Construction of the resolved shear stress. Click here to open in new window

To move a dislocation, and thus cause plastic deformation, a critical stress must be applied to overcome the resistance to dislocation movement (glide).

For an applied stress, σ, the resolved shear stress acting upon a dislocation

τ_R = σ cos φ cos λ

The resolved shear stress must exceed a critical value τ_C before shear can occur. This condition would define the proportional limit, or yield stress.

The microstructure of the metal defines τ_C.

The link between the two (for single crystals) is the Schmid factor, cos φ cos λ, which typically has values between 0.2 and 0.4. In a polycrystalline sheet of metal, each individual grain, or crystal, deforms on up to five slip systems, with the activity on each defined by the resolved shear stress, just as for the single crystal just discussed. However, the deformation of the individual grains is constrained by that of its neighbours.

Thus, the discrete nature of slip is the fundamental origin of plastic anisotropy: Slip activity is related to the way a crystal is oriented relative to the axes of deformation. Extension to multiaxial deformation is straight forward. One simply adds up the components of resolved shear stress due to each of the externally applied stresses and when the σ_crss is reached, deformation begins.