On the theoretical and phase field modeling of the stress state associated with ferroelastic twin nucleation and propagation near crack tip

Ferroelastic domain switching is considered as that a twin undergoes a lattice orientation described by a deviatoric eigenstrain in this work. A continuum analysis, based on Eshelby equivalent inclusion theory, is derived to study the nucleation and evolution of twin near a pre-existing crack tip (Mode-I) in a mono-phase tetragonal crystal, in an attempt to achieve a better understanding of the contribution that twinning makes to the toughening effect. The analysis develops progressively from the homogenous isotropic, inhomogeneous isotropic to the general anisotropic conditions; and several analytical solutions are evidenced through phase field simulations. Two important critical stresses, defined as the nucleation stress and thefracture-induced twinning (FIT) stress, are found to determine the twin evolution; specifically, the nucleation stress is related to the twinning in a defect-free solid, and the FIT stress is related to the twinning in the vicinity of crack tip. Moreover, considering the inhomogeneity effect, the change of the stress intensity factor (SIF) due to a ferroelastic twin is derived, which provides evidence that the modulus toughening effect originating from the elastic misfit between the parent and twin can be ignored at a relative low applied SIF. Theoretical analysis and phase field simulations both clarify that the shielding/anti-shielding effect to the crack tip is dependent on the nucleation location and the twin size during evolution. The best toughening formula of a ferroelastic ceramic can be obtained by decreasing the FIT stress and increasing the change of SIF simultaneously through adjusting certain material parameters.