Maximum Normal Stress Criterion 
The maximum stress criterion, also known as the normal stress, Coulomb, or Rankine criterion, is often used to predict the failure of brittle materials.
The maximum stress criterion states that failure occurs when the maximum (normal) principal stress reaches either the uniaxial tension strength s_{t}, or the uniaxial compression strength s_{c},
s_{c} < {s_{1}, s_{2}} < s_{t}
where s_{1} and s_{2} are the principal stresses for 2D stress. Graphically, the maximum stress criterion requires that the two principal stresses lie within the green zone depicted below,

Mohr's Theory  
The Mohr Theory of Failure, also known as the CoulombMohr criterion or internalfriction theory, is based on the famous Mohr's Circle. Mohr's theory is often used in predicting the failure of brittle materials, and is applied to cases of 2D stress.
Mohr's theory suggests that failure occurs when Mohr's Circle at a point in the body exceeds the envelope created by the two Mohr's circles for uniaxial tensile strength and uniaxial compression strength. This envelope is shown in the figure below,
The left circle is for uniaxial compression at the limiting compression stress s_{c} of the material. Likewise, the right circle is for uniaxial tension at the limiting tension stress s_{t}. The middle Mohr's Circle on the figure (dashdotdash line) represents the maximum allowable stress for an intermediate stress state. All intermediate stress states fall into one of the four categories in the following table. Each case defines the maximum allowable values for the two principal stresses to avoid failure.
 
Graphically, Mohr's theory requires that the two principal stresses lie within the green zone depicted below,
Also shown on the figure is the maximum stress criterion (dashed line). This theory is less conservative than Mohr's theory since it lies outside Mohr's boundary. 