Cohesionless materials

class groundhog.constitutivemodels.cohesionless.HardeningSoil(friction_angle, cohesion, Rf=0.9)

Class for setting up the equations for the hardening soil model and performing fitting of drained triaxial tests.

Reference - Schanz, T., Vermeer, P.A., Bonnier, P.G. (1999). The hardening soil model: Formulation and verification. Beyond 2000 in Computational Geotechnics - 10 years of PLAXIS.

__init__(friction_angle, cohesion, Rf=0.9)

Sets up the material using the strength parameters friction angle (\(\varphi_p\)), cohesion (\(c\)) and failure ratio (\(R_f\))

In absence of other guidance, :math:`R_f`=0.9 is a good default setting.

\[q_a = \frac{q_f}{R_f}\]

calculate_drainedtriaxial(sigma3, sigma1_0, m, N=100)

Calculates the response in a drained triaxial tests based on the parameters entered by the user.

Parameters:

• sigma3 – Radial consolidation stress [kPa]

• sigma1_0 – Initial vertical consolidation stress [kPa]

• m – Stress exponent governing the stress dependency of the stiffness [-]

• N – Number of points where \(q\) is calculated (\(\sigma_1\) is spaced evenly between \(\sigma_{1,0}\) and \(\sigma_{1,f}\))

\[ \begin{align}\begin{aligned}q_f = \frac{6 \sin \varphi}{3 - \sin \varphi} \left( p + c \cot \varphi \right)\\q_a = \frac{q_f}{R_f}\\\begin{split}\epsilon_1 = \frac{q_a}{2 E_{50}} \frac{(\sigma_1 - \sigma_3)}{q_a - (\sigma_1 - \sigma_3)} \\ \text{for} \\ q \leq q_f\end{split}\end{aligned}\end{align} \]

static calculate_stiffnesses(sigma_3, sigma_1, m, cohesion, friction_angle, E50_ref, Eur_ref, Eoed_ref, p_ref)

Calculates the stiffnesses \(E_{50}\), \(E_{ur}\) and \(E_{oed}\) for a given consolidation pressure \(\sigma_{3}\).

Parameters:

• sigma_3 – Consolidation pressure considered for shearing behaviour

• sigma_1 – Consolidation pressure considered for volumetric compression behaviour

• m – Stress exponent governing the stress dependence of the moduli.

\[ \begin{align}\begin{aligned}E_{50} = E_{50}^{ref} \left( \frac{\sigma_3 + c \cot \varphi}{p_{ref} + c \cot \varphi} \right)^m\\E_{ur} = E_{ur}^{ref} \left( \frac{\sigma_3 + c \cot \varphi}{p_{ref} + c \cot \varphi} \right)^m\\E_{oed} = E_{oed}^{ref} \left( \frac{\sigma_1 + c \cot \varphi}{p_{ref} + c \cot \varphi} \right)^m\end{aligned}\end{align} \]

set_reference_moduli(E50_ref, Eur_ref, Eoed_ref, p_ref)

Sets the stiffnesses for the hardening soil model using moduli identified at a reference pressure (typically 100kPa).

Parameters:

• E50_ref – \(E_{50}^{ref}\) [kPa] is the secant modulus at 50% of the maximum deviator stress for a drained triaxial test carried out at \(p_{ref}\)

• Eur_ref – \(E_{ur}^{ref}\) [kPa] is the unloading reloading stiffness for a drained triaxial test carried out at \(p_{ref}\)

• Eoef_ref – \(E_{oed}^{ref}\) [kPa] is the tangent stiffness for primary oedometric loading at \(p_{ref}\)

• p_ref – \(p_{ref}\) [kPa] is the reference pressure at which the tests are carried out. Note that the same pressure is used for all tests.