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Ta-Ger

Ta-Ger is an elastoplastic sand model based on the Ta-Ger formulation of Tasiopoulou and Gerolymos 12. It combines an explicit elastoplastic stiffness matrix with a Bouc-Wen type mapping variable, critical-state-compatible evolution of the bounding and phase transformation stress ratios, Rowe-type dilatancy and a stress-induced peak-strength correction.

Availability

The Ta-Ger model will be available with the upcoming release of numgeo

Input syntax

*Mechanical = Ta-Ger
G0, m, nu, Q, R, emin, emax, phics
delta, kappa, Gmodel

Material parameters

The material parameters for Ta-Ger are:

# Symbol Unit Description
1 \(G_0\) model-dependent Shear stiffness coefficient. Its interpretation depends on Gmodel.
2 \(m\) - Exponent controlling pressure dependency of the shear modulus.
3 \(\nu\) - Poisson's ratio used to compute the bulk modulus from the shear modulus.
4 \(Q\) - First parameter of Bolton's relative dilatancy index \(I_r=D_r[Q-\ln(p)]-R\).
5 \(R\) - Second parameter of Bolton's relative dilatancy index.
6 \(e_{min}\) - Minimum void ratio used to calculate relative density.
7 \(e_{max}\) - Maximum void ratio used to calculate relative density.
8 \(\varphi_{cs}\) deg Critical-state friction angle.
9 \(\delta\) - Intrinsic calibration coefficient for \(c=6+\delta I_{r0}\).
10 \(\kappa\) - Intrinsic calibration coefficient for \(\varphi_{s0}=\kappa\varphi_{cs}+5I_{r0}\).
11 Gmodel - Shear modulus option. 1: \(G=G_0(0.13D_{r0}+3.6)p^m\); otherwise: \(G=G_0p_a(2.97-e)^2/(1+e)(p/p_a)^m\).

Pressure units

The relative dilatancy index uses \(\ln(p)\) directly. The pressure unit convention must therefore be consistent with the calibration of \(Q\) and \(R\). The commonly used calibration is based on stresses in kPa.

Reference state

The initial void ratio must be provided as the first state variable before the first increment. At the first increment of the first step numgeo stores the reference state \(p_0\), \(e_0\), \(D_{r0}\) and \(I_{r0}\). These values remain fixed afterwards.

Typical values

For clean sands, values close to \(Q=9\)--\(10\) and \(R=0.5\)--\(1.0\) are commonly used. The coefficients \(\kappa\) and \(\delta\) are sand-dependent calibration parameters. Typical values reported in the Ta-Ger calibration study include approximately:

Sand \(\kappa\) \(\delta\)
Toyoura sand 0.9 1
Fontainebleau sand 0.8 4
Sacramento River sand 0.7 3

These values should be regarded as calibration guidance rather than universal defaults.

Example

*Mechanical = Ta-Ger
** G0, m, nu, Q, R, emin, emax, phics, delta, kappa, Gmodel
130, 0.8, 0.15, 9.15, 0.77, 0.597, 0.977, 32, 1, 0.9, 2

For the density-based shear modulus option, a possible input is:

*Mechanical = Ta-Ger
** G0, m, nu, Q, R, emin, emax, phics, delta, kappa, Gmodel
1000, 0.5, 0.15, 9.15, 0.77, 0.597, 0.977, 32, 1, 0.9, 1

State variables

At least 14 state variables are required. The state-variable layout is:

# Symbol Description
1 \(e\) Current void ratio. Must contain the initial void ratio before the first increment.
2 \(p_0\) Initial mean effective stress, stored at the first increment.
3 \(dW_{old}\) First-order work indicator from the previous substep.
4 - 9 \(\mathbf{r}_p\) Stress-ratio tensor at the last loading reversal.
10 \(\sum d\varepsilon_q\) Accumulated deviatoric strain measure.
11 \(K_w\) Auxiliary water bulk modulus for locally undrained calculations.
12 \(e_0\) Initial void ratio, stored at the first increment.
13 \(D_{r0}\) Initial relative density.
14 \(I_{r0}\) Initial relative dilatancy index.

The following state quantities can be accessed by their respective names instead of their index in the array statev

  • Void_Ratio: current void ratio \(e\).
  • Bulk_Water: Bulk modulus of water \(K_w\).

References

  1. Panagiota Tasiopoulou and Nikos Gerolymos. Constitutive modeling of sand: Formulation of a new plasticity approach. Soil Dynamics and Earthquake Engineering, 82:205–221, March 2016. doi:10.1016/j.soildyn.2015.12.014. ↩

  2. P. Tasiopoulou and N. Gerolymos. Constitutive modelling of sand: a progressive calibration procedure accounting for intrinsic and stress-induced anisotropy. Géotechnique, 66(9):754–770, September 2016. doi:10.1680/jgeot.15.P.284. ↩