Hardening-Soil (MN)
Availability
Available with the upcoming release.
Hardening-Soil-MN is an elastoplastic soil model follows the Hardening Soil (HS) concept with stress-dependent stiffness and a Matsuoka–Nakai (MN) failure surface. It combines:
- Stress-dependent stiffness governed by a power law with exponent \(m\) and reference moduli at \(p=p_{ref}\).
- Different stiffness for primary loading (\(E_{50}^{ref}\), \(E_{oed}^{ref}\)), unloading/reloading (\(E_{ur}^{ref}\)), and the initial tangent (\(E_i^{ref}\)).
- Frictional strength and dilatancy described by \(\varphi\) and \(\psi\).
- A cap mechanism controlled by \(\alpha\) and \(H_{pp}\) (pre-consolidation stress evolution).
The Matsuoka-Nakai surface introduces Lode-angle dependence and provides a smooth strength description beyond Mohr–Coulomb, which improves robustness for general 3D stress paths.
Credits
The initial implementation of this model was provided by Leonardo José Cocco. It has since undergone substantial refactoring and further development for integration into numgeo, including improvements to code structure, numerical robustness, and performance.
Input syntax
*Mechanical = Hardening-Soil-MN
E50 (F/A), Eoed (F/A), Eur (F/A), m (-), c (F/A), phi (deg), psi (deg), nu_ur (-)
pref (F/A), K0nc (-), Rf (-), Eiref (F/A), alpha (-), Hpp (F/A)
Material parameters
The material parameters for Hardening-Soil-MN are:
| # | Symbol | Unit | Description |
|---|---|---|---|
| 1 | \(E_{50}^{ref}\) | F/A | Triaxial secant stiffness for primary loading at \(p = p_{ref}\). |
| 2 | \(E_{oed}^{ref}\) | F/A | Oedometer tangent stiffness for primary loading at \(p = p_{ref}\). |
| 3 | \(E_{ur}^{ref}\) | F/A | Unloading/reloading stiffness at \(p = p_{ref}\). |
| 4 | \(m\) | – | Exponent controlling stress dependency of stiffness. |
| 5 | \(c_{eff}\) | F/A | Effective cohesion used in the strength formulation (typically \(0\) for sands). |
| 6 | \(\varphi\) | deg | Peak friction angle. |
| 7 | \(\psi\) | deg | Peak dilatancy angle. Use \(\psi = 0\) for no dilation. |
| 8 | \(\nu_{ur}\) | – | Poisson’s ratio for unloading/reloading. |
| 9 | \(p_{ref}\) | F/A | Reference mean effective stress (typical default: \(100\) kPa in kPa-based unit systems). |
| 10 | \(K_0^{nc}\) | – | At-rest earth pressure coefficient for normally consolidated conditions. |
| 11 | \(R_f\) | – | Failure ratio controlling the stress level at which the secant stiffness definition reaches the failure envelope (typical range: \(0.8\)–\(0.95\)). |
| 12 | \(E_i^{ref}\) | F/A | Initial tangent stiffness in triaxial loading at \(p = p_{ref}\). If not available, a common approximation is \(E_i^{ref} \approx 2E_{50}^{ref}/(2 - R_f)\). |
| 13 | \(\alpha\) | – | Cap shape parameter (controls steepness of the cap in \(p\)–\(q\) space). Set to 0 to let numgeo determine \(\alpha\) automatically from \(E_{oed}^{ref}\) and \(K_0^{nc}\). |
| 14 | \(H_{pp}\) | F/A | Hardening parameter relating plastic volumetric strain to the evolution of pre-consolidation stress. Set to 0 to let numgeo determine \(H_{pp}\) automatically from \(E_{oed}^{ref}\) and \(K_0^{nc}\). |
Automatic determination of cap parameters
If \(\alpha = 0\) and/or \(H_{pp} = 0\), numgeo determines the missing parameter(s) such that the model reproduces the target oedometric stiffness \(E_{oed}^{ref}\) and the target \(K_0^{nc}\) under oedometric loading at the reference stress level.
Example
*Mechanical = Hardening-Soil-MN
** E50, Eoed, Eur, me, c, phi, psi, nu_ur
30d3, 30d3, 90d3, 0.55, 0, 42, 16, 0.25
** pref, K0nc, Rf, Eiref, alpha, Hpp
100, 0.4, 0.9, 65d3, 1.46, 72028
Model-specific output parameters
Void_Ratio: void ratio (only meaningful if an initial value was prescribed at the beginning of an analysis)Strain-Dev-Pl: Accumulated plastic deviatoric strain.Stress-Precon: Pre-consolidation stress (cap hardening variable).