The Mohr Theory of Failure, also known as the Coulomb-Mohr criterion or internal-friction 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 sc of the material. Likewise, the right circle is for uniaxial tension at the limiting tension stress st.
The middle Mohr's Circle on the figure (dash-dot-dash 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.
||Both in tension
||s1 > 0, s2 > 0
||s1 < st, s2 < st
||Both in compression
||s1 < 0, s2 < 0
||s1 > -sc, s2 > -sc
||s1 in tension, s2 in compression
||s1 > 0, s2 < 0
||s1 in compression, s2 in tension
||s1 < 0, s2 > 0