Which one of the following samples of high-purity aluminium (Al) single crystal will plastically yield at the lowest applied load under ambient conditions? Loading axis is along the direction shown in the schematic.
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In FCC crystals, slip occurs on \{111\}<110> systems. The orientation with the highest Schmid factor yields first. Directions like [123] often maximize \(\cos\phi\cos\lambda\), giving the lowest yielding load.
To determine which single crystal of aluminium yields at the lowest load, we must analyze how plastic deformation begins in an FCC metal. Aluminium (Al) is a face-centered cubic material, and its slip systems are of the type \{111\}<110>, meaning that slip occurs on the close-packed \{111\} planes along the close-packed <110> directions. Plastic deformation begins when the resolved shear stress on one of these slip systems reaches the critical resolved shear stress (CRSS). The resolved shear stress acting on a slip system is given by Schmid's law: \[ \tau = \sigma \, m = \sigma (\cos \phi \cos \lambda) \] where \( \phi \) is the angle between the loading axis and the slip plane normal, and \( \lambda \) is the angle between the loading axis and slip direction. The quantity \( m = \cos \phi \cos \lambda \) is the Schmid factor. The crystal will yield first on the slip system having the highest Schmid factor, because it requires the smallest applied stress \( \sigma \) to reach the CRSS. We now compare the loading directions [100], [111], [101], and [123] with respect to their alignment with the FCC slip systems. In FCC materials, the direction [100] gives a moderate Schmid factor because it is symmetric with respect to multiple slip systems. The direction [111], however, is aligned with the slip plane normal and therefore gives a low Schmid factor; this means it requires a higher applied load to activate slip. On the other hand, the direction [101] gives a better alignment with <110> slip directions than [111], but still not the maximum possible. The direction [123] is known to give an unusually high Schmid factor for FCC materials because its orientation produces favorable angles between both the \(\{111\}\) slip plane normals and the <110> slip directions. This makes the product \(\cos \phi \cos \lambda\) largest among the given choices, and therefore this sample reaches the CRSS at the lowest external load. Thus, the sample oriented along the [123] direction (iv) yields first under the smallest applied stress.
