Tresca yield criterion

Remarks

The criterion is a constitutive assumption, the maximum shear stress hypothesis, not a theorem. The maximum shear stress at a point is half the largest difference of principal stresses, which explains the factor $2$. Its yield surface in principal stress space is an infinite regular hexagonal prism about the line $\sigma_1 = \sigma_2 = \sigma_3$, so hydrostatic pressure has no effect. Check in uniaxial tension, $\sigma_1 = \sigma$, $\sigma_2 = \sigma_3 = 0$: the maximum is $\abs{\sigma}$, so yielding starts at $\sigma = \sigma_y$. In pure shear, $\sigma_1 = \tau$, $\sigma_2 = 0$, $\sigma_3 = -\tau$: the maximum is $2\abs{\tau}$, so the shear yield stress is $\sigma_y / 2$, smaller than the value $\sigma_y/\sqrt{3}$ of the von Mises criterion. When the two are matched in uniaxial tension, the Tresca hexagon is inscribed in the von Mises circle.

Going deeper

For a student

The Tresca criterion is piecewise linear in the principal stresses. So it often gives simple closed-form answers in limit analysis and in hand calculations, for example for thick tubes. Its corners make the flow rule non-unique there, which is a nuisance in numerical work. It is the special case of the Mohr–Coulomb criterion with zero friction angle.