Creep, Anisotropy and Destructuration Dr Minna Karstunen with thanks to Prof. Pieter Vermeer & Dr Martino Leoni
Acknowledgements • Co-workers: – – – – – – – – – –
Dr Mike Kenny (USTRAT) (USTRAT) Dr Zhenyu Yin Yin (previously USTRAT, now now Nantes, soon soon ) Dr Martino Leoni (previously (previously USTUTT, Wechselwirkung Wechselwirkung Studio Italiano) Dr Mohammad Rezania (USTRAT) Ms Daniela Kamrat-Pietraszewska Kamrat-Pietraszewska (USTRAT/KELLER) (USTRAT/KELLER) Mr Igor Mataic (HUT) Prof. Pieter Vermeer (USTUTT) Dr Gustav Grimstad (NGI) (NGI) Dr Ronald Brinkgreve Brinkgreve (Plaxis bv/TUD) bv/TUD)
• Sponsors: – GEO-INSTALL “Modelling “Modelling Installation Effects in Geotechnical Engineering” IAPP project funded by the t he EC 2009-2014 – Experimental work work has been funded by the Academy of Finland (Grants 210744,128459) 210744,128459)
Acknowledgements • Co-workers: – – – – – – – – – –
Dr Mike Kenny (USTRAT) (USTRAT) Dr Zhenyu Yin Yin (previously USTRAT, now now Nantes, soon soon ) Dr Martino Leoni (previously (previously USTUTT, Wechselwirkung Wechselwirkung Studio Italiano) Dr Mohammad Rezania (USTRAT) Ms Daniela Kamrat-Pietraszewska Kamrat-Pietraszewska (USTRAT/KELLER) (USTRAT/KELLER) Mr Igor Mataic (HUT) Prof. Pieter Vermeer (USTUTT) Dr Gustav Grimstad (NGI) (NGI) Dr Ronald Brinkgreve Brinkgreve (Plaxis bv/TUD) bv/TUD)
• Sponsors: – GEO-INSTALL “Modelling “Modelling Installation Effects in Geotechnical Engineering” IAPP project funded by the t he EC 2009-2014 – Experimental work work has been funded by the Academy of Finland (Grants 210744,128459) 210744,128459)
Marie Curie IAPP GEO-INSTALL PIAG-GA-2009-230638
GEO-INSTALL IAPP Industry-Academia Partnerships and Pathways project funded by the EC/FP7 under programme ‘People’ •Total value €1.28M (2009-2013) for: • Secondments • Post Post-d -doc oc appo appoin intm tmen ents ts (Mar (Marie ie Cur Curie ie Fellowship) • Work Worksh shop ops, s, trai traini ning ng cou cours rses es and and othe otherr knowledge-exchange activities
Outline • 1D Creep model • Experimental data • 1D Creep model • Influence of temperature
•
f
il r
m
l
• 3D Soft Soil Creep model • SSC parameters
• Modelling of natural clays • Anisotropy & destructuration • S-CLAY1S and EVP-SCLAY1S • Application: Murro test embankment
Creep modelling with thanks to Prof. Pieter Vermeer, Dr Martino Leoni &
Load-controlled test: footing (3 x 3 m2) on dense sand
) m m ( t n e
24 hours
e l t t e S
Load (MN)
Briaud and Gibbens (1994)
Displacement-controlled load tests on floating micropiles
s& = constant Load (kN) 20
CSV micropiles Length: 8m Diameter: 17 cm
40
60
80
100
) 1 m c ( t 2 n e m e 3 l t t e S
s& = 10 a
4
10-4
s& = a
a s& = rate of penetration
Considering such a rate-type effect, it is important to do research on creep and stress relaxation.
Measured time – settlement curve near Friedrichshafen
Time [days] 0
1
10
100
103
104
Alluvium Kie s
Beckenton Fein NCSchluff
] m m [ t n e e l t t e S -450
Sand
s∞ = 600 mm Fels
Classical soil Mechanics assumed an S-curve with a clear transition between consolidation and creep.
Settlement curves of buildings in Drammen do not show S-shape (Bjerrum ,1967) time [years]
SKOGER SPAREBANK
OCReoc 1.2 SCHEITLIES GATE 1 OCReoc 1.1
SAND PLASTIC CLAY
w = 28%
w = 51.5%
w = 28.2%
Thickness varies with location LEAN CLAY
] m c [ t e m e l t t e s
KONNERUD GATE 16
OCReoc 1.05
wP = 21.4% w = 34.6% wL = 35.5% OCReoc = Thickness varies with location
σ po σ'o + ∆σ
TURNHALLEN
OCReoc 0.9
Often OCR varies with depth. The indicated OCR values refer to the bottom of the clay layer.
1D-consolidation and 1D-creep in single load step σ ∆σ 0
Load step time e i t a r
d i o V
Bjerrum (1967):
t
e = e eoc - C α log
t t eoc
eeoc 1 C = secondary compression index α
teoc eeoc = void ratio at end of consolidation
log t
or creep index
teoc = time at end of consolidation
Data from 24-hours-load-stepping test on a NC-peat
total time of test (days)
load step time (logscale)
1
den Haan (1994)
Cα
1 day
Load is daily doubled. First 3 load steps do not show S-curves. Except step 1, all slopes show Cα α.
Typical 24 hours load stepping
1 day
Each day: CC = compression index
∆σ = constant ⋅ σ
CS = swelling index
Cα = creep index
Increase of preconsolidation stress due to creep after Bjerrum (1967)
e NC-Line
creep
Cs
1
σ o′
σ p = preconsolidation stress log σ′
State of overconsolidation can be reached both by creep and unloading
Data can be modelled by relating the creep rate to OCR
e e& ≡
de dt
NCL
OCR = 1
e&
=
e& nc
OCR = 1.3
e&
≈
e& nc / 1000
OCR = 1.7
e&
≈
e& nc / 1000 / 1000
log σ’
Den Haan (1994)
after Hanzawa (1989)
MSL24 = usual 24 h Multi – Stage Loading CRS = Constant Rate of Strain test
1D creep model
e& =
e& nc
e& nc = −
OCRβ
τ
Cα 1 ⋅ ln 10 τ
= reference time
β=
Cc − Cs Cα
Classical creep rate concepts: Norton (1929):
ε&
Prandtl (1928):
& = ε
Soderberg (1936):
=
τ 1
τ
ε& =
(σ − σ o )α
τ = temperature dependent
⋅ sinh (ασ − ασ o ) 1
τ
⋅ ( exp (ασ − ασo ) − 1)
for σ > σο for σ > σο
The reference time τ depends on the definition of the NC-line. When NCL is based on the usual one-day load stepping test, we have τ = 1 day. This is a default in PLAXIS.
1D creep model Summation of elastic rate and creep rate
σ
σ
Typical soil data:
e& = e& e + e& c =
Cs ≈ Cc / 10
−Cs σ& ′ −C α 1 1 + ln10 σ′ ln10 τ OCR β
and Cα ≈ Cc / 30 ⇒ β =
− Cα
c
s
This implies that the creep rate is negligibly small for OCR well beyond unity.
σ’p = function of void ratio and temperature
≈ 27
1D creep model Tremendous influence of overconsolidation ratio
Example :
Cc = 0.15
e& c =
Cs = 0.015
Cα = 0.005
⇒
β = 27
e& nc OCR 27
e OCR
e& c
1.0
e& nc
1.3
e& nc ⋅ 10 −3
1.7
e& nc ⋅ 10 −6
NCL
log σ′
1D creep model NC soils show S-shaped settlement curve; OC soils do not.
8 days
1 day
e
∆σ
e Step A
Step A Cα
σpo
Step B
Step B
Cα
log σ’
Treporti Silt
( wP = 25%
w = 32%
wL = 38%)
log t = log ( teoc + t ‘ )
data after Simonini et al. (2006); Berengo (2006)
Influence of temperature
Preconsolidation pressure from oedometer tests at various temperatures
Leroueil (2006):
ln 10 ⋅ ∆ e c σp = σpo ⋅ exp − α ∆ T ⋅ exp − − Cc Cs α = 0,01 per °C
ln 10 σ& p = − σp e& c + α T& Cc − Cs
after Eriksson (1989)
Soft Soil Creep model
1. 3D Soft Soil Creep model 2. SSC parameters .
p ons o
4. Application of SSC: Leaning tower of Pisa 5. Anisotropic creep model
Martino Leoni
3D Soft Soil Creep model
β
p& ' µ peq ' *
e v
p v
*
ε &v = ε & + ε & = κ
p '
+
τ p p '
Modified compression index:
λ * =
Modified swelling index:
κ * ≈
Modified creep index:
λ * − κ * β = µ *
µ * =
C c
ln 10 2C s
q
ln 10 C α
NCS
ln 10 p′eq
Isotropic preconsolidation pressure
ε &vp p' p = p' exp * * λ − κ
p'p
p´
3D Soft Soil Creep model Ellipses of Modified Cam Clay are taken as contours of volumetric strain rate
q
c e&&vc ==aa ε
urrent stress
c ε <
NCS: p′eq = p′p
p′eq
p′p
q2
p 'eq = p '+ * M p' p' =
1 3
(σ '1 + σ ' 2 + σ '3 )
q =
1 2
(σ '1 − σ ' 2 ) 2 + (σ ' 2 − σ '3 ) 2 + (σ '3 − σ '1 ) 2
p´
3D Soft Soil Creep model Comparison with Modified Cam Clay
Both models have summation: σ
σ
Both models have hardening: (Isothermal case is considered)
Both models have flow rule:
ε &v = ε &ve + ε &v p ε &vp p ' p = p ' exp * * − λ κ
&c = Λ
ε
∂p′eq ∂σ
Modified Cam Clay: increase of density by primary loading
NCS is yield locus
Creep model:
NCS is iso-creep rate locus
ε& volumetric
increase of density by creep
∂ p ′eq ∂ p ′eq ∂ p ′eq q2 = ε& 1 + ε& 2 + ε& 3 = Λ ⋅ + + = Λ ⋅ 1 − 2 2 = Λ ⋅ d M p′ ∂σ 2 ∂σ 3 ∂ σ 1
Soft Soil Creep model 3D model:
Division of strains: e
c
p
d ε = d ε + d ε + d ε e
−1
d ε = D d σ ' c
d ε = d λ 1 p
d ε = d λ 2
g d σ ' dg f d σ '
Elastic strains according to Hooke’s law Creep strains (viscoplastic, time-dependent)
Plastic strains according to MC (at failure)
3D Soft Soil Creep model Typical performance of model for drained triaxial tests on NC-soil
q
σ1 − σ3 NCS-slow
•
•
pp 0
•
NCS-fast
•
pp-fast
•
p'
pp-slow
•
slow test
ε1
3D Soft Soil Creep model Performance of IC Model for undrained triaxial tests on NC-soils
σ1 − σ3
q/2
•
•
FAST SHEARING
•
•
Cu FAST
SLOW
Cu SLOW
p´
ε1
3D Soft Soil Creep model SSC Model 1.5
Cu C u (ε& = 1%/h)
= 1.00 + 0.10 log ε&
1.0
26 CLAYS
0.5
10-3
10-2
10-1
100
101
102
103
104
& (% /h )
ε
Kulhawy & Mayne (1990): Manual on Estimating Soil Properties for Foundation Design
105
SSC parameters Cc λ = 1 + e0 (1 + e0 ) ln10
Modified compression index:
λ* =
Modified swelling index:
κ * ≈ λ* / 10
Modified creep index:
µ * ≈ λ* / 30
Poisson’s ratio:
ν ′ = 0.15 ÷ 0.25
Critical state friction angle:
′ ϕcs
Initial conditions:
FROM PLAXIS MANUAL, NOT SUITABLE FOR STRUCTURED CLAYS
′ (Jaky) → K 0NC ≈ 1 − sin ϕ cs
POP or OCR
Parameters of the SSC model time (ln-scale)
p’ (ln-scale)
consolidation ε v
λ* ε v
creep µ*
κ *
time (logarithmic)
σ1’ (logarithmic)
consolidation e
C c
C s
e
creep C α
Soft Soil Creep model Relationships with other compression indices:
Cam-Clay uc
era ure
Den Haan International lit.
λ λ = 1+ e *
*
1
*
=
2
'
λ *= A + B λ *=
C c
2.3(1 + e)
1+ e
*
C p
*
2.3 C 's
C p ( ontl .) *
*
≈ 2A ≈
C s
1+ e
1
µ * = C µ * =
C α
2.3 (1 + e)
Options of SSC model Effective cohesion and steep cap
q
Failure MC
1 M*
M MC 1 cap
p pe '
c´cot ϕ´
Usually assume c’= 0 kPa
M =
6 sin φ cv ' 3 − sin φ cv '
Value of M* can be selected such that K 0nc = 1 – sin ϕ´ The above picture would suggest the possibility of tensile stresses, but these can be omitted by using a „tension cut-off“
p´
Options of SSC model Initial stress state for overconsolidated soils
POP = σp − σ′0
y x
input of OCR σ′0
input of POP σ′0 σp
σp
σ′y σp K
1 σ′y 0
nc 0
1- ν
POP
ν
This procedure for estimating the initial horizontal stress gives results that are in good agreement with the correlation: nc
σ′x 0
σ′x
K 0 ≈ K 0 ⋅ OCR
SSC model The role of OCR in self-weight loading and creep: Settlement [m] 0.00
OCR0=2.0
-0.05 -0.10
= .
* κ * µ*
-0.15 -0.20
νur OCR0=1.1
-0.25 -0.30 -0.35 0
Settlement of 10m thick layer
OCR0=1.0 200
400 Time
600
800
1000
c’
ϕ’ ψ K 0nc
. 0.02 0.005 0.15 0.0 kPa 25° 0° 0.677 (1-sin ϕ’)
References Creep in soft soils 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 Bjerrum, L. 1967. Engineering geology of norwegian normally-consolidated marine clays as related to settlements of buildings. Géotechnique, 17: 81118. Boudali, M. 1995. Comportement tridimensionnel et visqueux des argiles naturelles. PhD Thesis, Université Laval, Québec. Buisman (1936): Results of long duration settlement tests, Proc. 1st Int. Conf. Soil Mech. and Found. Eng., Vol. 1, pp. 103-107. Claesson, P. 2006. Creep around the preconsolidation pressure – a laboratory and field study. In CREBS Workshop. Edited by N.G.I. Oslo. 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. Den Haan, E.J. 1996. A compression model for non-brittle soft clays and peat. Géotechnique, 46: 1-16. Garlanger, J.E. 1972. The consolidation of soils ex hibiting creep under constant effective stress. Géotechnique, 22: 71-78. Janbu, N. 1969. The resistance concept applied to deformations of soils. In 7th ICSMFE. Mexico City, Vol.1. Leroueil, S. 1987. Tenth Canadian Geotechnical Colloquium: Recent developments in consolidation of natural clays. Canadian Geotechnical Journal, 25: 85-107. Leroueil, S. 2006. The isotache approach. W here are we 50 years after its development by Professor Šukljie? Malvern, L.E. 1951. The propagation of longitudinal waves of plastic deformation in a bar of metal exhibiting a strain rate effect. Journal of Applied Mechanics, 18: 203-208. Mesri (2006), Primary and secondary compression, In CREBS Workshop. Edited by N.G.I. Oslo. Mesri & Godlewski (1977): Time and stress compressibility interrelationship, J. Geot. Eng. Div., ASCE 103, GT5, pp.417-430. Odqvist, F.K.J: Mathematical Theory of Creep and Creep Rupture, Clarendon Press, Oxford, 1966 Perzyna P.: Fundamental Problems in Viscoplasticity, Advan. Appl. Mech., 9, 243-377, 1966 Šukljie, L. 1957. The analysis of the consolidation process by the isotaches method. In 4th ICSMFE, Vol.1, pp. 200-206.
References Yin, J.-H. 1999. Nonlinear creep of soils in oedometer tests. Géotechnique, 49(2): 699-707. Yin, J.-H., and Graham, J. 1999. Elastic viscoplastic modelling of the time dependent stress-strain behaviour of soils. Canadian Geotechnical Journal, 36: 736-745. Isotropic (Soft Soil Creep model) creep model: Stolle, D.F.E., Bonnier, P.G., and Vermeer, P.A. 1997. A soft soil model and experiences with two integration schemes. In NUMOG VI. Edited by Pietruszczak S. and Pande G.N. Montreal. 2-4 July 1997. Balkema, Rotterdam. Vermeer, P.A., and Neher, H.P. 1999. A soft soil model that accounts for creep. In Int.Symp. "Beyond 2000 in Computational Geotechnics". Edited by R.B.J. Brinkgreve. Amsterdam. Balkema, Rotterdam, pp. 249-261. Vermeer, P.A., Stolle, D.F.E., and Bonnier, P.G. 1998. From the classical theory of secondary com pression to modern creep analysis. In Computer Methods and Advances in Geomechanics. Edited by Yuan. Balkema, Rotterdam. 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 Desai et 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 Anisotropic (creep) modelling: Anandarajah, A., Kuganenthira, N., and Zhao, D. 1996. Variation of Fabric Anisotropy of Kaolinite in Triaxial Loading. Journal of Geotechnical Engineering, 122(8): 633-640. Leoni, M., Karstunen M. and Vermeer P.A. 2007. Anisotropic creep model for soft soils. Submitted for publication Näätänen, A., Wheeler, S.J., Karstunen, M., and Lojander, M. 1999. Experimental investigation of an anisotropic hardening model for soft clays. In 2nd International Symposium on Pre-failure Deformation characteristics of Geomaterials. Edited by M. Jamiolkowski, R. Lancellotta, and D. Lo Presti. Torino, Italy, pp. 541-548. Vermeer, P.A., Leoni, M., Karstunen, M., and Neher, H.P. 2006. Modelling and numerical simulation of creep in soft soils. In ICMSSE Conference, Vancouver, p. 57-71 Wheeler, S.J., Näätänen, A., Karstunen, M., and Lojander, M. 2003. An anisotropic elastoplastic model for soft clays. Canadian Geotechnical Journal, 40: 403-418.
Anisotropy and Destructuration - Modellin of natural cla s Dr Minna Karstunen With thanks to Mirva Koskinen, Zhenyu Yin, Martino Leoni & many others
Outline
• Introduction: some key features of natural clays • – Large strain anisotropy → S-CLAY1 – Bonding and destructuration → S-CLAY1S – Viscosity and time-dependence → ACM & EVPSCLAY1S & AniCreep
Structure of Natural Clays • Soil structure consists of:
For a constant η stress path:
– fabric (anisotropy ) – interparticle bonding (sensitivity ) v
λ
Reconstituted soil
soil
1
λi
Due to plastic straining gradual degradation of bonding (destructuration ) and changes in fabric
1
ln p'
Structure of Natural Clays • Fabric of clay:
e.g. Craig (1974): (a) dispersed; (b) flocculated; (c) bookhouse; (d) turbostratic (e) natural clay with silt particles
Leroueil & Vaughan (1990)
1D Compression σ'pi = 6 kPa
2.4
4
σ'p = 45 kPa
σ'pi = 0.37 kPa
σ'p = 29 kPa Vanttila clay
3.2
Intact
2
Remoulded
e
e
1.6
2.4
Murro clay
1.2
1.6
Intact Remoulded
0.8
0.8 1
(a)
10
100
σ 'v (kPa)
1000
0.1 (d) (d)
1
10
100
σ 'v (kPa)
1000
10000
1D Compression 0.1
0.04
Intact
Intact Remoulded 0.03
Remoulded
0.08
Vanttila clay
Murro clay
. e
e
C
C
0.02
α α
α α
0.04 0.01
0.02 0
0 1
(b)
10
100
σ 'v (kPa)
1000
1 (e)
10
100
σ 'v (kPa)
1000
10000
After Leroueil & Vaughan (1990)
After Leroueil & Vaughan (1990)
Mexico City Clay (Mesri et al. 1975)
The Grande Baleine clay (Locat & Lefebre 1982)
Triaxial Tests with Constant Stress Ratio Vanttila clay 2.3-3.1 m St>30
POKO clay 8.5-10.0 m St=12 1.2
1.4
1.0
1.2
λ
0.8
λ
1.0 0.8
λ 1 , κ
0.6
λ 1 , κ
0.4
λi
0.2
0.4
λi
λi
0.2 0.0
0.0 -1.0
0.6
-0.5
0.0
η1
0.5
1.0
-1.0
-0.5
0.0
η1
0.5
1.0
80
1 Winnipeg clay 0.75
a P k , '
191 kPa
241 kPa
310 kPa
380 kPa
Marjamäki clay Depth 5.5-6.1 m
60
0.5
40
M = 0.67
a P k , q
c v
/ q
0.25
M = 0.84
20
0
0 0
0.25
0.5
0.75
1
0
p'/ σv c ', kPa -0.25
20
80 p', kPa
90 Mexico City clay Depth 1.7 m
Bothkennar clay Depth 5.3 - 6.3 m
M=1.75
M = 1.4
60
60 a P k , q
a P k , q
Wheeler et al. (2003)
60
-20
90
φ’ = 18-43 degrees
40
30
30
yield points from p'-εv from q-εs
undrained failure
0
0
0
30
60
90
0
p', kPa -30
yield points
30
60
90 p', kP a
-30
On modelling anisotropy • Elastic anisotropy (Ev, Eh…) easy to model but values for parameters very difficult to measure • Plastic anisotropy relates to anisotropy associated with LARGE (irrecoverable) strains
Modelling Plastic Anisotropy 1. Standard elasto-plastic framework (kinematic or translational hardening laws) – Note: cannot use invariants •
Nova (1985), Banerjee & Yousif (1986), Dafalias (1986), Davies & Newson (1993), Whittle & Kavvadas (1994), Wheeler & al. (2003)
2. Micromechanical models i. Multilaminate framework •
Zienkiewicz & Pande (1977), Pande & Sharma (1983), Pietruszczak & Pande (1987), Karstunen (1998), Wiltafsky (2003), Neher et al. (2001, 2002), Mahin Roosta et al. (2004)
ii. Microplane models •
Bazant (995), Chang & Liao (1990), Chang & Gao (1995), Chang & Hicher (2005), Yin et al. (2009)
Modelling Destructuration • Concept of an intrinsic yield surface proposed by Gens & Nova (1993) – Lagioia & Nova (1995), Rouainia & Muir Wood (2000), Kavvadas & Amorosi (2000), Gajo & Muir Wood (2001), Liu & Carter (2002), Karstunen et al. (2005) CSL
q
M 1
1
pmi’ 1 CSL
M
pm’
p’
α
Modelling Time-Dependence & Creep – Creep models: • ACM (Leoni et al. 2008) and ACM-S (under development) • AniCreep (Yin et al., submitted for publication) • Time-resistance S-CLAY1S (Grimstad & al. in press)
– • EVP-SCLAY1S (Karstunen & Yin, in press) σ’y
CSL
q
α
M 1
p’
1
pmi’
σ’x σ’z
1 CSL
M
pm’
p’
α
Definitions: Deviatoric stress vector
Deviatoric fabric tensor (in vector form)
σ'x −p' σ' −p'
αx −1 α −1
σ' z − p ' σd = τ 2 xy 2τ yz τ 2 zx
αz −1 αd = 2 α xy 2α yz 2 α zx
p' =
σ' x + σ' y + σ' z
αx + αy + αz
3
3
=1
S-CLAY1S Model (Karstunen et al. 2005) α
0 20 40
p’
σ’y
60 80
80
60
40
20
0
σ’x
0 20
σ’z
40 60 80
2 3 3 T T F = [{σ d − p' α d } {σ d − p' α d }]− M − {α d } {α d } [p'm − p']p' = 0 2 2 α 14243
S-CLAY1S Model (Karstunen et al. 2005) “intrinsic” yield curve CSL
q
α σ’y
Natural yield curve
M 1
p’
1
pmi’
σ’x σ’z
pm’
α
p’
1 CSL
Intrinsic yield surface F=
3 2
[{σ
p'm = (1 + x )p'mi
]
3
− p' α d } {σ d − p' α d } − M 2 − {α d } {α d } [p'm −p']p' = 0 T
d
M
2
T
Hardening Laws: 1) Size of the intrinsic yield surface dp'mi =
vp 'mi dε pv
λ i − κ
p p dx = −ax dε v + b (dεd ) 3) Rotation of the yield surface η 3η p p dα d = µ ( − αd )〈dε v 〉 + β( − αd )dεd 3 4
η=
σd p'
Fabric Tensor (Assuming Initial Cross Anisotropy) αx −1 α −1 y αz −1 αd = 2α x 2α yz 2α zx
1 ( ) α − α x y 3 2 − (α x − α y ) 3 1 α d = (α − α ) x y
α xy = α yz = α zx = 0
αx = αz
0 0 0
1 3 (3α x − 3) 2 − (3α x − 3) 3 1 αd = (3α x − 3)
αx + αy + αz 3
=1
0
α − 3 2α 3 α αd = − 0 0 0
0 0
3
T
α = {α d } {α d } 2
2
Elasto-Plastic Matrix By following the standard procedure T
∂Q ∂F [D ] = [ D ] − [D ] [D e ] β ∂ σ' ∂ σ' ep
e
1
e
T
∂F ∂Q [ D e ] β = H + ' ' ∂ σ ∂ σ
For S-CLAY1S: T T T 2 ∂αd ∂Q ∂Q ∂F ∂x ∂Q 2 ∂x ∂Q ∂Q ∂F ∂p'mi ∂Q ∂F ∂αd ∂Q + + + + H=− ∂p'mi ∂εpv ∂p' ∂αd ∂εpv ∂p' 3 ∂εdp ∂σ'd ∂σ'd ∂x ∂εpv ∂p' 3 ∂εdp ∂σ'd ∂σ'd
S-CLAY1S has been implemented in SAGE Crisp (by Zentar at GU) and PLAXIS 2D v8.2 (by Wiltafsky at GU) and PLAXIS 2D Version 9, using NR (by Sivasithamparam at USTRAT/PLAXIS) in 2010.
S-CLAY1 and MCC • By setting x to zero and using an oedometric value (1D) for com ressibilit λ S-CLAY1 model (anisotropy only) • By setting, in addition, α and µ to zero MCC model (isotropy only)
Additional State Variables and Soil Constants S-CLAY1
Symbol α0
Definition
Method
Initial inclination of the yield curve Proportion constant
Estimated via φ’ Estimated via φ’
µ
Rate of rotation
≈
x0 λi b a
Initial amount of bonding
≈ St -1
Slope of intrinsic compression line Proportion constant
Oedometer test on reconstituted soil For most clays 0.2-0.3
Rate of destructuration
Typically 8-11
(10…20)/ λK0
S-CLAY1S
Tests on Reconstituted Clays a) Murr o clay
Yield points M=1.6 α=0.46 max Max. stress during η0 loading
40 30 20 ) a 10 P k ( q 0
-10
p'm=35.5 kPa 0
10
20
30
40
50
60
70
p' (kPa)
b) POKO clay
40 30 20 ) a 10 P k ( q 0
-10
-20
-20
-30
-30
c) Otaniemi clay
Yield points M=1.3 α=0.42 Max. max stress during η0 loading
40 30
20
30
40
50
60 70 p' (kPa)
20
30
40
50
60 70 p' (kPa)
Yield points
α=0.40 Max. max stress during η0 loading
30 20
10
10
40
) 10 a P k ( q 0
0
0
M=1.35
20
-10
p'm =42.0 kPa
d) Vanttila clay
) a 10 P k ( q 0
p'm=26.0 kPa
Yield points M=1.2 α=0.43 Max. max stress during η0 loading
-10
-20
-20
-30
-30
p'm=26.0 kPa 0
10
20
30
40
50
60 70 p' (kPa)
Simulations with S-CLAY1 1.5
2.5
3.5
4.5
-0.05
5.5
ln p'
0.0
-50
0
50
100
150
q (kPa) 0.05
0.1
ε ε d
ε ε v
0.15
0.2
0.3
0.25
-0.05 0.0
0.1
0.2
0.3
ε ε v 0.05
ε ε d 0.15
0.25
CAD 3216R Reconstituted Murro clay 6.9-7.6 m η0=0.98, η1=-0.62, η2=0.60 CAE 3216R S-CLAY1 MCC
& Koskinen (2004) For Karstunen full validation, see Karstunen & Koskinen (2008), Can. Geotech. J.
Tests on Natural Clay Samples M=1.6 α=0.63 p'm =34.5 kPa
a) Murro clay 50
b) POKO clay 50
40
40
30 ) a P k ( q
M=1.2 α=0.46 p'm=49 kPa
30 ) a P k (
20 10
q
0
20 10 0
0
10
20
30
40
50
60
70
0
-10
10
20
30
40
50
60
70
-10
-20
-20
p' (kPa)
p' (kPa)
c) Otaniemi clay
d) Vant tila clay
M=1.3 α=0.50 p'm =19.5 kPa
20
M=1.35 α=0.52 p'm=18.5 kPa
20
10
10
) a P k (
) a P k (
q
q
0
0 0
10
20
-10
30
0
10
20
-10
p' (kPa)
p' (kPa)
30
Viscosity of Natural Clays 2
Murro clay
) % e1 a C
0 10
100 σ 'v (kPa)
1000
Principle of EVP-SCLAY1S q
ε&ij = ε&ije + ε &ijvp
ε&ijvp = µ φ ( F ) 1
Mc ∂ f d ∂σ ij B
Static yield surface
p’ mi s
1
Dynamic loading surface
A
p’ m s
Intrinsic yield surface
∂ f d ∂σ ij
p’ m d p’
pmd µ φ (F ) = µ exp N ⋅ s −1 −1 pm Me
Model Parameters • Anisotropy parameters – 1 additional state variable (of tensorial form) describing anisotropy – 2 additional soil constants – All can be estimated based on standard oedometer and triaxial tests for soil with previous K0 history
• Destructuration arameters – 1 additional state variable describing the amount of bonding – 2 additional soil constants – Ideally need oedometer tests on reconstituted sample, but can be estimate based on standard oedometer test at high stresses
• Viscosity parameters – 2 additional soil constants that need to be optimised – Optimization requires either: • Oedometer/triaxial tests with two different strain rates or • Long term oedometer/triaxial tests or • Pressometer tests with different strain rates etc.
CSR oedometer test (Batiscan Clay) 0 Exp. 1.43x10-5 s-1 Exp. 2.13x10-6 s-1 Exp. 1.07x10-7 s-1 EVP model
5
130
120
Without destructuration
10 ) % (
v
ε ε
15
) a P k ( 100 ' p
20 (a) α=0 & χ=0 25 60
80
110
Experiment EVP model with α & χ EVP model without α & χ
With destructuration
σ σ
100
120
140 160 σ ' (kPa)
180
200
220
240
Without destructuration
90
v
0 Exp. 1.43x10-5 s-1 Exp. 2.13x10-6 s-1 Exp. 1.07x10-7 s-1 EVP model
5
80
With destructuration
10 ) % (
v
ε ε
10
-8
10
-7
10
-6
10
-5
10
-4
dε /dt v
15
20 (d) Anisotropy & destructuration 25 50
70 -9 10
100
150
200 σ ' (kPa) v
250
300
Destructuration can improve predictions on CSR oedometer test on structured clays
