Hypoplasticity + Intergranular Strain Anisotropy (HYPO+ISA):

Hypoplastic model for sand according to von Wollfersdorff (1996)¹ with intergranular strain anisotropy extension (Fuentes et al., 2020)². The numgeo keyword reads:

*Mechanical = Hypo-Isa   
phi (degree or rad), h_s (F/A), n (-), e_d0 (-), e_c0 (-), e_i0 (-), alpha (-), beta (-) 
m_R (-), R (-), beta_h0 (-), chi^0 (-), chi^max (-), epsilon^acc (-), c^z (-), beta_hmax (-)

The material parameters of the hypoplastic model with intergranular strain are given in the following. Further information on the model formulation and implementation are provided in the Theory Manual.

\(\varphi_c\) (degree or rad) is the critical friction angle and has to be provided in radian.
\(h_s\) (F/A) and \(n\) (-) describe the decrease of the maximum void ratio \(e_i\) with increasing pressure during isotropic compression. This parameter has no unit.
The void ratios \(e_{d0}\) (-), \(e_{c0}\) (-) and \(e_{i0}\) (-) correspond to the densest, the critical and the loosest state at pressure \(p\) = 0 F/A. These parameters have no unit.
The material constant \(\alpha\) describes the influence of density on the peak friction angle. This parameter has no unit.
\(\beta\) mainly influences the stiffness of dense soil skeletons. This parameter has no unit.
\(m_R\) governs the increase of the stiffness due to changes in the direction of the strain path. The parameter has no unit.
\(R\) is the radius of intergranular strain. This parameter has no unit.
\(\beta_{h0}\) and \(\beta_{hmax}\) are the minimum and maximum hardening parameters of the intergranular strain and describe the evolution of the intergranular strain tensor along a strain path. These parameters have no unit.
\(\chi^0\) and \(\chi^{max}\) are the minimum and maximum intergranular strain exponent. These parameters have no unit.
\(\epsilon^{acc}\) is the accumulation rate factor. This parameter has no unit.
\(c_z\) is the cyclic mobility factor. This parameter has no unit.

Changes with the upcoming release

*Mechanical = Hypo-Isa   
phi [rad or degree], h_s (F/A), n (-), e_d0 (-), e_c0 (-), e_i0 (-), alpha (-), beta (-) 
m_R (-), R (-), beta_h0 (-), chi^0 (-), chi^max (-), epsilon^acc (-), c^z (-), beta_hmax (-)

The material parameters of the hypoplastic model with intergranular strain are given in the following.

\(\varphi_c\) (degree or rad) is the critical friction angle and has to be provided in degree or in radian. Values larger than 1 are considered to be provided in degree.
\(h_s\) (F/A) and \(n\) (-) describe the decrease of the maximum void ratio \(e_i\) with increasing pressure during isotropic compression. This parameter has no unit.
The void ratios \(e_{d0}\) (-), \(e_{c0}\) (-) and \(e_{i0}\) (-) correspond to the densest, the critical and the loosest state at pressure \(p\) = 0 F/A. These parameters have no unit.
The material constant \(\alpha\) describes the influence of density on the peak friction angle. This parameter has no unit.
\(\beta\) mainly influences the stiffness of dense soil skeletons. This parameter has no unit.
\(m_R\) governs the increase of the stiffness due to changes in the direction of the strain path. The parameter has no unit.
\(R\) is the radius of intergranular strain. This parameter has no unit.
\(\beta_{h0}\) and \(\beta_{hmax}\) are the minimum and maximum hardening parameters of the intergranular strain and describe the evolution of the intergranular strain tensor along a strain path. These parameters have no unit.
\(\chi^0\) and \(\chi^{max}\) are the minimum and maximum intergranular strain exponent. These parameters have no unit.
\(\epsilon^{acc}\) is the accumulation rate factor. This parameter has no unit.
\(c_z\) is the cyclic mobility factor. This parameter has no unit.

Optional parameters

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

Optional mechanical parameters can be used to enable locally undrained conditions, change some default settings or activate some stabilisation methods. Their use is optional.

phantom_E, E_ph and phantom_nu, nu_ph are the parameters of the phantom elasticity. The Hypo-ISA implementation available from soilmodels.com uses a phantom linear elasticity overlying the constitutive model response of the constitutive model with \(E_{ph}=20\) kPa and \(\nu_{ph}=0.45\) (as of 22.05.2025). We have also included this in our implementation, but it is deactivated by default and if activated, the parameters must be specified in the input for transparency. We recommend turning them off by default (\(E_{ph}=0\)). Deviating from the original implementation from W. Fuentes, the phantom elasticity in the numgeo implementation is only active when the mean effective stress is equal to or drops below \(p_{min,ph}\). For more information see the Theory Manual.
bulk_water, Kw: Bulk modulus of pore water for locally undrained simulations (and only for those). Per default \(K^w=0\) F/A. To activate locally undrained conditions, \(K^w>0\) must be set by the user.
projection_dev, value: specifies the friction angle for the projection of stress states onto the Matsuoka-Nakai failure surface defined by a user defined friction angle \(\varphi^{cut}\) in degree. Per default value=-1 (no projection). For value=0 the cut-off friction angle is automatically determined by numgeo and for value=>0°, \(\varphi^{cut}=\) value is used. Consultation of the Theory Manual is strongly advised before use.
min_pressure, value: Minimum mean effective stress in kPa (compression = positive). The default is value=0.01 kPa.
integrator: a flag controlling which integration method should be used to advance the stress from time \(t_0\) to time \(t=t_0 + \Delta t\). integrator=1 (default) corresponds to a Forward Euler (FE) integration scheme, with integrator=2 a Euler-Richards integration scheme with adaptive time incrementation is used.

State variables

In addition to the (effective) stress, the hypoplastic constitutive model takes additional state variables. Performing simulations with this model requires the prescription of those. The following state variables are contained in the model:

Void ratio: \(e\), void_ratio. The prescription of the void ratio is mandatory, if the initialization is omitted or wrong values are prescribed the simulation will abort.
Intergranular strain: \(\mathbf{h}\), int_strain11, int_strain22, int_strain33, int_strain12, int_strain23, int_strain13. Only in combination with the intergranular strain extension. Prescribing \(\mathbf{h}_0\) is not mandatory, if omitted, \(\mathbf{h}=\mathbf{0}\) is assumed. \(h_{ii} = -R / \sqrt{3}\) for \(i=\{x,y,z\}\) corresponds to an isotropically consolidated state.
Intergranular back strain: \(\mathbf{c}\), int_back_strain11, int_back_strain22, int_back_strain33, int_back_strain12, int_back_strain23, int_back_strain13. Only in combination with the intergranular strain extension. \(\mathbf{c}_0=\mathbf{h}_0/2\), however, prescribing \(\mathbf{c}_0\) is not mandatory, if omitted, \(\mathbf{c}=\mathbf{0}\) is assumed.

The state variables can be initialized in two different ways:

Using the *Initial conditions, type = state variables command in the input file, e.g.

*Initial conditions, type = state variables
element_set_name, void_ratio, <value>
element_set_name, int_strain11, <value>
element_set_name, int_strain22, <value>
element_set_name, int_strain33, <value>
element_set_name, int_back_strain11, <value>
element_set_name, int_back_strain22, <value>
element_set_name, int_back_strain33, <value>

* Using the interface for user-defined initial state variables. In this approach, the vector holding the state variables is filled directly in a fortran subroutine as described in Section User defined state variables. The position of the state variables in the associated vector are as follows:

statev(1) = \(e\)
statev(3) = \(h_{11}\)
statev(4) = \(h_{22}\)
statev(5) = \(h_{33}\)
statev(6) = \(h_{12}\)
statev(7) = \(h_{23}\)
statev(8) = \(h_{13}\)
statev(9) = \(c_{11}\)
statev(10) = \(c_{22}\)
statev(11) = \(c_{33}\)
statev(12) = \(c_{12}\)
statev(13) = \(c_{23}\)
statev(14) = \(c_{13}\)

Changes with the upcoming release

With the upcoming release a new state variable is available: relative density \(D_r = \dfrac{e_c-e}{e_c-e_d}\). Note that \(e_c\) and \(e_d\) are pressure dependent (see the Theory manual). The relative density can also be used to initialise the void ratio \(e_0\). If \(D_{r0}\) is provided by the user, e.g. by the following input command:

*Initial conditions, type = state variables
element_set_name, rel-density, <value>

and no initial void ratio is specified, numgeo will calculate the initial void ratio according to the following equation

\[ e_0(p') = \left( e_{c0} - D_{r0}(e_{c0}-e_{d0}) \right) \exp\left(\dfrac{-3p'}{h_s}\right)^n \]

where \(p'\) is the mean effective stress (compression is positive), \(e_{c0}\) and \(e_{d0}\) are the void ratios at the critical and densest state for \(p'=0\) (material parameters).

Additional output variables

The following additional output variables are available in the *Output command:

Void ratio: void_ratio
Intergranular strain: int_strain11, int_strain22, int_strain33, int_strain12, int_strain23, int_strain13.
Intergranular back strain: int_back_strain11, int_back_strain22, int_back_strain33, int_back_strain12, int_back_strain23, int_back_strain13.
Stress projection: iproj. Flag that indicates if the stress correction is active (see Theory Manual). iproj takes the value of 0 if the stress projection is inactive.
Hypoplastic factors: fz, fb, fd, fe
Mobilised friction angle: phi_mob
Relative density: rel-density

P.-A. Wolffersdorff. A hypoplastic relation for granular materials with a predefined limit state surface. Mechanics of Cohesive-frictional Materials, 1(3):251–271, 1996. doi:10.1002/(SICI)1099-1484(199607)1:3<251::AID-CFM13>3.0.CO;2-3. ↩
William Fuentes, Torsten Wichtmann, Melany Gil, and Carlos Lascarro. ISA-Hypoplasticity accounting for cyclic mobility effects for liquefaction analysis. Acta Geotechnica, 15(6):1513–1531, 2016. URL: http://link.springer.com/10.1007/s11440-019-00846-2, doi:10.1007/s11440-019-00846-2. ↩