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Sanisand-MSf

*Mechanical = Sanisand-MSf
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 (-)
n_g (-), mue_0 (-), u (-), c_l (-), c_l (-), x (-), c_c (-), zeta (-)
m^s (-), n_l (-), c_r (-), p_th (F/A), f_l (-)

The extension of the well known Sanisand-04 model1 to incorporate effects of memory surface and semi-fluidized states according to Yang et al. (2022)2. The implementation was created in collaboration with Mahdi Taiebat and Sheng Zheng. The material parameters of Sanisand-MSf are given in the following.

  • Parameters of the Sanisand-04 model
    • \(p_{atm}\) (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. 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.
  • Parameters of the MSf extension
    • \(n_g\) - triaxial shear strain constant, controlling the shift of the stress-strain loops during cyclic triaxial loading
    • \(\mu_0\) - memory surface constant
    • \(u\) - memory surface constant
    • \(c_l\) - semi-fluidised state constant
    • \(x\) - semi-fluidised state constant

  • Additional constants

    The Sanisand-MSf model has some additional constants. However, as suggested in Yang et al. (2022), their suggested default values do not need to change in most cases. The additional constants are:

    • \(c_c\) - memory surface constant, default value is 1.0
    • \(\zeta\) - memory surface constant, default value is 0.00001
    • \(m^{s,\varepsilon}\) - memory surface constant, default value is 0.01
    • \(w\) - memory surface constant, default value is 2.0
    • \(n_l\) - semi-fluidised state constant, default value is 8.0
    • \(c_r\) - semi-fluidised state constant, default value is 0.0
    • \(p_{th}\) - semi-fluidised state constant, default value is 10.0
    • \(f_l\) - semi-fluidised state constant, default value is 0.01

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 (default), 1 = projection along deviatoric plane, else = drift correction based on Sloan et al. (2001)3.
  • 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 (default), 2 = elasto-plastic Jacobian.
  • 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.

State variables

The following state variables are contained in the model and can (must) be initialized:

  • 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.

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. M. Yang, M. Taiebat, and Y. F. Dafalias, ‘SANISAND-MSf: a sand plasticity model with memory surface and semifluidised state’, Géotechnique, vol. 72, no. 3, pp. 227–246, Mar. 2022, doi: 10.1680/jgeot.19.P.363. ↩

  3. 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. ↩