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Sanisand-2

*Mechanical = Sanisand-2 
p_atm [F/A], e_0 [-], lambda_c [-], xi [-], M_c [-], M_e  [-], m [-],  G_0 [-] 
nu [-], h_0 [-], c_h [-], n_b [-], A_0 [-], n^d [-], z_max [-], c_z [-]

The second implementation of the well-known Sanisand model (Dafalias & Manzari (2004)1) follows closely the implementation of the Sanisand model available in OpenSees and was created in collaboration with Mahdi Taiebat and Sheng Zheng. The material parameters of Sanisand are given in the following.

  • \(p_{atm}\) [F/A] is the atmospheric pressure usually given by 100 kPa.
  • \(e_0\) is the critical void ratio at zero mean effective stress and can for instance be obtained from (several) undrained monotonic triaxial tests reaching the critical state. It will be in the range of \(e_{c0}\) used by the hypoplastic model. The parameters are not identical, however.
  • \(\lambda_c\) and \(\xi\) define the relation between critical void ratio and mean effective stress. Again, they can for instance be obtained from (several) undrained monotonic triaxial tests reaching the critical state.
  • \(M_c\) and \(M_e\) are the critical stress ratios for compression and extension, respectively. They can be calculated using the critical friction angle.
  • \(m\) defines the elastic range of the model. Conventionally, it is chosen to 0.05.
  • \(G_0\) and \(\nu\) are the elastic parameters of the model.
  • \(h_0\) and \(c_h\) are obtained from the fitting of monotonic triaxial tests. They have no physical meaning.
  • \(n_b\) influences the peak in the \(q\)-\(\varepsilon_1\) plot. Again, it has no physical meaning.
  • \(A_0\) and \(n^d\) influence the dilatancy. They also have no physical meaning.
  • \(z_{\max}\) and \(c_z\) are the parameters influencing the cyclic mobility effect. They are obtained by fitting of cyclic triaxial test data.

Optional Parameters

*Optional mechanical parameter
<property 1>, <value 1>...
<property 2>, ...
Definition of optional parameters is not mandatory. The specification can also be made individually:

  • integration Flag controlling the integration method, available are: 0 = explicit with constant subincrement size, 1 = Modified-Euler (default), 2 = Forward Euler
  • min_pressure Minimum mean effective stress in kPa (compression = positive). The default is 0.5 kPa.
  • tol_stress Tolerance for stress error of the explicit integration scheme, default is \(1\cdot 10^{-4}\).
  • minimum_dh Minimum subincrement size for the integration scheme, default is \(10^{-6}\).
  • allow_cutback Controlling if the constitutive model is allowed to request a ''global cutback'' from numgeo: 0 = not allowed (default), 1 = allowed.
  • drift_correction Method for drift correction: 0 = map yield surface by adjusting alpha, 1 = projection along deviatoric plane, else = drift correction based on Sloan et al. (2001)2 (default).
  • drift_maxiter Maximum number of iterations for drift correction (only for the OpenSees drift correction, ignored otherwise). The default is 50.
  • tol_yield Tolerance for yield surface. Default is \(10^{-7}\).
  • tangent_stiffness When the contribution of the constitutive model \(J=\dfrac{\partial \sigma}{\partial \varepsilon}\) to the element stiffness is computed. 1 = tangent stiffness (evaluated at the end of the increment for the updated stress state), else = "incremental" stiffness (evaluated at the end of each subincrement, a weighted average is returned to the element, default).
  • jacobi How \(J=\dfrac{\partial \sigma}{\partial \varepsilon}\) is evaluated: 1 = elastic Jacobian, 2 = elasto-plastic Jacobian (default).
  • maximum_strain Maximum strain \(\Delta \varepsilon^{sub}\) for subincrement size in explicit integration, default is \(\Delta \varepsilon^{sub} = 5 \cdot 10^{-7}\). This option is only relevant for the Forward Euler (FE) integration.
  • maximum_voidratio, maximum_voidratio Maximum and minimum allowed void ratios \(e^{max}\) and \(e^{min}\), respectively (off per default). If active (for any value larger than zero), \(e=min(e,e^{max})\) and/or \(e=max(e,e^{min})\) is/are enforced.
  • bulk_water, 0.0 - Bulk modulus of pore water \(K^w\) for locally undrained simulations \(tr(\dot{\boldsymbol{\varepsilon}})=0\). For any value of \(K^w > 0\) the rate of pore water pressure is calculated as follows \(\dot{p}^w = - K^w tr(\dot{\boldsymbol{\varepsilon}}) (1+e)/e\). The constitutive behaviour is governed by the effective stress \(\boldsymbol{\sigma} = \boldsymbol{\sigma}^{tot}-p^w \boldsymbol{\delta}\), where \(\boldsymbol{\sigma}^{tot}\) is the total stress and \(\boldsymbol{\delta}\) is the Kronecker delta.
  • mb_formulation - 0 = original formulation of \(M_b\), otherwise: \(M_b\) cannot fall below \(M_c\)

State variables

From the state variables of Sanisand, only the void ratio can be initialised analogously to the procedures described for the hypoplastic model. The initial back stress tensor is always defined such that the initial stress state is in the centre of the elastic range of the model.

Additional output variables

The following additional output variables are available:

  • Void ratio: void_ratio
  • Back stress tensor: alpha_11, alpha_22, alpha_33.
  • Value of the yield surface: Fyield. Should always be less than zero. For values close to zero, the model will most likely produce plastic strain.
  • State variable \(\psi = e - e_c\): psi
  • Mean effective stress: stress-p
  • Deviatoric stress: stress-q
  • Pore water pressure: stress-pw (only for locally undrained simulations by setting bulk_water > 0)

  1. Y. F. Dafalias and M. T. Manzari, ‘Simple plasticity sand model accounting for fabric change effects’, Journal of Engineering mechanics, vol. 130, no. 6, pp. 622–634, 2004. ↩

  2. S. W. Sloan, A. J. Abbo, and D. Sheng, ‘Refined explicit integration of elastoplastic models with automatic error control’, Engineering Computations, vol. 18, no. 1/2, pp. 121–194, Feb. 2001, doi: 10.1108/02644400110365842. ↩