Creep in Soft SoilsW(H)YDOC 05

Paris, 23-25 November, 2005

Pieter A. Vermeer & Martino Leoni

University of Stuttgart, Germany

Outline:

- Introduction & Motivation

- 1D creep in oedometer tests

- 3D isotropic creep model

- 3D anisotropic creep model (for specialists)

(end of presentation)

Abstract:The well-known logarithmic creep law for secondary compression is transformed into a differential form in order to include transient loading conditions. This 1-D creep model for oedometer-type strain conditions is then extended towards general 3-D states of stress and strain by incorporating concepts of Modified Cam-Clay and viscoplasticity.

In this way a robust constitutive model is formulated that captures the stress-strain-time behaviour of isotropically consolidated clay samples. Considering lab test data for isotropically consolidated clay samples, it is shown that phenomena such as undrained creep, overconsolidation and aging are well captured by the model.

Both natural and reconstituted clays possess an anisotropic fabric which is coaxial to the direction of sedimentation. This is very pronounced in natural clays, as inter-particle bonds tend to develop during the diagenesis of clay layers. In order to model this anisotropy, one has to depart from the isotropic ellipses of the Modified Cam Clay model, as yield surfaces or contour lines for a constant rate of creep. Using rotated ovals in p-q plane, as justified by data from creep tests, an anisotropic creep model is formulated.

Displacement-controlled load tests on floating micropiles

Load (kN)20 40 60 80 100

= 10-4 a= a

= 10 a

CSV micropilesLength: 8mDiameter: 17 cm

1

Set

tlem

ent

(cm

)2

3

4

s&s&

s&

Load

rate of penetration

Considering such a rate-type effect, it is important to do research on creep and stress relaxation.

Constant rate of deformation oedometer tests after Marques (1996)

ε(%

)σ´ (kPa)

0

∆eε1 e−

=+

1.7 x 10-5 s-1

2.0 x 10-6 s-1

1.7 x 10-7 s-1

Marques (1996): Influencia da velocidade de deformacao e da temperatura no adensamento de argilasnaturais (in Portuguese), M. Sc. Thesis, University of Laval, Canada

Sällfors (1975): Preconsolidation pressure of soft high plastic clays, Ph. D. Thesis, Chalmers University of Technology, Goteborg, Sweden

Load-controlled test: footing (3.0 x 3.0 m2) on dense sand

Load (MN)S

ettle

men

t(m

m)

24 hours

Briaud and Gibbens (1994): Test and Prediction Results for Five Large Spread Footings on Sand, Proc. Spread Footing Prediction Symposium (Fed. High. Adm.) Eds J.L. Briaud, M.Gibbens, pp.92-128

Load-controlled tests: cast-in-place concrete micropiles in dense sand

Load

Load

Length: 6.2 to 6.3 m

Diameter: 18 cm

Creep is even observed for piles in dense sand.

Part 1

Creep in oedometer tests

Important publications:

Buisman (1936): Results of long duration settlement tests, Proc. 1st Int. Conf. Soil Mech. and Found. Eng., Vol. 1, pp. 103-107.

Bjerrum (1967): 7th Rankine Lecture, Géotechnique, Vol. 17, pp. 81-118.

Garlanger (1972): The consolidation of soils exhibiting creep under constant effective stress, Géotechnique 22 (1), pp. 71-78.

Mesri & Godlewski (1977): Time and stress compressibility interrelationship, J. Geot. Eng. Div., ASCE 103, GT5, pp.417-430.

Den Haan (1994): Stress-independent parameter for primary and secondary compression, Proc. 13th Int. Conf. Soil Mech. and Found. Eng., New Delhi, Vol 1, pp 65-70.

1D-Consolidation and 1D-Creep in single loading step

Cα = secondary compression index

teoclog t

1eeocV

oid

ratio

eσ

t

eeoc = void ratio at end of consolidation

teoc = time at end of consolidation

Early literature on secondary compression

Buisman (1936) used ε instead of e:

eocBeoc B eoc

eoc eoc

t + ttε = ε + C log = ε + logCt t

′

eocεteoc = end of consolidation time

= end of consolidation strain

t‘ = t - teoc creep time

Bjerrum (1967) and Garlanger (1972):

αeocτ + te e - logC τ

′= B0α C)e +(1 C ⋅= = creep indexwith

τ = extra parameter

Increase of OCR due to creep after Bjerrum (1967)

State of overconsolidation can be reached both by creep and unloading

e

NC-Line

Cs1

3 years

30300

3000

creep

oσ ′pσ

σ′log

p

o

σOCR

σ=

′

CS = swelling index

OCR-contours are lines for constant rate of creep (isotaches)

e

NCL

OCR = 1

OCR = 1.3

e a

e dedt

≡&

= −&

3e small, e.g. - 10a −= ⋅&

-6e very small, e.g. - 10a= ⋅&

log σ‘

OCR = 1.7

Den Haan (1994)

Oedometer creep model versus classical creep concepts

for all values of σc 1e constant OCR

β⎛ ⎞= ⎜ ⎟⎝ ⎠

& .Oedometer creep model:

Classical concepts:

Norton (1929):

Prandtl (1928):

Soderberg (1936):

co

1ε (σ σ )β

τ= −&

c

o1ε sin (βσ βσ )τ

= ⋅ −&

c

o1ε ( exp (β β ) 1)σ στ

= ⋅ − −&

for σ > σο

for σ > σο

τ = temperature dependent

Also read:

Odqvist, F.K.J: Mathematical Theory of Creep and Creep Rupture, Clarendon Press, Oxford, 1966Perzyna P.: Fundamental Problems in Viscoplasticity, Advan. Appl. Mech., 9, 243-377, 1966

Oedometer creep model

c s

α

C CC

e c s α

p

C Cσ 1 σe e eln10 σ ln10 τ σ

−

⎛ ⎞′ ′− −= + = + ⎜ ⎟⎜ ⎟′ ⎝ ⎠

&& & &

σ σ

Typical soil data: 10/CC cs ≈ and 27C

CC30/CC scc ≈

−⇒≈

αα

27

27

p OCR1

=⎟⎟⎠

⎞⎜⎜⎝

⎛

σσ′

It follows that the creep rate is proportional to

It follows that the creep rate is negligibly small for OCR well beyond unity.

Cc = compression index

τ = usually 1 day, but temperature dependent

Tremendous influence of overconsolidation ratio on creep rate

Example : Cc = 0.15 Cs = 0.015 Cα = 0.005

c s

α

C CC

c α27

p

C σ´ - eτ ln10 σ OCR

a−

⎛ ⎞−= =⎜ ⎟⎜ ⎟

⎝ ⎠&

e

ecOCR ec

1.0 - a

1.3

1.7

3- 10a −⋅

6- 10a −⋅

NCL

σ′log

Creep model for isotropic loading ( σ1 = σ2 = σ3 )

e c

p

p 1 pe e ep τ p

λ κµ

κ µ

−

⎛ ⎞′ ′= + = − − ⎜ ⎟⎜ ⎟′ ⎝ ⎠

&& & & ( )1 2 3

1p 3

σ σ σ′ ′ ′ ′= + +

λ = Cc / ln10 = modified compression index

κ ≈ 2Cs / ln10 = modified swelling index

µ = Cα / ln10 = modified creep index

The preconsolidation pressure pp must be continuously updated by using

p cp

- pp e

- λ κ=& &

po c cp o

pp exp (e -e )

-λ κ= ⋅pp = isotropic preconsolidation pressure

Part 2

3D isotropic creep modelor

Soft Soil Creep Model

Publications on this model:

Vermeer P.A., Neher H.P. (1999): A Soft Soil Model that Accounts for Creep. Proc. Int. Symp. “Beyond2000 in Computational Geotechnics”, Amsterdam, pp. 249-261, Balkema, Rotterdam.

Stolle D.F.E., Vermeer P.A., Bonnier P.G. (1999): Time Integration of a Constitutive Law for Soft Clays. Communications in Numerical Methods in Engineering, No.15, pp. 603-609.

Neher H.P., Wehnert M., Bonnier, P.G. (2001): An Evaluation of Soft Soil Models Based on Trial Embankments. Proc. 10° Int. Conf. on Computer Methods and Advances in Geomechanics (Eds Desaiet al.), Vol.1, pp. 373-378, Balkema, Rotterdam.

Neher H.P., Vogler U., Vermeer P.A., Viggiani C. (2003): 3D Creep Analysis of the Leaning Tower of Pisa. Proc. Int. Workshop on Geotechnics of Soft Soils (Eds Vermeer et al.), pp. 607-612. Noordwijkerhout, The Netherlands

Ellipses of Modified Cam Clay are contours for constant rate of volumetric strain

q

p´

ce = a&

pp

ce a& <<

2

eq equivalent 2

qp p pM p

′ ′ ′= = +′

NCL

eq pNCL: p p′ =

MCC model has yield function eq p p pf ′= −M = slope of critical state line2 2 2

1 2 2 3 3 11q (σ σ ) (σ σ ) (σ σ )2

= − + − + −

3D creep model with isotropy

eqe c

p

ppe e e p p

λ κµµκ

τ

−

⎛ ⎞′′= + = − − ⎜ ⎟⎜ ⎟

⎝ ⎠

&& & p c

p

-pp e

- λ κ=& &with

2

eq 2

qp pM p

′ ′= +′

For 3D we use the equivalent mean stress:

q

eqp′pp

iσ′ NCL

p´

Modified Cam Clay model versus Soft Soil Creep model

Both models have : e = ee + ec. ..

σ σ p cp

pp e

λ κ−

=−

& &Both models have hardening:

eqci

pε Λ

iσ′∂

=∂

&Both models have flow rule:

eqc

p

pe

p ii

λ κ σσ′∂−

= −∂

& & (summationconvention)MCC: increase of density by primary loading:

NCL is yield locus

eqc

p

pe

p

λ κµµ

τ

−

⎛ ⎞′= − ⎜ ⎟⎜ ⎟

⎝ ⎠&SSC: increase of density by creep:

NCL is creep rate locus

2eq eq eq

1 2 3 2 21 2 3

p p p qε ε ε ε 1 dσ σ σ M pvolumetric

′ ′ ′∂ ∂ ∂ ⎞⎛ ⎛ ⎞= + + = Λ ⋅ + + = Λ ⋅ − = Λ ⋅⎟⎜ ⎜ ⎟′∂ ∂ ∂ ⎝ ⎠⎝ ⎠

& & & &

The equations of the Soft Soil Creep model

eq

p

p1 µΛd τ p

λ κµ−

∗ ⎛ ⎞′= ⎜ ⎟⎜ ⎟

⎝ ⎠

2q pd 1 M

′⎛ ⎞= − ⎜ ⎟⎝ ⎠i

eqj

eij

ci

eii σ

pΛσCεεε

∂

′∂⋅+′⋅=+= &&&&

1 ν νe

ν 1 νijν ν 1

1C E

⎡ ⎤− −⎢ ⎥

− −⎢ ⎥⎢ ⎥− −⎣ ⎦

= pE 3(1 2 )κ

ν ∗

′= −elasticity matrix

q

p´eqp′

pp

iσ′

ciε&

2

eq 2

qp pM p

′ ′= +′

0

c c0

p pe ep p expλ κ

⎛ ⎞−= ⎜ ⎟−⎝ ⎠

NCL

0µ µ/(1 e )∗ = + 0κ κ/(1 e )∗ = + ν Poisson ratio=Model parameters:

Initial conditions:

, , , , Mλ κ µ ν

1 2 3 p o, , , p , eo o o oσ σ σ′ ′ ′

(τ = 1 day for temperature of 10°C)

Performance of SSC model for drained triaxial tests on NC-soil

q

p‘

0pp slow-ppfast-pp •

•

•

•NCL-slow

NCL-fast

31 σσ −

fast test

slow test

• •

31 εε −

CS

CS

6 sinM3 sin

ϕϕ

=−

Performance of SSC model for undrained triaxial tests on NC-soil

p´

fast shearing

slow

fast

slow

Cu - slow

1ε

q/2

Cu - fast•

• ••

q/2

Influence of strain rate on the undrained shear strength in triaxial compression

Kulhawy & Mayne (1990): Manual on Estimating Soil Properties for Foundation Design

Part 3

3D anisotropic creep model

Important publications:

Gens & Nova (1993): Conceptual bases for a constitutive model for bonded soils and weak rocks. Proc. Int. Symp. on Geotech. Eng. of Hard Soils and Soft Rocks (Eds Anagnostopoulos et al.), Vol.1, pp. 485-494, Balkema, Rotterdam

Wheeler (1997): A rotational hardening elasto-plastic model for clays. Proc.14th Int. Conf. on SoilMech. and Found. Eng. (in Hamburg), Vol.1, pp.431-434, Balkema, Rotterdam

Wheeler, Näätänen, Karstunen, Lojander (2003): An isotropic elastoplastic model for soft clays. Canadian Geotechnical Journal, Vol.40, pp. 403-418

Tests on anisotropically consolidated clays

Contour lines for constant rate of volumetric creep strain rate after Boudali (1995)

Boudali (1995): Comportément tridimensionnel et visqueux des argiles naturelles, Ph.D. Thesis, University of Laval, Canada

Equivalent mean stress for anistropic creep model

q

p‘

1α

1M

eqp ′

eqc

p

pe

p

λ κµµ

τ

−

⎛ ⎞′= − ⎜ ⎟⎜ ⎟

⎝ ⎠&

( )2

eq 2 2

q αp 1p p M α p

′−′ ′= +

′−

α = 0 gives isotropic model

αο relates to Ko

p pv d

3q qα m α ε β α ε4p' 3p'

⎡ ⎤⎛ ⎞ ⎛ ⎞= − + −⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎣ ⎦& & &Wheeler et al. (1993):

The equations of the anisotropic creep model

eq

p

p1 µΛd τ p

λ κµ−

∗ ⎛ ⎞′= ⎜ ⎟⎜ ⎟

⎝ ⎠

eqe c ei i i ij j

i

pε ε ε C σ Λ

σ′∂

′= + = ⋅ + ⋅∂

& & & &2 2 2

2 2 2

1 q α pd 1 p M α

′−= −

′ −

1 ν νe

ν 1 νijν ν 1

1C E

⎡ ⎤− −⎢ ⎥

− −⎢ ⎥⎢ ⎥− −⎣ ⎦

= pE 3(1 2 )κ

ν ∗

′= −elasticity matrix

q

p‘eqp ′ pp

• •

( )2

eq 2 2

q αp 1p p M α p

′−′ ′= +

′−NCL

p cp

-pp e

- λ κ=& &

Model parameters:

Initial conditions:

, , , , Mλ κ µ ν

1 2 3 p o, , , , p , eo o o o oσ σ σ α′ ′ ′

Practical difference between isotropic and anisotropic model

1σ ′

3 2σ ′ ⋅

0K

stress path in 1D creep

isotropic modelanisotropic

Isotropic creep model predicts a relatively low creep rate in oedometer tests

Anisotropic creep model predicts a relatively high creep rate in oedometer tests

On calibrating the isotropic model by oedometer data one overestimates the creep in all otherstress paths. This is exactly our experience with the isotropic creep model. On using the anisotropic creep model we expect realistic predictions.

Concluding remarks

1. The isotropic creep model is a straightforward generalizationof MCC to include creep

2. Creep tends to be overpredicted by the isotropic creep model

3. Anisotropic creep model would seem to be more realistic

4. Laboratory creep tends to be larger than field creep. Therefore lab data may need corrections (Leroueil, 1997)

We are working on numerical implementation, validation and verification of the anisotropic model.

S. Leroueil (1997): Critical State Soil Mechanics and the Behaviour of Real Soils, Proc. Symp. On Recent Developments in Soil and Pavement Mechanics, Rio de Janeiro, Brazil

Undrained triaxial tests after Vaid et al. (1977) and simulation with SSC model

q

fast shearingslow

CS-Line

q

p´

undrained

• •