Mohr–Coulomb Failure Criterion

Abstract

a (m-1)/(m + 1) b 1/(m + 1) c Cohesion C 0 Uniaxial compressive strength m (1 + sin /)/(1-sin /) S 0 Inherent shear strength (cohesion) T Uniaxial tensile strength T 0 Theoretical MC uniaxial tensile strength / Angle of internal friction l = tan / Coefficient of internal friction r Normal stress on plane s Shear stress on plane r 1 , r 2 , r 3 Principal stresses, with no regard to order r I , r II , r III Major, intermediate, minor principal stresses r m (r I + r III)/2 s m (r I-r III)/2 r I * C 0-mT r III *-T 1 Description The Mohr-Coulomb (MC) failure criterion is a set of linear equations in principal stress space describing the conditions for which an isotropic material will fail, with any effect from the intermediate principal stress r II being neglected. MC can be written as a function of (1) major r I and minor r III principal stresses, or (2) normal stress r and shear stress s on the failure plane (Jaeger and Cook 1979). When all principal stresses are compressive, experiments demonstrate that the criterion applies reasonably well to rock, where the uniaxial compressive strength C 0 is much greater than the uniaxial tensile strength T, e.g. C 0 /T [ 10; some modification is needed when tensile stresses act, because the (the-oretical) uniaxial tensile strength T 0 predicted from MC is not measured in experiments. The MC criterion can be considered as a contribution from Mohr and Coulomb (Nadai 1950). Mohr's condition is based on the assumption that failure depends only on r I and r III , and the shape of the failure envelope, the loci of r, s acting on a failure plane, can be linear or nonlinear (Mohr 1900). Coulomb's condition is based on a linear failure envelope to determine the critical combination of r, s that will cause failure on some plane (Coulomb 1776). A linear failure criterion with an intermediate stress effect was described by Paul (1968) and implemented by Meyer and Labuz (2012). 2 Background Coulomb, in his investigations of retaining walls (Heyman 1972), proposed the relationship jsj ¼ S 0 þ r tan / ð1Þ where S 0 is the inherent shear strength, also known as cohesion c, and / is the angle of internal friction, with the coefficient of internal friction l = tan /. The criterion