Mohr-Coulomb Failure Criterion
Mohr-Coulomb Failure Criterion: Theoretical Foundations
What is the Mohr-Coulomb Criterion?
Professor, the Mohr-Coulomb failure criterion is fundamental in soil mechanics, right?
The Mohr-Coulomb (MC) criterion is the most classical criterion describing the shear failure of soil and rock. Proposed by Coulomb in 1773.
- $\tau$ — Shear stress (on the failure plane)
- $c$ — Cohesion
- $\sigma_n$ — Normal stress (compression positive)
- $\phi$ — Internal friction angle
How does it differ from von Mises?
von Mises is independent of hydrostatic pressure (mean stress). The MC criterion depends on hydrostatic pressure (contains normal stress $\sigma_n$). Soil's shear strength increases with greater confining pressure. This is the essence of the MC criterion.
Principal Stress Representation
In deviatoric stress space, it forms an irregular hexagon (different from von Mises's cylinder).
Settings in FEM
Summary
Key Points:
- $\tau = c + \sigma_n \tan\phi$ — Shear strength depends on normal stress
- Two parameters: $c$ (cohesion) and $\phi$ (friction angle)
- Hydrostatic pressure dependence — Fundamental difference from von Mises
- Failure criterion for soil, rock, concrete — Fundamental in geotechnical engineering
Origin of Coulomb's Friction Law
Charles-Augustin de Coulomb organized experimental data on landslides in 1776, showing that shear strength can be expressed as τ=c+σtanφ. Later in 1900, Otto Mohr combined it with a geometric interpretation in principal stress space (Mohr's circle), systematizing it as the Mohr-Coulomb failure criterion. It has been used continuously in rock/soil mechanics for nearly 250 years.
Computational Methods for Mohr-Coulomb Failure Criterion
FEM Treatment of MC Criterion
The MC criterion's yield surface has corners. Stress return mapping at corners is numerically challenging.
Countermeasures:
- Approximation with Drucker-Prager (DP) criterion — Approximation with a conical surface (no corners). Good convergence.
- Exact treatment of MC criterion — Special handling at corners. Abaqus supports exact MC.
- Plaxis — Fully supports MC criterion. Strength of specialized geotechnical software.
Dilation Angle $\psi$
The dilation angle $\psi$ determines the direction of plastic flow. If $\psi = \phi$ (associated flow), volumetric expansion is overestimated. Usually $\psi < \phi$ (non-associated flow).
Associated vs. non-associated?
Associated means the yield surface and plastic potential are the same ($\psi = \phi$). Non-associated means they are different ($\psi < \phi$). For soil, $\psi = 0 \sim \phi/3$ is practical.
Summary
Triaxial Test Identification of c and φ
Cohesion c and internal friction angle φ are identified from triaxial compression tests (CU or CD tests). Confining pressure σ₃ is varied over three or more stages, plotted on the τ-σ plane, and the slope (tanφ) and intercept (c) of the common tangent to the Mohr circles are determined by least squares. Typical ranges: φ for sandy soil is 28–40°, c for clay is 0–100 kPa.
Mohr-Coulomb Failure Criterion in Practice
MC Criterion in Practice
Used in geotechnical analysis for excavation, slope stability, retaining walls, tunnels, and dam foundations.
Typical Geotechnical Parameter Values
| Soil/Rock | $c$ (kPa) | $\phi$ (°) |
|---|---|---|
| Soft clay | 10–25 | 0–5 |
| Medium clay | 25–50 | 15–25 |
| Sand (loose) | 0–5 | 28–32 |
| Sand (dense) | 0–5 | 35–42 |
| Rock (weak) | 100–500 | 25–35 |
| Rock (hard) | 1000–5000 | 35–55 |
Practical Checklist
Tunnel Excavation Analysis Track Record
For the design of excavation support for the Gotthard Base Tunnel (Switzerland, 57 km total length) completed in 2016, Mohr-Coulomb parameters φ and c for granite rock mass were analyzed using Phase2 (now Rocscience RS2). It was reported that the prediction accuracy of shear failure zones under high confining pressure matched field measurements within ±10%.