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.
Understanding Failure Criteria (von Mises, Tresca, Mohr-Coulomb)
The key difference between these models is what they treat as dangerous. In a ductile metal, yielding starts when the shape of a tiny material element is distorted enough for crystal defects to move. Equal pressure applied from every direction changes volume but does not create much shape change.
This is why the von Mises model is largely unaffected by hydrostatic stress. A deeply pressurized metal vessel can carry large normal stresses without yielding if those stresses are nearly equal in all directions.
A shaft under torque is different. Its shear stress creates strong distortion, so it can yield at a much lower applied load.
Tresca is based on the largest shear stress present inside the material. It is simple to visualize using the largest and smallest principal stresses. The difference between them controls the most severe internal sliding tendency.
Tresca usually predicts yield slightly earlier than von Mises for the same multiaxial stress state. Engineers may choose it when a cautious estimate is useful or when a design rule specifically requires it. Neither model says that a metal has snapped.
Yielding means permanent deformation has begun. A part can still support load after first yield, depending on its geometry, hardening behavior, and the way the load is applied.
Mohr-Coulomb describes a different physical process. Granular and cracked materials often fail because surfaces inside them slide past each other. Compression across a potential sliding surface increases friction and makes sliding harder.
Tension reduces contact between particles or crack faces, so failure can occur more easily. Cohesion represents shear strength that remains even with little normal compression. The friction angle describes how strongly strength rises as compression rises.
This behavior appears in soil slopes, retaining walls, foundations, rock tunnels, concrete, and powder handling. For these materials, treating tensile and compressive strength as equal can give a very poor prediction.
Before calculating anything, check the sign convention used by the course, textbook, or software. Some fields call tension positive, while many geotechnical formulas use compression positive. A correct equation with the wrong sign convention produces a wrong result.
Students should separate total stress from effective stress when studying saturated soil. Pore water pressure carries part of the normal load, reducing grain to grain contact and therefore reducing frictional resistance. Material data matters just as much as the chosen criterion.
Yield strength, cohesion, and friction angle come from tests performed under particular temperatures, loading rates, moisture conditions, and sample histories. A failure criterion is a model built from those measurements, not a guarantee that every real component fails in exactly the predicted way.