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CREEP School 1 – Creep in Soft Soils September 10-11, 2012

Trondheim, Norway

Course Handouts

This material may not be published, reproduced, rewritten, or redistributed without permission. CREEP is funded by the People Programme (Marie Curie Actions) of the

European Union's Seventh Framework Programme FP7/2007-2013/ under REA grant agreement n° PIAP-GA-2011-286397.

 

Monday, 10.09.12

Time Topic Lecturer

10:00 - 10:15 Welcome - CREEP project Benz

10:15 - 11:15 Experimental observations and 1D empirical models for creep in soft soils (1) Den Haan 11:15 - 12:00 Experimental observations and 1D empirical models for creep in soft soils (2) Den Haan 12:00 - 13:15 Break

13:15 - 14:00 Viscoplasticity - Posible creep formulations - Soft Soil Creep in 3D Benz 14:15 - 15:00 Surcharging effects and 1D isotache behaviour Den Haan 15:00 - 15:15 Break

15:00 15:15 Break

15:15 - 16:45 Performance of different creep models in comparison Jostad 16:45 - 18:00 Performance of different creep models in comparison - Exercise Jostad T esda 11 09 12

Tuesday, 11.09.12

Time Topic Lecturer

8:15 - 9:00 Anisotropy and destructuration - Experimental observations Karstunen 9:00 - 9:45 Anisotropy and destructuration - Aspects of modelling Karstunen 9:45 - 10:00 Break

10:00 - 11:00 Some creep models that incorporate anisotropy and destructuration Grimstad 11:00 - 12:00 Exercise: Test fill - Parameter determination from Oedometer test Grimstad 12:00 - 13:15 Break

12:00 13:15 Break

13:15 - 14:45 Exercise: Test fill - Numerical analysis with different models Grimstad 14:45 - 15:00 Break

15:00 - 15:45 Creep in various geomaterials -

C d ll t i tiff

Creep and small strain stiffness -

Discussions Benz

15:45 - 16:00 Closure

8 september 2012

1

Time scales in creep compression

(and a few more additional slides)

Trondheim CREEP course

E.J. den Haan 10-11.09.2012

secondary primary

u log t

compression 

slope c

The well-known eps - log t graph gradually attains a straight tail with slope c. But primary has usually already passed before this. Why isn’t the creep tail straight straight after primary?

8 september 2012

2

secondary primary

u log t

compression 

t_r

= cst + c log (t-t_r) d/dt = c/(t-t_r)

Resistance concept H. Lundgren - N. Janbu R = dt/d= 1/c (t-tr)

slope c

t R

t_r

slope 1/c

intrinsic time isobar for acting '_v intrinsic time = t - t_r

By reducing time t with a constant time shift t_r the whole secondary curve can be straightened. Obviously the soil is obeying another time: its in its own time zone. This is the intrinsic time. Or Bjerrum's equivalent time, but it has turned out to be a

fundamental concept hence 'intrinsic‘, following

Schiffman.

secondary primary

u log t

compression 

t_r

= cst + c log (t-t_r) d/dt = c/(t-t_r)

Resistance concept H. Lundgren - N. Janbu R = dt/d= 1/c (t-tr)

slope c

t R

t_r

slope 1/c

intrinsic time isobar for acting '_v intrinsic time = t - t_r

t_rnow smaller than 0

Negative time shift can occur. Your time is smaller than the soil's intrinsic time.

This occurs when you apply a small load and reset your clock. Nothing much happens in terms of primary compression, and

secondary continues as before. Only now you've reset your time to smaller values. On the log t scale this distorts the shape of the creep curve.

8 september 2012

3

secondary primary

u log t

compression  _t

r= t_transfer

slope c t_transfer

The negative time-shift is well illustrated by resetting the time when a project is

transferred from contractor to client. The secondary slope has already been reached and the soil has found it’s intrinsic eps – log t creep tail. Putting t_client = 0 at t = t_transfer, the curve is pulled back to negative infinity, and then gradually curves back to the intrinsic creep tail. The time difference remains t_r = t_transfer.

Resistance concept applied to stiffness

= cst + b log (-_s) M = d/d= 1/b (-_s)



M

_s

slope 1/b

thesis den Haan 1994

The Scandinavian resistance concept is often applied to 1D stiffness M, and in the nc region an offset linear relation with sigma is often found. This resemble the time shift in the creep resistance, and is the structural bond strength

8 september 2012

4

log 



slope c

_v

creep isobars (_vcst)

log _v



slope b

 

creep isochrones

creep isotaches: d^vp/dt = c/

Creep isobars and Creep isochrones (isotaches) cover the whole eps – sigma – intrinsic time or creep rate of strain space.

log 



slope c

v

creep isobars (_vcst)

log _v



slope b

 

creep isochrones

creep isotaches: d^vp/dt = c/

_v0

Creep is synonomous with visco-plastic. In addition there are elastic strains which are uniquely related to change of effective vertical stress.

Total strains are the sum of vp and el strains. These are used in a Darcy type consolidation calculation. In each time increment, both strain types occur

8 september 2012

5

0 5 10 15 20 25 30

0.0 0.1 0.2 0.3 0.4

b v1

11.7 16.7

v1 = 1.336 exp(9.26 b)

b = 0.0622(v1 - 1.54)^0.605 (remoulded clay - eq. 15, Den Haan 1992)

Sliedrecht

y = 1.04x^1.37

0 5 10 15 20 25 30

0 5 10 15

vo

v1

11.7 16.7 Sliedrecht

_v0 lnv

ln v v₀

(_v = 1 kPa, v₁)

b 1

specific volume – an alternative for strain

Plot deformation as ln v (with v = specific volume), and it turns out that normally

consolidated states plot on a unique line for a given soil. v_1 and b are interrelated, and for remoulded clays and natural organic Dutch clays, the relationship is unique.

This depiction gives extra information which should not be ignored. Many modellers however work only with epsilon

8 september 2012

6

using v allows identification of similar soil at different density

1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3

1 10 100 1000

'V [kN/m²]

ln (v)

20ab 33a 55b 112da

b16d 1 achter b15d 1 sloot b10d 3 achter b22d 4 achter

e0 = 14.2 14.5 19.4 21.3 0

0.2

0.4

0.6 0.8

1

1.2

1 10 100 1000

'V [kN/m²]

natural

20ab 33a 55b 112da

b16d 1 achter b15d 1 sloot b10d 3 achter b22d 4 achter

0 0.1 0.2 0.3 0.4 0.5

0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

t / w b [-]

b = 0.326 (t /_w)^-2.11

0 5 10 15 20 25 30

0.9 1.1 1.3 1.5 1.7 1.9

nat /w

b/c

boring 11.7 boring 16.7

Sliedrecht

correlations, organic clay

Compression and Creep CIE4367

E.J. den Haan 2012

Compression and Creep: History

Terzaghi 1918 – 1925 (Istanbul)

permeability, stiffness, consolidation of clay - k depends on voids ratio

- effective stress principle (but earlier also Fillunger 1915) - flow of water analogous to flow of heat

(Forchheimer)

- k determined from hydro-dynamic phase

1925 Erdbaumechanik

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01

T

U

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.5 1 1.5 2 2.5 3 3.5

T

U

U-logT first term 1/6th power U-sqrt T

wortelbenadering U = {4T/}

T/4=0.886

U = 1 - 2M^x exp(-M²T) M=/2, 3/2, 5/2, ..

solution to Terzaghi's consolidation equation:

6 3

3

5 .

0

 T U T

best approximation (relatively unknown):

U = Z/Z_max = 1 - u_av/

T = t c_v / L²

sqrt approach

Terzaghi’s consolidation:

- 1d - saturated - homogeneous

- incompressible water and grains

- compressibility of grain skeleton a constant - permeability a constant

- small strain Here:

- natural strain

- compressibility of grain skeleton: elasto-viscoplastic - permeability: f(e)

- consolidation equation large strain

v = 1 + e =>^H= -ln v/v₀ v = soortelijk volume

1 e

₀

e 1

h

_o

h

h

Natuurlijke rek

gewone rek ^C=h/ho (Cauchy) Almansi rek = h/h = h/(h-h) Green ...

natuurlijke rek (Hencky, eerder Röntgen):

^H = ^



^h

h_o h dh

= -ln(h/ho) = -ln((ho-h)/ho) = -ln(1-^C)

hoogte

h_o

h=0

^C

^H

"zakking groter dan laagdikte"

 rek

Natural strain (Hencky, earlier Röntgen) linear strain

height

strain

settlement exceeds initial layer thickness!!

NATURAL STRAIN

Almansi Green

v

= specific volume

h dh

Not only no settlement larger than layer thickness, but also better fits to



^H

- ln 

_v

for most soils without brittle, cemented structure, especially for soft soils.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 10 100 1000 10000

^v' kPa

rek

^H

^C

Veen Zegveldpolder

Terzaghi’s consolidation:

- 1d - saturated - homogeneous

- incompressible water and grains

- compressibility of grain skeleton a constant - permeability a constant

- small strain Here:

- natural strain

- compressibility of grain skeleton: elasto-viscoplastic - permeability: f(e)

- consolidation equation large strain

Elasto-viscoplastic behaviour - history:

- “creep”

- Buisman pre 1936: direct and secular

- North America pre 1936: primary and secondary (Gray 1936, Mesri from 1970’s)

- Koppejan 1948: time lines model

- Bjerrum 1967: conceptual isotache model

- Leroueil 1985: first isotache model + Darcy

- Mesri ~1985: EOP principle + secondary + Darcy

- Delft abc model 1994: isotache + Darcy + large strain

- Mesri / Leroueil ‘roadshow’: EOP versus isotache concept

10 kPa 22 kPa

34 kPa

68 kPa

134 kPa

270 kPa

600 kPa

1200 kPa

2400 kPa

0.0

0.2

0.4

0.6

0.8

1.0

1.2

10 100 1000 10000 100000 1000000 10000000

time [sec.]

natural strain [-]

Zegveldpolder peat 25D

نيف

ثخلا торф

tørv Torf turve

τύρφη לובכपीट

tőzeg gambut

泥炭

turba veen

10 kPa 22 kPa

34 kPa

68 kPa

134 kPa

270 kPa

600 kPa

1200 kPa

2400 kPa 0.0

0.2

0.4

0.6

0.8

1.0

1.2

10 100 1000 10000 100000 1000000

time in step [sec.]

natural strain [-]

Zegveldpolder peat 25D

34 kPa

68 kPa

134 kPa 270 kPa

0.2

0.4

0.6

ural strain [ ]

134 kPa primary secondary

0 0

10 100 1000 10000 100000 1000000

time in step [sec.]

t

_p

strain at t

_p

independent of layer thickness.

Mesri:

End of Primary EOP concept

Creep models based on time

) / log(

) / log(

) /

log( ₀ b c t t₀

a _{p v v p}

H 







 







 

 e.g. Maxwell element

H



s H



d



H



^v

^

Elastic (direct) compression

Creep models based on visco-plasticity

log_v

^

_v0 _p

t₀ 10t₀ 100t₀ a

b

c 1 1

Visco-plastic (secular) compression

H s H d

H

 

 _  _  _

H s H d

H

 

  

Buisman & Koppejan: divergence

t

₀

: 1 day (rest of the world)

: Mesri: t

_p

10 kPa 22 kPa

34 kPa

68 kPa

134 kPa

270 kPa

600 kPa

1200 kPa

2400 kPa 0.0

0.2

0.4

0.6

0.8

1.0

1.2

10 100 1000 10000 100000 1000000

time in step [sec.]

natural strain [-]

Zegveldpolder peat 25D

10 kPa 22 kPa

34 kPa

68 kPa

134 kPa

270 kPa

600 kPa

1200 kPa

2400 kPa 0.0

0.2

0.4

0.6

0.8

1.0

1.2

10 100 1000 10000 100000 1000000

time in step [sec.]

natural strain [-]

Zegveldpolder peat 25D

1e-4 1/s 1e-4.5 1/s 1e-5 1/s 1e-5.5 1/s 1e-6 1/s 1e-6.5 1/s 1e-7 1/s

10 kPa 22 kPa

34 kPa

68 kPa

134 kPa

270 kPa

600 kPa

1200 kPa

2400 kPa 0.0

0.2 0.4 0.6 0.8 1.0 1.2

10 100 1000 10000 100000 1000000

time in step [sec.]

natural strain [-]

1 10 100 1000 10000

effective vertical stress [kPa]

Zegveldpolder peat 25D

1e-4 1/s 1e-4.5 1/s 1e-5 1/s 1e-5.5 1/s 1e-6 1/s 1e-6.5 1/s 1e-7 1/s

0.0 0.2 0.4 0.6 0.8 1.0 1.2

10 100 1000 10000 100000 1000000

time in step [sec.]

natural strain [-]

1 10 100 1000 10000

effective vertical stress [kPa]

Zegveldpolder peat 25D

1e-4 1/s 1e-4.5 1/s 1e-5 1/s 1e-5.5 1/s 1e-6 1/s 1e-6.5 1/s 1e-7 1/s

0.0 0.2 0.4 0.6 0.8 1.0 1.2

10 100 1000 10000 100000 1000000

time in step [sec.]

natural strain [-]

1 10 100 1000 10000

effective vertical stress [kPa]

Zegveldpolder peat 25D

1e-4 1/s 1e-4.5 1/s 1e-5 1/s 1e-5.5 1/s 1e-6 1/s 1e-6.5 1/s 1e-7 1/s

0.0 0.2 0.4 0.6 0.8 1.0 1.2

10 100 1000 10000 100000 1000000

time in step [sec.]

natural strain [-]

1 10 100 1000 10000

effective vertical stress [kPa]

Zegveldpolder peat 25D

1e-4 1/s 1e-4.5 1/s 1e-5 1/s 1e-5.5 1/s 1e-6 1/s 1e-6.5 1/s 1e-7 1/s

0.0 0.2 0.4 0.6 0.8 1.0 1.2

10 100 1000 10000 100000 1000000

time in step [sec.]

natural strain [-]

1 10 100 1000 10000

effective vertical stress [kPa]

Zegveldpolder peat 25D

1e-4 1/s 1e-4.5 1/s 1e-5 1/s 1e-5.5 1/s 1e-6 1/s 1e-6.5 1/s 1e-7 1/s

0.0 0.2 0.4 0.6 0.8 1.0 1.2

10 100 1000 10000 100000 1000000

time in step [sec.]

natural strain [-]

1 10 100 1000 10000

effective vertical stress [kPa]

Zegveldpolder peat 25D

1e-4 1/s 1e-4.5 1/s 1e-5 1/s 1e-5.5 1/s 1e-6 1/s 1e-6.5 1/s 1e-7 1/s

1e-4 1/s 1e-4.5 1/s 1e-5 1/s 1e-5.5 1/s 1e-6 1/s 1e-6.5 1/s 1e-7 1/s

0.0 0.2 0.4 0.6 0.8 1.0 1.2

10 100 1000 10000 100000 1000000

time in step [sec.]

natural strain [-]

1 10 100 1000 10000

effective vertical stress [kPa]

Zegveldpolder peat 25D

0.0 0.2 0.4 0.6 0.8 1.0 1.2

10 100 1000 10000 100000 1000000

time in step [sec.]

natural strain [-]

1 10 100 1000 10000

effective vertical stress [kPa]

Zegveldpolder peat 25D

1e-4 1/s 1e-4.5 1/s 1e-5 1/s 1e-5.5 1/s 1e-6 1/s 1e-6.5 1/s 1e-7 1/s

0.0 0.2 0.4 0.6 0.8 1.0 1.2

10 100 1000 10000 100000 1000000

time in step [sec.]

natural strain [-]

1 10 100 1000 10000

effective vertical stress [kPa]

Zegveldpolder peat 25D

1e-4 1/s 1e-4.5 1/s 1e-5 1/s 1e-5.5 1/s 1e-6 1/s 1e-6.5 1/s 1e-7 1/s

creep isotaches

creep v

v

 and  

   _  _

distorted o.c. isotaches

b

ln(10)

vertical spacing c= ln(d/dt)

0.0 0.2 0.4 0.6 0.8 1.0 1.2

10 100 1000 10000 100000 1000000

tijd na begin stap [sec.]

natuurlijke rek [-]

1 10 100 1000 10000

spanning [kPa]

Zegveldpolder veen 25D

24 uur

1e-4 1/s 1e-4.5 1/s 1e-5 1/s 1e-5.5 1/s 1e-6 1/s 1e-6.5 1/s 1e-7 1/s

time in step [sec]

stress [kPa]

natural strain [-]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 10 100 1000 10000

spanning [kPa]

natuurlijke rek [-]

0.01 0.10 1.00 10.00

tijd in de stap [dagen]

Oostvaardersplassenklei (gehomogeniseerd)

1e-4 1/s 1e-4.5 1/s 1e-5 1/s 1e-5.5 1/s 1e-6 1/s 1e-6.5 1/s

1e-7 1/s 1e-7.5 1/s 1e-8 1/s

stress [kPa]

time in step [days]

natural strain [-]

Elastic-Viscoplastic model equations

) / ln(

) /

ln( _{v p s}^H_{,ref s}^H

H b

 

c

 



    _{ }

) / exp(

) /

( ^/

, _{v p} ^{b c H} c

H ref s H

s

   



     

v v H d

vo v H

d

a a













 



  /

) / ln(





H s H d

H

 



_  _  _

Secular strain (visco-plastic component)

Direct strain (elastic component)

Summation

H s H d

H

 



 

Maxwell element

H



s H



d



H



^v

^

ln_v

^

_v0 _p a

b

c 1 1

H

ref s,

 _

Lines of equal rate of creep strain:

isotaches

onderbouwing   ^



 



 





 

 





 

 

 ₀ ⁰

0 c ln ln c ln

 



 ^{ }



 



 



 

 



 

 

 



 

 

 



 

t t

dt c d

dt c d

c

0 0

0 0 0 0

0 0

exp exp exp



 





 













 







ln ’

_v



^H



₀^H

 

0 0

0 0 0

0

0 1

exp 











   









 

 



  t t

t c t

c



 

 t c t  _r 

 



 







 

 

 



 



r r

t t

t c t

c

0

0 ln 0 ln



 





₀

 t

₀

t

 

0

-t

_r

1/



_

 

 



 

 

 



 

 

 







 

 





 

 

 ₀

0

0 ln 0 c ln c ln

t t

t c t

r r

t r

t 

 

Intrinsic time

Time registration

time shift

time shift makes good the difference between the intrinsic time and

the value of t resulting from the chosen time zero.

ln(t), ln()



^H

c

t’

₀



₀

t’’

₀

• 

^H

- ln is linear, not 

^H

– ln(t)

• 

₀

< t

₀

(t

_r

> 0) for large 

_v

• 

₀

> t

₀

(t

_r

< 0) for small 

_v

shape of secondary phase curve depends on sign of t

_r

ln ’

_v

₀

,t

₀

log t log t



₁

For each arbitrary choice of time zero in a settlement calculation, t_rgives the difference between t and : t_r = t -

t

i

 c log

 



t c /

 _ 

) log( _r

i c t t

Integrate









creep tail:

( - t

_r

)





 

_i

 c log _

The intrinsic creep time

is

 

 t c t 

_r





ln(10) c

datum

dz

z 

 d

initial state deformed state Fig. 3. Coordinates of an element of soil in the initial and deformed states.

z v d

=v d

o



v v -

=

q 

v -

= u ) -

z( _o

w s s



 





u ) - p - ( z z =

ue  s











 











 



 









z p v kv z v v + 1 v k z v v - -

=

q ^{o o}

w o

w w s













 H ^Hs H= d +



 



















 



 







 c

z - p v k v z v v + 1 v k z v v - - a

= p t d

p

d _{o o}

w o

w w s

























 













































w e o o

w e

u z v kv z v - v

=

k u -

= q



 



Finite strain consolidation + isotache compression

v -v

H= 



) (p=



_v

mass conservation

finite strain consolidation equation for elasto-viscoplastic solid

Note: MSettle now superseded by D-Settlement

time in seconds load in kPa

2

settlement in mm 4

20 40 80 60

model effects, demonstrated by Consef Consef: consolidation and secular effect. Solves

finite strain consolidation equation including a visco-plastic term

for single, double draining layer, and single load

 

 



  



 







 



 

 

 









 

 

















 c

z v k v z v v v

k z v v a

dt

d

_{o o v}

w o

w w s

v

v

1

elastic Darcy, self weight Darcy, proper vp

=10⁴

d

_ref

=1d

=10⁴

d

slope a

sample

K_o-ring with insulation

load cell

_{v top}

piston

_h

backpressured triaxial cell

load cell

_{v bottom}

soil sample platen and drainage filter

Ko ring

excess pore pressure

K _o - C.R.S. oedometer

strain gauges

strain gauges

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 10 100 1000



v

' kPa



H

[-]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

K

o

Ko - C.R.S. oedometer test 710402 46A 152/.011-066



 ^

Sliedrecht Gorcum Licht clay _wet = 1.29 t/m³

0.0

0.1

0.2

0.3

0.4

0.5

0.6

1 10 100 1000



_v

' kPa

 ]

1.E-11 1.E-10 1.E-09 1.E-08 1.E-07 1.E-06

k [m/s]

710402 46A 152/.011-066

log

_v

'

logk

 = -C

_k

log (k/k

_o

)

k

_o

slope C_k

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 10 100 1000



v

' kPa



H

[-]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

K

o

Ko - C.R.S. oedometer test 710402 46A 152/.011-066

K _o



 ^

slope a

slope b

Sliedrecht Gorcum Licht clay _wet = 1.29 t/m³

K

_o,nc

72 77 82 87 92 97 102

0 2 4 6 8 10 12 14

relaxation time in hours

v' in kPa

_vR'

creep parameter c from relaxation phase

Ko - C.R.S. oedometer test 710402 46A Sliedrecht Gorcum Licht clay _wet= 1.29 t/m³ relaxation

b c

vR vR vR

v t

c

b ^{ /}

 

 







 

 

 

 

 1 ^

The Hypothesis A vs.

Hypothesis B controversy EOP vs. Isotache models?

Imai 1989

Exercise:

Use Consef to show that the isotache model

can yield both Hypothesis A and Hypothesis B

type behaviour

CREEP – School1 Trondheim, September 10^th2012

Viscoplasticity - Posible creep formulations - Soft Soil Creep in 3D

Soft Soil Creep in 3D

Thomas Benz

With thanks to Valentina Berengo, Martino Leoni & Pieter A. Vermeer With thanks to Valentina Berengo, Martino Leoni & Pieter A. Vermeer

Table of contents

Part 1 Possible approaches to creep modeling Part 2 Possible creep formulations

Part 2 Possible creep formulations

Part 3 Soft Soil Creep 3D as an example

Part 1

Possible approaches to creep modeling

Part 1: Possible approaches to creep modeling

RHEOLOGICAL MODELS

Uniaxial condition aiming to Uniaxial condition, aiming to

conceptual understanding

1D EMPIRICAL MODELS

Data-fitting of experimental data Data fitting of experimental data Specific boundary conditions

GENERAL STRESS-STRAIN-TIME MODELS

Mostly 3D models in incremental form

Liingaard et al. (2004) Characterization of Models for Time-Dependent Behavior of Soils

Basic components (solid mechanics)

Part 1: Possible approaches to creep modeling – Rheological models

Basic components (solid mechanics)

Constitutive models can be built by combining these basic elements, e.g.:

(Maxwell model)

( )

Part 1: Possible approaches to creep modeling

RHEOLOGICAL MODELS

Uniaxial condition aiming to Uniaxial condition, aiming to

conceptual understanding

1D EMPIRICAL MODELS

Data-fitting of experimental data Data fitting of experimental data Specific boundary conditions

GENERAL STRESS-STRAIN-TIME MODELS

Mostly 3D models in incremental form

Liingaard et al. (2004) Characterization of Models for Time-Dependent Behavior of Soils

Part 1: Possible approaches to creep modeling – Empirical models

Well known creep rate concepts:

Norton (1929):

1

_o α

ε (σ σ )

  



Prandtl (1928):



o

ε 1 sinh (ασ ασ )

   

 ^{for }



Soderberg (1936):

1

_o

ε ( exp (ασ ασ ) 1)

    



^{for }

 = reference time, may be temperature dependnet

Bj (1967)

Part 1: Possible approaches to creep modeling – Empirical models

C

_α= SECONDARY COMPRESSION INDEX eo c α

e = e - C lo g t t Bjerrum (1967)

e

atio e

e o c

t

1 e

_eoc

Void ra

C t

_eoc

log t

C

_α

log t

e

_eoceoc = void ratio at end of consolidation

t

_eoc = time at end of consolidation

eoc α

τ + t e e - C log

τ

  τ

Garlanger (1972):

t’= t

-

t_eoc

= extra parameter with:

Part 1: Possible approaches to creep modeling – Empirical models

Bj (1967)

e σ

Bjerrum (1967)

NC-Line

p o

OCR σ

 σ



C

_s

1 σ

p

C

_S = SWELLING INDEX



o

σ

AN OVERCONSOLIDATED STATE CAN BE REACHED BOTH BY CREEP

log 

AN OVERCONSOLIDATED STATE CAN BE REACHED BOTH BY CREEP AND UNLOADING

Šuklje (1957): Isotache model

Part 1: Possible approaches to creep modeling – Empirical models Šuklje (1957): Isotache model

Unique relationship between e, log and e

e

e   a1

 2

e  a

e   a3

log 

Further example: Den Haan (1994)

Part 1: Possible approaches to creep modeling – Empirical models

Incomplete list of 1D empirical models Incomplete list of 1D empirical models

Sing and Mitchell (1968) – 3 parameter, constant stress, primary loading, ...

Lacerda and Houston relaxation model - Relaxation, undrained, ...

Prevost relaxation model (1976) Triaxial undrained Prevost relaxation model (1976) – Triaxial, undrained Strain rate approach:

Sukjle (1957) Leroueil et al. (1985)

Viad and Campanella (1977) Vermeer, Stolle & Bonnier (1998) Vermeer & Neher, (1999)

Vermeer, Leoni, Karstunen & Neher (2006)

Part 1: Possible approaches to creep modeling – General models

RHEOLOGICAL MODELS

Uniaxial condition aiming to Uniaxial condition, aiming to

conceptual understanding

1D EMPIRICAL MODELS

Data-fitting of experimental data Data fitting of experimental data Specific boundary conditions

GENERAL STRESS-STRAIN-TIME MODELS

Mostly 3D models in incremental form

Liingaard et al. (2004) Characterization of Models for Time-Dependent Behavior of Soils

1 O (M l 19 1 P 1963) Part 1: Possible approaches to creep modeling – General models

1. Overstress concept (Malvern, 1951; Perzyna, 1963)

“Overstress models”

2. Nonstationary Flow Surface theory (NSFS) (Sekiguchi & Ohta,1977);

Nova, 1982) 3. Others

Note:

General 3D creep models typically belong to the class of viscoplastic General 3D creep models typically belong to the class of viscoplastic models. Differences and similarities to elastoplastic models see next section. The word "viscosity" is derived from the Latin "viscum",

i i tl t A i l ll d bi dli d f

meaning mistletoe. A viscous glue called birdlime was made from mistletoe berries and was used for lime-twigs to catch birds

[http://www.etymonline.com].

Part 2

Elastoplasticity - Viscoplasticity

Part 2: Elastoplasticty – Viscoplasticity

Elastoplasticity

Part 2: Elastoplasticty – Viscoplasticity

Viscoplasticity – Possibility 1: Overstress concept

• No viscous strain inside the yield surface

N i t diti

• No consistency condition

• Non-associated flow rule possible

Creep strain rate is function of overstress

• Creep strain rate is function of overstress

Part 2: Elastoplasticty – Viscoplasticity

Viscoplasticity – Possibility 2: NSFS

Define yield function as function of:

Define yield function as function of:

• Stress

• Internal variables including viscoplastic strains

• Internal variables, including viscoplastic strains

• Time  = (t)

d th f i t diti th t

and then enforce consistency condition so that:

• No viscous strain inside the yield surface

C i t diti f d

• Consistency condition enforced

• Non-associated flow rule possible

Creep strain rate is function of overstress

• Creep strain rate is function of overstress

Part 2: Elastoplasticty – Viscoplasticity

Viscoplasticity – Comparison

Part 2: Elastoplasticty – Viscoplasticity

Viscoplasticity – Comparison

Part 2: Elastoplasticty – Viscoplasticity

Viscoplasticity – Other possibilities exist, too.

Example: Soft Soil Creep 3D

 

e c eq

p

p p e e e

p p





 



 

           

   

2

eq 2

p p q

    M p



2

eq 2

p p q

    M p



q

NCL NCL σi ^NCLNCL

p´

p

eq

p

_p

Part 3

Soft Soil Creep 3D

1 + e

Part 3: Soft Soil Creep 3D

e

1

1 + e

1

e

C

_c





1

C

_s 

C

1 1

C

_α

log V′ ln p′



C / ln10

difi d i i d

log t



= C

_c

/ ln10 =

modified compression index

 

2C

_s

/ ln10 =

modified swelling index



= C

_α

/ ln10 =

modified creep index

CREEP MODEL FOR ISOTROPIC LOADING ( = = )

Part 3: Soft Soil Creep 3D

e c

p 1 p

 



 



 

 

   

CREEP MODEL FOR ISOTROPIC LOADING ( ₁= ₂= ₃ )

 

 1   

e c

p

p 1 p

e e e

p τ p

 

           

  



C / l 10

f



1 2 3



p

3   

      



= C

_c

/ ln10 =

modified compression index

 

2C

_s

/ ln10 =

modified swelling index



= C

_α

/ ln10 =

modified creep index

e

c

 T 

   



The preconsolidation pressure p_p is continuously updated by using:

p p

p p T

- a

      

 

CREEP MODEL FOR ISOTROPIC LOADING (Model derivation)

Part 3: Soft Soil Creep 3D

CREEP MODEL FOR ISOTROPIC LOADING (Model derivation)

' '

' p pc eop t

p





Assume:

, , ,

0 0

*ln ( * *) ln *ln

' '

pc eop

e p creep c

v v eop v eop v

pc c

p t

p

p p

        



       

Hardening rule:

 

0

' ' exp and thus * * '

* * '

creep

creep pc

pc pc v v

pc

p p d dp

p

   

 

 

     

’ ( )

A creep model based on Bjerrum’s idea (e.g. above hardening rule):

' '

*ln ( * *) ln

' '

e creep pc

v v v

p p

p p























0 pc0

p p

Equate the first and third equation above:

' ' '

' p t ' p

p ,



 p

0 0 0 0

*ln ( * *) ln *ln *ln ( * *) ln

' ' ' '

pc eop c pc

v pc c pc

p t p

p p

p p p p

        



       

' p'pc



t

,

*ln ( * *) ln

'

c pc

c pc eop

t p

p

   



  

(1)

CREEP MODEL FOR ISOTROPIC LOADING (Model derivation)

Part 3: Soft Soil Creep 3D

CREEP MODEL FOR ISOTROPIC LOADING (Model derivation) Assume and then: p'_pc  p' t' 



t_EOP

'

t p







Furthermore if then:

,

*ln ( * *) ln

'

c EOP

c pc eop

t p

p

 

  



   

c tEOP



 



( * *)

' *

' p p

 







  

  

 

( * *) , *

' '

pc eop

c eop

p OCR

p

 

 

  

  

    

 

thus

(2)

, c p pc eop



^_ ^_{ p}



^p



 

Finally, develop model formulation from:

' *

* ' '

e creep

v v v

c

dp

p t

    

   



   

( * *)

' *

' *

d

 

 

(1)

_*

, ,

' * '

* ' '

pc eop

e creep

v v eop v

c pc

dp p

d d d

p p





   



 

     

( * ^ *) ( *  *)

 

(2) (1)

( )

* *

' * ' ' * 1

* *

' ' '

v pc

dp p dp

d p p p OCR

 

 

  

 

   

       

(2)

ELLIPSES OF MODIFIED CAM CLAY ARE CONTOURS FOR Part 3: Soft Soil Creep 3D

q

ELLIPSES OF MODIFIED CAM CLAY ARE CONTOURS FOR CONSTANT RATE OF VOLUMETRIC CREEP STRAIN

q

e = ac

e c^<<a

p´

NCL σi

p

p

p

q

2

p

eq

eq equivalent 2

p p p q

      M p



_{NCL :}

p = p 

^{eq p}