Orowan bypass ($\Delta \tau \simeq 2T_{\ell }/(b\ell )$)

A dislocation unable to shear an obstacle can bow around it and leave a loop. Balancing the Peach-Koehler force against the tension of a curved dislocation gives the critical scale $\Delta \tau \simeq 2T_{\ell }/(b\ell )$, where $\ell$ is the free obstacle span on the slip plane. Centre pitch and free span are not interchangeable when particle size is appreciable.

An obstacle calculation produces a resolved shear-stress increment. A single crystal needs its Schmid factor, while a polycrystal needs a deformation-geometry or Taylor factor. The relation $\Delta \sigma =M\Delta \tau$ therefore requires an explicit orientation assumption rather than a universal tensile prefactor.