A common model used in geotechnics to describe the failure of materials like soil and rock under shear stress. It is based on the relationship between normal stress and shear stress at failure.

Mohr-Coulomb Model

The Mohr-Coulomb Model is a widely used constitutive model in geotechnical engineering that describes the behavior of materials such as soils, rocks, and concrete under shear stress. It is particularly useful for predicting the failure of these materials when they are subjected to stresses in different directions. The model is based on the principles of shear strength and normal stress, providing a criterion for when a material will fail.

Key Points about the Mohr-Coulomb Model:

1. Yield Criterion:The Mohr-Coulomb yield criterion is expressed as:

   τ = c + σn * tan(ϕ)

   where:

   • τ is the shear stress on the failure plane,
   • σ_n is the normal stress on the failure plane,
   • c is the cohesion of the material,
   • ϕ is the internal friction angle of the material.

   This criterion defines a linear relationship between the shear stress and the normal stress at the point of failure.

2. Failure Envelope:The Mohr-Coulomb model describes a failure envelope in the stress space, typically represented in a plot of shear stress versus normal stress. The line defined by the equation τ = c + σn * tan(ϕ) represents the failure envelope, beyond which the material will fail.
3. Mohr’s Circle:The model is often illustrated using Mohr’s Circle, a graphical representation of the state of stress at a point. Mohr’s Circle helps visualize the relationship between normal and shear stresses, and the failure envelope intersects the circle at the points of potential failure.
4. Material Parameters:The Mohr-Coulomb model requires the following material parameters:
   • Cohesion (c): Represents the inherent shear strength of the material due to intermolecular forces, independent of the normal stress.
   • Friction Angle (ϕ): The angle that represents the material’s resistance to shear stress, based on the sliding friction between particles.
5. Applicability:The Mohr-Coulomb model is widely used for:
   • Slope Stability Analysis: Assessing the stability of slopes under different loading conditions and predicting potential slip surfaces.
   • Foundation Design: Calculating the bearing capacity of soils and the potential for settlement or shear failure in foundations.
   • Retaining Structures: Designing retaining walls, embankments, and other structures that must resist lateral earth pressures.
   • Tunnel and Excavation Stability: Predicting the stability of tunnels and other underground excavations by analyzing the stresses around the excavation boundaries.
6. Limitations:While the Mohr-Coulomb model is widely used, it has some limitations:
   • Linear Assumption: The model assumes a linear relationship between shear stress and normal stress, which may not be accurate for all materials, especially under high stress levels.
   • No Tensile Strength: The model does not account for tensile strength, making it less accurate for materials that can sustain tensile stresses.
   • Dependence on Stress Path: The model’s predictions can be sensitive to the stress path, making it less reliable for complex loading conditions.

Summary:

The Mohr-Coulomb Model is a fundamental tool in geotechnical engineering for predicting the shear strength and failure conditions of materials such as soils and rocks. Its simplicity and practical applicability make it a cornerstone in the design and analysis of foundations, slopes, retaining walls, and other structures. However, engineers must be aware of its limitations and consider more advanced models for complex materials or loading conditions.