FLAC3D Theory and Background • Constitutive Models

Soft-Soil-Creep Model*

Note

*This model is available in FLAC3D only.

The Soft-Soil (SS) model cannot simulate time-dependent behavior such as the creep that develops during the secondary compression. The creep behavior of soft soil can be significant for some geotechnical engineering problems, e.g., the embankment construction. The Soft-Soil-Creep (SSC) model (Vermeer and Neher, 1999) takes time-dependence into account so that the volumetric cap expands to a new position within a specific time (called reference time in the model) instead of instantaneously in the SS model. Furthermore, the cap will never stop expanding although with a continuously decreasing expansion rate. The expansion rate will be determined by the value of the current OCR. When OCR is high, the creep deformation is negligible. The SSC model has the following features:

1. secondary time-dependent compression;
2. pressure-dependent moduli;
3. unloading-reloading distinct from the virgin loading;
4. memory of the equivalent pre-consolidation pressure; and
5. degenerates into a model with Mohr-Coulomb failure criterion and with pressure-dependent moduli (and OCR is meaningless) when the creep is off (creep time step is zero). Be aware that it does not degenerate into the Soft-Soil model when the creep is off.

In the SSC model, most formula are the same as those in the SS model.

The volumetric creep strain rate in the SSC model is defined as

(1)\[\dot{\epsilon}_v^c = \frac{\mu^*}{\tau} \left( \frac{p_{eq}}{p_c} \right)^{\frac{\lambda^*-\kappa^*}{\mu^*}}\]

where \(\mu^*\) is a material parameter called modified creep index, \(\tau\) is a reference time, \(p_{eq}\) is the equivalent pressure, and \(p_c\) is the current generalized pre-consolidation pressure, or cap pressure for short.

If we define \(OCR=p_c/p_{eq}\) in this model (Note: \(OCR\) possibly can be less than 1 in the SSC model based on this specific definition), Equation (1) can be rewritten as

(2)\[\dot{\epsilon}_v^c = \frac{\mu^*}{\tau} OCR^{- \frac{\lambda^*-\kappa^*}{\mu^*} }\]

The rate of the cap pressure is

(3)\[\dot{p}_c = p_c \frac{-\dot{\epsilon}_v^c}{\lambda^*-\kappa^*}\]

Integration of the equation above gives:

(4)\[p_c = p_{c}^0 exp \left( \frac{-\epsilon_v^c}{\lambda^*-\kappa^*} \right)\]

where \(p_c^0\) is the cap pressure at the beginning of the analysis, which could be related to the initial \(OCR^0\) at \(t=0\) by \(p_c^0=OCR^0 \times p_{eq}^0\).

The rate of general creep strain is of the form

(5)\[\dot{\epsilon}_{ij}^c = \frac{\mu^*}{\tau} OCR^{- \frac{\lambda^*-\kappa^*}{\mu^*}} \left( \dfrac{\frac{\partial p_{eq}}{\partial \sigma_{ij}}}{\frac{\partial p_{eq}}{\partial p}} \right)\]

The total strain rate in the SSC model is the sum of the elastic strain rate and the creep strain rate, or \(\dot{\epsilon}_{ij} = \dot{\epsilon}_{ij}^e + \dot{\epsilon}_{ij}^c\)

For constant \(p_{eq}\), the creep volumetric strain increment can be integrated (Stolle, Vermeer, & Bonnier, 1999) as

(6)\[\Delta {\epsilon}_{ij}^c = -\mu^* \ln{\left( 1+\frac{\Delta t}{\tau} OCR^{-\frac{\lambda^*-\kappa^*}{\mu^*}} \right)}\]

Since the current implementation uses an explicit time integration method, careful consideration should be taken to the choice of creep time step in the simulation, as it can affect numerical stability, and the solution accuracy. The time step limitation for creep compaction is estimated as the ratio of apparent viscosity to bulk modulus. The viscosity may be expressed as the ratio of \(p+c\cot{\phi}\) to the volumetric creep compaction rate \(\dot{\epsilon}_v^c\). The maximum creep time step for creep compaction is:

(7)\[\Delta t_{max} = \tau \frac{\kappa^*}{\mu^*} OCR^{\frac{\lambda^*-\kappa^*}{\mu^*}}\]

where \(OCR\) is the minimum value expected to occur in the simulation. A rule-of-thumb is to begin the creep analysis with a creep time step approximatively two orders of magnitude smaller than \(\Delta t_{max}\):

(8)\[\Delta t \leq \frac{\tau}{100} \frac{\kappa^*}{\mu^*} OCR^{\frac{\lambda^*-\kappa^*}{\mu^*}}\]

As a rule, the maximum value for the time step should not exceed the value derived for \(\Delta t_{max}\).

For \(OCR\) = 1, the creep time step could be less than \(0.04\tau\) if \(\kappa^*/\mu^*\) = 4; but for a high value of \(OCR\), i.e., \(OCR\) = 1.5, \(OCR^{(\lambda^*-\kappa^*)/\mu^*}\) could be as big as 1000 if \((\lambda^*-\kappa^*)/\mu^*\) = 17 (which is in the typical range of 10 to 50) and the creep time step could be around \(40\tau\); for \(OCR\) = 2.0, the creep time step could be around \(5000\tau\).

The automatic creep time step technique that has been in FLAC3D can be used to control the time step during the calculation. The user should pay attention to the possible numerical stability issue, which is intrinsic to the explicit time integration method. Equation (8) provides a good first-step estimation.

Acknowledgment

The implementation of this model was partially sponsored by SAGE Engineers, Inc.

References

Stolle, D.F.E., P.A. Vermeer, and P.G. Bonnier. Time integration of a constitutive law for soft clays. Communications in Numerical Methods in Engineering, 15(8) 603-609 (1999).

Vermeer, P.A., and H.P. Neher. A soft soil model that accounts for creep. in Proceedings of the international symposium “Beyond 2000 in Computational Geotechnics”. (1999).

soft-soil-creep Model Properties

Use the following keywords with the zone property command to set these properties of the Soft-Soil-Creep model. Only two more material properties are added compared with the Soft-Soil model: time-reference math:\(\tau\) and index-creep \(\mu^*\).

soft-soil-creep

friction f

friction angle, \(\phi\), The default value is 30 [in degrees]. The recommended input value is the critical-state friction angle rather than a higher value at small strains. Input of a zero value is not allowed.

index-creep f

modified creep index, \(\mu^*\), which controls the secondary creep rate. \(\mu^* \approx C'_{\alpha}/[(\ln{10})] \approx C_{\alpha}/[(\ln{10})(1+e_p)]\), where \(e_p\) can be approximately the void ratio at the end of primary consolidation and \(C_{\alpha}\) is the 1-D creep index for secondary compression. Typical magnitudes of \(C'_{\alpha}\) in various natural deposits (Das & Sobhan, Principles of Geotechnical Engineering. Cengage Learning, 2013) are : 0.001 or less for overconsolidated clays, 0.005 to 0.03 for normally consolidated clays, and 0.04 or more for organic soils.

kappa-modified f

modified swelling index, or slope of elastic swelling line, \(\kappa^*=\kappa/(1+e)\), where \(\kappa\) is the swelling index in the Cam-Clay model, and \(e\) can be approximately the initial or average void ratio during a swelling path; or \(\kappa^* \approx [(1+2K_0)(1-\nu)/(1+\nu)] C_s/[(\ln{10})(1+e)]\), where \(C_s\) is the 1-D swelling index, \(\nu\) is the Poisson’s ratio and \(K_0\) is the at-rest earth pressure coefficient. Typical value of \(C_s\) is \(C_s \approx (1/5) \sim (1/10) C_c\). A good estimation is \(C_s \approx PI(\%)/370\).

lambda-modified f

modified compression index, or slope of elastic compression line, \(\lambda^*=\lambda/(1+e)\), where \(\lambda\) is the compression index in the Cam-Clay model, and \(e\) can be approximately the initial or average void ratio during a compression path; or \(\lambda^*=C_c/[(\ln{10})(1+e)]\), where \(C_c\) is the 1-D compression index. Typical value of \(C_c\) can be estimated (Terzaphi & Peck 1967) as \(C_c=0.009(LL-100)\) for normally consolidated clay where \(LL\) is the liquid limit. Based on the modified Cam-Clay model, it can estimated that \(C_c \approx PI(\%)/74\), where \(PI\) is the plastic index.

poisson f

Poisson’s ratio, \(\nu\), The default value is 0.15.

stress-1-effective f

initial minimum effective principal stress, \(\sigma^0_1\). Only for calculation of the initial moduli, will not be update once assigned.

stress-2-effective f

initial median effective principal stress, \(\sigma^0_2\). Only for calculation of the initial moduli, will not be update once assigned.

stress-3-effective f

initial maximum effective principal stress, \(\sigma^0_3\). Only for calculation of the initial moduli, will not be update once assigned.

coefficient-normally-consolidation f (a)

normal consolidation coefficient, \(K_{nc}\). It is not allowed to be less than \(\nu / (1-\nu)\), a range 0.5 to 0.7 is common. The default is \(K_{nc} = 1 - \sin \phi\).

cohesion f (a)

effective cohesion, \(c\). The default value is 0. Remember that this model is mainly for normally consolidated or slightly over-consolidated soft soils, so \(c\) should be zero or a small value, and a large value may be not realistic.

dilation f (a)

dilation angle, \(\psi\). The default value is 0 [in degrees]. The zero dilation angle is the standard setting in the SS model.

over-consolidation-ratio f (a)

over consolidation ratio, \(OCR\). The default is 1.0, which corresponds to the normally consolidated soils. The input \(OCR\) will be used to determine the initial cap pressure so that \(p_{c}^0=OCR \times p_{eq}^0\). \(OCR\) will be updated during calculation.

pressure-cutoff f (a)

cut-off pressure for cap pressure, \(p_{cut}\). The default value is 1.0 (of the stress unit used in the model). This is the lower-bound of the cap pressure. The actual minimum cap pressure used in the model is \(max(p_{cut}, c\cot{\phi})\).

tension f (a)

tensile strength, \(\sigma^t\). The default value is 0, which is for most cases.

time-reference f (a)

reference time, \(\tau\). The default value is 1.0, which should be compatible to the time unit used in the model. This parameter provides time scale. For example, if the time unit used in the model is day, denotes the soil takes an increment of 1 day to reach the normally consolidated state. The Typical value of 1 (if the time unit is in day) is compatible to the standard incremental oedometer tests which run with 1 (day) increments.

void-initial f (a)

initial void ratio, \(e_{ini}\). The default is 1.0.

bulk f (r)

current elastic bulk modulus, \(K\)

pressure-cap f (r)

current cap (apparent pre-consolidation) pressure, \(p_c\)

pressure-effective f (r)

current effective pressure, \(p\)

pressure-equivalent f (r)

current equivalent pressure, \(p_{eq}\)

ratio-radius f (r)

current elastic shear modulus, \(M\), which will be calculated internally.

shear f (r)

current elastic shear modulus, \(G\)

strain-volume-plastic f (r)

accumulated plastic volumetric strain, \(\epsilon_v^p\)

stress-deviatoric-equivalent f (r)

current equivalent deviatoric stress, \(\tilde{q}\)

void f (r)

current void ratio, \(e\)

Key

(a) Advanced property.

This property has a default value; simpler applications of the model do not need to provide a value for it.

(r) Read-only property.

This property cannot be set by the user. Instead, it can be listed, plotted, or accessed through FISH.