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Mohr Coulomb CriterionEdit

The Mohr-Coulomb criterion is a foundational, widely used constitutive model in geotechnical engineering. It provides a simple, practical relation for shear strength of soils and rocks by combining a cohesion term with an internal friction term. In its most common form, the shear strength τ on a potential failure plane is expressed as a linear function of the effective normal stress σ′ on that plane: τ = c′ + σ′ tan φ′ where c′ is the effective cohesion and φ′ is the effective internal friction angle. This linear envelope translates into a straight-line failure boundary in Mohr’s circle space, making it convenient for hand calculations and computer codes alike. See Mohr's circle and friction angle for background on the geometric and physical concepts involved. The effective-stress interpretation relies on the concept that pore pressure reduces the normal stress carrying capacity, so the relevant stress variable is σ′ = σ − u, with u denoting pore-water pressure, a relationship described in the effective stress principle.

The criterion has a long historical pedigree and stands at the intersection of classical friction theory and soil mechanics. The effective frictional behavior embodied in φ′ originates from Coulomb’s friction law, while its geometric interpretation and the idea of an envelope of failure come from Mohr’s circle representations. The combination was refined and popularized in geotechnical engineering during the 20th century, becoming a standard tool in design and analysis. See Charles-Augustin de Coulomb and Mohr's circle for the underlying ideas, and geotechnical engineering for the field where the criterion is most typically applied.

Despite its enduring practicality, the Mohr-Coulomb model is a simplification of real materials. Soils and rocks exhibit nonlinear strength envelopes, rate effects, stress-path dependence, dilatancy, fabric anisotropy, and post-peak softening in many cases. Engineers calibrate c′ and φ′ from laboratory tests (for example, triaxial tests) and apply them within the MC framework to predict failure surfaces in field problems such as slope stability, foundation engineering (bearing capacity), and retaining wall design. In some situations, more advanced constitutive models provide better fidelity, but the Mohr-Coulomb criterion remains a baseline due to its transparency and the breadth of empirical data supporting it. See bearing capacity and slope stability for related design applications.

Definition

The core concept is a linear failure envelope in terms of the shear strength on a plane and the normal stress on that plane. In drained, saturated, or effectively drained conditions, the criterion is commonly written with effective stress terms as τ = c′ + σ′ tan φ′. Here: - τ is the shear strength on the plane of interest. - σ′ is the effective normal stress acting on that plane (total normal stress minus pore pressure). - c′ is the material’s cohesive strength under effective stress. - φ′ is the internal friction angle, describing how friction mobilizes with increasing normal stress.

The parameters c′ and φ′ are material properties that depend on soil type, density, moisture, fabric, and stress history. They are typically determined from laboratory tests such as triaxial tests or from back-analysis of field performance. See cohesion and friction angle for deeper definitions, and effective stress principle for the underlying physics.

Mathematical formulation

The Mohr-Coulomb criterion corresponds to a straight-line boundary in the space of Mohr’s circles, characterized by intercept c′ and slope tan φ′. In this representation, the onset of yielding occurs once the state of stress reaches the line defined by τ = c′ + σ′ tan φ′. The same relation can be expressed in terms of principal stresses or in terms of the stress invariants often used in advanced soil mechanics. The effective-stress formulation acknowledges that pore pressure reduces the material’s resistance to shear, linking hydro-mechanical coupling to strength.

The model is widely used with the assumption of an associated flow rule in plasticity theory, although non-associative flow is not uncommon in soils and can influence dilatancy and volume changes during shearing. See Drucker–Prager criterion for a related, pressure-dependent envelope in a smooth form, and Cam-clay for an alternative framework that emphasizes critical-state behavior and nonlinear envelopes.

Applications

  • Slope stability: The MC envelope defines the resistive shear strength along potential slip surfaces, enabling factors of safety calculations in hillsides and embankments. See slope stability.
  • Foundation design: The criterion informs bearing-capacity estimates and ultimate shear resistance along soil–foundation interfaces. See bearing capacity and foundation engineering.
  • Retaining structures and earthworks: Stability checks for walls, excavations, and fills often employ MC parameters to assess potential failure along shear planes. See retaining wall.
  • Rock engineering: In rock masses, MC parameters are sometimes used as a first-order approximation, though often augmented with corrections for joints, rock quality, and blockiness. See Hoek-Brown criterion for rock-mass strength alternatives.

Limitations and debates

  • Linearity and stress range: Real soils may exhibit nonlinear strength envelopes, especially at low confining pressures or near the onset of yielding. The MC form can misrepresent such behavior unless c′ and φ′ are carefully interpreted or modified.
  • Rate effects and dynamics: The criterion is fundamentally quasi-static. Under rapid loading or seismic events, rate-dependent strength changes and inertial effects can dominate, requiring dynamic or visco-plastic models.
  • Dilatancy and flow rules: Many soils dilate or contract during shear, and the MC model’s conventional forms assume a particular flow rule. Non-associative flow or path-dependent behavior can lead to discrepancies between predicted and actual responses.
  • Effective-stress applicability: For unsaturated soils or complex pore-pressure histories, the simple σ′ concept may be insufficient. Extensions and modifications, such as Bishop’s effective-stress concept, are used in some analyses. See effective stress principle and unsaturated soil.
  • Anisotropy and fabric: Layered deposits, sedimentary structure, or anisotropic fabric can cause strength to vary with direction, which the isotropic MC envelope does not capture. More sophisticated models or site-specific calibrations are employed in such cases.
  • Extensions and alternatives: To address limitations, engineers may adopt refined criteria or entirely different constitutive frameworks. Notable alternatives and extensions include:
    • Drucker–Prager criterion: a smooth, pressure-dependent envelope that can be more convenient for numerical analysis.
    • Cam-clay: a critical-state soil mechanics model emphasizing nonlinear envelopes and path-dependent behavior.
    • Hoek-Brown criterion: a widely used model for rock masses that accounts for jointing and blockiness.
    • Dissipation and dilatancy considerations and various soil-specific calibrations.

History

The Mohr-Coulomb criterion sits at the intersection of classical friction theory and soil mechanics. Coulomb’s friction law provided the frictional basis, while Mohr’s circle offered a geometric interpretation of stress states. Over time, geotechnical engineers fused these ideas into a practical yield condition that could be calibrated from laboratory tests and used in hydraulic and structural design. The framework has evolved with the incorporation of effective-stress concepts, saturation effects, and the development of alternative constitutive models for complex ground conditions.

Extensions and alternatives

  • Drucker–Prager criterion: A widely used alternative that smooths the Mohr-Coulomb envelope, often preferred in numeric simulations.
  • Cam-clay: A nonlinear, critical-state model that emphasizes soil behavior under varying consolidation and shear paths.
  • Hoek-Brown criterion: A rock-mass strength criterion that accounts for rock mass quality and jointing.
  • Dilatancy and non-associative flow concepts: Extensions to better capture volumetric changes during shear.
  • Unsaturated soil behavior and effective-stress refinements: Modifications to account for matric suction and other pore-fluid effects.
  • Triaxial tests and other laboratory protocols: The empirical backbone for estimating c′ and φ′ values in various soils.

See also