Solution (source code)

= Solution

Initially, one <slip system> carries most of the deformation. Easy <dislocation glide> gives the nearly flat stage-I response, although lattice rotation and gradual <dislocation> interactions can already change the required tensile <stress>. In tension the loading direction expressed in the rotating crystal approaches the primary slip direction. Its path has the form $(5+t,1+t,4)$ until it reaches the symmetry boundary $y=z$, at $[844]=[211]$.

At that orientation, \b[$(1\bar11)[110]$ and $(11\bar1)[101]$] have equal <Schmid factors>, $1/\sqrt6$. Conjugate slip therefore becomes competitive. Intersections between the two <dislocation> families generate obstacles, junctions and <forest hardening>, giving the sharp stage-II rise in flow <stress>. This is the ideal single-slip-to-conjugate-slip construction; real activation stresses can be shifted by latent hardening and imperfections.

\Image[/past-exam-of-the-materials-science-course-of-the-university-of-cambridge/2019/paper-1-single-slip.png]
{title=Ideal single-slip rotation and the onset of conjugate slip}
{description=Original geometric construction. The stress trace before conjugate slip retains a constant critical resolved shear stress; the later hardening curve is schematic.}