A solid circular shaft of radius $a$ is twisted until the elastic-plastic boundary is at radius $c$. Find the Torque $T$.
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The distortion energy theory states that yielding occurs when: $$ ( \sigma_1 - \sigma_2 )^2 + ( \sigma_2 - \sigma_3 )^2 + ( \sigma_3 - \sigma_1 )^2 = 2Y^2 $$ For pure shear, the principal stresses are $\sigma_1 = \tau$, $\sigma_2 = -\tau$, $\sigma_3 = 0$. Substituting these in: $$ (\tau - (-\tau))^2 + (-\tau - 0)^2 + (0 - \tau)^2 = 2Y^2 $$ $$ (2\tau)^2 + (-\tau)^2 + (-\tau)^2 = 2Y^2 $$ $$ 4\tau^2 + \tau^2 + \tau^2 = 2Y^2 \Rightarrow 6\tau^2 = 2Y^2 $$ $$ \tau = \fracY\sqrt3 \approx 0.577Y $$
J. Chakrabarty’s text is prized for its rigorous approach to the mechanics of solids. Unlike introductory texts, it covers: Deep dives into Tresca and von Mises.