= Torsion of a solid circular shaft
{title2=$C=GJ_p\phi/L,\quad J_p=\pi a^4/2$}
A circular shaft twisted uniformly through small end angle $\phi$ over length $L$ has local engineering <shear strain> $r\phi/L$ at radius $r$. Its <shear stress> is $Gr\phi/L$, so the integrated <torque> is $C=\int_A r(Gr\phi/L)\,dA=GJ_p\phi/L$, using the <polar second moment of area>. The restoring torque has the opposite sign to the imposed twist. The stored elastic <energy> is $C\phi/2=GJ_p\phi^2/(2L)$. The circular cross-section is important: general noncircular sections can warp and have a different torsion constant.
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