MohrCoulomb

Mohr-Coulomb is a constitutive model used in soil and rock mechanics to describe shear strength and failure under stress. It combines Coulomb friction with cohesion to form a simple failure envelope that relates shear stress on a plane to the normal stress acting on that plane. The common form of the criterion is tau = c + sigma_n tan(phi), where tau is the shear stress, sigma_n is the normal stress on the failure plane, c is cohesion, and phi is the angle of internal friction.

In terms of principal stresses, failure is depicted using Mohr’s circle: when the state of stress reaches the Mohr-Coulomb envelope, the circle is tangent to the line tau = c + sigma_n tan(phi). This leads, for principal stresses sigma1 (major) and sigma3 (minor), to the relation sigma1 = sigma3 (1 + sin phi)/(1 - sin phi) + 2 c cos^2(phi)/(1 - sin phi). The parameters c and phi characterize the material’s shear strength.

Mohr-Coulomb is widely used in geotechnical and rock mechanics for applications such as slope stability analysis, foundation bearing capacity, tunnel and excavation design, and rock mass assessment. It remains a staple due to its simplicity and intuitive link to friction and cohesion, though it is an empirical model with limitations.

Limitations include its assumption of isotropy, rate and temperature independence, and a fixed friction angle; it may not capture strain softening, nonlinear hardening, or complex three-dimensional behavior. Alternatives include the Drucker-Prager model and other advanced constitutive frameworks.