Engineers use yield criteria to predict when a ductile material will begin to deform permanently under combined stresses. Simple tensile yield strength is measured in one direction, but real parts often experience tension, compression, torsion, bending, and pressure at the same time. The von Mises and Tresca criteria turn a three-dimensional stress state into a single comparison with the material yield strength.
This helps designers decide whether a shaft, bracket, pressure vessel, or machine part is safe before it is built.
Key Facts
- Von Mises yield criterion: σv = sqrt(((σ1 - σ2)^2 + (σ2 - σ3)^2 + (σ3 - σ1)^2)/2)
- Yield begins by von Mises when σv >= σy, where σy is the uniaxial yield strength.
- Tresca yield criterion: τmax = max(|σ1 - σ2|, |σ2 - σ3|, |σ3 - σ1|)/2
- Yield begins by Tresca when τmax >= σy/2, or equivalently max(|σi - σj|) >= σy.
- Hydrostatic stress σh = (σ1 + σ2 + σ3)/3 does not cause yielding in ideal ductile metal criteria.
- In pure shear, von Mises predicts τy = σy/sqrt(3), while Tresca predicts τy = σy/2.
Vocabulary
- Principal stress
- A normal stress acting on a plane where the shear stress is zero, usually written as σ1, σ2, and σ3.
- Yield criterion
- A mathematical rule that predicts when a material begins permanent plastic deformation under a multiaxial stress state.
- Von Mises stress
- An equivalent stress based on distortion energy that is compared with uniaxial yield strength.
- Tresca criterion
- A yield rule stating that yielding begins when the maximum shear stress reaches the shear stress at yield in a tensile test.
- Hydrostatic axis
- The line in principal stress space where σ1 = σ2 = σ3, representing equal pressure or tension in all directions.
Common Mistakes to Avoid
- Using hydrostatic stress alone to predict yielding is wrong because ideal von Mises and Tresca yielding depend on differences between principal stresses, not their average value.
- Forgetting to sort or compare all principal stress differences in Tresca is wrong because the maximum shear stress must use the largest absolute difference between any two principal stresses.
- Using σy instead of σy/2 for the Tresca shear limit is wrong because a uniaxial tensile test reaches yield when its maximum shear stress is σy/2.
- Assuming von Mises and Tresca always give the same answer is wrong because Tresca is usually more conservative, especially in pure shear and many combined-stress cases.
Practice Questions
- 1 A ductile steel has σy = 250 MPa and principal stresses σ1 = 180 MPa, σ2 = 60 MPa, and σ3 = 0 MPa. Compute the von Mises stress and decide whether yielding is predicted.
- 2 For σy = 300 MPa and principal stresses σ1 = 220 MPa, σ2 = -40 MPa, and σ3 = 20 MPa, use the Tresca criterion to determine whether yielding begins.
- 3 A part is under equal triaxial compression, with σ1 = σ2 = σ3 = -100 MPa. Explain why von Mises and Tresca criteria do not predict yielding for this ideal stress state, even though the pressure is large.