Creep of frozen soils
Prof. Jilin Qi
State Key Lab of Frozen Soil Engineering Cold & Arid Regions Env. & Eng. Res. Institute Chinese Academy of Sciences Merger
No.286397
Vienna: For the extension of the underground line U2 from the first to the second district of Vienna, the underground station Schottenring is constructed with low overburden right
beneath the Danube channel. Because of the low overburden, artificial ground freezing was employed as temporary support and sealing device. (http://www.imws.tuwien.ac.at)
Copenhagen: A pedestrian passage from a new metro station to an existing railway station was constructed underground. Since the existing rail traffic had to continue, the ground was frozen to avoid the
risk of collapse due to excavation of the transfer tunnel.
Artificial Ground Freezing Widely used in Europe
Berlin: unter den Linden Metro Station (2006)
The Netherlands: Artificial Ground Freezing at Sophiaspoortunnel
Soft clay: supporting during excavation; diaphragm wall
Permafrost distribution
Geographical Conditions
Altitude Latitude
Climate
Temperature Precipitation Wind speed etc.
Environment
Vegetation Geography Water supply etc.
Delft Svalbard
Permafrost table and Active layer
From Andersland and Ladanyi (1999)
Active layer
Permafrost table
Colder regions Warmer regions
Active layer: depth ranging from tens cm to several metres
AGENDA
Factors influencing creep of frozen soils
Creep of Warm Frozen Soils: Field Observations State of The Art: Creep Models for Frozen Soils
Our attempts on constitutive modeling
Concluding Remarks
AGENDA
Factors influencing creep of frozen soils
Creep of Warm Frozen Soils
State of The Art: Creep Models for Frozen Soils Our attempts on constitutive modeling
Concluding Remarks
Frozen soil displays features very similar to that of unfrozen soil
Strain vs. time
Strain rate vs. time dm
tf
TYPICAL creep curves for frozen soil (Ting 1983)
C REEP CURVES FOR FROZEN SOILS
From Wu and Ma.
1994
Uniaxial compression Frozen Lanzhou sand
S TRESS DEPENDENCE
0 300 600 900
Time/min
0 2 4 6 8 10 12 14 16 18 20
Temperature -1
1.5 MPa 2.5 MPa 4.5 MPa
Triaxial compression Frozen ISO Standard sand
3=0.5 MPa
-5 oC
Both very much stress dependent.
S TRAIN RATE DEPENDENCE
From Zhu and Carbee (1984)
-2 oC
Under the same temperature, stress increases with strain rate.
0 4 8 12 16
0 5 10 15 20 25 30
-2 oC
-5 oC
-10oC
Time / h
Creepstrain/%
FromWu and Ma.
1994
dm vs.
stress
tf vs.
stress
T HERMAL DEPENDENCE
5 MPa
Temperature plays a role similar to load
Minimal strain rate and time to failure all dependent on temperature
dm/mm.h-1
Water content / %
FromWu and Ma. 1994
1.-0.5oC, 1.4 MPa; 2. -1.0oC, 2.2 MPa;3.-2.0oC, 3.4 MPa
D EPENDENT ON TOTAL WATER CONTENT *
Generally, different water content range, different minimal strain rate development for frozen silty clay (samples from Lanzhou, China)
* Total water content: ice is counted as water together with unfrozen water
3-4
Solute (Nixon and Lem 1984):
S OLUTE DEPENDENCE
The higher solute content, the larger axial strain under a certain stress level.
T EMPERATURE & SOLUTE CHANGE UNFROZEN WATER CONTENT
Unfrozen water content plays important roles in mechanical properties of frozen soils
Tm
Warm
Frozen Soils
AGENDA
Factors influencing creep of frozen soils Creep of Warm Frozen Soils
State of The Art: Creep Models for Frozen Soils Our attempts on constitutive modeling
Concluding Remarks
Shields et al. (1985): -2.5 and -3.0 C.
Foster et al. (1991): -1 C
Tsytovich(1975): different temperature boundaries for different soils
D EFINE WARM FROZEN SOIL
Constant load and stepped temperature
Material: silty clay (CL) Qinghai-Tibet Plateau Different water content: 40%, 80%, 120%
Constant Loads: 0.1, 0.2 and 0.3 MPa, respectively Temperature steps: -1.5, -1.0, -0.6, -0.5 and -0.3 C (Qi and Zhang, 2008)
Temperature boundary:
Detecting unfrozen water content is rather difficult (NMR) Tend to use mechanical properties
2E+005 4E+005 6E+005 Time (S) 0
2 4 6 8
Strain(%)
40% - 0.1 MPa Strain vs. Time Temperature vs. Time
-2 -1.5 -1 -0.5
Temperature(C)
-2 -1.6 -1.2 -0.8 -0.4
Temperature ( C) 0
0.001 0.002
Strainrate(h-1 )
40%- 0.1 40%- 0.2 40%- 0.3 80%- 0.1 80%- 0.2 80%- 0.3 120%- 0.1 120%- 0.2 120%- 0.3
0
-1.0 -0.6 C
Strain rate vs. Temperature
like we applied higher loads in oedometer test
Compressive Coefficient vs. Temperature
Own test
Turning point around: -1 C
0 -2 -4 -6 -8
Temperature ( C) 0
1 2 3
Compressivecoefficient(10-2kPa-1)
Testing results by Zhu et al. (1982)
0 -0.4 -0.8 -1.2 -1.6 -2
Temperature ( C) 0
1 2 3 4
Compressivecoefficient(x10-2kPa-1)
w=40%
w=80%
w=120%
-1 C is defined as the temperature boundary for warm frozen soil for the silty clay we frequently encounter on the plateau.
Active Layer
Thaw Settlement Frozen Soil
Freeze-Thaw PF Table
Surcharge Load
Settlement of road embankment
Different layers, different physical and mechanical processes
Thaw Settlement Freeze-Thaw
Creep
Creep (Warm)
-2 0 2 4 Temperature ( C)
16 12 8 4 0
Depth(m)
2001.10.1 2003.10.9
Warmer PF layer
0 100 200 300
Duration (Days)
-0.09 -0.07 -0.05 -0.03 -0.01
Settlements(m)
Location: DK 1136+540 (QT Railway)
PF table Ebkmt Base Ebkmt Surf
Settl. of Warm PF
F IELD OBSERVATION Q INGHAI - TIBET R AILWAY
PF Table stable
Cap Steel Tube
Lubr. Oil
PVC Tube
Plat
Keep Vertical
Mitigate Friction
Convenience; Protection
Road Surface
No. 4: Nail
Original Surface
Permafrost Table No. 1
No. 2
No. 3
F IELD OBSERVATION Q INGHAI - TIBET H IGHWAY
Along the highway: 300 km, 10 observation sites, 4500 m a.s.l.
Beiluhe
Test section
On each site: we obtained total settlement and creep The warmer it is, the more creep occurred.
-0.5 oC
-1.2 oC
23/47
No.06
No.09 Working Site: Beiluhe field station
(by Dr. Zhang)
L ONG - TERM L OAD TEST
Permafrost zone
Seasonally frozen zone
24/47
p
Natural Surface
PF Talbe Glove PVC tube
Load plate Grease
Active layer
Permafrost
Natural ground, only cared about creep of frozen layer.
-3 -2 -1 0 1 2 3 4 5 6 0.00
0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04
5-8-2009 4-2-2010 6-8-2010 5-2-2011 7-8-2011 6-2-2012 7-8-2012
Temperature()
Settlement(m)
Date
09-1#
09-1#
50 kPa
250 kPa
-2.0 -1.5 -1.0 -0.5 0.0 0.00
0.02
0.04
0.06
0.08
2-7-2006 1-7-2008 1-7-2010 30-6-2012
Temperature()
Settlement(m)
Date
Settlement curve Temperature curve
100 kPa 200 kPa 300 kPa
06-1#
Settlement Temperature
No.06
No.09
AGENDA
Factors influencing creep of frozen soils Creep of Warm Frozen Soils
State of The Art: Creep Models for Frozen Soils Our attempts on constitutive modeling
Concluding Remarks
Microscopic view
The theory of rate process Damage creep model
C REEP MODELS FOR FROZEN SOILS
Phenomenological view
Empirical model
Time Hardening Theory
Elementary element based creep model
Model classification based on its scale of representation:
4-1
The theory of rate process: View deformation as a thermal activation process
= exp exp
Statistical thermodynamics
Activation energy
Equation for rate of deformation
Related studies: Andersland (1967), Assur (1980)
Displacement
Activationenergy Activation energy
C REEP MODELS FROM MICROSCOPIC VIEW _1
A reasonable physical description 4-2
Difficulty: Parameters obtained qualitatively by current testing technology
from Fish 1983
Thermodynamic constraints
Internal variable (Endochronic time)
Endochronic time theory
Constitutive relationship from Gopal et al. 1985
Definition z and z’
= +
Deviatoric: z
Volumetric:z’
= +
2 2
2
1 1
2 2
2
2 1
d dt
dz z
d dt
dz z
Parameters determined byBazant et al. 1983
1) Strain hardening and softening law;
2) Expansion and contraction 3) Hydrostatic pressure sensibility
Unnecessary to specify a yield surface
Difficulty: Parameters for this theory are too many
C REEP MODELS FROM MICROSCOPIC VIEW _2
Endochronic time theory: irreversible thermodynamic process of dissipative material
4-3
C REEP MODELS FROM MICROSCOPIC VIEW _2
The damage creep model: damage or recovery of soil structure
Damage mechanics for continuum medium
Damage variable
Creep damage equation
Following thermodynamics; Unique internal variable in frozen soil: ice content
Difficulty: Calibration of Parameters for thermodynamics and damage mechanics
From Miao et al. 1995
CT Damage mechanism
grain orientation; damaged area Ice content
Yield criterion Dissipation potential
= 3
2 + + 9
4-4
I. Primary creep model (Vyalov, 1966) II. Secondary creep model (Ladanyi,1972) III. Tertiary creep model (?)
Simple structure; conveniently applied in simple engineering analysis (first estimation) 1) Poor versatility; 2) do not reflect internal mechanism
Stress-strain-time
Stress-strain rate-time
1 2
Empirical methods a mathematical description of creep curve
Basic form: Classified by creep stages:
C REEP MODELS FROM PHENOMENOLOGICAL VIEW _1
Time hardening model
C REEP MODELS FROM PHENOMENOLOGICAL VIEW _2
( , ) ( ) ( )
i
cr
f m t
/ E
0A
b ct
Klein and Jessberger (1979) : Stress-strain-time:
a b
t dt
Herzog and Hofer (1981):
Simple; direct
Only available for constant temperature
4-6
C REEP MODELS FROM PHENOMENOLOGICAL VIEW _3
Elementary creep model: combination of mechanical elements Basic elements:
Typical
Combination:
Elastic Viscous Plastic
Maxwell body Kelvin body
Standard Viscoelastic body
Generalized Bingham body
Reasonable mechanical basis; simple structure; convenient in engineering design
S UMMARIZATION
Some are too complicated in form, too many
parameters, even impossible to be obtained from conventional tests
Some are lacking mechanism, just mathematical description
Some are difficult to accommodate different thermal or load conditions
We need something
new.
AGENDA
Factors influencing creep of frozen soils Creep of Warm Frozen Soils
State of The Art: Creep Models for Frozen Soils Our attempts on constitutive modeling
Concluding Remarks
B ORROW THEORIES FROM UNFROZEN SOILS
Ladanyi (1999): creep of frozen soils is not very much different from that of unfrozen soils
Warm frozen soil is between frozen and unfrozen soils, most
likely closer to unfrozen soils
After Bjerrum, 1973
—Creep of unfrozen soils based on p
c/
0 0
'
/ /
' exp
' '
c
ep z
z z z z
z p
k V V V
t
' '
'
B C
e c
p
A C
c exp
c
p p
B
Yin, et al. (1989):
Vermeer and Neher(1999):
Is there an index in frozen soils similar to p
c?
Questions
What is its relationship with creep?
If so
How is it possible to apply creep model of
unfrozen soils to frozen soils?
Looking for such an index Relationship with creep
K0
Compression Frozen
Soil Unfrozen
Soil
Establish a creep model for frozen soils based on this index
Settlement of embankments
Methodology
Phases Purpose Conditions Samples
1 Prove the existence of an index similar to Pc
Same T
Different d 4
Same d
Different T 4
2 Influence of creep on this index d, T, preload 30
3 Comprehensive analysis Relationship: Creep vs. Pc
Orthogonal design:
d, T, preload, creep time
10
48 creep tests so far
Testing
In “ln(1+e) lnP” coordinates, there is clearly such an index Pseudo preconsolidation pressure
PPC for frozen soils
1,05 MPa
Stress / kPa Time / Hours
Strain/%
K
0Test
PPC increases with the increase in d, then does not change obviously PPC increases linearly with the decrease in temperature
Well reflects the bonding in frozen soils
d/ kN/m3
PPC/kPa
Temperature / C
PPC: Mechanical behavior of frozen soils
PPC/kPa
Preload 0.815 MPa Preload 1.63 MPa
We are not ready to get a relationship between PPC and temperature, creep time; but we successfully proved the existence of such an index.
When preload is less than original PPC, PPC increases with time
When preload is larger than original PPC, PPC decreases with time
A N ELEMENT MODEL FOR CREEP OF FROZEN SOIL
The creep model is
1 exp ( ) 0
2 2 2
ij ij ij K
ij
M M K K
S S S G
e t t F
G H G H
1 exp 1 ( ) 0
2 2 2 2
ij ij ij K
ij
M M K K N
S S S G Q
e t t F t F
G H G H H
Instantaneous elastic
Viscous(strain rate approaches
zero)
viscous (strain rate increases with
stress)
Viscoplastic with a yield surface
2 2
3 tan tan
m 2 m
m
F J c
p
Yield criterion(Ma, et al. 1994)
Establishing the model
(Dr. Songhe Wang)
Only creep of underlying permafrost was considered
Field load test Loading pile Numerical model
100 kPa
Long-term load test
A simple model for creep of frozen soil might provide a way in engineering analysis.
After implementation of this model,
Thermal state analysis Deformation analysis
Numerical simulation
A
VISCO-
HYPOPLASTIC CONSTITUTIVE MODEL FOR FROZEN SOILs d
2 2
s
s 1 s 2 s cd 3 s 4 s d
s s
tr[( - ) ]
[tr( - )] ( - ) ( - ) ( - )
tr( - ) tr( - )
c c
c
f f cc
cc
c c
2 exp( )
f l
s is static stress, d is dynamic stress.
in which c is the cohesion of frozen soil, f is a scalar function of deformation, fcd is a factor of creep damage.
0
t
d l
tand are parameters, l is the accumulation of deformation.
Static part
(Dr. Guofang Xu, Prof. Wei Wu)
2 2 2 2
1 2 cd 3 4 d 1 2
tr[( - ) ]
[tr( - )] ( - ) ( - ) ( - ) tr( )
tr( - ) tr( - )
c c c f f c c c c
c c
2 1
cd
1
td
f
tis a parameter, is Macaulay brackets.
Dynamic part
2 2
d 1 2
tr( )
1 and 2 are parameters, is strain acceleration.
Complete constitutive model - Rate dependent
C ALIBRATION OF THE CONSTITUTIVE MODEL
When the two linear and nonlinear terms in the static part of the model are abbreviated as L1, L2, N1 and N2, this part can be rewritten as:
s
c
1 1L (
s): + c
2L
2(
s): c
3N
1(
s) c
4N
2(
s)
In a conventional triaxial test, owing to ,the above equation can be divided into two scalar equations as follows:
2 3
0
2 2 2 2
1
c L
1 11 1c L
2 12 3c N
3 11 12
3c N
4 12 12
32 2 2 2
3
c L
1 21 1c L
2 22 3c N
3 21 12
3c N
4 22 12
3Parameters in the static part
Owing to the radial stiffness EA3 = EB3 = 0, we have
The parameters ci (i = 1, …, 4) can be obtained by solving the above equation system with respect to the variables ci.
Parameters and can be obtained from the following expressions:
in which Tref is a reference temperature and can be regarded as -1°C (Zhu and Carbee, 1984).
Parameters 1 and 2 in the dynamical part can only be obtained by fitting the experimental data, as done by Hanes and Inman (1985).
1
1 (T Tref )n
2
( T T
ref)
nParameters and
V ERIFICATION OF THE CONSTITUTIVE MODEL
Uniaxial compression tests at different loading rates
Compressivestress(kPa)
Stress-strain relationship at different strain rates (Data from Zhu and Carbee, 1984)
Uniaxial creep tests at different stress levels
0 200 400 600 800 0
4 8 12 16
0 400 800 1200 1600 0
4 8 12 16
0 50 100 150 200 250 0
4 8 12 16
0 10 20 30 40
Time (min) 0
2 4 6 8 10
0 500 1000 1500 2000 2500 0
2 4 6
0 400 800 1200 1600 0
2 4 6
10000 kPa 9000 kPa
8000 kPa 7000 kPa
6000 kPa 3000 kPa
Numerical Experimental
1 10 100 1000
0.01 0.1 1
1 10 100 1000 10000 0.001
0.01 0.1 1
1 10 100 1000
0.01 0.1 1
1 10 100
Time (min) 0.1
1
1 10 100 1000 10000 0.0001
0.001 0.01 0.1
1 10 100 1000 10000 1E-005
0.0001 0.001 0.01 0.1
10000 kPa 9000 kPa
8000 kPa 7000 kPa
6000 kPa 3000 kPa
Numerical Experimental
Creep strain vs. time
(Test data from Orth (1986))
Creep strain rate vs. time
(Test data from Orth (1986))