Analysis and finite element simulations of the near-tip singular fields around a mode-3 stationary crack in plastically graded materials

The near-tip variable-separable singular series solution around a stationary crack embedded in plastically graded materials is derived and complementary modified-boundary-layer finite element simulations are performed. The stationary crack is subjected to an anti-plane shear (mode-3 or torsion) loading and the material nonlinearity is characterized by a power-law hardening model. The spatial variation of the material property (yield-stress) is allowed to be in any arbitrary direction around the crack and is characterized by another power-law form. The complementary modified-boundary-layer finite element simulations are performed using the newly developed nonlinear multiple-isoparametric finite element formulation. Our full-field finite element simulations validate the existence of the variable-separable form of solution. Our analysis and simulations predict that the most-singular term in the series solution remains unaffected by the inhomogeneity and its direction. The second-singular term of the series solution is however affected by the inhomogeneity and its direction. The second-term dominant region depends on the directionality of the inhomogeneity and is maximum for the x-gradation and minimum for the y-gradation. The contribution of the second-term is significant and hence a two-term based fracture mechanics methodology is required in designs using plastically graded materials.