Failure criteria help engineers predict when a material will yield, fracture, or slip under a combined stress state. This cheat sheet covers the stress quantities needed to apply von Mises, Tresca, and Mohr-Coulomb criteria. Students need these models because real parts rarely experience simple uniaxial loading.
A compact reference makes it easier to compare criteria and choose the right one for ductile, brittle, or frictional materials.
The setup begins with principal stresses, usually ordered as sigma1 >= sigma2 >= sigma3. Von Mises uses distortion energy and is common for ductile metals, while Tresca uses maximum shear stress and is more conservative in many cases. Mohr-Coulomb uses cohesion and internal friction angle, so it is widely used for rocks, soils, concrete, and other pressure-sensitive materials.
Each criterion compares an equivalent stress or failure function to a material strength limit.
Key Facts
- Principal stresses are the normal stresses on planes where shear stress is zero, and they are ordered as sigma1 >= sigma2 >= sigma3.
- For plane stress with sigma_x, sigma_y, and tau_xy, the principal stresses are sigma1,2 = (sigma_x + sigma_y)/2 +/- sqrt(((sigma_x - sigma_y)/2)^2 + tau_xy^2).
- The von Mises equivalent stress is sigma_vm = sqrt(0.5*((sigma1 - sigma2)^2 + (sigma2 - sigma3)^2 + (sigma3 - sigma1)^2)).
- Von Mises yielding occurs when sigma_vm >= sigma_y, where sigma_y is the uniaxial yield strength of the material.
- The Tresca maximum shear stress is tau_max = (sigma1 - sigma3)/2, and Tresca yielding occurs when sigma1 - sigma3 >= sigma_y.
- For pure shear, von Mises predicts yielding at tau = sigma_y/sqrt(3), while Tresca predicts yielding at tau = sigma_y/2.
- A common Mohr-Coulomb shear form is tau = c + sigma_n tan(phi), where c is cohesion, sigma_n is normal stress, and phi is the friction angle.
- In principal stress form for compression-positive convention, Mohr-Coulomb failure can be written as sigma1 = sigma3*(1 + sin(phi))/(1 - sin(phi)) + 2c cos(phi)/(1 - sin(phi)).
Vocabulary
- Principal stress
- A normal stress acting on a plane where the shear stress is zero.
- Equivalent stress
- A single calculated stress value used to compare a multiaxial stress state with a material strength limit.
- von Mises criterion
- A ductile yield criterion based on distortion energy, commonly used for metals.
- Tresca criterion
- A ductile yield criterion based on the maximum shear stress in the material.
- Cohesion
- The shear strength a material has when the normal stress on the failure plane is zero.
- Friction angle
- A material parameter that describes how much shear strength increases with normal compressive stress.
Common Mistakes to Avoid
- Using normal stress components directly instead of principal stresses is wrong because von Mises, Tresca, and Mohr-Coulomb are usually applied after finding the principal stress state.
- Forgetting to include sigma3 in plane stress problems is wrong because plane stress means sigma3 = 0, not that the third principal stress can be ignored.
- Mixing sign conventions in Mohr-Coulomb is wrong because compression-positive and tension-positive formulas have different signs and can give opposite conclusions.
- Treating von Mises as a brittle fracture criterion is wrong because it predicts ductile yielding and does not model pressure-sensitive cracking well.
- Comparing Tresca tau_max directly to sigma_y is wrong because yielding occurs when tau_max >= sigma_y/2, or equivalently when sigma1 - sigma3 >= sigma_y.
Practice Questions
- 1 A plane stress state has sigma_x = 80 MPa, sigma_y = 20 MPa, and tau_xy = 30 MPa. Find the two in-plane principal stresses.
- 2 For principal stresses sigma1 = 120 MPa, sigma2 = 40 MPa, and sigma3 = 0 MPa, calculate the von Mises equivalent stress.
- 3 For principal stresses sigma1 = 90 MPa, sigma2 = 30 MPa, and sigma3 = -10 MPa with sigma_y = 100 MPa, determine whether Tresca predicts yielding.
- 4 Explain why von Mises is usually preferred for ductile metals while Mohr-Coulomb is often preferred for soils, rocks, or concrete.