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• 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]