The orientation dependence of the plane strain Young’s modulus, $E~$, of cubic materials has been analysed as a function of the direction along which a uniaxial stress is applied to a single crystal and the perpendicular direction in the single crystal along which the strain is constrained to be zero. The locus of $E~$ in the plane perpendicular to the axis of uniaxial stress is shown to be a circle when this stress is applied along $\langle$111$\rangle$. For materials with anisotropy ratios A > 1, global minima in $E~$ occur when the stress is applied along $\langle$001$\rangle$ and when the strain along one of the two perpendicular $\langle$100$\rangle$ directions is set to zero. Identical global maxima in $E~$ are found when the stress is applied along two different families of $⟨$uuw$⟩$ directions and the direction of zero strain is along either a perpendicular $\langle$1$\bar 1$0$\rangle$ or $⟨$ww$2¯$$u¯$$⟩$ direction. For materials with A < 1, the global maxima in $E~$ occur when the stress is applied along $\langle$001$\rangle$ and when the strain along one of the two perpendicular $\langle$100$\rangle$ directions is set to zero, and identical global minima are found when the stress is applied along two different families of $⟨$uuw$⟩$ directions and the direction of zero strain is along either a perpendicular $\langle$1$\bar 1$0$\rangle$ or $⟨$ww$2¯$$u¯$$⟩$ direction.