Viscous Soft-soil Modeling

Geo-Denver 2007 ISBN 978-0-7844-0897-1

Viscous soft-soil modeling S.van Baars 1

VISCOUS SOFT-SOIL MODELING

S. van Baars1

ABSTRACT

Viscosity is in fact a relaxation of the shear stresses over time, which results

for constant stress states in a corresponding elastic strain relaxation. In other words, viscosity is a sort of slow plasticity before the Coulomb criterion is reached. Creep settlement can be regarded as one of the viscous soil phenomena, but also the slow movement (a few mm or cm every year) of many natural slopes in mountainous area’s. Therefore a viscous constitutive model is derived from the standard principles that the reduction rate of the relative shear stress (the viscosity) is a function of this relative shear stress and that the reduction rate approaches zero while the stress-state approaches the K0-state. As an example a User-Defined (Soft) Soil Model is developed in order to model the continuous moving slopes. Although the model works fine for viscous behavior and creep problems, it seems as if the slope movements do not depend on viscosity but on dilatancy.

Keywords: constitutive model, creep, slope movement, soil, viscosity 1. INTRODUCTION

In most everyday geotechnical consulting the applied soil behavior is kept constant with time, except for creep settlement calculations. Creep and slow slope movement are examples of viscous soil phenomena. Maybe the effect of viscosity is underestimated in soil mechanics. Even world spread and well-known finite element codes for geotechnical calculations (like Plaxis) don not provide a viscous constitutive model. The lack of such a model can lead to serious design problems. For example the lateral displacement versus time behavior of the recently constructed high speed train line in The Netherlands on very soft peat and clay layer, next to an existing high way, exceeds well all expectations, leading to large additional works

1 Assistant Professor Soil Mechanics, Delft University of Technology, Department of Geo-Engineering, Delft, The Netherlands & Visiting Assistant Professor, University Joseph Fourier, Department 3S, Grenoble, France

Geo-Denver 2007 ISBN 978-0-7844-0897-1


and delays. A difficult to model subject is the movement of slopes. According Vulliet (1997) many natural slopes in mountainous area’s show slow movement (a few mm or cm per year). In some cases, slope movements degenerate into fast-moving or catastrophic landslides. These constant moving slopes may have to be modeled with a viscous constitutive model. Therefore this paper derives a viscous constitutive model, starting from the Mohr-Coulomb model. As an example a Plaxis User-Defined (Soft) Soil Model is developed and applied to several viscous circumstances like oedometer (creep) testing, triaxial (strength) testing and continues slope movement. 2. MOHR-COULOMB MODELING

The Mohr-Coulomb constitutive model is used as a starting point for the viscous model and will therefore be firstly presented in brief. In this model the elasto-plastic (effective) stress increments are calculated from the total and plastic strain increments by (pressure and strain reduction taken positive):

( )tot plasDσ ε ε∆ = ⋅ ∆ + ∆ , (1)

with the constitutive matrix:

0 0 00 0 00 0 0

and 0 0 0 0 00 0 0 0 00 0 0 0 0

xx

yy

zz

xy

yz

xz

A B BB A BB B A

DG

GG

σσσ

σσσσ

∆ ∆ ∆

= ∆ = ∆ ∆ ∆

, (2)

in which the pivots are defined by: ( )1A F v= − , B Fν= and ( )1

2G F ν= − . The stiffness factor F is constant for Mohr-Coulomb type models, depending on the Young’s modulus E and the Poison’s ratio ν:

( )( )1 1 2

EFv v

=+ −

. (3)

The stiffness factor F can also be variable for Soft-Soil type models by depending on the maximum reached isotropic stress pmax (the pre-overburden pressure):

Geo-Denver 2007 ISBN 978-0-7844-0897-1


( )

max*

31

pFv λ

=+

, (4)

in which the isotropic stress p is defined as: ( )1

1 2 33p σ σ σ= + + ( )13 xx yy zzσ σ σ′ ′ ′= + + .

The stresses σ’xx, σ’yy and σ’ zz, are the effective horizontal en vertical stresses. For the calculation of the principal stresses σ1, σ 2 and σ 3, see Van Baars (2003). The soil stiffness λ* is derived from:

* lninitial

pp

ε λ

=

. (5)

The plastic stress increments follow from the stress state shown in Figure 1.

c·cot( φ )

c

p= p× × ××

rr*

*σ

ii× × × ×

σ

σ*

×3σ

ijσ

jiσ

*3σ *2σ *1σ*0σ

0σ 1σ

FIG 1. Effective stress state before plastic reduction The radius r of the Mohr-circle divided by the radius of the Mohr-circle during plastic failure r* = rmax gives the relative shear R:

( )( )1

1 3 22 sinsin cos

pR

p cσ σ φ σ

φ φ− + −

=⋅ + ⋅

. (6)

For the derivation of this equation see Van Baars (2003). In case of no shear (i.e. isotropic loading) R = 0 and at failure R = 1.

Geo-Denver 2007 ISBN 978-0-7844-0897-1


If the stresses in the calculation go beyond the Coulomb criterion (plasticity; so for: R > 1), then the stresses will be reduced in such a way that the size of the Mohr-circle is squeezed between the Coulomb lines (back-to-base method) and around the isotropic stress p to avoid volume changes, so (see Figure 1):

** 1ijii

ii ij

pp R

σσσ σ

−= =

−. (7)

Therefore this method contains no hardening or softening. The normal and shear stresses at plastic failure σ* are reached by adding incremental plastic stresses: *

,ii ii ii plasσ σ σ= + ∆ and *,ij ij ij plasσ σ σ= + ∆ , (8)

so the incremental plastic normal and shear stress will be:

( ),11ii plas ii pR

σ σ ∆ = − − −

and ,11ij plas ijR

σ σ ∆ = − −

for R > 1. (9)

In this way the elasto-plastic behavior of the Mohr-Coulomb model and the Soft-Soil model is fully described. 3. VISCOSITY MODELING

Viscosity is in fact a relaxation of the shear stresses over time, which results for constant stress states in a corresponding elastic strain relaxation. In other words, viscosity is a sort of slow and rate-depending plasticity without reaching the Coulomb criterion; therefore also the word visco-plasticity can be used. The logarithmic creep law from Keverling Buisman (1936) suggests there is no point at which creep and viscosity stop. But according Bishop (1961) there is “no very large strength reduction on a long term basis”, otherwise “most of the world would be as flat as Holland.” Therefore it might be right to assume that a visco-plastic soil will always creep towards the K0 stress state, and not beyond. This K0 stress state can be found by combining: ( ) ( )1 1

max 1 3 0 1 3 3 0 12 2sin cos , andr c r Kσ σ φ φ σ σ σ σ= + + ⋅ = − = . (10) This combination results in the neutral relative shear stress R0:

( )

0 00

max0

1

12sin 1 cos

r KR cr Kφ φσ

−= =

+ + ⋅. (11)

Geo-Denver 2007 ISBN 978-0-7844-0897-1


Approaching R0 the viscosity and creep will disappear. Such a viscosity halt, is in fact the same as the stick-slip stress limit σS used by Lemaitre (1996) in his viscosity model. The viscosity rate can be defined as a reduction over time of the relative shear stress (Y = -∂R/∂t). This rate depends on the relative shear stress R. Figure 2 shows such a relationship. Beside the R-axis also an X-axis can be defined. Using the neutral relative shear stress R0, a normalized relative shear stress X can be defined:

0

01R RX

R−

=−

. (12)

The maximum value for which viscosity can exist is R = 1. Here, by definition, X = 1. The minimum value of the relative shear stress R0 for which viscosity occurs is the stress state of K0. Here, by definition, X = 0. The K0 value is only used as an input soil parameter for the calculation of R0 (final equilibrium). For fully elastic unloading or reloading at the beginning, the Poisson’s ratio ν of Equations (2), (3) and (4) is replaced by the elastic an unloading/reloading Poisson’s ratio νur which has a lower value then ν (K < K0).

R = rrmaxR0 1

X = R-R01-R00 1

Cvp

mvp

R C Xt

∂= −

∂

RYt

∂= −

∂

00

1m <

1m >

1m =

Fig. 2. Reduction rate Y of the relative shear stress as a function of this relative

shear stress R According Figure 2 the visco-plasticity rate ∂R/∂t depends on the normalized shear stress X, the viscosity constant Cvp and the power m:

XR Yt

∂= −

∂ in which m

X vpY C X= for m > 0. (13)

The two parameters Cvp and m characterize the viscous behavior of the soil.

Geo-Denver 2007 ISBN 978-0-7844-0897-1


At X = 1 (or R = 1), the question arises whether the visco-plasticity reaches a maximum (the visco-plasticity constant Cvp in the figure) or even peaks sharply to infinity. Theoretically this difference is of fundamental interest, but for practical calculations the plasticity already limits this point, so the practical importance is small. Another fundamental point is the question whether the viscosity rate (∂R/∂t) remains constant over time if the stress state (σ ′ ) is kept constant. According the many constant moving slopes, it seems the strain rate can be constant for years. However, it might be that all these slopes have a safety factor close to one. Van Genuchten (1989) and Nieuwenhuis (1991) studied one of these moving slopes, which is a slope near La Mure (Grenoble) in the French Alps, consisting of alternating clay and silt laminae. Their conclusion was that this slope has a very low stability factor and moves therefore, depending on the amount of rainfall, about 700 mm per year. According Bishop and Lovenbury (1969), whose long term drained triaxial test results are also presented by Mitchell (1993), it is clear the viscosity rate declines over time. Their curves of undisturbed London Clay can be described by:

( )( )1

with 0.9n

Xn

YR nt t

−∂

= − =∂

. (14)

Also the undrained triaxial tests (with constant stress state) of Arulanandan et all (1971) give an almost similar correlation but a slightly different power (n = 0.7). These findings prove that the viscous behavior is not constant but declining over time. 4. DECLINING VISCOSITY RATE MODELLING

In fact the viscosity rate does not depend on time, but on the amount of viscosity which already occurred. This can be found by integrating Equation (14):

( )( )

( )

1

0

for 0 11

ntX

t

Y tRR dt nt n

−−∂∆ = = ≤ <

∂ −∫ . (15)

By implementing Equation (15) in Equation (14), time t can be erased:

( ) 1

for 0 1( 1 )

Xnn

t

R Y nt

n R −

∂ −= ≤ <

∂− − ∆

. (16)

In this way the viscosity rate depends on the amount of viscosity in the past. This equation does not depend on an increasing stress. In case of a new load step in an oedometer for example does not lead to a new viscous creep phase. A solution can

Geo-Denver 2007 ISBN 978-0-7844-0897-1


be to reduce the total amount of viscosity in the past for an increasing stress, for example:

( )( )

( )

1

0

for 0 11

ntt X

t t t t

p Y tRZ p R p dt nt n

−−∂= ∆ = = ≤ <

∂ −∫ . (17)

By implementing this in Equation (16), we find:

( )1

for 0 1

1

Xnn

t

R Y nt Zn

p

−

∂ −= ≤ <

∂ − −

. (18)

This causes dividing by zero at t = 0, because 0tR∆ = , but this is solved by using Equation (17) for the first time step:

( )( )

( )

1

1

nX

t

p Y tZ

n

−

∆

− ∆=

− with m

X vpY C X= , 0

01R RX

R−

=−

and max

rRr

= . (19)

For all other time steps, the sum Zt of the stress times viscosity rate is simply updated:

0

t

t t t t tR RZ p dt Z p tt t−∆

∂ ∂= = + ⋅∆ ⋅

∂ ∂∫ . (20)

0.0001

0.001

0.01

0.1

1

10

0.00001 0.0001 0.001 0.01 0.1 1 10

Time [day]

Def

orm

atio

nra

te d

Uy/

dt

n = 0.8

FIG. 3. Undrained triaxial test with declining deformation rate

Geo-Denver 2007 ISBN 978-0-7844-0897-1


The Equations (18) to (20) have been implemented in Plaxis as a User-Defined Soil Model. The declining deformation rate, which has been found by Bishop, Arulanandan and others, can be modeled with this constitutive model. The undrained triaxial test of Figure 3 has been produced with Plaxis and shows an identical declining deformation rate. Only constant slope movements are found in case n = 0, but this value does not fit the findings of Bishop and of Arulanandan. Based on the previous findings the concept arises that the continuous slope movement has little to do with viscous behavior. The safety factors of these natural slopes are around SF= 1.00, so failure takes place. This is slowed down by the dilatant behavior of the soil caused by large blocks, since pore water has to flow around these blocks. The discussed soil model does not contain this dilatant behavior and is therefore not equipped to model the constant moving slopes. The model however works fine for normal viscosity problems and creep behavior (see Fig. 4).

0.01 0.1 1 10 100 1000 10,000-1.5

-1.2

-0.9

-0.6

-0.3

0.0

Time [day]

uy [mm]

0

10 kPa

0.830.20.6

yy

ur

nm

K

σ

ν

∆ =

====

FIG. 4. Oedometer test result of the viscous model

Figure 5. to the left shows the vertical displacement during a multiple step oedometer test. The viscosity results in a logarithmic creep effect after each loading step. This is caused by the horizontal stress relaxation. Figure 5. to the right shows the horizontal total and effective stresses during this test. The vertical load increases from 0 to 10, 20 and 30 kPa, therefore the total horizontal stress starts every load step 10 kPa (the pore pressure) higher than the effective horizontal stress. Over time the effective stress “creeps” towards its K0-state ( 0xx yyKσ σ′ ′= ), but unfortunately only exactly in case n = 0, since the declining viscosity rate over time limits this process a little too fast.

Geo-Denver 2007 ISBN 978-0-7844-0897-1


0.01 0.1 1 10 10025

20

15

10

5

0

Time [day]

σxx & σ’xx [kPa]

σxx

σ’xx

0.01 0.1 1 10 100 1000-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

Time [day]

uy [mm]

10 kPayyσ =

20 kPayyσ =

30 kPayyσ =

10 kPayyσ =

20 kPayyσ =

30 kPayyσ =0

0.830.20.6

ur

nm

Kν

====

0xx yyKσ σ′ ′=

FIG. 5. Oedometer test: Consolidation and creep & Lateral stress development CONCLUSIONS

A viscous constitutive model is derived from the following standard principles: I. The reduction rate of the relative shear stress (the viscosity) is a function of this relative shear stress. II. The reduction rate approaches zero while the stress-state approaches the K0-state. III. The reduction rate decreases with an increasing total amount of viscosity. Looking at the results of the computations, the viscous model seems to model correctly the viscosity and creep behavior, but the constant slope movement can not be modeled without implementation of the dilatancy. The modeling of viscous behavior around plastic failure still needs more attention. ACKNOWLEDGEMENTS

The author wishes to thank the people of the research institute 3S of the University Joseph Fourier in Grenoble and especially Etienne Flavigny for offering their hospitality and scientific support for this research. LITERATURE Arulanandan, K., Shen, C.K. Young, R.B. (1971) “Undrained Creep behaviour of a

coastal organic silty clay.” Géotechnique 21, No 4, 359-375.

Geo-Denver 2007 ISBN 978-0-7844-0897-1


Bishop, A.W. (1961) “Discussions at conference”, 5th Int. Conf. on Soil Mech. and Found. Eng., Dunod, Paris, 1962, Vol III, 102

Bishop, A.W. and Lovenbury, H.T. (1969) “Creep characteristics of two undisturbed clays”, 7th Int. Conf. on Soil Mech. and Found. Eng., Mexico, Fig. 4., 32

Keverling Buisman, A.S. (1936) “Results of long duration settlement tests”, Int. Conf. on Soil Mech. and Found. Eng., Harvard, 103-106

Lemaitre, J. Chaboche, J-L. (1996) Méchanique des matérieaux solides, Dunod Mitchell, J.K. (1993) Fundamentals of Soil Behavior, Wiley, New York, Fig.14.73 Nieuwenhuis, J.D. (1991) Variations in stability and displacements of a shallow

seasonal landslide in varved clays, Dissertation, Balkema, Rotterdam/Brookfield

Van Baars, S. (2003) “Soft Soil Creep modelling of large settlements”, 2nd International Conference on Advances in Soft Soil Engineering and Technology, Kuala Lumpur, Malaysia, 361-371

Van Genuchten, P.M.B. (1989) Movement mechanisms and slide velocity variations of landslides in varved clays in the French Alps, Thesis, Utrecht University, The Netherlands

Vulliet, L. (1997) “Three families of models to predict slowly moving landslides”, 9th International Conference of the Association for Computer Methods and Advances in Geomechanics, Wuhan, China

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Viscous Soft-soil Modeling