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Hardening-Soil (HS / HSsmall)

Generalized Hardening-Soil model with stress-dependent stiffness and an optional HSsmall branch for small-strain shear stiffness degradation.

A benchmark simulation of the implementation is available in the Benchmark Manual.

Material Parameters

Keyword: *Mechanical = Hardening-Soil

Provide the following 12 values in the order:

# Name Symbol Unit Description
1 Friction angle \(\varphi\) deg Peak friction angle
2 Dilatancy angle \(\psi\) deg Peak dilatancy angle. Set to \(0\) for no dilation.
3 Effective cohesion \(c_\mathrm{eff}\) kPa Mohr–Coulomb cohesion used in HS strength envelope.
4 Secant modulus at 50% strength (ref.) \(E_{50}^{ref}\) kPa Controls initial shear stiffness in drained triaxial at \(p=p_{ref}\).
5 Oedometer modulus (ref.) \(E_{oed}^{ref}\) kPa Controls volumetric stiffness in one-dimensional compression at \(p=p_{ref}\).
6 Unloading/reloading modulus (ref.) \(E_{ur}^{ref}\) kPa Elastic unloading/reloading modulus at \(p=p_{ref}\).
7 Stress dependency exponent \(m\) – Stiffness scaling exponent.
8 Poisson ratio (un/reload) \(\nu_{ur}\) – Defines elastic bulk/shear splitting with \(E_{ur}\).
9 Small-strain shear modulus (ref.) \(G_0^{ref}\) kPa HSsmall parameter. Set \(0\) to disable HSsmall.
10 Shear strain at \(G/G_0=0.7\) \(\gamma_{0.7}\) – HSsmall parameter; \(\gamma_r=\gamma_{0.7}/0.7\).
11 Reference pressure \(p_{ref}\) kPa Reference mean effective stress (default \(100\)).
12 Water bulk modulus \(K_w\) kPa \(0\) for drained; \(>0\) for ideally undrained in solid formulation meaning that total stresses are used.

Example block

*Mechanical = Hardening-Soil
38,6,1,105000,105000,315000,0.55,0.2,0,0,100,0

Model-specific output parameters

Key Name Symbol / Formula Unit Sign / Range Meaning & Notes
VOID_RATIO Void ratio \(e\) – \(>0\) Current void ratio; advanced explicitly from the volumetric strain rate.
PCAP Cap size parameter \(p_c\) kPa \(\ge p_{\min}\) Controls onset of compressive cap flow; hardens with \(\dot p_c = H_c\,\dot{\varepsilon}_{v,\text{cap}}^p\).
EPS_PL_VOL Accumulated plastic volumetric strain \(\varepsilon_v^p = \int (\dot{\varepsilon}_{v,\text{dil}}^p + \dot{\varepsilon}_{v,\text{cap}}^p)\,\mathrm{d}t\) – Compression positive Sum of dilatancy- and cap-driven plastic volume change.
STRESS-PW Pore water pressure \(u\) kPa Compression positive Used only for undrained runs (\(K_w>0\)); total stress \(\boldsymbol{\sigma}=\boldsymbol{\sigma}' - u\,\boldsymbol{\delta}\).
EPS_PL_DEV Accumulated plastic shear strain \(\varepsilon_q^p\) (triaxial-equivalent plastic shear) – \(\ge 0\) Increases when \(q>q_h(\varepsilon_q^p)\); drives the hyperbola \(q_h\) and the dilatancy rule.
STRESS-P Mean effective stress (magnitude) \(p=\tfrac{1}{3}\,\mathrm{tr}(-\mathbf{T})\) kPa \(\ge 0\) Mean effective pressure; internal \(\mathbf{T}\) is compression-negative.
STRESS-Q Deviatoric stress invariant \(q=\sqrt{\tfrac{3}{2}\,\mathbf{S}:\mathbf{S}}\), with \(\mathbf{S}=\mathbf{T}-\tfrac{1}{3}\mathrm{tr}(\mathbf{T})\boldsymbol{\delta}\) kPa \(\ge 0\) Mises-type equivalent shear stress controlling shear hardening.
GAMMA_EQ Equivalent shear strain (HSsmall track) \(\gamma_{\text{eq}}\) – \(\ge 0\) Running measure for shear strain amplitude used in small-strain modulus reduction (if \(G_0^{\text{ref}}>0\)).

Optional Parameters

*Optional mechanical parameter
<property 1>, <value 1>...
<property 2>, ...`

A list of currently supported optional parameters is given below:

  • integrator: Numerical integrator:

    • integrator = 1: Modified-Euler

    • integrator = 2: Explicit Euler with Richards extrapolation of Order 2

    Per default, the Explicit Euler with Richards extrapolation method is used (integrator = 1).

  • tol_stress: Tolerance \(T^\sigma\) for error \(\epsilon^\sigma\) evaluated based on constitutive stress. The solution is accepted if \(\epsilon^\sigma < T^\sigma\). Per default \(T^\sigma = 10^{-4}\). Stress control can be deactivated by setting \(T^\sigma=1\) (not recommended!).

  • num_diff: Method for the numerical differentiation used for the estimation of the material Jacobian (tangent stiffness) \((\partial \boldsymbol{\sigma})/(\partial \boldsymbol{\varepsilon})\):

    • num_diff=1: Forward Euler of \(O(h)\), \(\dfrac{\partial f}{\partial x} \approx \dfrac{f(x_0+\theta)-f(x_0)}{\theta}\)

    • num_diff=2: Central Differences of \(O(h^2)\), \(\dfrac{\partial f}{\partial x} \approx \dfrac{f(x_0+\theta)-f(x_0-\theta)}{2\theta}\)

  • perturbation: The perturbation factor \(f^\theta\) used in the numerical differentiation to evaluate the material tangent stiffness \((\partial \boldsymbol{\sigma})/(\partial \boldsymbol{\varepsilon})\). The perturbation is \(\theta=\text{max}(||\dot{\varepsilon}||)f^\theta\), where \(||\dot{\varepsilon}||\) is the \(L_2\) norm of the strain rate. The default value is \(f^\Theta=10^{-7}\).

  • jacobi: Method to evaluate the Jacobian:

    • jacobi = 1 (Default): The material Jacobian is determined at the end of the numerical integration as the tangent at the final stress state.

    • jacobi = 2: The material Jacobian is calculated as the weighted sum of all Jacobians evaluated at the end of each converged Runge-Kutta substep.