Properties and a constitutive model for
frozen and unfrozen soil
Ronald Brinkgreve, Delft University of Technology / Plaxis
Authors: Manuel Aukenthaler, Ronald Brinkgreve, Adrien Haxaire
Content
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Introduction
Characteristics of frozen soils
Starting from particle size distribution…
Soil-Freezing Characteristic Curve
Hydraulic conductivity
Validations
Constitutive model for frozen / unfrozen soil
Model parameters
Applications
Conclusions
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Introduction
Frozen/unfrozen soils:
• Cold regions
• Climate change
• Artificial ground freezing
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Introduction
Research on modelling of frozen/unfrozen
soils i.c.w. NTNU
• Constitutive model (NTNU)
• Parameters and Validation
(Plaxis / TUDelft)
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Characteristics of frozen soils
Transition pore water > pore ice (and vice versa):
• Gradual process (in t and T)
• Transition zone: Frozen fringe
• Unfrozen water content θuw(T)
(Soil Freezing Characteristic Curve, SFCC)
• Pressure melting θuw(p)
• Change of hydraulic conductivity k(T,p)
• Change of stiffness and strength
• Expansion / compression
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Characteristics of frozen soils
Problem:
• Temperature-related parameters are
uncommon for geotechnical engineers
Solution:
• Derive them from common soil data
(Particle Size Distribution)
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Starting from particle size distribution…
Selection of soil type > PSD > SSA
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Soil-Freezing Characteristic Curve
SSA > Unfrozen water content θuw:
(after Anderson & Tice, 1972)
ρw = bulk density of water
ρb = bulk density of unfrozen soil
T
Tf
= temperature
= freezing temperature of water
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Soil-Freezing Characteristic Curve
Pressure-dependence of the freezing temperature:
(Clausius-Clapeyron equation)
Cryogenic suction sc:
pice =
pw =
ρice =
L
=
pore ice pressure
pore water pressure
bulk density of ice
latent heat
(1)
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Soil-Freezing Characteristic Curve
Melting pressure for ice pmelt:
(after Wagner et al., 2011)
(T = Tf)
(vapour-liquid-solid triple point)
(2)
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Soil-Freezing Characteristic Curve
Pressure melting:
(pressure in MPa)
• For a given pw and T:
sc, pice and Tf can be
calculated by solving
eq. (1) and (2) iteratively
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Soil-Freezing Characteristic Curve
Final result: SFCC for different soils at different pressures
Increasing pressure
Fine grained
Coarse grained
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Hydraulic conductivity
PSD > ksat > k(θuw):
(after Tarnawski & Wagner, 1996)
Permeability of partially frozen soil:
mi = mass fractions of clay, silt, sand
di = particle size limits
θuw = unfrozen water content
θsat = saturated water content
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Validations
(soil mass fractions after Smith & Tice, 1983)
SFCC based on SSA
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Validations
Pressure-dependence:
(after Zhang et al., 1998)
Lanzhou Loess:
mcl = 0.12
msi = 0.80
msa = 0.08
e0 = 0.7
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Validations
Hydraulic conductivity:
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(after Burt & Williams, 1976)
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Constitutive model for frozen/unfrozen soil
Solid state (grains + ice) stress σ*:
Similarities with BBM
In unfrozen state: MCC
σ
Suw
pw
εm
εs
= net stress
= unfrozen water saturation
= pore water pressure
= solid phase strain component
= cryogenic suction strain component
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Constitutive model for frozen/unfrozen soil
Elastic strains:
sc = cryogenic suction
ice saturation
Frozen soil properties:
Temperature-dependent (but stress-independent)
Young’s modulus of frozen soil
νf = frozen Poisson’s ratio
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Loading-collapse
Constitutive model for frozen/unfrozen soil
Plastic strains:
Segragation
M
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Model parameters
Elastic parameters
Strength
Primary loading
- isotropic stress
- freezing
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Model parameters
Parameter determination:
• From oedometer test (frozen/unfrozen):
(py0*)in, pc*, β, κ0, λ0, r
• From simple shear test:
G0, M
• From axial compression test:
Ef,ref, Ef,inc, νf, kt
• From frost heave test:
(sc,seg)in, κs, λs
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Applications – Frost heave
Chilled pipeline
Temperature
distribution
a. 40 days
b. Long time
253 K
Frost heave
after long time
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Applications – Thaw settlement
Footing on frozen soil
∆T = 2 K
500 kPa
Ice saturation
a. Initial
b. After warming
a. Initial
Settlement
b. After warming
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Conclusions
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Practical approach for θuw(T,p) and k(T,p) of frozen soils
Validated against data from literature
PSD seems to enable a good estimate of properties of frozen soils
Constitutive model for frozen / unfrozen soil
Typical phenomena of frozen soil
Successful applications:
• Frost heave
• Thaw settlement
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