensional crack propagation is successfully modeled by the proposed method and correct crack paths and load-displacement responses are obtained. Due to the implemented h-adaptivity, lots of approximation nodes are saved and the computational scale is significantly reduced. Improved accuracy of the proposed method in stress distribution and crack width is also demonstrated. [ABSTRACT FROM AUTHOR]