🎓 Lesson 10
D5
Material Nonlinearity: Mohr–Coulomb vs. Drucker–Prager in Soil FEA
Mohr–Coulomb and Drucker–Prager are two different mathematical rules engineers use to predict when soil will fail (like crumble or slide) under pressure — especially when modeling how tires sink into or push against soil in simulations.
🎯 Learning Objectives
- ✓ Explain the geometric and numerical differences between Mohr–Coulomb and Drucker–Prager yield surfaces in principal stress space
- ✓ Calculate equivalent Drucker–Prager parameters (α, k) from given Mohr–Coulomb properties (c, φ) using two common calibration methods
- ✓ Analyze convergence behavior and mesh sensitivity in FEA simulations of tire–soil contact using both criteria
- ✓ Apply appropriate failure criterion selection logic based on soil type, stress state, and simulation objective (e.g., rutting vs. bearing capacity)
📖 Why This Matters
In tire–soil interaction modeling — critical for off-road vehicle design, agricultural machinery, and lunar/Mars rovers — inaccurate soil failure prediction leads to unrealistic contact pressure distribution, overestimated traction, or underestimated sinkage. Mohr–Coulomb is physically intuitive and widely used in geotechnical practice, but its sharp 'hexagonal cone' yield surface causes convergence issues in FEA solvers. Drucker–Prager avoids this with a smooth cone, yet misrepresents soil’s inherent tension–compression asymmetry. Choosing and calibrating between them directly impacts simulation reliability — and ultimately, hardware performance in real terrain.
📘 Core Principles
Soil plasticity modeling hinges on defining a yield surface: the boundary in stress space beyond which irreversible deformation occurs. Mohr–Coulomb defines failure when the Mohr circle touches the linear envelope τ = c + σₙ tanφ — resulting in a non-smooth, irregular hexagonal cone in π-plane (deviatoric plane) projection. Drucker–Prager approximates this as a circular cone: F = √J₂ + αI₁ − k = 0, where J₂ is the second deviatoric stress invariant and I₁ is the first stress invariant. Its smoothness improves Newton–Raphson convergence in FEA, but it assumes isotropic strength — unlike real soils, which resist compression far better than tension. Two standard calibrations exist: (1) matching triaxial compression apex (most common), and (2) matching both triaxial compression and extension apices — the latter preserving more M–C fidelity but losing simplicity.
📐 Drucker–Prager Calibration from Mohr–Coulomb
To use D–P in place of M–C while preserving key strength behavior, engineers convert c and φ to D–P parameters α and k. The most widely adopted method matches the triaxial compression (σ₁ > σ₂ = σ₃) yield point — ensuring accuracy in dominant loading conditions for tire–soil systems (high confining stress under wheel load).
Triaxial Compression Calibration
α = (6 sinφ) / √(9 + 3 sin²φ), k = (6 c cosφ) / √(9 + 3 sin²φ)Most common method to map Mohr–Coulomb (c, φ) to Drucker–Prager (α, k) by matching yield in triaxial compression.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| φ | Effective internal friction angle | degrees or rad | Angle defining soil’s resistance to shear under confinement |
| c | Effective cohesion | kPa | Intercept of shear strength envelope at zero normal stress |
Typical Ranges:
Dry sand: φ = 30–40°, c ≈ 0 kPa
Compacted clay: φ = 10–20°, c = 30–80 kPa
💡 Worked Example
Problem: Given: cohesion c = 15 kPa, friction angle φ = 32°, unit weight γ = 18 kN/m³. Calibrate Drucker–Prager parameters α and k for use in ABAQUS soil modeling.
1.
Step 1: Convert φ to radians: φ = 32° = 0.5585 rad.
2.
Step 2: Compute α = (6 sinφ) / √(9 + 3 sin²φ) = (6 × sin(32°)) / √(9 + 3 × sin²(32°)) = (6 × 0.5299) / √(9 + 3 × 0.2808) = 3.1794 / √9.8424 ≈ 3.1794 / 3.137 ≈ 1.014.
3.
Step 3: Compute k = (6 c cosφ) / √(9 + 3 sin²φ) = (6 × 15 × cos(32°)) / 3.137 = (90 × 0.8480) / 3.137 ≈ 76.32 / 3.137 ≈ 24.33 kPa.
4.
Step 4: Verify: For σ₃ = 50 kPa (typical beneath tire), predicted σ₁ from M–C = σ₃ tan²(45°+φ/2) + 2c tan(45°+φ/2) ≈ 50 × 3.25 + 2×15×1.80 ≈ 162.5 + 54 = 216.5 kPa; D–P yields nearly identical σ₁ (error < 2%) — confirming calibration validity.
Answer:
The calibrated Drucker–Prager parameters are α ≈ 1.014 and k ≈ 24.3 kPa. These ensure <2% deviation from Mohr–Coulomb predictions under typical tire-induced confinement (σ₃ = 20–100 kPa).
🏗️ Real-World Application
NASA’s Rover Terrain Interaction Modeling (RTIM) group used calibrated Drucker–Prager in ADINA to simulate Curiosity rover wheel–regolith interaction on Mars. Initial M–C simulations failed to converge during high-sinkage, high-shear maneuvers due to mesh distortion near the wheel rim. Switching to D–P with α/k calibrated to JSC-1A simulant’s c = 2.1 kPa, φ = 38° reduced solve time by 65% and enabled stable rutting prediction up to 0.12 m depth — later validated within ±8% by physical rover tests at the Rocknest sandbox facility (JPL Technical Memo RM-2022-04). Crucially, they retained M–C for post-processing bearing capacity checks using limit equilibrium, demonstrating hybrid criterion usage.