Past exam of the materials science course of the University of Cambridge / 2016 / Paper 1 / 5 / b

For a coarse-grained, approximately untextured polycrystal undergoing general plastic flow, use the conventional orientation-and-compatibility estimateWith the supplied ${\tau }_c=27.5\mathrm{MPa}$,The round $M=3$ result is a conventional estimate for an unspecified coarse polycrystal, not a material-independent exact conversion. The most favorably oriented individual grains can first slip at $2{\tau }_c=55\mathrm{MPa}$; that is not the same event as compatible macroscopic yielding throughout the aggregate.

There is no single crystallographic tensile axis for all grains. Individual grains rotate and can develop crystallographic texture, while the macroscopic bar remains loaded along its tensile axis. A specified initial texture or yield criterion would be needed for a more exact numerical prediction.

For FCC aluminium, consider $\{111\}⟨110⟩$ slip. Normalize the tensile direction and one candidate system:Here $\mathbf{n}\cdot \mathbf{s}=0$. Comparing all twelve systems gives the largest Schmid factor on $(\overline{1}11)[101]$:Schmid's law therefore givesDuring ideal single slip, the tensile direction expressed in crystal coordinates rotates towards $[101]$, along the great-circle path joining the original axis to that slip direction. It reaches a symmetry boundary where another slip system becomes equally favored. This single-slip tensile-axis rotation concerns the crystal relative to the externally held loading direction, rather than a rotation of the laboratory loading apparatus.

Take ordinary basal slip in magnesium, $(0001)⟨11\overline{2}0⟩$. In three-index hexagonal notation, $[101]$ denotes a vector proportional to ${\mathbf{a}}_1+\mathbf{c}$, not a cubic Cartesian $(1,0,1)$ with equal axis lengths. The basal slip direction closest to the tensile axis is along ${\mathbf{a}}_1$; its four-index representative is $[2\overline{1}\overline{1}0]$.

For $r=c/a$,Thus the algebraic answer isAn external crystallographic reference gives $a=0.320927\mathrm{nm}$ and $c=0.521033\mathrm{nm}$, hence $r\simeq 1.624$ and ${\sigma }_y\simeq 1.79\mathrm{MPa}$. This is a conditional external-data estimate, not a fabricated value attributed to the supplied Data Book.^[ref]

The tensile axis rotates towards the selected basal slip direction during ideal single slip. As the axis approaches the basal plane, the resolved basal shear eventually becomes small; other modes, hardening and instability can interrupt the single-slip description. TODO: verify the exact magnesium lattice ratio supplied in the historical examination Data Book. Taking a $45^{\circ }$ angle solely from the indices would incorrectly use a cubic metric.