Hypoplasticity for Practical Applications
Part 4: Determination of material parameters
David Mašín
Charles University in Prague
Zhejiang University
PhD course on hypoplasticity
Zhejiang University, June 2015
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Outline
7
Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Clay hypoplastic model (Mašín, 2014)
Intergranular strain concept (Niemunis and Herle, 1997)
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameter determination of sand hypoplasticity
The model by von Wolffersdorff (1996) is now considered as a
reference hypoplastic model for granular materials. It requires 8
material parameters:
ϕc is the critical state friction angle
hs and n control the shape of limiting void ratio curves (normal
compression lines and critical state line)
ed0 , ec0 and ei0 are reference void ratios specifying positions of
limiting void ratio curves
α controls the dependency of peak friction angle on relative
density
β controls the dependency of soil stiffness on relative density
Calibration procedure for the von Wolffersdorff hypoplastic model
detailed in Herle and Gudehus (1999).
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
The critical state is reached during monotonic shearing while both
the stress rate and the volumetric deformation rate vanish.
However, problem of shear testing is localisation of deformation
into shear bands. A suitable way for determination of ϕc is
measurement of the angle of repose ϕrep .
Not suitable for materials with grain size below 0.1 mm (increase
of ϕrep due to capillary water resulting from air humidity).
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
Measurement of the angle of repose corresponds well to the
measurement of ϕc in shear tests (Herle and Gudehus, 1999):
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
Principle of the measurement of the angle of repose: simple shear
type of deformation occurs under quite low confining pressure
within the thin surface layer:
Miura et al. (1997)
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
Sliding in the thin surface layer may be explained by ordinary
stability analysis of infinite slope:
Miura et al. (1997)
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
The sand is necessarily in a loose state, because of dilatancy
caused by shear deformation within the thin layer.
The sand heap is loose. Because ϕrep is a good approximation of
the mobilised friction angle, it is then also a good approximation of
the critical state friction angle.
The most common way of measurement ⇒ the funnel filled with
dry sand lifted vertically and slowly
to form a sand heap. Funnel is in
contact with the heap
Miura et al. (2010)
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
Herle (2010)
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
Miura et al. (1997), however, observed, that the measured angle of
repose depends on a number of factors, such as amount of sand,
roughness of the base plate and funnel lifting rate.
Miura et al. (1997)
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
Based on the findings of Miura et al. (1997)
The base should be rough to prevent sliding along the base
causing non-planar shear planes within the cone
The lifting should be slow to ensure quasi-static conditions
The amount of sand should be approx. 50 g.
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
Measurement of ϕc as the angle of repose is possible also in the
case of gravel:
Herle (2010)
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Part 4: Calibration of material parameters
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
Friction angle of gravel is larger than friction angle of sand
Herle (2010)
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
Measurement of ϕc by means of measurement of the angle of
repose is unsuitable for material with grain sizes below 0.1 mm.
In the case the amount of fines
is low (below cca 20%), they do
not have substantial influence on
ϕc . They can thus be sievedout and the angle of repose can
be mesured on the coarse-grained
fraction.
Different percentages of Kaolin in sand.
From Pitman et al. (1994)
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
In the case of silty soils, ϕc cannot be measured as the angle of
repose due to capillary effects. For these soils, however, the von
Wolffersdorff hypoplastic model may still be adequate.
In this case, shear experiments are required. It is necessary to
minimise the effects of localisation of deformation into shear band.
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
If shear banding occurs, the overall stress and strain measured does
not correspond to the state within the shear band (critical state is
achieved inside the shear band only).
(stereofotogrammetric measurements by Desrues and Viggiani 2004)
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
Shear-banding affects the measured stress-strain relationship:
(from Desrues and Viggiani 2004)
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Critical state friction angle ϕc
To limit the influence of localisation of deformation in triaxial tests
The sample should be in the loosest state possible:
shear-banding is related to the post-peak softening, which in turn
depends on relative density.
Frictionless plattens should be used.
Shear box test can be used to estimate ϕc . Unsuitable to measure
critical state void ratio.
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameters hs and n
Limiting void ratios by Gudehus (1996):
ei is maximum void ratio
(isotropic normal
compression)
ec is critical state void ratio
(CSL)
ed is minimal void ratio at the
state of maximum density
ei
− tr σ n
ec
ed
=
=
= exp −
ei0
ec0
ed0
hs
hs , n, ei0 , ec0 and ed0 are model parameters
(equation by Bauer, 1996)
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameters hs and n
In fact, in any proportional compression (constant direction of
strain rate), a normal compression line (NCL)
n 3p
ep = ep0 exp −
hs
is followed after reaching normally consolidated state. ep0 controls
position of the given NCL with ec0 < ep0 < ei0 .
Any proportional compression test can thus be used for the
determination of hs and n. Oedometric test is the most easy to
perform.
Oedometric test on loose soil, either dry or fully saturated.
It is NOT RECCOMENDED TO CALIBRATE hs AND n BY DIRECT
REGRESSION , but rather adopt the physical meaning of the
parameters →
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameters hs and n
The slope of the oedometric curve plotted in the ln σa vs. e space
described by the compression index Cc :
Cc =
∆e
∆ ln σa
During proportional loading along NCL, K0 is constant
3
3
ln σa = ln
p = ln
+ ln p
1 + 2K0
1 + 2K0
we thus also have
Cc =
David Mašín
∆e
∆ ln p
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameters hs and n
Time-differentiation of the Bauer formula yields
ne 3p n
ṗ
ė = −
p
hs
Comparing with the above equation for Cc , we have
hs = 3p
ne
Cc
1/n
where Cc is a tangent compression index.
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameters hs and n
Considering the value of Cc at two
different values of p, the above
equation for hs can be rewritten to
Cc2
ln ee21 C
c1
n=
ln (p2 /p1 )
⇒ direct calculation of n from Cc1
and Cc2
hs can then be calculated from the secant compression index in
the range p1 to p2 using equation on the previous slide.
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameters hs and n
The influence of the parameters hs and n on the shape of compression
curves: hs controls the overall slope and n controls curvature.
Herle and Gudehus (1999)
.
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameter ec0
The parameter ec0 defines the position of the critical state line in the
ln p vs. e space:
n 3p
ec = ec0 exp −
hs
The most appropriate way for its
determination is thus through evaluation of undrained triaxial shear
tests.
It is advisable, however, to use the parameters hs and n evaluated
from the oedometric compression as explained above. Thus, only ec0
is varied while fitting the CSL data.
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameter ec0
As in the case of the ϕc determination, calibration of ec0 using
results of shear tests is problematic due to shear banding.
⇒ Sample as loose as possible, frictionless platens.
Simpler way of its determination is based on the following idea:
soil heap during evaluation of the angle of repose is close to the
critical state. Such a soil is close to the loosest state, and stresses
are due to small size of the heap small.
Soil for the oedometric test performed for the determination of hs
and n is also aimed to be in the loosest state:
Initial void ratio of the oedometric test on loose soil can be thus
considered as appropriate estimate of ec0 ⇒ no additional
experiment needed for the determination of ec0 .
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Calculation of hs , n and ec0 in real calibration
Now I explain how to calibrate the parameters hs , n and ec0 for a
real oedometric test on loose sample with 6 loading steps.
0.84
0.82
void ratio e [-]
0.8
experiment
calibration
a
b
0.78
c
0.76
d
0.74
e
0.72
0.7
0.68
f
0.66
100
1000
mean stress p [kPa]
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Calculation of hs , n and ec0 in real calibration
The following procedure leads to the best fit:
0.84
0.82
a
0.8
void ratio e [-]
eb − ea
ln pb − ln pa
ef − ee
Cc2 = −
ln pf − ln pe
ed − ec
Cc = −
ln pd − ln pc
Cc1 = −
experiment
calibration
b
c
0.78
0.76
d
0.74
0.72
e
e1 = (ea + eb )/2
0.7
0.68
e2 = (ee + ef )/2
f
0.66
100
e = (ec + ed )/2
1000
mean stress p [kPa]
p1 = exp
ln pa + ln pb
2
p2 = exp
ln pe + ln pf
2
p = exp
ln pc + ln pd
2
ec0 subsequently adjusted so the curve fits vertically.
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameter ec0
Comparison of "nominal" end predicted K0 test
Unification of ec line with K0 line for calibration purposes is
approximation only, but the discrepancy is relatively minor
(depends on the other model parameters)
0.95
simulation
direct calibration
0.9
void ratio e [-]
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
100
1000
mean stress p [kPa]
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameter ei0
Parameter ei0 controls the position of the isotropic compression
line. For the given mean stress, ei represents the theoretical
loosest possible state.
Difficult to measure experimentally: emax reached while preparing
the loose soil sample corresponds to ec0 . During isotropic
compression, the compression line converges towards the ideal
NCL very slowly. ei in fact represents theoretical emax in
gravity-free space.
Herle and Gudehus (1999) suggest
ei0 = 1.2ec0 , based on a study of
idealised loosest packing of
spherical particles.
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameter ed0
Parameter ed0 controls position of the minimum void ratio line.
The best densification can be obtained by means of cyclic
shearing with small amplitude at constant pressure.
ed0 then obtained by extrapolation using hs and n evaluated from
oedometric experiments results.
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameter ed0
Alternative empirical approach: Herle and Gudehus (1999)
evaluated ed0 and ec0 on 7 different granular soils. The ratio
ed0 /ec0 varied within the range 0.52-0.64.
ed0 = 0.5ec0 would be a reasonable estimate of ed0 in the case
the minimum void ratio test is not available.
Lower bound of the range by Herle and Gudehus (1999) to ensure
the inallowed state e < ed is not reached.
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameter α
Parameter α controls the dependency of peak friction angle ϕp on
relative void ratio
e − ed
re =
ec − ed
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Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameter α
Calibration by means of single element simulation of triaxial shear
test on soil with re < 1.
450
400
350
q [kPa]
300
250
200
150
α=0.05
α=0.10
α=0.13
α=0.15
α=0.20
100
50
0
0
David Mašín
0.05
0.1
0.15
0.2
εs [-]’
0.25
Part 4: Calibration of material parameters
0.3
0.35
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Parameter β
The parameter β enters the expression of fs ⇒ it influences the
size of the response envelope (both bulk and shear stiffness).
400
350
300
q [kPa]
It is best
calibrated by
means of fitting
shear stiffness
in triaxial shear
test.
250
200
150
100
β=0.5
β=1.0
β=1.5
β=2.0
50
0
0
David Mašín
0.02
0.04
0.06
0.08
εs [-]’
Part 4: Calibration of material parameters
0.1
0.12
0.14
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Minimalistic experimental programme
"Minimalistic" experimental programme for von Wolffersdorff
(1996) hypoplastic model:
Angle of repose test: Parameter ϕc .
One oedometric test on initially loose sample: hs , n, ec0 ; empirically
ei0 and ed0 .
One drained triaxial shear test on initially dense sample: α and β.
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Performance of the model
using minimalistic (3-test) experimental programme
Research project on applicability of hypoplastic model in
probabilistic numerical analyses: large number of specimens
needed, minimum number of experiments possible.
Suchomel and Mašín (2011)
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Performance of the model
using minimalistic (3-test) experimental programme
Predictions of oedometric tests [Mašín (2015). The influence of
experimental and sampling uncertainties on the probability of unsatisfactory
1
1
0.9
0.9
0.8
0.8
0.7
0.7
e [-]
e [-]
performance in geotechnical applications. Géotechnique (in print)].
0.6
0.5
0.6
0.5
0.4
natural variability
- experiments
0.4
0.3
natural variability
- simulations
0.3
10
100
1000
10000
10
p [kPa]
David Mašín
100
1000
10000
p [kPa]
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Performance of the model
using minimalistic (3-test) experimental programme
1000
1000
800
800
q [kPa]
q [kPa]
Predictions of drained triaxial tests (Mašín, 2015).
600
400
natural variability
- experiments
400
200
natural variability
- simulations
200
0
0
0
0.02
0.04
0.06
0.08
εs [-]
0.1
0.12
0.14
0
0.5
0.02
0.04
0.06
0.08
εs [-]
0.1
0.12
0.14
0.08
εs [-]
0.1
0.12
0.14
0.5
natural variability
- experiments
0.45
natural variability
- simulations
0.45
0.4
e [-]
0.4
e [-]
600
0.35
0.35
0.3
0.3
0.25
0.25
0
David Mašín
0.02
0.04
0.06
0.08
εs [-]
0.1
0.12
0.14
0
0.02
0.04
Part 4: Calibration of material parameters
0.06
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Range of parameters for different soils
Herle and Gudehus (1999) evaluated parameters of von Wolffersdorff
hypoplastic model for 7 different granular soils.
Hochstetten gravel
Hochstetten sand
Hostun sand
Karlsruhe sand
Lausitz sand
Toyoura sand
Zbraslav sand
min.
max.
David Mašín
ϕc
[◦ ]
36
33
31
30
33
30
31
30
36
hs
[GPa]
32
1.5
1.0
5.8
1.6
2.6
5.7
1.0
32
n
[-]
0.18
0.28
0.29
0.28
0.19
0.27
0.25
0.18
0.29
ed0
[-]
0.26
0.55
0.61
0.53
0.44
0.61
0.52
0.26
0.61
ec0
[-]
0.45
0.95
0.96
0.84
0.85
0.98
0.82
0.45
0.98
Part 4: Calibration of material parameters
ei0
[-]
0.5
1.05
1.09
1
1
1.1
0.95
0.5
1.1
α
[-]
0.1
0.25
0.13
0.13
0.25
0.18
0.13
0.1
0.25
β
[-]
1.9
1.5
2
1
1
1.1
1.0
1
2
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Determination of material parameters
Sand hypoplastic model (von Wolffersdorff, 1996)
Range of parameters for different soils
1.4
0.5
1.2
0.4
0.3
0.2
0.1
34
36
φc [°]
38
40
Rel. frequency
4
3.5
3
2.5
2
1.5
1
0.5
0
8
0.2
0.4
0.6
0.8
ec0
David Mašín
1
1.2
1.4
9
10 11 12 13 14 15 16 17
ln(hs/1 kPa)
0.6
0.4
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
-2.5
-2
-1.5
-1
ln(n)
-0.5
0
3
4
1.2
1
Rel. frequency
32
1
0.8
0.2
0
30
Rel. frequency
0.6
Rel. frequency
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Rel. frequency
Rel. frequency
Variability of parameters in 40 samples (Mašín, 2015):
0.8
0.6
0.4
0.2
0
-6
-5
-4
-3
ln(α)
-2
-1
0
Part 4: Calibration of material parameters
-1
0
1
2
β
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Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Parameter determination of clay hypoplasticity
CLAY HYPOPLASTICITY
The clay hypoplasticity model requires altogether 5 material
parameters. The parameters are equivalent (but not identical) to
the parameters of the Modified Cam-clay model.
ϕc is the critical state friction angle
N and λ∗ control the position and slope of the isotropic normal
compression line
κ∗ controls the slope of the isotropic unloading line
ν controls the shear stiffness
Here I describe only calibration of the parameters of the basic
model. Important is consideration of the effects of structre (Part 6).
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Critical state friction angle ϕc
Unlike in the case of hypoplasticity for granular materials, ϕc of
clays cannot be calibrated using the simple angle of repose test.
On the other hand, we may take advantage of the fact that the
natural and reconstituted soils have the same ϕc .
From the reconstituted soil we easily create "loose" (soft, normally
consolidated) sample ⇒ less susceptible to shear banding.
ϕc is thus preferably measured on reconstituted, normally
consolidated sample using undrained triaxial shear test
(undrained just because it is faster than drained).
To ensure as much as possible homogeneous deformation,
frictionless platens should preferably be used.
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
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Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Parameters controlling NCL (N and λ∗ )
Parameters N and λ∗ straightforward to calibrate using isotropic
compression test on reconstituted soil and natural soil with stable
structure. Much more complicated in natural soils with the effects
of structure – Part 6.
1.05
ln (1+e)
experiment
κ*=0.005
κ*=0.010
κ*=0.015
1
0.95
Isotr. normal compression line
current state
Isotr. unloading line
κ*
ln (1+e) [-]
N
0.9
0.85
0.8
0.75
0.7
1
0.65
λ*
Critical state line
pcr
David Mašín
p*e
0.6
1
0
1
ln p
Part 4: Calibration of material parameters
2
3
4
ln p/pr [-]
5
6
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Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Parameters controlling NCL (N and λ∗ )
Isotropic vs. oedometric compression
The parameter N specifies the position of the isotropic normal
compression line.
However, it is often useful to calibrate N using results of
oedometric compression test – easier and faster to perform.
q
State Boundary Surface
CSL
K0 NCL
p’
isotropic NCL
When projected onto
the p vs. e plane,
isotropic and
oedometric normal
compression lines are
shifted with respect to
each other
e
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
47 / 71
Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Parameters controlling NCL (N and λ∗ )
Isotropic vs. oedometric compression
When plotted in the ln p vs. ln(1 + e) plane, the isotropic and
oedometric normal compression lines are parallel.
0.65
0.6
ln (1+e)
0.55
0.5
∆N
0.45
0.4
0.35
0.3
isotropic compression
oedometric compression
0.25
2
2.5
3
3.5
4
David Mašín
4.5 5
ln p/pr
5.5
6
6.5
7
The simplest way to evaluate ∆N
corresponding to the particular soil
parameters is simulation of isotropic
and oedometric tests. ∆N is the
vertical offset of the normal
compression lines in the ln p vs.
ln(1 + e) plane.
Part 4: Calibration of material parameters
PhD course on hypoplasticity
48 / 71
Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Parameters controlling NCL (N and λ∗ )
Isotropic vs. oedometric compression
The oedometric results need to be plotted in terms of ln p, not
ln σv .
Calculate p from σv
1 + 2K0
3
using Jáky formula K0 = 1 − sin ϕc .
p = σv
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
49 / 71
Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Parameter κ∗
Within Modified Cam clay model, the parameter κ represents the
slope of unloading line in the ln p vs. e plane. The slope of
unloading line is constant.
In hypoplasticity, there is also parameter κ∗ , but the model
predicts non-linear behaviour even inside SBS.
This difference between Modified Cam-clay and hypoplasticity
must be considered in the model calibration.
κ∗ should preferably be calibrated using experiments on
undisturbed soil samples (they are performed anyway for the
calibration of N).
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
50 / 71
Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Parameter κ∗
ln(1+e) [-]
Unlike in elasto-plasticity, in hypoplasticity κ∗ represents exactly
the slope of unloading line in the ln p vs. ln(1 + e) plane at
unloading from normally consolidated state
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
elasto-plasticity
hypoplasticity
2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
ln(p/pr) [-]
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
51 / 71
Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Parameter κ∗
The slope of unloading (loading) line is controlled by the
parameters κ∗ , but the model takes over control of the
non-linearity
1.05
0.95
ln (1+e) [-]
κ∗ thus should be calibrated
by parametric study by
simulation of the isotropic /
oedometric test
experiment
κ*=0.005
κ*=0.010
κ*=0.015
1
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0
1
David Mašín
2
3
4
ln p/pr [-]
5
6
7
Not as λ∗ by a direct
measurement of the slope of
the loading/unloading line.
Part 4: Calibration of material parameters
PhD course on hypoplasticity
52 / 71
Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Parameter κ∗
-σa [kPa]
The ratio λ∗ /κ∗ controls size of response envelope. It thus,
indeed, controls also undrained stress paths.
240
220
200
180
160
140
120
100
κ*=0.007
κ*=0.010
κ*=0.015
0
David Mašín
50
100
150
200
-σr√2 [kPa]
250
Part 4: Calibration of material parameters
300
350
PhD course on hypoplasticity
53 / 71
Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Parameter ν
The parameter ν controls the shear modulus (increase of ν
decreases the shear modulus).
Preferable way of calibration of ν are undrained shear tests on
undisturbed soil.
500
450
400
q [kPa]
350
300
250
200
150
experiment
ν=0.2
ν=0.33
ν=0.4
100
50
0
0
David Mašín
0.02 0.04 0.06 0.08
εs [-]
0.1
0.12 0.14 0.16
Part 4: Calibration of material parameters
PhD course on hypoplasticity
54 / 71
Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Parameter ν
-σa [kPa]
The parameter ν controls shape of response envelope, and thus
also undrained stress path (similarly to parameter κ∗ ).
240
220
200
180
160
140
120
100
ν=0.10
ν=0.27
ν=0.34
0
David Mašín
50
100
150
200
-σr√2 [kPa]
250
Part 4: Calibration of material parameters
300
350
PhD course on hypoplasticity
55 / 71
Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Initial value of e
The initial void ratio in the simulation of the shear test must
correspond to the adopted position of NCL so that the implied
OCR is correct.
If e from compression and shear tests (for the same state) is
different due to the experimental scatter, use e from the
compression test.
Alternatively, it is possible to directly initialise OCR.
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
56 / 71
Determination of material parameters
Clay hypoplastic model (Mašín, 2014)
Range of parameters for different soils
Parameters of the clay hypoplastic model from different sources
Brno clay
London clay
Kaolin
Dortmund clay
Weald clay
Koper silt
Fujinomori clay
Pisa clay
Beaucaire clay
Trmice clay
min.
max.
David Mašín
ϕc
22◦
21.9◦
27.5◦
27.9◦
24◦
33◦
34◦
21.9◦
33◦
18.7◦
18.7◦
34◦
λ∗
0.128
0.095
0.065
0.057
0.059
0.103
0.045
0.14
0.06
0.09
0.045
0.14
κ∗
0.015
0.015
0.01
0.008
0.018
0.015
0.011
0.01
0.01
0.01
0.01
0.02
Part 4: Calibration of material parameters
N
1.51
1.19
0.918
0.749
0.8
1.31
0.887
1.56
0.85
1.09
0.85
1.51
ν
0.33
0.1
0.35
0.38
0.3
0.28
0.36
0.31
0.21
0.09
0.09
0.38
PhD course on hypoplasticity
57 / 71
Determination of material parameters
Intergranular strain concept (Niemunis and Herle, 1997)
Intergranular strain concept
INTERGRANULAR STRAIN CONCEPT
The intergranular strain concept Niemunis and Herle (1997) enables to
model small-strain-stiffness effects in hypoplasticity. It requires 5
material parameters:
mR : parameter controlling the initial (very-small-strain) shear
modulus upon 180◦ strain path reversal and in the initial loading
(supplemented by Ag and ng in the clay model).
mT : parameter controlling the initial shear modulus upon 90◦
strain path reversal (supplemented by mrat in the clay model)
R: The size of the elastic range (in the strain space)
βr and χ: control the rate of degradation of the stiffness with strain.
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
59 / 71
Determination of material parameters
Intergranular strain concept (Niemunis and Herle, 1997)
Intergranular strain concept
The intergranular strain concept may be used with both von
Wolffersdorff and clay hypoplasticity without any modification.
Response of the combined model then, however, depends on the
basic model used.
When used with clay hypoplasticity, model can be calibrated
directly to fit the initial stiffness G0 :
ng
p
G0 = pr Ag
pr
The parameters Ag and ng supplement the parameter mR of the
sand model.
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
60 / 71
Determination of material parameters
Intergranular strain concept (Niemunis and Herle, 1997)
Parameters Ag and ng (clay model)
Representation of the bender element measurements on clay:
160
140
Gvh0 [MPa]
120
100
80
G 0 = pr Ag
60
p
pr
ng
40
experiment
Gvh0=Ag (p/pr)ng
linear dependency
20
0
0
100
200
David Mašín
300
400 500
p [kPa]
600
700
800
Part 4: Calibration of material parameters
PhD course on hypoplasticity
61 / 71
Determination of material parameters
Intergranular strain concept (Niemunis and Herle, 1997)
Parameter mR (sand model)
Using von Wolffersdorff hypoplasticity, G0 is proportional to p(1−n)
G0 ∼ p(1−n)
1200
1000
G0 [MPa]
800
mR is a proportionality
constant. Calibrated by a
parametric study.
600
400
200
von Wolffersdorff, n=0.25
linear dependency
0
0
1000
David Mašín
2000
3000
p [kPa]
4000
5000
Part 4: Calibration of material parameters
PhD course on hypoplasticity
62 / 71
Determination of material parameters
Intergranular strain concept (Niemunis and Herle, 1997)
Parameter mT (mrat in clay model)
Parameter mT (or mrat = mT /mR in the clay model) is difficult to
calibrate.
The ratio mrat is the ratio G90 /G0 , where G90 is the initial shear
stiffness after 90◦ change of strain path direction.
G90 cannot be measured by bender element tests. We need
accurate strain measurements using local strain transducers. Still,
it is hard to estimate G0 /G90 .
Experiments with different strain path direction change with local
strain measurements allow us to evaluate the ratio of shear moduli
at larger strains. They may be used to estimate mrat .
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
63 / 71
Determination of material parameters
Intergranular strain concept (Niemunis and Herle, 1997)
Parameter mT (mrat in clay model)
Examples are experiments by Richardson (1988). They indicate
mrat ≈ 0.7
Richardson, 1988; Atkinson et al., 1990
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
64 / 71
Determination of material parameters
Intergranular strain concept (Niemunis and Herle, 1997)
Parameter R
The remaining parameters R, βr and χ need to be calibrated by
means of a parametric study by fitting the stiffness degradation
curve obtained using accurate local strain measurements
The influence of the parameter R (size of elastic range):
140
exp., 226gUC
exp., 25gUC
exp., 23gUE
R=1.e-5
R=5.e-5
R=1.e-4
120
G [MPa]
100
80
60
40
20
0
1e-06
David Mašín
1e-05
0.0001
εs [-]
0.001
Part 4: Calibration of material parameters
0.01
PhD course on hypoplasticity
65 / 71
Determination of material parameters
Intergranular strain concept (Niemunis and Herle, 1997)
Parameters βr and χ
βr and χ control the shape of the stiffness degradation curves
140
80
100
60
80
60
40
40
20
20
0
1e-06
1e-05
0.0001
εs [-]
0.001
exp., 226gUC
exp., 25gUC
exp., 23gUE
χ=1
χ=0.5
χ=2
120
G [MPa]
100
G [MPa]
140
exp., 226gUC
exp., 25gUC
exp., 23gUE
βr=0.1
βr=0.033
βr=0.3
120
0.01
0
1e-07
1e-06
1e-05
0.0001
εs [-]
0.001
0.01
We see that R has similar influence on the stiffness curve as βr . It
can then be advised to treat R = 10−4 as material independent
constant, and control the stiffness curve by βr and χ.
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
66 / 71
Determination of material parameters
Intergranular strain concept (Niemunis and Herle, 1997)
Calibration using results of cyclic tests
The above calibration procedure is suitable for simulation of
geotechnical problems with continuous loading or several strain
path reversals.
However, the hypoplastic model with intergranular strains also
enables us to model the effects of cyclic loading.
In the case of simulation of cyclic loading problem, it is preferable
to use cyclic loading tests also for the model calibration.
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
67 / 71
Determination of material parameters
Intergranular strain concept (Niemunis and Herle, 1997)
Calibration using results of cyclic tests
Using the results of cyclic tests only,
it is difficult to distingush the
influence of individual parameters
⇒ Cyclic undrained triaxial test, the
influence of mR
100
100
no istr.
mR=5
50
q [kPa]
q [kPa]
50
mR=8
mR=10
0
-50
0
-50
-100
-100
0
50
David Mašín
100
150
p [kPa]
200
250
0
50
Part 4: Calibration of material parameters
100
150
p [kPa]
200
PhD course on hypoplasticity
250
68 / 71
Determination of material parameters
Intergranular strain concept (Niemunis and Herle, 1997)
Calibration using results of cyclic tests
The parameters βr and χ, however, have similar influence on the
cyclic behaviour as mR
βr=0.033
βr=0.1
βr=0.3
100
50
q [kPa]
50
q [kPa]
χ=2
χ=1
χ=0.5
100
0
-50
0
-50
0
50
David Mašín
100
150
p [kPa]
200
250
0
50
Part 4: Calibration of material parameters
100
150
p [kPa]
200
PhD course on hypoplasticity
250
69 / 71
Determination of material parameters
Intergranular strain concept (Niemunis and Herle, 1997)
Calibration using results of cyclic tests
A possible simplified way to model calibration could then be
Treat R = 10−4 and χ = 1 as material independent constants.
Calibrate mR (Ag and ng for the clay model) using bender element
measurements (they are relatively easy to perform).
Set mT = 0.7mR (mrat = 0.7).
Control the cyclic behaviour using the parameter βr only.
David Mašín
Part 4: Calibration of material parameters
PhD course on hypoplasticity
70 / 71
Determination of material parameters
Intergranular strain concept (Niemunis and Herle, 1997)
Parameters for different soils
Not many rigorously calibrated parameters sets for the intergranular
strain concept available. Even some parameters in the following tables
are estimates only.
Hochstetten sand
Karlsruhe sand
R
1x10−4
5x10−5
London clay (data Gasparre)
Brno clay (nat.)
David Mašín
mR
5
6.7-12 (p dep.)
R
5.e-5
1e-4
βr
0.08
0.2
χ
0.9
0.8
Part 4: Calibration of material parameters
mT
2
-
Ag
270
5300
βr
0.5
0.3
ng
1
0.5
χ
6
1
mrat
0.5
0.5
PhD course on hypoplasticity
71 / 71
Hypoplasticity for Practical Applications
Part 6: Advanced modelling approaches
David Mašín
Charles University in Prague
Zhejiang University
PhD course on hypoplasticity
Zhejiang University, June 2015
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
1 / 163
Outline
10
Modelling the effects of structure (fabric, bonding, crushing)
11
Stiffness anisotropy
12
Unsaturated soils
13
Expansive soils
14
Thermal effects
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
2 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
To quantify the effects of structure on mechanical behaviour, we
first need to define a reference material - material without
inter-particle bonds and with sort-of "standard" structure.
As an international consensus, so-called reconstituted soil as
defined by Burland (1990) is used. By Burland’s definition, it is
created by thorough mixing of natural soil at water content 1 to 1.5
times higher then liquid limit wL . Mix in distilled de-aired water.
Mixed with water to form slurry without drying prior to mixing.
Samples then prepared by consolidation up to stresses high
enough so they can be handled (cca 70 kPa) under 1D conditions.
Low stress enough so permanent effects of cross-anisotropy are
not induced.
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
4 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Structure:
Fabric – arrangement of particles. Predominantly created during
soil sedimentation. Influenced greatly by the chemistry of
sedimentation environment.
Bonding – cementation bonds between individual particles. Formed
by precipitation of bonding chemicals during subsequent
diagenetical processes.
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
5 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Dispersed and flocculated fabric
Clay fabric depends on the sedimentation environment:
In neutral environment (pH=7), edges and
faces are neutrally charged ⇒ dispersed
structure.
Feda (1982)
In acid environment (pH<7), edge charge
is positive and face charge is negative ⇒
edge-to-face flocculated structure.
Feda (1982)
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
6 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Dispersed and flocculated fabric
In alkaline environment (pH>7), edge
charge is negative and face positive.
However, repulsive double layer forming
around faces is very thin, and attractive
van der Waals-London forces prevail ⇒
"salty" flocculation.
Feda (1982)
Distilled water has pH=7 ⇒ Fabric of reconstituted soil will tend to
be dispersed.
Sea water has 7.9 < pH < 9 ⇒ tends to create "salty flocculated"
fabric.
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
7 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Dispersed and flocculated fabric
Fabric of a real soil is much more complex. In any case, it turns
out that the undisturbed soil has at the same stress more open
structure than the reconstituted soil
Reconstituted soil
Undisturbed soil
Sides and Barden (1970)
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
8 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Dispersed and flocculated fabric
• The more open structure of undisturbed soil is manifested in different
positions of normal compression lines in the ln(1 + e) vs. ln p space:
e
natural, only fabric
reconstituted
ln p’
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
9 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Dispersed and flocculated fabric
Let’s interpret this behaviour using the Cam-clay concept.
q/pe*
standard CC
1
2
p/pe*
Natural soil should have higher undrained shear strength than the
reconstituted one at the same void ratio.
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
10 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Dispersed and flocculated fabric
Let’s interpret this behaviour using the Cam-clay concept.
q/pe*
structured CC, s=2
standard CC
1
2
p/pe*
Natural soil should have higher undrained shear strength than the
reconstituted one at the same void ratio.
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
11 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Dispersed and flocculated fabric
This really is the case. Recall
the well-known example of
quick clay.
Sedimented in salty
environment, but then uplift
caused the quick clay to rise
above current water level and
the salt was leached out from
the pores.
Structure broken more easily
than by thorough
reconstitution.
David Mašín
Mitchell
(1993)
course on hypoplasticity
Part 6: Advanced modelling approaches
12 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Dispersed and flocculated fabric
Typically, soils with fabric-dominated structure have normal
compression lines parallel to NCL of reconstituted soil. Structure
remains = stable structure:
Gullá et al. (2006)
David Mašín
Takahashi et al. (2006)
Part 6: Advanced modelling approaches
course on hypoplasticity
13 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Dispersed and flocculated fabric
It is important to point out that also reconstituted soil has its
structure. When the sample is prepared in different way, different
response. Reconstitution using Burland’s definition = standard.
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
14 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
Behaviour of fine-grained soils
Also structure, as observed by scanning electron
microphotography, is different: (Fearon and Coop, 2000).
minced
reconstituted
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
15 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Bonding
• Diagenesis (post-sedimentation processes) often causes
precipitation of chemical agents between particle contacts =
cementation, bonding
e
natural, only fabric
reconstituted
ln p’
Bonding increases the resistance of the skeleton, but up to certain
stress only. Then, bonds break.
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
16 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Bonding
• Diagenesis (post-sedimentation processes) often causes
precipitation of chemical agents between particle contacts =
cementation, bonding
e
natural, fabric and bonding
natural, only fabric
reconstituted
ln p’
Bonding increases the resistance of the skeleton, but up to certain
stress only. Then, bonds break.
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
17 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Bonding
Typical example is compression behaviour of soft clay.
Nash et al. (1992)
David Mašín
Callisto and Rampello (2004)
Part 6: Advanced modelling approaches
course on hypoplasticity
18 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Sensitivity framework
Framework for the behaviour of
structured soils – sensitivity framework by Cotecchia and Chandler
(2000).
Undisturbed soil has larger
undrained shear strength (cu ) than
the reconstituted soil at the same
void ratio. Their ratio = strength
sensitivity
Su =
David Mašín
cunat
curec
Mitchell
(1993)
course on hypoplasticity
Part 6: Advanced modelling approaches
19 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Sensitivity framework
• Compression behaviour is used to define stress sensitivity:
e
natural, fabric and bonding
Stress sensitivity:
natural, only fabric
Sσ =
pnat
prec
reconstituted
prec
David Mašín
pnat
ln p’
Part 6: Advanced modelling approaches
course on hypoplasticity
20 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
The influence of structure on clay behaviour
Sensitivity framework
Sensitivity framework (Cotecchia and Chandler, 2000):
Strength sensitivity is equal to stress sensitivity, i.e. Sσ = Su .
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
21 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
Modelling the effects of clay structure
The primary features of structured soil behaviour are modelled by
variation of the size of state boundary surface (Rouainia and Muir
Wood, 2000; Kavvadas and Amorosi, 2000; Baudet and
Stallebrass, 2004).
ln (1+ e )
N nat
Due to the effects of structure,
two different values of N may be
distinguished:
λ* ln s
N rec
current state
κ*
Current SBS, nat.
1
Isot. unl.
SBS for s= 1
ln s
Clearly, Nnat = Nrec + λ∗ ln s
Sensitivity s is additional state
variable.
λ*
1
0
p*e
David Mašín
Nrec : Reconstituted soil
Nnat : Natural soil
s p*e
ln p
Part 6: Advanced modelling approaches
course on hypoplasticity
22 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
Modelling the effects of clay structure
In models for clays with meta-stable structure, sensitivity
decreases with loading.
ln (1+ e )
N rec
NCL nat. (UCL)
ln s
ln s
NCL rec.
λ*
1
0
David Mašín
Part 6: Advanced modelling approaches
ln p
course on hypoplasticity
23 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
Modelling the effects of clay structure
For demonstration purposes, hypoplastic model for structured
clays (Mašín, 2007), but the other models similar in principle.
Sensitivity additional state variable, such that Nnat = N + λ∗ ln s
Evolution equation for sensitivity:
ṡ = −
k
(s − sf )˙d
λ∗
where k and sf are additional parameters. ˙d is a "damage strain
rate", defined as
r
A
d
˙ = (˙v )2 +
(˙s )2
1−A
Parameter A specifies the relative influence of volumetric and
shear strain rate on the rate of structure degradation.
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
24 / 163
Fabric, bonding, crushing
Structure of fine-grained soils
Modelling the effects of clay structure
The parameter k specifies the rate of structure degradation.
1.1
1
ln (1+e) [-]
0.9
0.8
0.7
k=0
0.6
k=0.4
k=0.7
k=1
0.5
4.5
David Mašín
5
5.5
6
6.5
ln (p/pr) [-]
7
Part 6: Advanced modelling approaches
7.5
8
course on hypoplasticity
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Fabric, bonding, crushing
Structure of fine-grained soils
Modelling the effects of clay structure
A controls the influence of shear strains on structure degradation.
Calibration by simulation of the shear test.
60
50
q [kPa]
40
30
20
stable structure
A=0.1
A=0.2
A=0.5
10
0
0
20
40
60
80
100
p [kPa]
David Mašín
Part 6: Advanced modelling approaches
course on hypoplasticity
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Fabric, bonding, crushing
Structure of fine-grained soils
Modelling the effects of clay structure
The parameter sf quantifies stable elements of structure (Baudet
and Stallebrass 2004). In many soft clays, sf = 1.
ln (1+ e )
N rec
NCL nat. final
NCL rec.
λ*
1
0
David Mašín
ln sf
ln p
Part 6: Advanced modelling approaches
course on hypoplasticity
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Fabric, bonding, crushing
Structure of fine-grained soils
Modelling the effects of clay structure
Example of predictions: stress paths normalised with respect to pe∗ of
experiments of natural and reconstituted Pisa clay (experiments by
Callisto and Calabresi, 1998)
2
2
A135
1.5
A90
A30
1
A180
1
A0
R0
0
SOMS rec.
R315
-1
A280
0
0.5
David Mašín
1
1.5
2
p/p*e
2.5
R90
R60 R30
R0
0
R315
-0.5
A315
-1
hypo., reconst.
hypo., nat.
-1.5
A60
A180
0.5
R90 R60 R30
-0.5
A90
A30
SOMS nat.
A0
q/p*e
q/p*e
0.5
A135
1.5
A60
3.5
A315
experiment, reconst.
experiment, nat.
-1.5
3
A280
0
0.5
Part 6: Advanced modelling approaches
1
1.5
2
p/p*e
2.5
3
course on hypoplasticity
3.5
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