Drucker-Prager降伏基準
Theory and Physics
What is the Drucker-Prager Criterion?
Professor, is the Drucker-Prager (DP) criterion an improved version of Mohr-Coulomb?
The Drucker-Prager criterion is a "cone" approximation of Mohr-Coulomb. It replaces the irregular hexagon of the MC criterion with a smooth cone. Numerically stable (no corners).
$p$ is the mean stress, $t$ is a function of deviatoric stress. $\beta$ corresponds to the friction angle, $d$ corresponds to cohesion.
Correspondence with MC Criterion
Methods to correlate DP parameters with MC's $c, \phi$ (three ways: inscribed, circumscribed, equal area). In Abaqus, MC-compatible settings are possible with *DRUCKER PRAGER.
Summary
The Drucker-Prager Paper is 4 Pages
The paper "Soil Mechanics and Plastic Analysis or Limit Design" published by Daniel C. Drucker and William Prager in 1952 was a short note of only 4 pages. However, the idea of extending von Mises' circular yield surface to a pressure-dependent conical shape was revolutionary, making plastic analysis of soil/concrete/rock practically usable at once. Drucker was from Brown University, and Prager was a German-born mechanician active at the University of Basel in Switzerland.
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, which is the assumption that "acceleration can be ignored because the force is applied slowly". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return", right? That is Hooke's law $F=kx$, the essence of the stiffness term. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber. 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 force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire contents" (body force), 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 typical pitfall here: getting the load direction wrong. Intending "tension" but it becomes "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, right? Because the vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is important.
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, stress-strain relationship is linear
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definition)
- Quasi-static assumption (for static analysis): Ignores inertial/damping forces, considers only balance between external and internal forces
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity/creep, constitutive law extension is needed
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. Be careful 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 deformation) |
| 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 of DP Criterion
```
*DRUCKER PRAGER
beta, K, psi
*DRUCKER PRAGER HARDENING
yield_stress, plastic_strain
```
Ansys: TB, DP. Nastran: SOL 400 + Drucker-Prager support.
Extended Drucker-Prager (Cap Model)
A model that adds a cap (yield surface on the compression side) to the DP criterion. Represents consolidation (volumetric plasticity) under high hydrostatic compressive states. Used for powder compaction, ground consolidation.
Summary
A model that adds a cap (yield surface on the compression side) to the DP criterion. Represents consolidation (volumetric plasticity) under high hydrostatic compressive states. Used for powder compaction, ground consolidation.
Handling Singularities at the DP Cone Apex
At the apex of the Drucker-Prager yield surface, the gradient cannot be defined, risking the Return-Mapping algorithm falling into a singularity. Abaqus avoids this with "Modified Drucker-Prager/Cap", smoothly replacing the apex in the low-stress region with a cap to guarantee a unique normal direction. This improvement was proposed in the 1980s by former students of Drucker himself, enabling simultaneous modeling of compressive yield and dilatant behavior in soil.
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 about 2-3x. Recommended: when stress evaluation is important.
Full integration vs Reduced integration
Full integration: Risk of over-constraint (locking). Reduced integration: Risk of hourglass mode (zero-energy mode). 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 every iteration. Quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix with initial value or every few iterations. Cost per iteration is low, but convergence speed 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 not all at once, but in small increments. 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 accuracy improves 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 at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
DP Criterion in Practice
Used in geotechnical analysis, concrete plasticity, rock shear failure, powder forming.
Practical Checklist
Standard Method for Tunnel Excavation Analysis
In road/railway tunnel design, the Drucker-Prager model is widely used for evaluating ground stability during excavation. The Japan Society of Civil Engineers' "Standard Specifications for Tunnels" (2016 edition) explicitly states the procedure for converting DP strength parameters from Mohr-Coulomb cohesion c and internal friction angle φ using equal area circle conversion. For the deep Tokyo Outer Ring Road tunnel (approx. 16m diameter), DP analysis was performed based on Kanto loam layer c=15 kPa, φ=30°, and the design estimating maximum ground surface settlement of 30mm matched well with construction results.
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 (visualize in post-processing). Here's an important question—which step is most prone to failure in cooking? Actually, it's the "prep work". If mesh quality is poor, results will be a mess no matter how good the solver is.
Pitfalls Beginners Easily Fall Into
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. Verifying that results stabilize across at least 3 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 the same as "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 constraint conditions is actually the most important step in the entire analysis.
Software Comparison
Tools for DP Criterion
Selection Guide
Implementation Comparison: Plaxis, Midas, Abaqus
The Drucker-Prager model is implemented in almost all geotechnical FEM software, but parameter definitions differ. Plaxis uses it internally as "Extended Mohr-Coulomb (EMC)", automatically converting DP constants from c, φ input. Abaqus requests direct input of DP angle β and cohesion d. Midas automatically applies DP equal area conversion after selecting "Mohr-Coulomb". Even with the same ground data, limit loads can differ by 5-12% across the three software, so benchmark comparison is recommended.
The 3 Most Important Questions for Selection
- "What to solve?": Does it support the physical models/element types required for the Drucker-Prager yield criterion? For example, in fluids, presence of LES support; in structures, contact/large deformation capability makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUI are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic transmission (GUI) and manual transmission (script) in cars.
- "How far to extend?": Future analysis scale expansion (HPC support), expansion to other departments, integration with other tools...
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