Drucker-Prager Yield Criterion
Drucker-Prager Yield Criterion: Theoretical Foundations
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.
Computational Methods for the Drucker-Prager Yield Criterion
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.
Drucker-Prager Yield Criterion in Practice
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.
Drucker-Prager Yield Criterion: Software & Solver 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.
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