The mechanics of crack-tip dislocation emission and twinning

Dislocation emission from a crack tip is a necessary mechanism for crack tip blunting and toughening. A material is intrinsically ductile under Mode I loading when the critical stress intensity $K_{Ie}$ for dislocation emission is lower than the critical stress intensity $K_{Ic}$ for cleavage. In intrinsically ductile fcc metals, a first partial dislocation is emitted, followed either by a trailing partial dislocation (''ductile'' behavior) or a twinning partial dislocation (''quasi-brittle'').

$K_{Ie}^{first}$ for the first partial emission is usually evaluated using the approximate Rice theory, which predicts a dependence on the elastic constants and the unstable stacking fault energy $\gamma_{usf}$. Here, atomistic simulations across a wide range of fcc metals show that $K_{Ie}^{first}$ is systematically larger (10-30%) than predicted. However, the critical crack-tip shear displacement is up to 40% smaller than predicted. The discrepancy arises because Mode I emission is accompanied by the formation of a surface step that is not considered in the Rice theory. A new theory for Mode I emission is presented based on the ideas that (i) the stress resisting step formation at the crack tip creates ''lattice trapping'' against dislocation emission such that (ii) emission is due to a mechanical instability at the crack tip. The new theory naturally includes the energy to form the step, and reduces to the Rice theory (no trapping) when the step energy is small. The new theory predicts a higher $K_{Ie}^{first}$ at a smaller critical shear displacement, rationalizing deviations of simulations from the Rice theory.

The twinning tendency is estimated using the Tadmor and Hai extension of the Rice theory. Atomistic simulations reveal that the predictions of the critical stress intensity factor $K_{Ie}^{twin}$ for crack tip twinning are also systematically lower (20-35%) than observed. Energy change during nucleation reveal that twining partial emission is not accompanied by creation of a surface step while emission of the trailing partial creates a step. The absence of the step during twinning motivates a model for twinning nucleation that accounts for the fact that nucleation does not occur directly at the crack tip. New predictions are in excellent agreement with all simulations that show twinning. A second mode of twinning is found wherein the crack first advances by cleavage and then emits the twinning partial at the new crack tip.

The stacking fault stress dependence is analyzed through (i) the generalized stacking fault potential energy (GSFE) and (ii) the generalized stacking fault enthalpy (GSFH). At an imposed shear displacement, there is also an associated inelastic normal displacement $\Delta_{n}$ around the fault. Atomistic simulations with interatomic potentials and/or first principle calculations reveal that GSFE and $\Delta_n$ both increase with tensile stress. An increasing GSFE contradicts long-standing wisdom and previous studies. Positive $\Delta_{n}$ coupled to the applied normal stress decreases the GSFH, but GSFH is not useful for general mechanics problems. ''Opening softening'' effects are not universal, and so the analysis of any particular nanomechanics problem requires precise implementation of the combination of GSFE and $\Delta_{n}$ rather than the GSFH.