Unified Critical State Sand Model in Flow-Liquefaction Deformation Analysis
X.S. LI
Hong Kong University of Science & Technology
Abstract
The extent of flow liquefaction deformation depends on the stress and material states of the
soil in question. Knowing the fact that the stress state and the material state in an earth structure
vary from location to location as well as from time to time during an earthquake, a constitutive
model adopted for earthquake response analysis involving flow-liquefaction deformation should
take care of the spatial and temporal variations of the soil behavior associated with the changes
in stress and material states, in particular the state of soil dilatancy, which plays a significant role
in flow liquefaction and is a strong function of soil density, confining pressure, and stress ratio.
In this regard, traditional sand models with model constants dependent on soil initial state are a
bit farfetched for post-liquefaction deformation analysis.
Furthermore, one of the major obstacles to using sophisticated analysis procedure in practice
is the complexity involved in determination of constitutive model constants. For models whose
constants are sensitive to soil state, it is unavoidable to involve multiple sets of model constants,
that are site/location-specific, to cover different characteristics of possible responses attributed to
different in situ states. This feature renders using such models in practice impractical.
This paper describes a framework for unified modeling of sand behavior. In this modeling
framework, model constants are considered as intrinsic soil properties, independent on stress
state and material state. A single set of model constants, once calibrated, automatically takes care
of a wide range of ‘on-process’ state variation. As a result, for a given soil or similar soils, either
in a very loose liquefiable state or in a dense dilative state, the responses from essentially elastic
response at the beginning up to critical state failure, including flow liquefaction response, are
modeled in a unified manner, without intervention from users through adjusting model constants.
The case of the Upper San Fernando Dam during the 1971 earthquake is used as an example of
numerical analyses using such a model.
Introduction
Deformations and displacements induced by flow liquefaction are central considerations in
assessment of seismic performance of earth structures and for recommendation of remediation
measures. In this regard, flow-liquefaction displacement analysis has already been a significant
part of geotechnical earthquake engineering practice. The predictive capability of displacement,
however, is largely restricted by the availability of a reliable model that can faithfully describe
the stress-strain-strength behavior of granular soils during the entire earthquake, starting from an
essentially elastic response up to the steady state failure, accompanied with an excessive amount
of deformation.
Whether a soil is flow liquefiable, as well as the extent of flow deformation, is dependent on
the soil density state and on the magnitude and direction of the driving shear stress. For a sand
subjected to shear, it contracts (a necessary condition for flow liquefaction) if it is in a loose state
and dilates if it is in a dense state. Whether a sand is in a loose or a dense state depends not only
on the physical density but also on the effective confining pressure applied. Furthermore, for a
sand, which initially is either in the loose or dense state, the behavior may change considerably
during loading, due to possible loading-induced changes in its density state.
In an earth structure the in situ density states of a soil are in general heterogeneous, due to the
spatial variations in density and confining pressure. During a strong earthquake, accompanying
the generation, redistribution, and dissipation of excess pore pressures, the effective confining
pressures in soils surely change significantly, and the soil densities in liquefiable zones may also
change considerably, rendering the density state in an earth structure not only a spatial but also a
temporal function. Moreover, when large deformation occurs after liquefaction is triggered, the
configuration of an earth structure may change noticeably, resulting in certain changes in the
driving stresses. Changes in soil density and stress states during an earthquake may result in possible switches of soil response from initially dilative to contractive (flow liquefiable), or vice
versa. For those sand models whose responses, either contractive or dilative, are predetermined
by the given initial state through picking specific material constants, the above described ‘onprocess’ variation of material behavior cannot be properly captured. In addition, depending on
the spatial distribution of the initial states in an earth structure, such models need multiple sets of
model constants individually calibrated for the same physical material but in different states.
To fully capture the impact of the stress and density states, and their spatial and temporal
variations, it is essential for a constitutive model to clearly separate the influences of stress and
material states from those of intrinsic soil properties. The former should be characterized by a
group of state variables and the latter by a group of material constants, independent of the stress
and material internal states. This approach not only allows an automatic tracing of the temporal
changes in stress and material states, but also eliminates the need of multiple sets of model constants for spatial variations of initial states. It is clear that this approach unifies the modeling of
state-dependent soil responses.
It has been identified that the key issue in such a unified approach is to properly formulate
soil dilatancy. It has been pointed out by Li and Dafalias (2000) that the classical stress dilatancy
theory in its exact form is incapable of doing this, and the dilatancy of a granular soil must be a
function of stress state as well as of the internal state of the material. Such a state dependent dilatancy in conjunction with the critical state framework forms a platform that can effectively
model both the contractive and dilative behavior of granular soils over a wide range of material
densities and confining pressures, using a single set of material constants.
A sand model of this type has been integrated into a fully-coupled finite-element procedure
of Biot type to analyze various flow-liquefaction cases, including the cases of the well-known
San Fernando Dams during the 1971 earthquake.
State-dependent dilatancy
In this paper, it is assumed that there exists a unique critical state line in the e-p plane, where
e is the void ratio and p is the effective mean normal stress. The critical void ratio, ec , decreases
when p increases. Been and Jefferies (1985) introduced a state parameter, ψ = e − ec , which
measures the difference between the current void ratio and the critical void ratio at same p. When
ψ is positive, the material is in a loose state (looser than critical state), when it is negative, the
material is in a dense state (denser than critical state). When the critical state is reached, ψ = 0 .
Fig. 1 shows typical undrained shear
η=M
responses of sand with different densities
B
Critical state
but the same initial confining pressure.
Specimen A, which has a low relative denq
q
B
sity, is in a loose state (ψ > 0 ). The sand
tends to contract when sheared, resulting
A
A
in a reduction of the effective confining
p
εq
pressure and a significant drop in shear
strength, yielding flow liquefaction. SpeciFigure 1. Influence of density on undrained shear
men B with a higher relative density is in a
is in a dense state. The sand tends to dilate
η=M
after a phase transformation state (Ishihara
et al. 1975), at which the reversal from
contractive to dilative response takes
q
q
A
place. Both the loose and the dense speciA
B
mens finally reach a state called the critiCritical state
B
cal or steady state, at which ψ = 0 ,
p
εq
dp = dq = dε v = 0 while dε q ≠ 0 , where q
is the deviatoric stress, ε v and ε q are the
Figure 2. Influence of initial confining pressure
volumetric strain and deviatoric strain, respectively. Fig. 2 shows the typical undrained responses of two specimens of same density but
with different initial p. The specimen A with the higher p is in a loose state (ψ > 0 ), which tends
to contract towards flow liquefaction. The specimen B with the lower p is in a dense state
(ψ < 0 ), tending to dilate. The two specimens finally reach an identical critical state because
they have the same e.
The soil dilatancy is defined as the ratio of the plastic volumetric strain increment to the
magnitude of the plastic deviatoric strain increment, as follows:
D=
dε vp
2 p p
deij deij
3
=
dε vp
| dε qp |
(1)
where the superscript ‘p’ stands for plastic; eij = ε ij − ε kk δ ij / 3 are the components of the deviatoric strain tensor. In Eq. 1 the deviatoric strain ε q is consistent with its definition in the triaxial
space. Rowe (1962) showed, based on the hypothesis of least rate of internal work, that D could
be expressed, in a simplified manner, as a function of a stress ratio and the true angle of friction
between the mineral surfaces of the particles. Following Rowe’s original proposition, there exist
various dilatancy functions that teat D as a unique function of the stress ratio, η = q / p , i.e.,
D = D(η , C )
(2)
where C denotes a set of intrinsic material constants. Examples of Eq. 2 include the dilatancy
functions in two versions of Camclay models (Roscoe and Schofield 1963; Roscoe and Burland
1968), where D = M − η and D = ( M 2 − η 2 ) / 2η , respectively, where M is the critical stress ra-
tio, an intrinsic material constant. Adopting a dilatancy as a function of η only forms a major obstacle to unified description of loose and dense sand behavior, because such a dilatancy function
identify contractive or dilative response solely based on a stress variable, contradictory to the
observations as shown in Figs. 1 and 2. As a result, whether a soil is flow liquefiable is largely
predetermined by model constants, independent of evolving material states.
To successfully model sand behavior during the entire process of earthquake, including flow
liquefaction, it is essential to have a modeling framework in which the same sand is identified as
possessing a set of unique intrinsic properties, irrespective of its initial state. The models so developed should be able to reproduce all the significant stress-strain responses of sand, subjected
to all possible changes in stress state and material density during loading, while involving only a
single set of model constants, for which a consistent calibration procedure is available. Attempts
have been made to establish such a framework from the perspectives of peak dilatancy (Jefferies
1993), drained peak strength (Wood et al. 1994), and phase transformation stress ratio (Manzari
and Dafalias 1997). Li and Dafalias (2000) showed, based on soil mechanics theory, that the key
issue in such a much-desired modeling framework is to make the dilatancy, D, dependent of the
stress state as well as of the current material internal state, in reference to the critical state in
e − p − η space. It is interesting to see that when Rowe (1962) proposed his stress-dilatancy theory, he had also pointed out that a further amount of energy is absorbed depending on the sample
state and the stress history, and proposed to take it into account by replacing the frictional angle
between the mineral surfaces of the particles, an intrinsic material constant, by a quantity depending on the degree of remolding and associated energy loss. This proposition is in essence to
include a dependence of D on the internal state of the material. Based on a simple microstructural model, Li and Dafalias (2000) concluded that Eq. 2 is a result of an unconstrained minimization of the rate of internal work. This approach does not consider the static and kinematic constraints at the particle contacts that are associated with the packing of the particles, which may
vary drastically as the material state changes. A dilatancy function depending not only on stress
state but also on material internal state is referred to as the state-dependent dilatancy (Li and
Dafalias 2000), expressed in the following general form:
D = D(η , e, Q, C )
(3)
where Q denotes all internal state variables other than e that may affect D. This includes fabric
anisotropy, either inherent or loading induced, as elaborated later. Based on the definition, Eq. 1,
D > 0 and D < 0 represent the contractive and dilative responses, respectively; and D = 0
represents either a critical state or a phase transformation state (Ishihara et al. 1975). The dilatancy formulated according to Eq. 3 is uniquely related to an existing state, a combination of the
external stress state and the internal material state. Note that in a critical state, the void ratio
e = ec and the stress ratio η = M . Therefore, in critical states, D(η = M , e = ec , Q, C ) = 0 , which
is a basic rule for a particular form of the state-dependent dilatancy. One of such functions in the
triaxial compression space (Q is not involved) has been proposed by Li and Dafalias (2000), as
follows:
η 

D = d 1  e mψ − 
M

(4)
where d 1 and m are two positive material constants. Observe that at the same stress ratio η, when
the state parameter ψ increases, D becomes more positive (more contractive), in agreement with
common experimental results. At critical state, ψ = 0 and η = M simultaneously, hence, Eq. 4
yields a zero dilatancy, which is consistent with the basic concept of critical state. Furthermore,
Eq. 4 shows that there exists another type of zero dilatancy state, in which ψ ≠ 0 and η ≠ M but
η = Me mψ . This is the well-known phase transformation state (Ishihara et al. 1975). The performance of Eq. 4 has been shown in Li and Dafalias (2000).
To extend Eq. 4 to multiaxial space and to include cyclic response, Eq. 4 needs to be modified. The following equation shows such a modification.

D = d 1  e mψ

ρ
R 
−

ρ M (θ ) 
(5)
where the stress ratio R = 3J 2 D / p ( J 2 D = sij sij / 2 is the second invariant of the deviatoric
stress tensor with components sij = σ ij − σ kk δ ij / 3 ) is the counterpart of the triaxial stress ratio η ;
ρ and ρ , belonging to Q in Eq. 3, are projection “distances” in bounding surface formulation. It
is sufficient to point out here that ρ / ρ varies from zero upon unloading (changing shearing direction) to unity upon virgin plastic loading. Refer to Wang et al. (1990) for more details about
ρ and ρ . Note that in Eq. 5 the critical stress ratio M is a function of the Lode angle θ.
Bounding surface model framework
The model briefly described here is based on the bounding surface framework of Dafalias
(1986a, 1986b). The basic model structure is attributable to the hypoplasticity model developed
by Wang et al. (1990). As a notation convention, all the quantities evaluated on the bounding
surfaces are distinguished with a superposed bar. The bounding surface in the deviatoric space is
a cone with straight surface meridians in J 2 D − θ − p space, analytically expressed as
F = R / g (θ ) − Η = 0
(6)
where R = R ( rij ) , θ = θ ( rij ) are the two nontrivial invariants of the image stress ratio tensor rij
on F , and Η is a function of the internal state variables that are associated with the evolution of
F . Here the stress ratio rij = sij / p . Η defines the size of the cone. The function g (θ ) interpolates the stress ratio invariant, R , on the bounding surface, F , based on the Lode angle, θ .
The condition of consistency for F1 yields the following equation:
pnij drij − K p dL = 0
(7)
where nij is a zero-trace unit tensor normal to F at the image stress ratio rij , K p is the plastic
modulus controlling the evolution of F , and dL is a scalar loading index.
By considering the plastic deviatoric strain as an internal state variable, and assuming that the
associative flow rule applies to a constant p subspace, one has
deijp = nij dL
(8)
where deijp is the plastic deviatoric strain increment. Based on the definition of dilatancy, Eq. 1,
the plastic volumetric strain can be written as
d ε vp =
2
D deijp deijp
3
(9)
where D is the dilatancy function, given by Eq. 5. It is this state-dependent dilatancy function
that unifies the contractive and dilative responses of the same sand, based on the current stress
and material internal states.
Assuming the elastic response being Hookean, a straightforward algebraic manipulation
yields an incremental elastoplastic stress-strain relationship depending on the following quantities: (i) the elastic constants such as the shear and bulk moduli G and K; (ii) the unit tensor nij ;
(iii) the plastic modulus K p ; and (iv) the dilatancy function D . To determine nij , K p , and D ,
one needs to define the image stress ratio tensor, rij , via a mapping rule. The details are omitted
here. Refer to Li (2001) for detailed formulation of the model.
SUMDES2D
To directly calculate flow-liquefaction deformation, it is necessary to use a fully-coupled effective stress procedure based on the physical laws of balance of momentum and conservation of
mass, expressed as follows:
σ ij , j − ρ bi = − ρ u&&i
(10)
and
n (vi ,i + ε&vwc ) + u&i ,i = 0
(11)
In Eq. 10, σ ij represents the total stress, ρ is the bulk mass density of the soil, bi is the body
force function, and ui denotes the displacement of the soil skeleton (here the acceleration of the
pore fluid relative to the solids are considered negligible). In Eq. 11, v is the seepage velocity,
ε vwc denotes the volumetric strain of the pore fluid due to compression, and n is the porosity.
Eqs 10 and 11 are coupled in that they both depend on the state variables of the soil skeleton and
the pore fluid. The above physical laws (Eqs 10 and 11) must be joined with the constitutive laws
for the soil skeleton, the pore fluid, and the seepage flow, to yield the governing field equations,
which, in conjunction with specified boundary and initial conditions, are solved by numerical
techniques.
For the analyses reported in this paper, a two-dimensional finite element procedure, SUMDES2D (Ming and Li 2001), was used. The procedure solves Eqs 10 and 11, based on the wellknown Galerkin’s approximation in space and the Newmark integration scheme, enhanced by a
Hilber- α numerical damper (Hilber et al. 1977), over time steps. Nine-node quadrilateral ele-
ments and six-node triangle elements have been coded in the procedure for both the displacement
and the pore pressure fields. Reduced integration is used for the pore pressure field to minimize
the locking phenomenon. The above described bounding surface sand model (Eqs 6 to 9), with
the state-dependent dilatancy, Eq. 5, has been integrated into the procedure to describe the responses of soil skeleton.
Case of Upper San Fernando Dam during 1971 earthquake
The finite element mesh of the Upper San Fernando Dam is shown in Fig. 3. As shown, the
mesh can be grouped into a number of subregions according to soil description. They are a zone
of rolled fill (at the top of the dam), two zones of hydraulic fill (extended from both the upstream
and downstream slopes), together with a clay core and two layers of alluvium (foundation). An
extensive study on the Lower San Fernando dam was performed during the period 1985-1987
(Castro et al. 1989, 1990; Seed et al. 1989). The Upper San Fernando dam, however, was not included in that investigation and, hence, no detailed information about the soils in the Upper San
Fernando Dam was reported. However, the Upper and Lower dams are located within about
three km of each other. Both are founded on similar natural alluvial soils and, of most importance, both were constructed primarily by hydraulic filling using similar borrow materials. The
results of drilling, sampling, and trenching at both dams indicated no major difference in the type
or quality of finished product in either of the dams (Seed et al. 1973). Hence, it seems appropriate to deem the soil data for the Lower dam applicable to the Upper dam, as well.
Rolled Fill
G1022
Water Table
Hydraulic Fill
Clay
Core
20
40
unit: m
Hydraulic Fill
2.3
15.3
Upper Alluvium
0
Phreatic Line
Lower Alluvium
95
G968
131
G1535
95
Fig. 3 Finite element meshes for the Upper San Fernando Dam
In a parametric study, it was found that the response of the soil model as described above at
large strains was most sensitive to the critical state line but not very sensitive to other parameters. Based on the available data in the literature, only the group of model constants that defines
the critical state line was specifically calibrated according to the test results of the sample of
Batch Mix 7 in Lower San Fernando Dam (Castro et al., 1989, 1990). All other model constants
were simply copied from those calibrated for Toyoura sand (Li 2001). The Batch Mix 7 was considered representative of the soils from the critical hydraulic-fill layer. This set of model constants was used for all the soils in the dam. As mentioned earlier, the soil response will depend
on the stress and material states, in which the information about the in situ soil density is crucial.
It has been reported that the mean of the 1985 in situ void ratios in the hydraulic fill soils within
the critical layer (zone 5 in the Lower San Fernando Dam) is about 0.660 (Castro et al. 1989,
1990). The in situ void ratios of the Alluvium and Rolled fill are estimated to be about 0.567 and
0.500, respectively, based on their representative unit weights of 20.26 kN/m3 and 20.73 kN/m3,
and an assumed specific gravity of 2.69.
The initial fields of stresses, including driving stresses, were established by performing a
static analysis with a slowly increasing gravitational acceleration from zero to g (9.81 kg-m/s2).
This approach of establishing initial stress fields is for convenience only. It simulates the process
of stress buildup in a centrifuge model but not of that established during the real construction
process. By the same argument used for physical modeling in centrifuge, it was assumed that the
discrepancy in the initial stress fields associated with their establishment procedures would not
significantly alter the deformation and failure mechanisms during an earthquake.
It is observed that the difference between the upstream and downstream water tables, as
shown in Fig. 8a, produces a downstream-directed seepage force field, which acts as driving
forces, pushing the dam moving downstream upon liquefaction. This effect on flow-liquefaction
deformation could be significant but was rarely discussed in the literature. The computed deformed mesh at the end of the earth shaking ( t = 40sec ), as well as the observed displacements
reported by Seed et al. (1973, 1975), is shown in Fig. 4. A downstream movement is clearly seen
from the deformed mesh. The computed and measured displacements match reasonably well except where the computed settlement at the crest is 80% larger than the observed value (1.61 m
vs. 0.91 m). This discrepancy could arise from the uncertainties involved in soil characterization.
(0.91m)
1.61m
0.04m
0.46m
(1.52m)
1.31m
2.76m
(x.xx m) Measured (Seed et al. 1973)
x.xx m Calculated Displacement
Direction of Displacement
(0.61m)
0.73m
0.59m
2.19m
0.10m
0.53m
Fig. 4 Deformed mesh after earthquake
0
0.3
0.5
0.7
0.9
Pore Pressure Ratio Contours
negative excess pwp
Fig. 5 Pore pressure ratio contour at t = 6sec
Fig. 5 shows the calculated excess pore pressure ratio in terms of ∆uw σ v′0 at t = 6 s , where
∆uw is the excess pore water pressure and σ v′0 is the initial effective vertical stress. The pore
pressure ratio in the lower part of the hydraulic fill is quite high, indicating that soils in this region may have been either liquefied or weakened severely. At both the upstream and down-
stream toe of the dam, the pore pressure ratio also goes up to or beyond 0.9, in agreement with
the field observations of sandboils at the downstream toe.
To show the state dependent soil behavior, the local soil responses at typical locations are
plotted. Fig. 6 shows the stress paths and the shear stress-strain response of the soil at location
G1022 (refer to Fig. 3 for the location), which is in the subregion of loose hydraulic fill. In the
figure, q = 3J 2 D is a deviatoric stress invariant representing the magnitude of all shear components and is compatible with its definition in the triaxial space. Fig. 6c shows clearly that flow
liquefaction has occurred there, accompanying a relatively large shear strain (Fig. 6b). It is found
that the biased shear strain is consistent with the downstream movement. Fig. 7 shows the soil
responses at location G968 (refer to Fig. 3 for the location), which is located in the dam foundation, i.e., in the alluvium layer. Because of a relatively high density ( e0 = 0.567 in the analyses),
flow type deformation does not occur, and the shear strains are limited. Comparing Fig. 7 with
Fig. 6, one can see clearly two different types of soil responses. Note that the same set of material constants was assigned to the soils in the two locations, only the different stress and material
states made the difference.
100
100
(a) G1022 σ13 vs. p'
0
-50
-100
0
-50
0
50
100 150
p' (kPa)
100
50
-100
-12
200
(c) G1022 q vs. p'
150
q (kPa)
50
σ13 (kPa)
σ13 (kPa)
50
200
(b) G1022 σ13 vs. γ13
-9
-6
-3
γ 13 (%)
0
0
3
0
50
100 150
p' (kPa)
200
Fig. 6 Stress paths and shear stress-strain behavior at location G1022 (hydraulic fill)
100
100
50
50
0
-50
-100
-150
300
(b) G968 σ13 vs γ 13
q (kPa)
150
(a) G968 σ 13 vs. p'
σ 13 (kPa)
σ 13 (kPa)
150
0
-50
(c) G968 q vs. p'
200
100
-100
0
Fig. 7
100
200
p' (kPa)
300
-150
-2
-1
0
γ 13 (%)
1
0
0
100
200
p' (kPa)
300
Stress paths and shear stress-strain behavior at location 968 (alluvium foundation)
Summary
The concept of state-dependent dilatancy, which is consistent with the framework of critical
state soil mechanics, is described. The state-dependent dilatancy play a key role in unified mod-
eling of sand behavior, either contractive or dilative, up to failure. This feature is significant in
flow-liquefaction deformation analysis. An additional advantage of this unified modeling framework, which is important in practice, is it eliminates the need of multiple sets of model constants
for the same sand. Numerical analyses using a model of this type yield not only reasonable results, but also detailed information of soil responses, which is valuable for better understanding
of deformation and failure mechanisms of earth structures during earthquakes.
Acknowledgement
The research was partially support by the Research Grants Council of Hong Kong under
Grant HKUST6111/99E. This support is gratefully acknowledged.
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