Failure theories are used to predict when a material will fail due to static loading. They do this by comparing the stress state at a point with material properties that can be easily determined from testing, like the yield or ultimate strengths of the material determined from a uniaxial test.
The video below covers some of the main theories for ductile and for brittle materials, including the Rankine failure theory (based on the maximum principal stress), Tresca failure theory (based on the maximum shear stress) and von Mises failure theory (based on the maximum distortion energy), as well as the Coulomb-Mohr and Modified Mohr theories.