Estimates from atomic models of tension-shear coupling in dislocation nucleation from a crack tip

The normal stress distribution across a slip plane has the effect of reducing the critical loading required for dislocation emission from a crack tip. The reduction by normal stresses was found to be very significant for Si, based on properties estimated for it using density functional theory, to be large for Fe as modeled by the embedded atom method (EAM), and to be smaller in Al, Ni and ordered Ni3Al, estimated using the EAM. The general dependence over a wide range on parameters characterizing the tension-shear coupling was also determined. In the context of a Peierls model for dislocation nucleation at a crack tip (J.R. Rice, J. Mech. Phys. Solids, 40 (1992) 239), our approach was to search for onset of the dislocation nucleation instability based on the numerical solution of the system of non-linear integral equations describing an incipient dislocation. The incipient dislocation consists of a distribution of sliding and opening displacements along a slip plane emanating from the crack tip; these displacements are related to the shear and tensile stresses across the slip plane by constitutive relations based on the atomic models mentioned. Results from the atomic models are used to parametrize constitutive relations involving a Frenkel sinusoidal dependence of shear stress on sliding displacement at any fixed opening displacement, and a Rose-Ferrante-Smith universal binding form of dependence of tensile stress on opening displacement at any fixed shear displacement. These relations then enter the system of integral equations, solved numerically, which describe the elasticity solution for a non-uniform distribution of sliding and opening along the slip plane. The results show that tension-shear coupling will often significantly reduce the loading for dislocation emission from the value estimated on the basis of an unstable stacking energy γ us determined with neglect of such coupling, in a shear-only type analysis. For the EAM models of the metals considered, a simple and approximate method to account for the tension effects is to use a modified quantity γ us (u ∗) , which is an unstable stacking energy for lattice planes which are constrained to a fixed opening Δθ ∗ , corresponding to that for vinishing normal stress at the unstable shear equilibrium position. Moreover, it is found that the normal stress effect can be described well in these cases by replacing the unstable stacking energy γ us in the shear-only model by a tension softened γ us(Ψ), which depends on the phase angle Ψ of the combined tension-shear loading along the slip plane according to the stress intensity factors of the elastic singular solution. The same simple procedures for accounting for tension effects on nucleation are less suitable for lattices with strong coupling such as Si.