Introduction to Static Failure Theories

In mechanical component design, engineers must ensure that the induced stresses do not cause the material to fail. For ductile materials (like most structural steels and aluminum alloys) under static loading, yielding is considered the primary mode of failure.

To evaluate complex loading states, we first extract the Principal Stresses ($\sigma_1, \sigma_2, \sigma_3$), which are the maximum and minimum normal stresses acting on planes with zero shear stress. Once known, we apply a static failure criterion to determine if the material will yield.

Maximum Shear Stress Theory (Tresca Criterion)

The Maximum Shear Stress theory, or Tresca criterion, states that yielding begins when the maximum shear stress in a complex stress state equals the maximum shear stress at the yield point in a simple tension test.

For a principal stress state where $\sigma_1 \ge \sigma_2 \ge \sigma_3$, the criterion for a safe design is:

Distortion Energy Theory (von Mises Criterion)

The Distortion Energy theory, commonly known as the von Mises criterion, provides a more accurate prediction for the yielding of ductile materials. It states that yielding occurs when the distortion energy per unit volume equals the distortion energy at yield in a simple tension test.

The equivalent von Mises stress ($\sigma'$), also known as effective stress, is calculated by:

$\sigma' = \sqrt{\frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2}{2}}$

For a safe design, the equivalent stress must be less than the allowable stress:

$\sigma' < \frac{S_y}{n}$

Engineering Check

The von Mises criterion generally predicts a slightly larger safe region than the Tresca criterion. When evaluating brittle materials, or cyclic (fatigue) loading, entirely different theories must be used (e.g., Mohr's theory or Goodman diagrams).