Tresca Yield Criterion Explained

The Tresca criterion, also known as the maximum shear stress theory, is a failure criterion used in mechanics of materials to predict when a material will start to yield under plane stress or general stress conditions. It postulates that yielding occurs when the maximum shear stress ($ \tau_{max} $) experienced by the material reaches a critical value.

Understanding the Yield Locus

The yield locus is a boundary in stress space that separates elastic behavior (where the material returns to its original shape upon unloading) from plastic behavior (where permanent deformation occurs). For a given material, the shape of this locus depends on the yield criterion used.

Tresca Criterion and the Hexagonal Locus

The Tresca criterion relates the maximum shear stress to the shear stress at yield in a uniaxial tension test. Mathematically, it can be expressed as:

$$ \tau_{max} = \frac{\sigma_1 - \sigma_3}{2} \le \tau_{yield} $$

where $ \sigma_1 $ is the maximum principal stress and $ \sigma_3 $ is the minimum principal stress. $ \tau_{yield} $ is the shear stress at yield in uniaxial tension.

When this criterion is represented graphically in terms of principal stresses, particularly in the deviatoric stress plane (a plane perpendicular to the hydrostatic axis in principal stress space), the Tresca yield surface forms a regular hexagon.

Each side of the hexagon corresponds to a condition where the maximum shear stress occurs between a different pair of principal stresses:

• $ \sigma_1 - \sigma_2 = 2 \tau_{yield} $
• $ \sigma_2 - \sigma_3 = 2 \tau_{yield} $
• $ \sigma_1 - \sigma_3 = 2 \tau_{yield} $
• And corresponding negative stress difference conditions.

These conditions represent the boundaries where yielding initiates according to the maximum shear stress theory.

Conclusion

Based on the Tresca criterion, the yield locus, when properly visualized in the deviatoric stress plane, is a hexagon.