Mohr-Coulomb破壊基準
Theory and Physics
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.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly enough to ignore acceleration". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. So here's a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber band. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong". Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body forces $f_b$ (gravity, etc.) and surface forces $f_s$ (pressure, contact forces, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common pitfall here: getting the load direction wrong. Intending "tension" but ending up with "compression"—sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—deliberately absorbing vibrational energy for a smoother ride. What if damping were zero? Buildings would keep shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and stress-strain relationship is linear
- Isotropic material (unless specified otherwise): Material properties are independent of direction (anisotropic materials require separate tensor definitions)
- Quasi-static assumption (for static analysis): Ignores inertial/damping forces, considering only equilibrium between external and internal forces
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Plasticity, creep, and other nonlinear material behaviors require constitutive law extensions
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads/elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Beware of unit system inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformations) |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
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.
Linear Elements (1st-order elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with mid-side nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2–3 times. Recommended when stress evaluation is important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix each iteration. Quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Lower cost per iteration, but convergence is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies total load in small increments rather than all at once. The arc-length method (Riks method) can trace beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: opening to an estimated page and adjusting forward/backward (iterative) is more efficient than searching sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
1st-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd-order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even with the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
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%.
Analogy of Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, results will be a mess no matter how good the solver is.
Common Pitfalls for Beginners
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer is far from reality. Confirm that results stabilize with at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct".
Thinking About Boundary Conditions
Setting boundary conditions is like "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling real-world constraints is often the most critical step in the entire analysis.
Software Comparison
Tools for MC Criterion
Selection Guide
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