Definition
The Mohr–Coulomb theory is a mathematical model that describes the failure behavior of brittle materials such as soils, rocks, and concrete. It defines a linear relationship between the shear stress (τ) and the normal stress (σ) on a failure plane, expressed as τ = c + σ tan φ, where c is the material’s cohesion and φ is its angle of internal friction.
Overview
Developed for geotechnical and structural engineering, the Mohr–Coulomb theory is employed to predict the onset of shear failure under various loading conditions. By combining the concepts of Coulomb’s friction law with Mohr’s stress circle representation, the theory provides a criterion that can be plotted on a σ–τ diagram. It is widely used in the design of foundations, retaining structures, slopes, and tunnels, as well as in the analysis of rock masses and earth dams.
Etymology/Origin
The theory is named after two scientists:
- Christian Otto Mohr (1835–1918), a German civil engineer who introduced the graphical method of Mohr’s circle for visualizing stress states.
- Charles Augustin de Coulomb (1736–1806), a French physicist known for Coulomb’s law of friction, which relates shear stress to normal stress through a friction angle.
Their combined contributions were formalized into a unified failure criterion in the late 19th and early 20th centuries, becoming a standard in soil and rock mechanics.
Characteristics
- Linear Failure Envelope: The relationship τ = c + σ tan φ yields a straight line on a σ–τ plot, implying constant cohesion and friction angle across stress levels.
- Parameters:
- c (cohesion) – intrinsic shear strength independent of normal stress.
- φ (internal friction angle) – reflects the material’s resistance to sliding.
- Assumptions: The material is considered homogeneous, isotropic, and fails in a brittle manner; plastic deformation prior to failure is not accounted for.
- Applicability: Best suited for materials where shear failure dominates and where the stress path does not significantly deviate from the assumed linear envelope.
- Limitations: Does not capture strain-softening, hardening, or complex non‑linear behavior observed in some soils and rocks under high confining pressures.
Related Topics
- Mohr’s Circle – a graphical method for representing the state of stress at a point.
- Coulomb’s Law of Friction – the foundational principle relating shear and normal forces.
- Drucker–Prager Criterion – a non‑linear, pressure‑dependent failure model often used as a smoother alternative to Mohr–Coulomb.
- Hoek–Brown Failure Criterion – a rock‑mass specific failure model that incorporates material disturbance.
- Limit Equilibrium Methods – engineering techniques that apply the Mohr–Coulomb criterion to analyze slope stability and bearing capacity.