Question
Show that in a cubic crystal, the effective elastic constant for a shear across the (110) plane in the $[1 \overline{1} 0]$ direction is equal to $\frac{\left(C_{11}-C_{12}\right)}{2}$.
Step 1
Step 1: Define cubic elastic tensor components: nonzero independent constants are C11, C12, C44, and stiffness in Voigt notation has C11 on normal diagonals, C12 off-diagonals, C44 shear entries. Show more…
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Effective shear constant. Show that the shear constant $\frac{1}{2}\left(C_{11}-C_{12}\right)$ in a cubic crystal is defined by setting $e_{\mathrm{r}}=-e_{y y}=\frac{1}{2} e$ and all other strains equal to zero, as in Fig. 22. Hint: Consider the energy density (43); look for a $C^{\prime}$ such that $U=\frac{1}{2} C^{\prime} e^{2}$.
Consider a single crystal of nickel oriented such that a tensile stress is applied along a [001] direction. If slip occurs on a (111) plane and in a $[\overline{1} 01]$ direction, and is initiated at an applied tensile stress of $13.9 \mathrm{MPa}(2020 \mathrm{psi})$ compute the critical resolved shear stress.
Consider a single crystal of nickel oriented such that a tensile stress is applied along a [001] direction. If slip occurs on a (111) plane and in a $[\overline{101}]$ direction and is initiated at an applied tensile stress of $13.9 \mathrm{MPa}$ (2020 psi), compute the critical resolved shear stress.
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