Drucker-Prager: four ways to match Mohr-Coulomb

The Mohr-Coulomb criterion takes no account of the intermediate principal stress, and in principal stress space that makes its surface a hexagonal pyramid with six sharp edges and a point. The edges are what makes the return mapping awkward, and the standard escape is to replace the hexagon with a circle — the Drucker-Prager cone,

which is smooth everywhere except its apex and has a gradient defined everywhere on it.

The question is what $\alpha$ and $k$ should be, and the honest answer is that there is no choice that reproduces Mohr-Coulomb, because a circle is not a hexagon. Every choice is right in some directions and wrong in the others, and the size of the error is worth knowing before picking one.

Every number in this article is recomputed from its inputs by a script that runs on each build of this site.

The hexagon, measuredPermalink to “The hexagon, measured”#

Take $\varphi' = 30°$, $c' = 10$ kPa, and look at a deviatoric plane at a mean stress of 100 kPa. Two states define the hexagon’s vertices.

Triaxial compression, $\sigma_1 > \sigma_2 = \sigma_3$. Solving Mohr-Coulomb together with the mean stress constraint gives $\sigma_3 = 53.0718$ and $\sigma_1 = 193.8564$ kPa, so $q = 140.785$ kPa and

Triaxial extension, $\sigma_1 = \sigma_2 > \sigma_3$. The same two conditions give $\sigma_1 = 133.5201$ and $\sigma_3 = 32.9597$ kPa, $q = 100.560$ kPa and

The ratio is 0.7143, and it is exactly $(3 - \sin\varphi)/(3 + \sin\varphi)$ — a property of the friction angle alone, independent of cohesion and of mean stress. Mohr-Coulomb says the soil is 40% stronger in compression than in extension, in the deviatoric sense, and it says so at every stress level.

The four matchingsPermalink to “The four matchings”#

A cone has one radius. Setting it equal to one of the two vertex radii, or fitting it between them, gives the choices in use:

The inscribed circle is the one tangent to the hexagon’s sides rather than passing through its vertices; its radius is the perpendicular distance from the hydrostatic axis to a side, which for these numbers is 56.359 against a side length of 72.515 in the same units.

A fourth matching exists for plane strain problems specifically, chosen so the cone gives the same limit load as Mohr-Coulomb under plane-strain conditions. It is the right choice for a strip footing or a long wall, and it is the wrong choice for anything axisymmetric — which makes it a matching that has to be re-selected when the idealisation changes, and the idealisation is not a detail.

Which error is dangerousPermalink to “Which error is dangerous”#

The numbers above are large, and their direction matters more than their size.

The outer cone is unconservative and it is the default in many places. It is the easiest matching to derive, it reproduces the triaxial compression test that most laboratories actually run, and it overstates the strength by 40% in extension. Extension states are not exotic: the soil behind a retaining wall, beneath the heave zone of an excavation, and on the passive side of most failure mechanisms is in extension or close to it. A model that is 40% strong there will find a mechanism that avoids those regions, which is to say it will find the wrong mechanism.

The inner cone is conservative everywhere and by a lot. 28.6% below the compression strength is a large penalty to pay on a design governed by bearing capacity.

The inscribed cone is the least bad compromise — within 3% at one vertex and 31% low at the other — but “least bad” is doing a lot of work in that sentence.

What this argues forPermalink to “What this argues for”#

The alternative to matching is not to match: implement the hexagon, handle the edges and the apex exactly, and pay the cost of a branch and an occasional two-surface return. That cost is bounded, it is paid once by the implementer, and it removes a decision the user would otherwise have to make correctly and would have no way of knowing they had made.

Where a smooth surface is genuinely needed — for a formulation whose derivation requires differentiability, or where the corner treatment cannot be made robust — the honest practice is to state which matching was used, and to state it alongside the friction angle rather than in an appendix. “Drucker-Prager, $\varphi' = 30°$” is not a specification. The same three words with “outer” or “inscribed” attached differ by a third in the strength they describe, which is the same order as the difference associated flow makes, and for the same reason: both are choices made in the plasticity formulation that never appear in the soil report.