RESEARCH PAPER

Large post-liquefaction deformation of sand, part I: physicalmechanism, constitutive description and numerical algorithm

Jian-Min Zhang • Gang Wang

Received: 22 October 2011 / Accepted: 28 October 2011 / Published online: 3 March 2012

� Springer-Verlag 2012

Abstract This paper presents a theoretical framework for

predicting the post-liquefaction deformation of saturated

sand under undrained cyclic loading with emphasis on the

mechanical laws, physical mechanism, constitutive model

and numerical algorithm as well as practical applicability.

The revealing mechanism behind the complex behavior in

the post-liquefaction regime can be appreciated by decom-

posing the volumetric strain into three components with

distinctive physical background. The interplay among these

three components governs the post-liquefaction shear

deformation and characterizes three physical states alter-

nating in the liquefaction process. This assumption sheds

some light on the intricate transition from small pre-lique-

faction deformation to large post-liquefaction deformation

and provides a rational explanation to the triggering of

unstable flow slide and the post-liquefaction reconsolidation.

Based on this assumption, a constitutive model is developed

within the framework of bounding surface plasticity. This

model is capable of reproducing small to large deformation

in the pre- to post-liquefaction regime. The model perfor-

mance is confirmed by simulating laboratory tests. The

constitutive model is implemented in a finite element code

together with a robust numerical algorithm to circumvent

numerical instability in the vicinity of vanishing effective

stress. This numerical model is validated by fully coupled

numerical analyses of two well-instrumented dynamic cen-

trifuge model tests. Finally, numerical simulation of lique-

faction-related site response is performed for the Daikai

subway station damaged during the 1995 Hyogoken-Nambu

earthquake in Japan.

Keywords Centrifuge tests � Constitutive model �Earthquake � Liquefaction � Large deformation �Numerical analysis � Site response

List of symbols

e, Dr Void ratio and relative density

pa Atmospheric pressure

s Simple shear stress

pe, ru Excess pore water pressure and excess

pore water pressure ratio

r0c,r0m Initial effective consolidation stress and

mean effective stress

p, q Mean effective stress and deviatoric

stress invariant

g, gm Shear stress ratio (g = q/p) and its

maximum value in loading history

c Total shear strain

cd Solid-like shear strain that occurs in non-

zero effective confining stress state

co Fluid-like shear strain that occurs in zero

effective confining stress state

cmax Preceding maximum cyclic shear strain

_ceff Effective shear strain rate

cmono Monotonic shear strain length

cd,r Reference shear strain length

cr Residual shear strain

ev Total volumetric strain

ev;recon Reconsolidation volumetric strain

J.-M. Zhang (&)

Institute of Geotechnical Engineering, School of Civil

Engineering/State Key Laboratory of Hydroscience and

Engineering, Tsinghua University, Beijing 100084, China

e-mail: zhangjm@mail.tsinghua.edu.cn

G. Wang

Ertan Hydropower Development Company Limited,

Chengdu 610051, China

123

Acta Geotechnica (2012) 7:69–113

DOI 10.1007/s11440-011-0150-7

evc Volumetric strain component due to the

change in p

evc,o Threshold volumetric strain to delimit

whether the effective confining stress

reaches zero, determined as evc value at

zero effective confining stress state

pmin Threshold pressure for numerical

calculation to delimit whether the

effective confining stress reaches zero

evd Volumetric strain due to dilatancy

evd,ir Irreversible dilatancy component

evd;re Reversible dilatancy component

r(rij), s(sij) Effective stress tensor and its deviatoric

part

e(eij), e(eij) Strain tensor and its deviatoric part

r(rij) Deviatoric shear stress ratio tensor

I(ij) Identity tensor of rank 2 (Kronecker

delta)

n Loading direction in stress ratio space

m Flowing direction of plastic deviatoric

strain increment

a Projection center

f ðrÞ, �f ð�rÞ Failure surface and maximum prestress

memory surface serving as bounding

surfaces

L Plastic loading intensity

G, K, H Elastic shear modulus, elastic bulk

modulus and plastic modulus

D, Dir, Dre Total, irreversible and reversible

dilatancy rates

Dre,gen, Dre,rel Reversible dilatancy rates in dilative and

contractive phases

Mf,c, Mf,o Failure stress ratios in triaxial

compression stress state and torsional

shear stress state

Go, n, h, j Modulus parameters

Md,c, dre,1, dre,2 Reversible dilatancy parameters

dir; a; cd;r Irreversible dilatancy parameters

hr Lode angle

q; �q Mapping distances in stress ratio space

1 Introduction

Saturated sands subjected to undrained cyclic loading can

arrive at failure through either liquefaction or cyclic

mobility, depending on their initial density and effective

stresses. The terms ‘‘liquefaction’’ and ‘‘cyclic mobility’’

were first introduced by Castro [16] and Castro and Poulos

[17] as two clearly different phenomena in saturated sands.

Liquefaction is referred to the steady flow of saturated sand

as a result of high excess pore water pressure and sudden

reduction and even loss of shear strength. For cyclic

mobility, large but limited shear deformation develops as a

result of progressive degradation of sand stiffness due to

excess pore water pressure. In each loading cycle, large

shear deformation is induced at instantaneous ‘‘liquefaction

state’’ with zero effective confining stress, i.e., at the

instants when the effective confining stress becomes zero

instantaneously and repeatedly during cyclic mobility.

Liquefaction can be triggered only in extremely loose

sand deposits with SPT-N values of about one or less

[115, 154]. Cyclic mobility may develop in saturated

dilative sands over a wide density range from loose,

medium-to-dense and even the densest state, which cover

almost all foundations of structures built on and in the

natural or improved sandy soil deposits. Many experi-

mental and theoretical studies have been carried out to

investigate the liquefaction-induced flow deformation

behavior of extremely loose sands. However, such extre-

mely loose sand deposits can be rarely found in practice.

For safety of structures against earthquakes, it is more

important to understand the deformation behavior related

to the cyclic mobility of dilative sands. Therefore, we will

focus on cyclic mobility of dilative sands in this paper.

In cyclic mobility, the point where the effective stress

vanishes for the first time in cyclic undrained shearing is

termed as ‘‘initial liquefaction’’ [106], which separates the

whole liquefaction process into ‘‘pre-liquefaction’’ stage

and ‘‘post-liquefaction’’ stage. Since the disastrous 1964

Niigata earthquake, many theoretical and experimental

studies have been focused mainly on the likelihood of the

initial liquefaction and the pre-liquefaction stress–strain

response, and large shear and volumetric deformations in

saturated sands are found to take place mainly after the

initial liquefaction. Large post-liquefaction ground settle-

ments and lateral spreading of saturated sandy deposits

were observed in almost all strong earthquakes (e.g., [36,

37]), which often caused heavy damage to various struc-

tures such as building foundations, infrastructures and

lifeline systems. Evaluation of large post-liquefaction

ground deformation has become an ever-increasingly

important issue in seismic design of various structures built

on and in liquefiable sandy deposits. Several approaches

based on empirical criteria and correlations of in situ and

laboratory tests have been developed to evaluate lique-

faction-induced ground settlements and lateral spreading.

However, most approaches provide only a rough estimate

under limited conditions for saturated sandy soil ground

with level or gently sloping surface. In order to meet the

need for accurate evaluation of earthquake damages, great

efforts have been made to reveal the physical fundamentals

of post-liquefaction stress–strain response for saturated

sand under cyclic loading and to develop reliable and

70 Acta Geotechnica (2012) 7:69–113

123

pragmatic constitutive models and related numerical

methods to describe such complex behavior. However,

there still exist large discrepancies between actual post-

liquefaction behavior and our acquired capabilities of

predictions.

The study presented in this paper has arisen from

practical needs to seek more accurate theory and method-

ology for predicting complex post-liquefaction deformation

and recognizing liquefaction hazards and also to develop

the technology concerning protection against large post-

liquefaction deformation. Experimental observations,

physical explanations and theoretical descriptions are made

for the attempts to establish a theoretical framework of

evaluating large post-liquefaction deformation of saturated

sand. Special emphasis is mainly placed on the mechanical

laws, physical mechanism, constitutive modeling and

numerical algorithm as well as practical applicability

related to the above issue.

The specific objectives of this study are (1) to give a

brief critical review for previous studies on the lique-

faction behavior of saturated sand with emphasis on the

three main aspects concerning the physical mechanisms,

constitutive models and approaches of evaluation, (2) to

reveal the mechanism regarding small-to-large deforma-

tion process of saturated sand from the pre- to post-

liquefaction on the basis of observations, explanation and

analysis for experimental facts obtained from various

laboratory tests, (3) to propose a general approach

originated from the above mechanism, develop a cyclic

constitutive model within the theoretical frame of

bounding surface plasticity for the prediction of small

pre-liquefaction to large post-liquefaction deformation

and to confirm their essential effectiveness through

comparing the predicted and tested results, (4) to develop

a robust numerical algorithm using the present model to

circumvent numerical instability in the vicinity of zero

effective confining stress state alternating in the post-

liquefaction regime, (5) to implement the present model

into a fully coupled finite element code to develop a

numerical model and then examine its applicability by

simulating previous dynamic centrifuge model test results

and finally, (6) to provide a typical example of practical

application on numerical analysis of liquefaction-related

seismic response for the famous Daikai subway station

damaged in the 1995 Hyogoken-Nambu earthquake,

showing good ability of the presented model and corre-

sponding numerical algorithm in the analysis of practical

liquefiable soil–structure interaction problems.

As for notation, the volumetric strain takes either a

positive or negative sign, depending on contraction or

expansion of the soil volume. All stresses are effective

stresses. Bold-faced letter denotes vector and tensor.

Effective confining stress indicates mean effective stress.

2 A critical review of previous studies and basic issues

2.1 Experimental observations of post-liquefaction

deformation

Actual manifestation regarding post-liquefaction behavior

of sands can be observed in the laboratory tests. A number

of cyclic or monotonic undrained torsional shear tests after

initial liquefaction was run on hollow cylinder specimens

of Toyoura sand (qs = 2.65 g/cm3, d50 = 0.18 mm,

emax = 0.973, emin = 0.635) by Zhang [162] and Zhang

et al. [163]. The specimens were prepared by pluviating dry

sand through air and saturated by circulating CO2 gas,

percolating de-aired water and then by applying a back-

pressure of 100 kPa. B values of more than 0.95 were

obtained for all the saturated specimens used in the tests.

The specimens were consolidated isotropically under an

initial effective consolidation stress r0c = 100 kPa. Sinu-

soidal cyclic shear stress with constant or varying ampli-

tudes was applied on the specimens. A very low frequency

of 0.01 Hz was used in order to reduce the effect of vis-

cosity of both specimens and equipment system on the

measured stress–strain response and obtain experimental

data with high quality.

Shown in Figs. 1 and 2 are time histories of the cyclic

shear stress s, excess pore water pressure pe and shear

strain c as well as the effective stress path and s–c rela-

tionship in a typical cyclic undrained torsional test for

saturated Toyoura sand with a specified initial density and

constant cyclic shear stress amplitude. Other two typical

test results with irregular cyclic shear stress amplitudes of

two patterns are drawn in Figs. 3, 4, 5, 6. As can be seen

clearly from all these figures, the effective stress paths

following the initial liquefaction show similar dilative and

contractive behaviors for each cycle. The corresponding

shear strain amplitude, however, gradually increases with

the increasing number of cyclic loading. Particularly, large

post-liquefaction deformation is induced mainly at the

moments when the effective stress vanishes at the lique-

faction state during undrained cyclic loading with either

constant or varying amplitude.

Similar phenomena have been confirmed by many other

experimental studies performed by Castro et al. [16, 17],

Katada et al. [56], Yasuda et al. [149], Yoshida et al. [152],

Vaid and Thomas [133], Shamoto et al. [114, 116–118],

Shamoto and Zhang [115], Zhang et al. [164] and Pan et al.

[89].

Acta Geotechnica (2012) 7:69–113 71

123

2.2 Preliminaries of post-liquefaction mechanisms

Deep understanding of the physical mechanisms behind the

large deformation that occurs under undrained cyclic

loading after the initial liquefaction is crucial for rationally

modeling large post-liquefaction deformation. Since satu-

rated sand is a two-phase porous media composed of sand

skeleton and pore water, the basic mechanisms concerning

onset of initial liquefaction and occurrence of cyclic

mobility are necessarily linked to the excess pore water

pressure generation under undrained cyclic loading. This

can be understood by examining the features of excess pore

pressures and shear strains in cyclic undrained tests, such

as in the earliest attempts by Wang [134, 135] and Seed

and Lee [106].

-30

0

30

60τ

(kP

a) (a)

0

40

80

120 Initial liquefaction

p e(k

Pa)

(b)

0 1 2 3 4 5 6 7 8 9 10 11 12-10

-5

0

5 (c)

γ( %

)

Number of cycles, N

Pre- Post-liquefaction'σc =100 kPa

Fig. 1 Time history of Toyoura sand of Dr = 70% in an undrained

torsional test, (a) shear stress, (b) excess pore water pressure and

(c) shear strain

-10 -8 -6 -4 -2 0 2 4 6 8 10-60

-40

-20

0

20

40

60

Before initial liquefaction After initial liquefaction

Toyoura sand Dr=70%

τ (k

Pa)

γ (%)

(a)

0 20 40 60 80 100-60

-40

-20

0

20

40

60

τ (k

Pa)

σ 'm (kPa)

(b)CSL

Fig. 2 Stress–strain hysteresis curve and effective stress path of

Toyoura sand of Dr = 70% in an undrained torsional test

0

20

40

60

80

100Initial liquefaction (b)

Toyoura sandDr=75%

-60-30

030

60(a)

0 10 40 50 60 70-8-404 (c)

Number of cycles, N

Pre- Post-liquefaction

=100 kPaσ'c

τ ( k

Pa)

γ (%

)p e

( kP

a)

Fig. 3 Time history of Toyoura sand of Dr = 75% in an undrained

torsional test (Pattern 1), (a) shear stress, (b) excess pore water

pressure and (c) shear strain

-6 -4 -2 0 2 4 6-40

-20

0

20

40

Before initial liquefaction After initial liquefaction

Toyoura sand Dr=75%

τ (k

Pa)

γ (%)

(a)

0 20 40 60 80 100-40

-20

0

20

40

τ (k

Pa)

σ 'm (kPa)

(b)CSL

Fig. 4 Stress–strain hysteresis curve and effective stress path of

Toyoura sand of Dr = 75% in an undrained torsional test (Pattern 1)

72 Acta Geotechnica (2012) 7:69–113

123

In the early studies, two gedanken models for the gen-

eration of excess pore water pressure were proposed by

Wang [134, 135] and Martin et al. [72], respectively. In

general, application of cyclic loading in drained tests can

result in progressive decrease in volume, in the case of

either dense or loose sand. For undrained conditions, the

tendency for volume reduction can result in a progressive

increase in pore water pressure. Although this qualitative

explanation was widely accepted in the earlier studies, the

most important issue is how to establish the quantitative

relationship between the volume reduction tendency and

corresponding pore pressure increase in the undrained

cyclic loading. A triaxial apparatus for cyclic loading was

developed in China in 1959 [43, 44]. Through the experi-

mental observation using this apparatus, Wang [134, 135]

found that the compression curves from consolidation tests

on sandy soils with and without cyclic loading history are

parallel, as shown in Fig. 7, depending on the initial den-

sity. This is related to the orientation and rearrangement of

grains in the soil skeleton structure. Based on such exper-

imental facts, it is considered that the compression curve

always keeps a parallel translation in position during cyclic

loading. As seen from Fig. 7, it is supposed that point A on

initial compression curve is the initial state and then the

initial compression curve shifts gradually toward another

compression curve after cyclic loading. If cyclic loading is

imposed under drained condition, then point A will move

toward point B, while no change in excess pore water

pressure pe will generate, which will lead to a void ratio

decrease De. On the contrary, if cyclic loading is applied

under undrained condition, then point A will move toward

point C, and consequently, pe will increase with a quantity

of Dpe. Based on the considerations just made above, it is

easy to establish the relationship between De and Dpe,

which provides a mechanism regarding generation of Dpe

under undrained cyclic loading and an approach to predict

Dpe. According to this mechanism and approach, the

developing process of pe under undrained cyclic loading

can be predicted through first determining volumetric strain

increment Dev depending on De and then determining Dpe

by the product of Dev and compression bulk modulus. On

the basis of extension of the above mechanism and further

research results, Wang [134–138] suggested approaches to

predict the generation, dissipation and diffusion of excess

pore pressure for saturated sandy soil ground and slopes

with more complicated conditions of drainage, which was

partially introduced by Finn [34].

-40-20

02040 (a)

0

20

40

60

80

100Initial liquefaction (b)

Toyoura sandDr=75%

0 10 60 70 80-4-202

(c)

Number of cycles, N

Pre- Post-liquefaction=100 kPaσ'c

5 65 75

τ ( k

Pa)

γ (%

)p e

(kP

a)

Fig. 5 Time history of Toyoura sand of Dr = 75% in an undrained

torsional test (Pattern 2), (a) shear stress, (b) excess pore water

pressure and (c) shear strain

-4 -2 0 2 4-40

-20

0

20

40

Before initial liquefaction

After initial liquefaction

Toyoura sand Dr=75%

τ (k

Pa)

γ (%)

(a)

0 20 40 60 80 100-40

-20

0

20

40

τ (k

Pa)

σ 'm

(kPa)

(b)CSL

Fig. 6 Stress–strain hysteresis curve and effective stress path of

Toyoura sand of Dr = 75% in an undrained torsional test (Pattern 2)

Initial Compression CurveA

B

C

e

pe

eA

Compression Curve after ShakingeB

mσ ′mAσ ′

mCσ ′

e

peApeC

pe pe=induced excess pore water pressure

e=distance of compression curve movement

Fig. 7 Mechanism of pore pressure generation during cyclic loading

proposed by Wang [134, 135]

Acta Geotechnica (2012) 7:69–113 73

123

Another well-known point of view regarding the

mechanism of excess pore pressure generation under

undrained cyclic loading was proposed by Martin et al.

[72], which is schematically illustrated in Fig. 8. Point D is

initially on compression curve before cyclic loading. For a

saturated sand in drained condition, shearing effect of

cyclic loading can lead to a void ratio decrease De due to

contraction of the sand skeleton and correspondingly a

plastic volumetric strain Depv . As a result, point D moves

toward point E. For undrained condition, the volume

decrease caused by the cyclic loading cannot occur since

the movement of the incompressible pore water within the

sand is prevented. Instead, the tendency to decrease in

volume is counteracted through a decrease in mean effec-

tive stress. i.e., the plastic volumetric strain Depv is coun-

terbalanced by an elastic volumetric strain Deev due to

reduction in mean effective stress r0m. Consequently, a

volumetric expansion of the sand skeleton takes place with

a quantity of De, thereby causing the movement of point E

toward point F and an increase in excess pore pressure Dpe.

As a result, Dpe can be determined by the product of Depv

and rebound bulk modulus. In the above considerations, the

De - Dpe relation and correspondingly the Depv � Dpe

relation are established, which provides another mecha-

nism concerning the pore pressure generation and an

approach to predict the pore pressure. Initial liquefaction

likelihood could be evaluated through assessment whether

the predicted pore pressure reaches its limit that the pore

water carries the entire confining stress.

The above two approaches differ in several aspects such

as selection of compression or rebound bulk modulus and

determination of drained volumetric strains due to cyclic

shearing. Both of them are available under the two pre-

requisites that (1) the total normal stress remains constant

and (2) there exists close relation between pore pressure

generation under undrained condition and densification

under drained condition during cyclic loading, irrespective

of obvious difference in effective stress paths. They have in

common that they are both based on the average value of

pore pressure before the initial liquefaction, without con-

siderations of its transient change in fluctuation at different

instants in the course from pre- to post-liquefaction cyclic

loading, which provides a rough approximation for the

actual generation process of pore pressures as shown in

Figs. 1, 3 and 5. As a matter of fact, they are more useful as

an approach to predict approximately rather than as a

mechanism regarding the pore pressure generation under

the cyclic loading before the initial liquefaction.

Early studies on liquefaction were mainly based on

cyclic loading tests to establish pore pressure generation

models, such as the endochronic model [33], the stress path

model [51], the dissipated energy model [27, 61, 83] and

other well-known models (e.g., [66, 108]. Application of

these models is also limited to the prediction of the average

monotonic generation of pore pressure before the initial

liquefaction. Xie et al. [141, 142] developed an approach

for predicting the transient pore pressure generation, but its

use and parameter determination are very complicated,

since the pore pressure development actually belongs to an

integral part of the whole complicated stress–strain

response. In fact, a prediction with high accuracy for actual

change in the pore pressure strongly depends on under-

standing and description of cyclic constitutive laws.

Existing pore pressure models are therefore adequate to

simplified and pragmatic assessments of initial liquefaction

and pre-liquefaction behavior.

Shamoto et al. [114] provided for the first time a rational

explanation and mechanism of large post-liquefaction

deformation. In their study, large post-liquefaction shear

strain is classified into two components. One component

depends on the effective stress, but the other is independent

of effective stress. The two shear strain components during

cyclic shearing application are controlled by two types of

volumetric strain due to stress-dilatancy, i.e., an irrevers-

ible and a reversible component. The shear strain compo-

nent independent of effective stress is triggered only at the

liquefaction state of zero effective confining stress, and its

magnitude has a unique relationship with the preceding

maximum shear strain. The above observations may help to

explain large deformation after the initial liquefaction.

However, this explanation is over-simplified and cannot

account for the complicated behavior of large deformation

induced at the liquefaction states of zero effective confin-

ing stress.

Zhang and Wang [168, 169] developed a more rational

explanation and formulation to large post-liquefaction

deformation, which will be illustrated later as an integral

part of the theoretical framework of this paper.

E

e

pe

Void ratio decrease due to cyclic

shear application under drained

loading

Compression Curve

F D

Rebound Curve

pe

e

mFσ ′ mDσ ′peDpeF

pe=induced excess porewater pressure

mσ ′

eD

eE

Fig. 8 Mechanism of pore pressure generation during cyclic loading

proposed by Martin et al. [72]

74 Acta Geotechnica (2012) 7:69–113

123

2.3 Existing constitutive models for post-liquefaction

deformation

Boundary-value problems involving large post-liquefaction

deformation require realistic yet conceptually simple and

computationally efficient cyclic constitutive models. A

number of cyclic constitutive models for sands were pro-

posed in the past decades. They may be classified into two

main categories, namely nonlinear elastic models and

plasticity models.

Cyclic nonlinear elastic models mainly include the

Ramberg–Osgood model and its modifications (e.g.,

[47, 50, 99, 101]) as well as the hyperbolic model and its

modifications (e.g., [32, 38, 58, 97]). Saturated sand sub-

jected to cyclic loadings usually shows different stress–

strain responses during initial loading, unloading and

reloading. Nonlinear elastic description of such cyclic

stress–strain responses is made for each phase, respec-

tively. A nonlinear elastic model is thus generally com-

posed of one skeleton curve for initial loading and a series

of hysteresis curves for subsequent unloading and reload-

ing loops. Frequently, the hysteresis curves are defined

according to Masing criterion [73]. The Masing criterion

postulates that the tangent shear moduli at the reversal

points of unloading and reloading branches are identical to

the initial shear modulus. Moreover, the shape of the

hysteresis curves is similar to that of the skeleton curve

enlarged by a factor of two. In principle, the Masing cri-

terion is valid only for cyclic loading without degradation

of soil stiffness. In fact, stiffness degradation occurs due to

build-up of pore pressure, which causes the decrease in

mean effective stress and consequently degradation of

shear modulus. The shear modulus is found to be a function

of the mean effective stress and shear strain. When a

nonlinear elastic model is used in liquefaction-induced

deformation analysis, the following main factors must be

considered [32]: (1) the initial shear modulus, (2) the

variation of shear modulus with shear strain, (3) damping

and its variation with shear strain, (4) hardening or soft-

ening, (5) changes in mean effective stress and (6) con-

temporaneous generation and dissipation of pore water

pressures. Among these factors, the pore water pressure

generation and dissipation are most difficult to model,

because the pore pressure is an integral part of the con-

stitutive description.

Finn et al. [32] considered these factors in their hyper-

bolic nonlinear model with the Masing criterion and made

use of the model in effective stress analysis of liquefaction.

Shamoto et al. [114] developed a nonlinear constitutive

model for evaluating large post-liquefaction deformation

under undrained monotonic loading. In their model, the

shear strain increment induced at zero effective confining

stress state is supposed to have a linear relationship with a

reversible volumetric strain. Zhang and Wang [167]

developed a cyclic nonlinear elastic model based on

Ramberg–Osgood model and the mechanism proposed by

Shamoto et al. [114]. The above nonlinear elastic models

are suitable only for the dynamic analysis of saturated

sandy deposits with level or nearly level ground. They

cannot be used to predict large post-liquefaction deforma-

tion with more complicated boundary-value conditions.

In comparison with the cyclic nonlinear elastic models,

cyclic plasticity models may provide more satisfactory

prediction of the stress–strain behavior for saturated sands

subjected to cyclic loadings. Some early models were

developed by modifying classical plasticity models (e.g.,

[29, 87, 102, 103]), but their prediction capacity is rather

limited. Many other well-known cyclic plasticity models

have been proposed in the literature, and they can be fur-

ther divided into multisurface models (e.g., [31, 53, 76, 77,

78–80, 81, 95, 96, 97, 146, 147]), single-surface models

(e.g., [23]), two-surface models (e.g., [22, 59, 67, 71, 93]),

bounding surface models (e.g., [7, 24, 25, 26]) and gen-

eralized plasticity models (e.g., [65, 92, 175]), subloading

surface models (e.g., [39, 40, 41, 42]), and multimecha-

nisms models (e.g., [3, 45, 55, 86]). The multi-, single- and

two-surface models are regarded as extension of the non-

linear kinematic hardening model proposed by Armstrong

and Frederick [6] and extended by Chaboche [18] and

Chaboche and Rousselier [19]. They stipulate the transla-

tion of multi-, single- or two-yield or loading surface(s) to

model the cyclic loading behavior. The subloading surface

model reproduces the cyclic stress–strain response through

the expansion and contraction of a subloading surface,

while the surface keeps a similarity to the conventional

yield surface. Recently, Zhang et al. [172, 173] develop an

elastoplastic model for the description of cyclic mobility,

by incorporating the subloading surface [39] and the su-

perloading surface [2] into the Cam-clay model [102].

Among the above models, the multisurface and bounding

surface models have been more successfully used for sands

because of their flexibility and simplified treatment of the

complex mechanisms that govern the deformation of sands

under cyclic loading.

Several cyclic plasticity models have been proposed to

model large deformation induced by liquefaction and

cyclic mobility. A representative constitutive model was

proposed as development of an existing multisurface

plasticity formulation by Parra [91], Yang [145], Yang and

Elgamal [147] and Elgamal et al. [30, 31]. In this model, an

additional shear strain is introduced when the effective

stress path crossed the phase transformation line (PTL) that

is defined by Ishihara et al. [51]. As illustrated in Figs. 9

and 10, the additional shear strain is added, on the point of

the stress–strain hysteresis curve that corresponds to the

crossing point of the stress path with the PTL, as long as

Acta Geotechnica (2012) 7:69–113 75

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the stress path crosses the PTL, for example, from point 1

to point 2 or from point 7 to point 8. The calculated shear

strain is thus gradually increased with the increasing

number of cyclic loading. The test results in Figs. 2, 4 and

6 show that large shear strain does not occur along the PTL

before the initial liquefaction, but it is always induced at

zero effective confining stress state after the initial lique-

faction. It is clear by comparing the model assumption

(Figs. 9, 10) and the test results (Figs. 2, 4 and 6) that this

additional strain is an artifice and falls short of the physical

mechanism behind the post-liquefaction deformation.

Moreover, the determination of this additional strain is

rather difficult. In the strict sense, this model is more useful

for the evaluation of pre-liquefaction deformation.

Within the framework of the bounding surface model

[139], Wang and Dafalias [140] simulated the progressive

shear strain increase under the same cyclic stress paths by

reducing plastic shear modulus with the accumulation of

plastic shear strain. Some typical result of this model is

shown in Fig. 11. This model, too, is based on an artifice,

since the large post-liquefaction deformation is caused by

vanishing effective stress rather than diminishing shear

modulus. Therefore, this model is more suitable for the pre-

liquefaction regime.

The simulation of both pre- and post-liquefaction

deformation poses high challenge for constitutive models.

Most constitutive models are not well suited for the post-

liquefaction regime. Often, some numerical artifices are

introduced as saving grace, which lack the sound physical

background. Some recent developments in constitutive

modeling of sand include hypoplastic model [105, 121] and

plastic model with state parameters [110]. However, the

suitability of such models for cyclic loading need be

investigated.

2.4 Current approaches to predict post-liquefaction

deformation

Existing approaches to predict large liquefaction-induced

deformation may be distinguished into two main catego-

ries, which are based on empirical criteria and numerical

analysis, respectively.

In the first category, simplified but pragmatic approa-

ches have been developed to estimate the liquefaction-

induced ground settlement and lateral spreading and par-

ticularly liquefaction potential. They are based on geo-

logical criteria (e.g., [156]), simple geotechnical criteria

(e.g., [49, 136]) or empirical correlations using in situ

investigations and laboratory observations (e.g., [15, 48,

52, 62, 82, 107, 109, 112, 113, 115–118, 124, 131, 148,

150, 151, 153, 155, 159, 160]).

Liquefaction-induced ground surface settlements are

essentially vertical deformation that results from densifi-

cation due to dissipation of excess pore water pressures.

Fig. 9 Calculation mechanism of models by Parra and Yang et al.

(after [91])

Fig. 10 Performance of models by Parra and Yang et al. (after [146])

-3 -2 -1 0 1 2 30 10 20 30 40 50

Test resultsModel Simulation

0

10

20

30

-10

-20

-30

Shear strain (%)Mean effective stress (kPa)

She

ar s

tres

s (k

Pa)

Fig. 11 Performance of cyclic constitutive model (after [140])

76 Acta Geotechnica (2012) 7:69–113

123

The earliest study on the deformation behavior of soil

under vibration may be traced back to the publication of

Mogami and Kubo [75]. Since the 1970s, several cyclic

undrained tests followed by drained reconsolidation have

been performed by Lee and Albaisa [62], Yoshimi et al.

[153], Tatsuoka et al. [124], Nagase and Ishihara [82],

Ishihara and Yoshimine [52], Shamoto et al. [112, 113,

116–118], and Shamoto and Zhang [115]. The volumetric

strain induced by drained reconsolidation following

undrained cyclic loading is found to increase abruptly after

the initial liquefaction, and is closely related to the expe-

rienced maximum shear strain. According to laboratory

experiments and case histories of seismic liquefaction,

Tokimatsu and Seed [129] were the first to develop an

STP-based method to calculate liquefaction-induced

ground settlements. Based mainly on the laboratory results

of Nagase and Ishihara [82], Ishihara and Yoshimine [52]

established a family of curves to estimate post-liquefaction

volumetric strain for clean sands. Shamoto and Zhang

[115] provided some improved laboratory results together

with a theoretical description of the mechanisms behind the

post-liquefaction volume change. They found that the

residual volumetric strain, induced by drained reconsoli-

dation following undrained cyclic loading, can be well

described using an index of ‘‘relative compression’’ for

different sands over a wide range of density. Moreover, a

close correlation was obtained between this index and

maximum shear strain induced by preceding undrained

cyclic loading. Based on this correlation, a simplified

method was proposed to determine liquefaction-induced

ground surface settlement for level sandy deposits, and its

effectiveness was confirmed through comparing with

observations made in the past earthquakes. Most of the

currently published methods for evaluating liquefaction-

induced settlements are based on either the standard pen-

etration tests (SPT) ([52, 116–118, 129]) or the cone pen-

etration tests (CPT) ([48, 159]). Recently, an approach

using shear wave velocity was proposed by Yi [150].

Ground lateral spreading is another pervasive type of

liquefaction-induced ground deformation that was widely

observed in ground with gently sloping surface and near

riverbanks during the past strong earthquakes. Several

empirical or semi-empirical methods have been proposed

to estimate liquefaction-induced ground lateral spreading

by using empirical formula based on the database of

observed case histories [9, 10, 36]), SPT data [8, 100, 116,

117, 157, 160]), CPT data [160] or shear wave velocity

[151]. It is worth noting that liquefaction-induced ground

settlement and lateral spreading are separately evaluated in

most methods. Shamoto et al. [116, 117] indicated that

ground settlement and horizontal displacement may occur

not only in liquefied sandy ground with gently sloping

surface and near riverbanks but also in liquefied level

ground with a sufficient lateral extent. Moreover, the post-

liquefaction vertical ground settlement and horizontal dis-

placement are related to each other, and their magnitudes

depend on the irreversible volume change induced by

earthquake shaking. They suggested a new method and

related charts for concurrently estimating the post-lique-

faction ground settlement and lateral spreading.

The empirical or semi-empirical approaches have been

widely used as a cost-effective tool, but their limitations in

application and accuracy are quite insurmountable. The

numerical approaches have been continuously developed to

improve their prediction capacity. In particular, pragmatic

equivalent linear analyses using cyclic nonlinear elastic

models and pore pressure generation empirical models

have been widely accepted in practice (e.g., [32, 107, 119,

143, 174]). As discussed before, however, there exist

obvious deficiencies that the induced pore pressure and

permanent deformation cannot be calculated directly, and

consequently, large post-liquefaction deformation cannot

be rationally predicted.

Effective stress dynamic response analysis using cyclic

plasticity constitutive model has now become the major

approach to evaluate post-liquefaction response and assess

the adequacy of proposed remediation measures. In the

numerical analysis, saturated sand is usually treated as a

two-phase material based on the Biot theory [11, 12].

Coupling the Biot field equations and complex plasticity

constitutive models gives rise to large nonlinear equation

system. Numerical efficiency and stability play an impor-

tant role in the simulation of large-scale problems. Many

efforts have been made to develop more efficient numerical

techniques and better cyclic plasticity model (e.g., [13, 14,

30, 35, 68, 69, 91, 145, 146, 147, 175–177]), but there have

been few direct validations against field data in the past

earthquakes. Case histories with well-documented damage

to high embankment dams and building foundations, which

were observed in the May 12, 2008, Wenchuan earthquake

of China with magnitude Ms 8.0 (e.g., [170]), provide an

excellent opportunity for validating earthquake-induced

deformation analysis.

The investigation into and analysis of the damage to

high embankment dams in the great Wenchuan earth-

quake in China provide sufficient evidence for the fol-

lowing recognition and challenging problems: (1)

evaluation of earthquake-induced deformation has become

the most important issue in the seismic design of high

embankment dams; (2) numerical analysis of dynamic

response is playing an indispensable role in the evaluation

of the seismic deformation behavior and aseismic safety

of high embankment dams; and (3) control of seismically

induced deformation is one of the most challenging

problems of high embankment dams. These imply coex-

istence of opportunities and challenges in developing

Acta Geotechnica (2012) 7:69–113 77

123

theories and techniques from evaluation to control of

earthquake-induced deformation, including post-liquefac-

tion deformation.

It is known from the above review that modeling large

post-liquefaction deformation still remains a challenging

problem, and more research is needed to develop more

satisfactory numerical models for large liquefaction-

induced deformation with emphasis placed on the follow-

ing aspects: (1) rational description of the cyclic behavior

of loose to dense sands over wide range from small to large

strains and particularly of gradually accumulated volu-

metric strains induced by cyclic shearing; (2) deep under-

standing of large post-liquefaction deformation mechanism

as well as conditions of unstable flow slide and large post-

liquefaction reconsolidation deformation; (3) robust

numerical algorithm in the vicinity of zero effective con-

fining stress state that appears repeatedly in the post-liq-

uefaction regime; (4) efficient numerical techniques to

reduce computation time for large-scale problems with

coupled field equations, strongly nonlinear constitutive

model and geometrical nonlinearity; and (5) essential

verification of numerical models against laboratory test

results and case studies.

Usually, sand deposits show anisotropy to some extent

[28], which is thought to have little effect on the behavior

in the post-liquefaction regime. Moreover, sand often

deforms in a localized manner [128], which gives rise to

local drainage and pattern formation of shear bands. This

will be investigated in a companion paper under prepara-

tion. The post-liquefaction behavior of sand has also

important bearing on tsunami-induced scour and liquefac-

tion [158].

3 Basic feature of post-liquefaction deformation

3.1 Two kinds of post-liquefaction shear strain

The mechanism behind large post-liquefaction deformation

can be best appreciated by studying the undrained behavior

under cyclic loading in laboratory. Some typical results of

cyclic undrained torsional shear tests are shown in Figs. 1,

2, 3, 4, 5, 6. It can be seen from Fig. 1 that cyclic shearing

gives rise to the accumulation of excess pore pressure with

considerable fluctuations. In this test, the excess pore

pressure reached the initial effective confining stress at the

4.5th cycle for the first time. The specimen reached the

initial liquefaction and entered the post-liquefaction

regime. To examine the post-liquefaction behavior, the

data in the post-liquefaction regime are extracted from

Fig. 2 and plotted in Fig. 12. The following observations

can be made: (1) The effective stress path of all loading

cycles follows the same patterns, and the effective

confining stress becomes zero twice in each loading cycle.

(2) The amplitude of cyclic shear strain develops rapidly

along with the increasing number of loading cycles, and the

difference in shear strain change in each loading cycle

mainly occurs at the liquefaction state of zero effective

confining stress. Similar observations can be made in other

two cyclic test results with varying amplitude in Figs. 4

and 6.

As schematically illustrated in Fig. 13, the post-lique-

faction shear strain in each loading cycle c can be

decomposed into two components [114], i.e., the one

occurred in non-zero effective stress state, denoted as cd,

and the other occurred in zero effective stress state,

denoted as co, namely

-8 -6 -4 -2 0 2 4 6 8 10-60

-40

-20

0

20

40

6011th5th cycle

f=0.01Hz

Post-liquefaction undrained cyclic shearing

τ (kP

a)

γ (%)

(a)

-10

Toyoura sandDr=70%

0 20 40 60 80 100-60

-40

-20

0

20

40

60

5th to 11th cycle

τ (k

Pa)

(b)

CSL

'σc =100 kPa

σm' (kPa)

Fig. 12 Typical shear stress–shear strain curve and effective stress

path during post-liquefaction undrained cyclic shearing (the same test

as in Fig. 1)

-10 -8 -6 -4 -2 0 2 4 6 8 10-60

-40

-20

0

20

40

60

11th cycle

τ (kP

a)

γ (%)

Zone of zero effective confining stress state

γο γd

γογd

Fig. 13 Illustration of the decomposition of post-liquefaction shear

strain

78 Acta Geotechnica (2012) 7:69–113

123

c ¼ cd þ co ð1Þ

Considering their basic feature and triggering mechanism,

we define cd as the ‘‘solid-like shear strain’’ and co as the

‘‘fluid-like shear strain’’.

The stress–strain curve in each loading cycle can also be

divided into two parts of non-zero and zero effective con-

fining stress. A closer look at the stress–strain curves in

Figs. 2 and 12 shows that the stress–strain hysteresis curves

associated with non-zero effective confining stress are sim-

ilar and parallel to each other. Consequently, if we extract the

fluid-like shear strain co from the total shear strain c, the

obtained new hysteresis curves, i.e., s vs. cd curves, would

almost coincide for all loading cycles. It is further worth

noting that the effective stress paths in the post-liquefaction

regime are nearly the same. This implies that the change in

the solid-like shear strain cd depends only on the current

effective stress, independent of the loading history.

Figure 14 illustrates the increase in the fluid-like shear

strain co with the increasing number of loading cycles,

which is extracted from the same test data in Fig. 1. Based

on the experimental observations as demonstrated in

Figs. 12 and 14, the feature of co can be concluded as

follows:

1. The fluid-like shear strain co is triggered only when

stress path crosses zero effective confining stress.

2. The fluid-like shear strain co depends only on the

shearing history, which will be elucidated later.

3. The fluid-like shear strain co increases monotonically

with increasing number of loading cycles, but the

increasing rate gradually reduces for cyclic loading

with constant amplitude.

4. Flow direction of co is identical to the direction of

shear stress.

5. The fluid-like shear strain co has a large share in the

large post-liquefaction deformation.

A similar phenomenon can also be observed in

undrained monotonic shear tests after complete or incom-

plete liquefaction shown in Fig. 15, where curve 3 to curve

6 correspond to the cases of shearing after complete liq-

uefaction and curves 1 and 2 to those after incomplete

liquefaction. The maximum double-amplitude shear strain

induced in preceding cyclic undrained loading, denoted as

cmax, is labeled to each curve in order to indicate different

loading histories. It can be seen that (1) all the effective

stress paths almost coincide except for the initial loading

phase for curves 1 and 2, (2) all the shear stress–shear

strain relations are similar and parallel to each other, and

the fluid-like shear strain co is the main cause of the dif-

ference in the total shear strain values at the same shear

stress level and (3) the larger the preceding maximum shear

strain cmax is, the larger the total shear strain c induced at

the same shear stress level. The above experimental facts

show that for the case of undrained monotonic shearing, the

solid-like shear strain cd also depends only on the current

effective stress, but the fluid-like shear strain co is governed

by the shearing history.

Note that the fluid-like shear strain co is often neglected

in most of the past studies. In most constitutive models, the

post-liquefaction shear strain is obtained by reducing the

shear modulus or employing a technique of reducing shear

modulus in the vicinity of zero effective confining stress

state, which falls short of the physical mechanism of the

post-liquefaction response.

3.2 Post-liquefaction reconsolidation volumetric strain

Reconsolidation volumetric strain is the volumetric strain

induced by dissipation of excess pore pressure that

0 1 2 3 4 5 6 7 8 9 10 11 12

0

2

4

6

8

10

Undrained conditionCyclic torsion testf=0.01Hz

Toyoura sandDr=75%

γ o(%

)

Number of cycles, N

Initial liquefaction(NL=4.5)

Fig. 14 Development of shear strain component co (data obtained

from the test shown in Fig. 1)

0

20

40

60

80

100

120

0 5 10 15

τ (k

Pa)

(a)

γmax= 1.5%

18.6%

11.6%

13.4%

6.8%2.5%

Curve-1

Curve-6

5432

Induced by preceding cyclic undrained loading

0

20

40

60

80

100

120

0 40 80 120 160

CSL

(b)

Dr = 70%

1Mcs

τ (kP

a)

Toyoura Sand

Post-liquefaction monotonic loading in undrained condition

'σc =100 kPa

σm' (kPa)

Fig. 15 The post-liquefaction stress–strain behavior of saturated sand

in undrained conditions (after [114])

Acta Geotechnica (2012) 7:69–113 79

123

generates under undrained cyclic loading. Lee and Albaisa

[62] showed that the pre-liquefaction reconsolidation vol-

umetric strain increases with the increasing excess pore

pressure, and this relationship is almost independent of the

way how the excess pore pressure was generated. However,

they found that the post-liquefaction reconsolidation vol-

umetric strain is obviously larger than that before the initial

liquefaction. Shamoto et al. [112, 113, 116–118] and

Shamoto and Zhang [115] further found that the volume

change in different sands over a wide density range can be

uniquely related to an index ‘‘relative compression’’. A

close correlation between the relative compression and the

maximum shear strain induced during preceding undrained

cyclic loading was experimentally obtained by evaluating

numerous test data over a range of maximum double-

amplitude shear strain from 0.01 to 10% and relative

densities from 20 to 90% for five sands.

Shown in Fig. 16 are the relationships of the reconsol-

idation volumetric strain ev;recon with the excess pore

pressure ratio pe

�r0c and maximum double-amplitude

cyclic shear strain cmax induced during undrained cyclic

loading. The relationships were obtained from undrained

cyclic triaxial tests followed by drained reconsolidation

under constant confining pressure conditions for isotropi-

cally consolidated specimens of saturated sand. As can be

seen from Fig. 16a, ev;recon increases as pe

�r0c increases

before the initial liquefaction; however, ev;recon increases

abruptly, but pe

�r0c remains at 1.0 after the initial lique-

faction. This fact implies that it is difficult to reasonably

evaluate the liquefaction-induced settlements of sand

deposits by using the excess pore pressure ratio. Moreover,

Fig. 16b shows that there exists a close correlation between

ev;recon and cmax both before and after the initial

liquefaction. This confirms that the maximum shear strain

can be used as an important factor to evaluate the lique-

faction-induced level ground settlements (e.g., [52, 115,

124]).

4 Physics of post-liquefaction deformation

4.1 Three volumetric strain components

It is generally accepted that the total volumetric strain ev of

sand subjected to a general loading can be divided into two

parts, i.e., the one induced by the change in mean effective

stress, denoted as evc, and the other due to dilatancy,

denoted as evd. Figure 17 shows the change in shear strain

and volumetric strain with the increasing number of load-

ing cycles, which was obtained from a drained cyclic tor-

sional test under constant isotropic effective confining

stress of r0c = 30 kPa. Based on the experimental facts as

illustrated in Fig. 17, Shamoto et al. [114] and Zhang [166]

found that the volumetric strain due to dilatancy can be

decomposed into two different components during drained

shearing, a reversible dilatancy component evd;ir and an

irreversible dilatancy component evd;re, namely

evd ¼ evd;ir þ evd;re ð2Þ

Hence, the total volumetric strain ev can be expressed in

general as

ev ¼ evc þ evd ¼ evc þ evd;ir þ evd;re ð3ÞFigure 17 shows that the irreversible dilatancy compo-

nent evd;ir always remains positive, corresponding to vol-

ume contraction, and conversely, the reversible dilatancy

component evd;re always keeps less than or equal to zero,

corresponding to volume expansion. The irreversible

dilatancy component evd;ir increases monotonically with the

number of loading cycles and depends mainly on the shear

history. The accumulation rate of evd;ir gradually becomes

0 0.4 0.8 1.20.01

0.1

1

10

0.01 0.1 1 10 100

Dr=50%

(a)ε v

, rec

on(%

)

pe /σc'

Toyoura sand

γ = 1.5εa

(b)

γmax (%)

Cyclic triaxial

Pre-liquefaction

Post-liquefaction

'σc =100 kPa

Fig. 16 Relationship of reconsolidation volumetric strain with excess

pore water pressure ratio and maximum shear strain (data from

[112, 113, 116–118])2.5

2.0

1.5

1.0

0.5

0.0

0 5 10 15 20

0.0

-0.4

-0.8

-3

0

3-0.4

0.0

0.4

Dr=70%

Drained torsional test

εvd

(c)

ε vd

& ε

vd,ir

( %) SaturatedToyoura sand

εvd,re = εvd - εvd,ir(d)

ε vd,

re(%

)

Number of cycles, N

(b)

γ(%

)σ/τ

' c

(a)

σ 'c = 30 kPa

εvd,ir

Fig. 17 Two dilatancy components in cyclic drained shear test (data

from [114])

80 Acta Geotechnica (2012) 7:69–113

123

smaller and smaller with increasing number of cycles.

Based on the test results in Fig. 17, typical relationships of

the reversible dilatancy components evd;re with the shear

strain c and the shear stress ratio s�r0c for the data of 18th

to 20th cycles are plotted in Fig. 18. The reversible dilat-

ancy component evd;re shows a reversible and synchronous

change with the cyclic changes in shear strain c and shear

stress ratio s�r0c. The s

�r0c-c curves almost coincide for the

cycles from 18th to 20th, indicating that the loading history

has little effect on the stress–strain response.

Further studies on physical mechanism and feature of

the reversible and irreversible dilatancy were conducted by

Zhang [166] and Zhang et al. [171].

4.2 Three physical states

According to the above decomposition of the volumetric

strain, the volumetric strain component induced by the

change in mean effective stress evc can be expressed as

evc ¼ ev � evd;ir þ evd;re

� �ð4Þ

Under undrained conditions, we have ev ¼ 0, and Eq. 4

becomes

evc ¼ � evd;ir þ evd;re

� �ð5Þ

During undrained cyclic loading, evd;ir and evd;re will

change with the cyclic shearing, and evc will also change

according to the constraint condition of Eq. 5. As seen

from Fig. 17, evd;ir increases monotonically, and con-

versely, evd;re changes reversibly along with the cyclic

shear stress. Since evd;ir always keeps lager than evd;re in the

entire cyclic shearing, as an inevitable consequence, the

absolute value of evc increases averagely in a swelling

manner cycle by cycle but also fluctuates with the suc-

cessive cyclic shear stress. evc reaches its minimum value,

denoted as evc;o, when the effective confining stress

decreases from its initial value to zero.

In this study, evc;o is used as an important threshold

value to judge whether the effective confining stress

reaches zero by comparing evc and evc;o. The effective

confining stress vanishes when evc� evc;o. For convenience,

evc;o is defined as the threshold volumetric strain.

Because of the changes in evd;ir and evd;re, the value of

evc, calculated by Eqs. 4 or 5, can be less than, greater than

or equal to evc;o in the entire undrained cyclic loading

process, which corresponds, respectively, to three physical

states in which saturated sand may behave as follows:

(1) State of contact and friction between particles:

Saturated sand exists in the state of contact and friction

between their particles when evc [ evc;o or � evd;re þ evd;ir

� �

[ evc;o under undrained condition. In this state, the parti-

cles of sand contact each other and form a compressible

soil skeleton structure. As a result, the saturated sand has a

non-zero effective confining stress and can sustain shear

stress. Saturated sand behaves as a granular frictional

material. The shear strain developed in this state is the

solid-like shear strain cd defined before.

(2) State of particles in suspension: Saturated sand exists

in the state of no contact and friction between their parti-

cles when evc\evc;o or � evd;re þ evd;ir

� �\evc;o under

undrained condition. In this state, the particles of saturated

sand are in suspension, as illustrated in Fig. 19b, and

therefore, saturated sand behaves instantaneously like a

viscous fluid and cannot bear shear stress. This state is the

so-called liquefaction state. The shear strain developed at

this state is the fluid-like shear strain co. It should be noted

that although a soil skeleton does not exist in such a state of

evc\evc;o and thus the premises of the abstract concept of

zero effective confining stress are in no existence, we still

regard this state as zero effective confining stress state from

macroscopic aspect. In this state, the change in evc is

irrespective of the change in effective confining stress.

Only in the range of evc� evc;o, a change in evc can be

linked to a change in the effective confining stress. Under

the liquefaction state, i.e., the state of particles in suspen-

sion, therefore, the value of evc can be calculated only by

Eq. 5.

Note that although the saturated sand in the state of

particles in suspension behaves like a fluid, it is essentially

different from a pure fluid. The two volumetric strain

components due to dilatancy are still assumed to exist and

evolve along themselves way in this state, and the particles

of saturated sand in this state will return to contact each

other and reform a skeleton after experiencing a

ε vd,

re(%

)

-0.2

d

c

b

a0.1

0

-0.1

-0.2

-0.3

-0.4

-0.5

d

c

b

a

τ/σc'γ (%)

-2 -1 0 0.5-0.5-1.5-2.5 1 -0.1 0 0.1 0.2 0.3

(a) 18th to 20th cycles (b) 18th to 20th cycles

Fig. 18 Change in reversible volumetric strain component with

cyclic shear strain

γο

(a) Critical contact state (b) Suspension state (c) Critical contact state

Fig. 19 Illustration of probable arrangement of particle assembly of

two physical states existing in zero effective confining stress state

Acta Geotechnica (2012) 7:69–113 81

123

sufficiently large shear strain. In other words, the sand in

the state of particles in suspension is still a fluid-like

granular material with dilatancy characteristics.

(3) State of critical contact and friction between parti-

cles: Saturated sand exists in the critical contact and fric-

tion between their particles, when evc ¼ evc;o or

� evd;re þ evd;ir

� �¼ evc;o under undrained condition. This

state was the boundary state between states (1) and (2). In

this state, the particles of sand is in a critical contact with

each other and form a soil skeleton, but there is no contact

stress between the particles and the effective stress is zero.

Because of the directional rearrangement of particles dur-

ing cyclic undrained shearing, the sand may either enter

into the liquefaction state due to the contraction of soil

skeleton when shearing along a certain direction as illus-

trated in Fig. 19a, b or enter into the state of contact due to

dilatancy of the soil skeleton when shearing in the opposite

direction as shown in Fig. 19b, c.

4.3 Mechanism of post-liquefaction shear strain

As stated above, the volumetric strain evc has to change in

undrained cyclic shearing due to the changes in evd;ir and

evd;re, and its change is linked to the change in effective

confining stress when evc is in the range greater than evc;o.

As a result, excess pore water pressure pe builds up in the

sand when the total confining stress remains constant.

For a saturated sand subjected to undrained cyclic

shearing, evd;ir always increases, while evd;re varies revers-

ibly, thereby resulting in that the average value of evc

always decreases from cycle to cycle with the decreasing

effective confining stress. When evc first satisfies the con-

dition of zero effective confining stress state, we have

evc ¼ ev � evd;ir þ evd;re

� �� evc;o ð6Þ

The effective confining stress vanishes for the first time,

i.e., the saturated sand reaches the initial liquefaction. With

the continuation of shearing, the decrease in evd;re that leads

to dilation of the soil skeleton tends to prevail and induces

an increase in evc. When evc satisfies the condition of non-

zero effective confining stress state again

evc ¼ ev � evd;ir þ evd;re

� �[ evc;o ð7Þ

the saturated sand will leave the zero effective confining

stress state, and the shear resistance will recover. In the

post-liquefaction regime, the inequalities of Eqs. 6 and 7

are satisfied alternatively with the change in the reversible

component evd;re and its progressive increase in amplitude.

Correspondingly, the saturated sand experiences alterna-

tively a change of state between zero and non-zero effec-

tive confining stress in each loading cycle. Since evd;ir

increases monotonically during the entire loading process,

the amplitude of evd;re needs to increase from cycle to cycle

in order to balance evd;ir and satisfy Eq. 7. Moreover,

generation of sufficiently large evd;re requires large fluid-

like shear strain co. Since the increment of evd;re induced by

the solid-like shear strain cd is nearly the same for all

shearing cycles with constant amplitude, a further increase

in evd;re can only be induced by an increase in co. Thus, the

fluid-like shear strain co triggered in zero effective con-

fining stress state increases from cycle to cycle, as can be

observed in the post-liquefaction regime in Fig. 12. In

addition, the accumulation rate of evd;ir gradually tends to

reduce along with the increasing number of loading cycles.

As a result, the rate of co shows similar tendency with

increasing loading cycles.

Based on the above mechanism, Fig. 20 provides a ge-

danken experiment for the evolution of the two dilatancy

components, which are presumed from the test data shown

in Fig. 1. Correspondingly, Fig. 21 shows a gedanken

experiment on the change in the reversible dilatancy

component evd;re with the number of loading cycles and

cyclic shear strain, c, respectively, with the data extracted

from Fig. 20. The following can be seen clearly from the

two figures: (1) The irreversible dilatancy component evd;ir

increases monotonically during the whole loading process;

(2) The reversible dilatancy component evd;re is completely

reversible and its value becomes zero twice in each loading

cycle, while its amplitude gradually tends to increase with

the increasing number of cycles and this amplitude increase

in each cycle occurs only at the liquefaction state of zero

0 2 4 6 8 10 12 144

3

2

1

0

-1

(%)

,vd

irε

vdε&

Number of cycles, N

needed to leave zero effective confi-ning stress state

,vd reε−

,vc

oε−vdε

Pre- Post-liquefaction,vd irε

Non-zero effectvie confining stress state

-2

Zero effectvie confining stress state

Fig. 20 Gedanken experiment on developments of dilatancy com-

ponents based on test result shown in Fig. 1

0 2 4 6 8 10 12

0

-1

-2

-3

-4

-10 -5 0 5 10

(%)

Number of cycles, N γ (%)

( ),vd re fε γ=(b)(a) εvd,re = εvd - εvd,ir

Fig. 21 Gedanken experiment on development of reversible dilat-

ancy component and its change with cyclic shear strain based on test

result shown from Fig. 20

82 Acta Geotechnica (2012) 7:69–113

123

effective confining state; (3) The fluid-like shear strain co

increases with the increasing number of cycles, thereby

triggering sufficient increase in evd;re each time when the

effective stress vanishes; (4) The change in cd and also

evd;re are nearly the same for all loading cycles in the post-

liquefaction regime, since all the effective stress paths

nearly coincide; and (5) A sufficiently large fluid-like shear

strain co is needed at zero effective confining stress to

generate sufficiently large reversible dilatancy component

evd;re so as to satisfy the zero volume change condition of

Eq. 5 and counterbalance the increasing irreversible dilat-

ancy component evd;ir.

Comparing Figs. 17 and 20 reveals that the evolution of

evd;ir and evd;re is quite different for drained and undrained

cyclic shearing process. This is caused by the difference in

the volumetric constraint conditions of the two types of

tests. This is the reason for the obvious deficiency when the

volumetric strains obtained from cyclic drained tests are

used for predicting the excess pore water pressure gener-

ation, as discussed before.

4.4 Mechanism of volumetric strain during

post-liquefaction reconsolidation

Some basic laws of the volumetric strain during the post-

liquefaction reconsolidation summarized from previous

experimental observations can also be well explained. As

known from the above mechanism, reconsolidation volu-

metric strain, denoted as ev;recon before, is mainly composed

of irreversible dilatancy component evd;ir and residual

reversible dilatancy component evd;re

� �residu

and can be

written as

ev;recon ¼ evd;ir þ evd;re

� �residu

ð8Þ

Since evd;ir always monotonically increases both before and

after the initial liquefaction, ev;recon has a continuous

increase in the entire cyclic shearing process. In addition,

evd;ir has close correlation with cmax, which was theoreti-

cally explained and experimentally confirmed [115]. The

maximum cyclic shear strain cmax induced after the initial

liquefaction is much larger than that before the initial liq-

uefaction, resulting in larger evd;ir and ev;recon.

In undrained cyclic loading, evd;re increases with

increasing evd;ir so as to satisfy the volumetric consistent

condition of Eq. 5, but the relationship between evd;re and cis independent of loading history. In the subsequent

reconsolidation, evd;re

� �residu

depends only on the residual

shear strain, denoted as cr. The magnitude of evd;re

� �residu

is

determined only by the residual shear strain cr. In partic-

ular, evd;re

� �residu

= 0 when cr = 0. In other words, the

larger cr is, the lager evd;re

� �residu

is, the smaller ev;recon is.

Certainly, the smaller cr is, the larger ev;recon is, and ev;recon

reaches its maximum value when cr = 0.

Equation 8 and the above mechanism indicate that there

is an intrinsic relationship between the reconsolidation

volumetric strain ev;recon, the maximum cyclic shear strain

cmax and the residual shear strain cr. They are interdepen-

dent one upon another. It is worth noting that the lique-

faction-induced ground surface settlement for level ground

can be evaluated as one-dimensional reconsolidation. It can

be computed by equating the vertical strain to the recon-

solidation volumetric strain ev;recon and then integrating

ev;recon over the depth interval of liquefied soil layer. In

addition, case studies [118] show that lateral spreading can

occur in liquefiable level ground without initial driving

shear stresses. In this situation, liquefaction-induced lateral

spreading can be determined as the integration of the

residual shear strain cr over the depth interval of liquefied

soil layer. Since no initial driving shear stresses are present

in level ground during the post-liquefaction reconsolida-

tion, the residual shear strain cr is attributed to the value of

the fluid-like shear strain co remained at the end of the

reconsolidation. It can be inferred from the above analysis

that liquefaction-induced ground settlement and lateral

spreading for level ground are interdependent of each

other, and neither can be predicted separately in principle.

This conclusion is also appropriate for liquefiable ground

with gently inclined surface and relatively shallow

groundwater, but the residual shear strain cr in this case is

produced as the values of both the fluid-like shear strain co

and the solid-like shear strain cd remained at the end of the

post-liquefaction reconsolidation.

4.5 Mechanism of post-liquefaction flow slides

The criterion triggering the post-liquefaction flow slides in

saturated sand can be written in terms of the volumetric

strain components. Since saturated sand cannot dilate to an

unlimited extent, the reversible dilatancy component evd;re

has a minimum value, i.e., the maximum volumetric strain

in expansion due to dilatancy, denoted as evd;re;min here. If

evd;re is mobilized to evd;re;min and conditions of Eq. 7

cannot be satisfied, namely,

�evd;re;min\ evd;ir þ evc

� �� ev ð9Þ

then the saturated sand will behave in two ways. One is for

the sand that exists in the liquefaction state at which evc ¼evc;o and ev ¼ 0 in Eq. 9. The sand will continue staying in

zero effective confining stress state and exhibits a larger

flow deformation with instability. The other one is for the

sand that exists in non-zero effective confining stress states

at which evc [ evc;o and ev ¼ 0 in Eq. 9. The sand will

appear a large collapse-type flow deformation. The two

Acta Geotechnica (2012) 7:69–113 83

123

kinds of flow deformation are so-called post-liquefaction

flow slide.

Figure 22 shows a gedanken experiment on evolution of

reversible and irreversible dilatancy components when the

post-liquefaction flow slide is triggered. It can be under-

stood that the smaller the initial sand density is, the smaller

�evd;re;min is, but the larger evd;ir is. As a result, inequality of

Eq. 9 is more easily satisfied. Therefore, post-liquefaction

flow slides are usually easier to occur in looser saturated

contractive sand subjected to cyclic or monotonic loading.

Equation 9 also indicates that the post-liquefaction flow

slides can occur in denser saturated dilative sand as long as

sufficiently large shearing-induced water absorption takes

place in naturally drained condition and leads to ev\0. This

conclusion was experimentally confirmed by Zhang [162],

Zhang et al. [165] and Tokimatsu et al. [132]. Post-lique-

faction flow slide is an instable deformation in substance.

Once flow slide occurs, whether catastrophic deformation is

induced or not depends mainly on boundary conditions of

practical problems. Equation 9 provides a criterion for

assessing whether post-liquefaction flow slide occurs.

4.6 A general approach to post-liquefaction

deformation

The assumption described in the previous sections play an

important role in understanding the physics of the post-

liquefaction regime and in developing the constitutive

model. The main ingredients of this assumption are reca-

pitulated below.

(1) Development of post-liquefaction deformation is

accompanied by phase transformation between the solid-

like and fluid-like state. This phase transformation occurs

twice in each loading cycle. The solid-like and fluid-like

state corresponds, respectively, to the state of contact and

friction between sand particles and the state of particle

suspension. The threshold volumetric strain evc;o is used to

delimit the two states by comparing evc and evc;o. When

evc [ evc;o, saturated sand is in the solid-like state and

behaves like a biphasic porous medium. When evc� evc;o,

the same sand is in the fluid-like state and behaves like a

quasi-viscous fluid. It should be noted that this quasi-vis-

cous fluid differs from conventional fluids in that it may

show dilatancy. The solid-like shear strain cd and the fluid-

like shear strain co are actually two kinds of shear strain

induced in the solid-like and fluid-like states, respectively.

These two shear strains compose the post-liquefaction

shear strain, i.e., c = cd ? co as stated in Eq. 1.

(2) There are two kinds of post-liquefaction flow

deformation after the initial liquefaction. The one takes

place in the liquefaction state at which evc ¼ evc;o and

ev ¼ 0 in Eq. 9. The other occurs in non-zero effective

confining stress states at which evc [ evc;o and ev ¼ 0 in

Eq. 9. Both deformations are triggered under the condition

that the reversible dilatancy evd;re is not large enough to

satisfy the constraint condition of Eq. 5. These two

deformations are characterized by instable flow. Although

exact evaluation of these deformations is rather difficult,

Eq. 9 provides a criterion for assessing whether the two

kinds of post-liquefaction deformation are induced.

(3) The fluid-like shear strain co is actually the minimum

shear strain length required to generate sufficiently large

evd;re so as to satisfy the volume constraint condition of

ev � evd;re þ evd;ir

� �� evc;o and to enable the phase transfor-

mation from the fluid-like to solid-like state. In other words,

the maximum absolute value of evd;re required to leave the

zero effective confining stress state depends on the length of

co, which can be expressed mathematically as follows.

evd;re ¼Z co

0

Dre;odc ¼ ev � evd;ir þ evc;o

� �ð10Þ

devd;re=dc ¼ Dre;o ð11Þ

in which Dre,o is the reversible dilatancy rate at zero effective

confining stress. Assuming that the average value of Dre,o is

Dre;o, the value of co can be solved from Eq. 10 as

co ¼ev � evd;ir þ evc;o

� �

Dre;o

ð12Þ

(4) The solid-like shear strain cd may be determined by

commonly used constitutive models for pre-liquefaction

stress–strain response. Particularly, the post-liquefaction

solid-like shear strain cd is much more easily described

than pre-liquefaction shear strain. This is because the

experienced stress–strain history is completely swept out of

memory when the effective stress vanishes each time in the

post-liquefaction cyclic shearing process. For the case of

monotonically undrained shearing application from zero

effective confining stress state after the initial liquefaction,

the effective stress path always undergoes along the critical

stress state line as illustrated in Fig. 15, which is the

simplest proportional loading. In addition, the sand having

0 2 4 6 8 10 12 144

3

2

1

0

-1

(%)

,vd

irε

vdε&

Number of cycles, N

needed to leave zero effective confi-ning stress state

,vd reε−

,vc

oε −vdε

Pre- Post-

,vd irε

Non-zero effectvie confining stress state

-2

Zero effectvie confining stress state

, ,minvd reε−

liquefactionPoint to trigger

flow slides

Fig. 22 Gedanken experiment on evolution of dilatancy when onset

of flow slides

84 Acta Geotechnica (2012) 7:69–113

123

been swept out of memory will act like a simple material.

Determination of cd therefore becomes much easier. We

take this condition as an example to illustrate a method of

calculating cd.

In the absence of rate-dependent behavior, the strain

rates can equivalently be used instead of strain increments.

According to elastoplasticity theory, the solid-like shear

strain increment _cd can be decomposed into an elastic shear

strain increment _ced and a plastic one _cp

d, i.e.,

_cd ¼ _ced þ _cp

d ð13Þ

The elastic shear strain increment _ced can be determined

by

_ced ¼ _q=3G ð14Þ

in which G is the elastic shear modulus and _q is the de-

viatoric stress invariant increment.

The plastic shear strain increment _cpd can be determined

by the stress-dilatancy relationship and volumetric consis-

tency condition. The total stress-dilatancy relationship is

assumed as

_evd

�_cp

d ¼ D ð15Þ

where D is the total dilatancy rate including irreversible

and reversible components. The volumetric consistency

condition is

_evc þ _evd ¼ _ev ð16Þ

The strain rate _evc can be determined by the mean

effective stress increment _p as follows:

_evc ¼_p

K¼ _q

KMcs

ð17Þ

where K is the bulk modulus, and Mcs is the slope of critical

stress state line in p - q plane. Substituting Eqs. 15 and 17

into Eq. 16 leads to

_cpd ¼

_ev

D� _q

KMcsDð18Þ

Thus, the solid-like shear strain increment _cd can be

written as

_cd ¼ _ced þ _cp

d ¼_q

3Gþ _ev

D� _q

KMcsDð19Þ

Equations 10 and 19 provide a general formulation for

conditions of monotonic shearing after the initial

liquefaction. From the two equations, the post-liquefaction

shear strains can be determined once Dre,o and D are

calculated from Eqs. 11 and 15. Obviously, one of the most

important issues concerning prediction of co and cd is how to

determine the two dilatancy equations.

The above constitutive relations provide a general

framework to evaluate the post-liquefaction stress–strain

response. These expressions are also suitable for cyclic

loading. Combining Eqs. 10 and 19 with commonly used

theories of cyclic plasticity may develop specific constitutive

models for the two shear strain components co and cd.

As an example of application, presented in the next

section are specific formulations that are defined to estab-

lish a specific cyclic elastoplasticity model for large post-

liquefaction deformation.

5 Cyclic model for post-liquefaction deformation

5.1 Stress and strain variables

Tensor quantities are denoted either by bold-faced letters in

direct notation or by letters with indices with summation

over repeated indices. Colon denotes inner product. The

stress and strain variables are defined as follows. Effective

stresses and their increments are taken to be positive in

compression.

The mean effective stress p and the volumetric strain ev

are given by:

p ¼ 1

3trr ¼ 1

3rii ð20Þ

ev ¼ tre ¼ eii ð21Þ

where r is the effective stress tensor, e is strain tensor.

The deviatoric stress and strain tensors are defined as

s ¼ r� pI ð22Þ

e ¼ e� ev

3I ð23Þ

where I ¼ dij denotes the identity tensor of rank two, i.e.,

the Kronecker delta. A useful variable is the deviatoric

stress ratio tensor r with

r ¼ rij ¼sij

pð24Þ

Moreover, the following invariants will be frequently

used in the model definition:

q ¼ffiffiffiffiffiffiffiffiffiffiffi3

2s : s

r

ð25Þ

g ¼ q

pð26Þ

S ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

3s : s : s

3

r

ð27Þ

hr ¼1

3sin�1 � 1

2

S

q

� �3 !

ð28Þ

c ¼ffiffiffiffiffiffiffiffiffiffiffiffi2

3e : e

r

ð29Þ

Acta Geotechnica (2012) 7:69–113 85

123

in which hr denotes the Lode angle. Here, hr ¼ �30�

represents triaxial compression, and hr ¼ 30� represents

triaxial extension.

5.2 General formulation

We use the additive decomposition of strain increment into

an elastic and a plastic part. In the absence of rate-depen-

dent behavior, the strain rates can equivalently be used

instead of strain increments, i.e.,

_e ¼ _ee þ _ep ð30Þ

where _e, _ee, _ep; denote the strain rate tensor and its elastic

and plastic parts, respectively.The elastic strain can be

determined by a generalized Hooke’s law, i.e.,

_ee ¼ _ee þ 1

3_evI ¼ 1

2G_sþ 1

3K_pI ð31Þ

in which G and K are, respectively, elastic shear and bulk

modulus. Their dependence both on material state and on

effective stress state will be specified in the later section.

The plastic response can be induced due to two different

and superposed mechanisms [139]. The first mechanism is

associated with the shear stress ratio r, known as shear

yielding mechanism, and the second, with change in mean

effective stress p, known as compression mechanism. In the

present model, only the first mechanism is taken into

consideration for simplicity. To this end, the following

plastic loading intensity L can be defined.

L ¼ p _r : n ð32Þ

in which n is the loading direction in stress ratio space.

Loading or unloading is defined by the sign of L, i.e.,

positive for loading and negative for unloading. In case of

loading, plastic deformation will be generated. The plastic

hardening criterion is assumed as

_k ¼ _cp ¼ Lh iH¼ p _r : nh i

Hð33Þ

where k is the plastic multiplier, and H is the plastic

modulus. hi is the Macauley brackets. The rate of the

plastic deviatoric strain can be written as

_ep ¼ m _cp ¼ mp _r : nh i

Hð34Þ

in which m is a zero trace tensor defining the direction of

plastic deviatoric strain increment.

As discussed in Sect. 4.1, the volumetric strain induced

due to shear is composed of an irreversible component and

a reversible component. But according to current laws of

elasticity, elastic shearing does not cause volumetric

change. Therefore, both dilatancy components are assumed

as plastic ones in the present model. They can be written as

_evd;re ¼ _epvd;re ¼ Dre _cp ¼ Dre

p _r : nh iH

ð35Þ

_evd;ir ¼ _epvd;ir ¼ Dir _c

p ¼ Dir

p _r : nh iH

ð36Þ

where Dre and Dir are the reversible and irreversible

dilatancy rates, respectively.

The total strain rate can be obtained by adding Eqs. 31,

34, 35 and 36 to give

e ¼ 1

2Gp _rþ r

2Gþ I

3K

� �_pþ mþ Dre þ Dir

3I

� �p _r : nh i

H

ð37Þ

The above constitutive equation has the following

ingredients: (a) elastic shear and bulk moduli G and K; (b)

loading and flow direction n and m in the stress ratio space;

(c) plastic modulus H; and (d) dilatancy function Dre and Dir.

These quantities will be specified in the following section.

5.3 Model ingredients

The bounding surface model of Dafalias [26] is adopted to

define the model operation and give the specific formula-

tion of the model ingredients. Some model features are also

attributable to the hypoplasticity model developed by

Wang et al. [139] and the model developed by Li and

Dafalias [63].

5.3.1 Definition of surfaces and mapping rule

The cone-shaped failure surface is defined by

f rð Þ ¼ g�Mf;cg hr

� ¼ 0 ð38Þ

A superposed hat is used to signify that the stress quantities are

associated with the failure surface. Mf,c is the stress ratio at

failure in triaxial compression. The function g hrð Þ interpolates

the failure stress ratio between triaxial compression and

extension stress state. In this model, the following

interpolation function proposed by Zhang [162] is adopted:

g hrð Þ ¼1

1þMf;c=6 1þ sin 3hr � cos2 3hrð Þ þ 1=Mf;o Mf;c �Mf;o

� �cos2 3hr

ð39Þ

Mf;c ¼6 sin /f;c

3� sin /f;c

ð40Þ

Mf;o ¼2ffiffiffi3p

tan /f;cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3þ 4 tan2 /f;c

q ð41Þ

The coefficients Mf,c and Mf,o represent the failure stress

ratios at hr ¼ �30� (triaxial compression stress state) and

86 Acta Geotechnica (2012) 7:69–113

123

hr ¼ 0 (torsional shear stress state after isotropic

consolidation). Both coefficients depend on the failure

friction angle at triaxial compression /f;c. Shown in Fig. 23

are the failure loci in the p plane by Eq. 39 and the

Matsuoka-Nakai criterion, respectively, together with some

test data in the literature ([122, 98]). It can be seen that a

good agreement exists between the tested data and the two

calculated curves. Note that the Matsuoka-Nakai criterion

is not adopted here since it is not well defined for vanishing

effective stress which appears repeatedly in the post-

liquefaction regime.

The maximum prestress memory surface, which records

the maximum r in the loading history and expands if

pushed further outwards until it meets failure surface

f ¼ 0, serves as bounding surface and is analytically

expressed as

�f �rð Þ ¼ �g� gmg �hr� �

¼ 0 ð42Þ

in which a superposed bar symbolizes their association

with the bounding surface �f ¼ 0. gm is a history parameter

remembering the maximum prestress level in terms of g at

triaxial compression stress state. g hrð Þ serves again for

interpolation.

5.3.2 Mapping rules and flow direction

The mapping rule, proposed by Wang et al. [139] and also

employed by Li [64], is adopted for the present model.

Figure 24 provides an explanation of the mapping rules in

the deviatoric stress ratio space. In the figure, a is the

relocatable projection center, r is the current stress ratio,

and �r is the image stress ratio on the bounding surface�f ¼ 0. As shown in the figure, the image stress ratio �r is

obtained as linear projection from r onto the bounding

surface with respect to the projection center a, i.e.,

�r ¼ aþ b r� að Þ ð43Þ

where the scalar variable b can be determined by

substituting �r into Eq. 42 and then solving for it. The

loading direction �n is defined as a unit deviatoric tensors

normal to �f ¼ 0 at �r. The q is the distance between the

projection center a and the current stress ratio r, and �q is

the distance between the projection center a and the image

stress ratio �r. The two scalar distances can be determined,

respectively, as follows:

q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2r� að Þ : r� að Þ

r

ð44Þ

�q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2�r� að Þ : �r� að Þ

r

ð45Þ

Loading direction n and flowing direction m in the

deviatoric stress ratio space are assumed the same as the

normal direction of the bounding surface at the image

stress, i.e.,

n ¼ m ¼ �n ð46ÞThe projection center is initially assumed as the origin

and subsequently as the stress ratio at the last stress

reversal. The relocation mechanism of the projection center

and the following mapping rule are illustrated in Fig. 25.

Zhang’s simple criterionMatsuoka-Nakai criterionSutherland et al. (1969)

Ramamurthy et al. (1973)

(q/p´)1.5

1.0

0.5

0

3σ2σ

1σ

o30σθ = −

o0σθ =

o30σθ =

o40fφ =o35fφ =

Fig. 23 A comparison of previous test data with the model failure

criterion

33r

p

σ=2

2rp

σ=

11r

p

σ=

o

r

r

ρ

ρ

n

0f =Bounding surface

Fig. 24 Mapping rules of the model and definition of quantities (after

[63])

3r

1r

90oθ >

r

2r

r

n

=n

n

0f

=

3r

1r

r

2rn

r

r

r

old

(a) Before stress reversal (b) After stress reversal

0f

=

Fig. 25 Illustration of stress reverse mechanism (after [63])

Acta Geotechnica (2012) 7:69–113 87

123

The projection center a is relocated upon stress reversal and

the sign of L changes from positive to negative. This relo-

cation gives rise to a new loading direction n and brings

L back into the positive realm again. The loading intensity

L serves as the loading criterion and enables the description

of plastic deformation inside the bounding surface.

5.3.3 Elastic moduli

The elastic shear and bulk moduli G and K depend both on

the sand density and on mean effective stress. They are

given by the following relations:

G ¼ Go

2:973� eð Þ2

1þ epa

p

pa

� �n

ð47Þ

K ¼ 1þ e

jpa

p

pa

� �n

ð48Þ

where e is the current void ratio, pa is the atmospheric

pressure and Go, j and n are the material parameters among

which n is about 0.5 for most granular soils.

5.3.4 Plastic modulus

A simplified formulation proposed by Wang et al. [139] is

adopted for the plastic modulus for the deviatoric plastic

strain rate due to the action of p _r : n:

H ¼ hg �h� �

GMf;c

gm

�qq

� �� 1

� �ð49Þ

where h is a model parameter, and q and �q are two scalar

distances defined by Eqs. 44 and 45 respectively.

5.3.5 Reversible dilatancy component

From the experimental facts as shown in Fig. 18, each half

loading cycle can be divided into two phases: one for the

generation of the reversible dilatancy component, named as

the dilative phase (phase a–b and c–d in Fig. 18), and the

other for the release of the reversible dilatancy component,

named as the contractive phase (phase b–c and d–a in

Fig. 18). In these two phases, the reversible dilatancy

component exhibits substantially different characteristics.

Therefore, different functions are suggested for the dilative

and contractive phase, respectively, i.e.,

Dre ¼_epvd;re

_cp¼

Dre;gen; g�Md;cg hrð Þ and _g[ 0� �

Dre;rel; g\Md;cg hrð Þ or _g [ 0� �

(

ð50Þ

The parameter Md,c is the stress ratio invariant value at

which the reversible dilatancy changes from contraction to

expansion in triaxial compression. It should be noted that Md,c

is different from the stress ratio at the phase transformation

line proposed by Ishihara et al. [51]. The latter is defined with

reference to total dilatancy including both irreversible and

reversible components, while the former is associated with the

reversible dilatancy component.

Previous experimental studies show that there exists a

linear correlation between the stress-dilatancy ratio and

shear stress ratio of sand in the dilative phase. Therefore, a

linear function is suggested to describe the generation rate

of reversible dilatancy as follows:

Dre;gen ¼ dre;1ðMd;cg hrð Þ � gÞ ð51Þ

in which dre,1 is a material parameter. In dilative phase, we

have g C Md,cg(hr), and thus, Dre B 0. The above equation

is similar to the stress-dilatancy equation proposed by

Rowe [104], which was derived based on the static equi-

librium of an assembly of particles in contact. The suit-

ability of Eq. 51 for evaluating the reversible dilatancy was

confirmed by laboratory tests [166] and supported by

microscopic consideration [74].

Since saturated sand at vanishing effective stress state

also shows dilatancy characteristics, Eq. 51 is assumed to

be applicable to both zero and non-zero effective stress

state. Although the mean effective stress is equal to zero in

the zero effective confining stress state, the shear stress

ratio may not be zero. The shear stress ratio is calculated

according to g ¼ Mf;cg hrð Þ and changes with shear strain.

In order to avoid numerical difficulties at vanishing

effective stress, a small positive value is assigned to the

mean effective stress, which will be explained later.

Shamoto et al. [114] assumed that the reversible shear-

dilatancy rate in zero effective confining stress state is con-

stant and is equal to the stress ratio Mf,c based on associated

flow rule. In our model, the reversible dilatancy rate in zero

effective confining stress state varies with the shear stress

ratio g as described by Eq. 51. For monotonic loading after

the initial liquefaction, a large portion of the post-liquefac-

tion shear strain is generated when the stress state moves

along the failure surface, i.e., g ¼ Mf;cg hrð Þ. The average

shear-dilatancy rate in zero effective confining stress state in

Eq. 12 can be approximately estimated as follows:

�Dre;o ¼R co

0Dre;odc

co

¼R co

0Dre;gendc

co

¼R co

0dre;1 Md;cg hrð Þ � g

� �dc

co

� dre;1 Md;cg hrð Þ �Mf;cg hrð Þ� �

ð52Þ

For triaxial stress state with g(hr) = 1, the above equation

becomes

�Dre;o � dre;1 Md;c �Mf;c

� �ð53Þ

This equation indicates that the reversible dilatancy rate

given by Shamoto et al. [114] may be a little larger than the

value calculated by the presented model.

88 Acta Geotechnica (2012) 7:69–113

123

The release rate of the cumulated volumetric expansion

due to reversible dilatancy can be obtained with the fol-

lowing form:

Dre;rel ¼ dre;2epvd;re

� 2

ð54Þ

in which dre,2 is a coefficient and epvd;re is the reversible

dilatancy component. Obviously, the above expression

remains non-negative, i.e., Dre,rel C 0 and Dre,rel is zero

when the cumulated reversible component epvd;re is com-

pletely released, i.e., epvd;re ¼ 0: Eq. 54 implies that larger

volumetric expansion in the preceding dilative phase

results in larger volumetric contraction in the following

contractive phase. This tendency is qualitatively corrobo-

rated with the experimental observations (e.g., [51, 60, 111,

114, 166]) and micromechanical explanations ([84, 90]).

Equation 50 together with Eqs. 51 and 54 ensures that

epvd;re remains non-positive (i.e., volumetric expansion) and

shows reversible change within each loading cycle. The

performance of our model for the reversible dilatancy

component is shown in Fig. 26. As can be seen from the

figure, epvd;re is assumed to be zero at the beginning of the

virgin loading, and thus, Dre,rel is equal to zero. When the

stress point enters the dilative phase, epvd;re is generated

followed by the subsequent release. In the subsequent

loading process, epvd;re shows completely reversible change

for both symmetrical and non-symmetrical stress cycles.

5.3.6 Irreversible dilatancy component

Previous experimental studies [166, 171–173] show that

the evolution of the irreversible dilatancy is mainly influ-

enced by its two tendencies of decrease. First, the irre-

versible dilatancy rate obviously decreases with increasing

shear strain for each monotonic shearing that commences

from the last stress reversal. Second, the irreversible

dilatancy rate decreases gradually with increasing number

of loading cycles.

In order to account for the first tendency of the decrease

in the irreversible dilatancy rate with the shear strain, a new

variable, named as the effective shear strain rate _ceff , is

introduced to reflect this phenomenon and determined by

_ceff ¼ _cp�

1þ cmono

�cd;r

� �2 ð55Þ

in which cmono is the monotonic shear strain length origi-

nating from the last stress reversal point to current stress

point and cd,r is a model parameter named as the reference

shear strain length.

For the second tendency of the decrease in the irre-

versible dilatancy rate with the number of loading cycles,

the following formula is suggested:

_epvd;ir ¼ dir exp �aep

vd;ir

� _ceff ð56Þ

in which dir is a material parameter determining the irre-

versible dilatancy potential and a is another material

parameter controlling the decreasing rate of the irreversible

dilatancy.

By combining Eqs. 55 and 56, the equation for the

irreversible dilatancy rate can be derived as

Dir ¼_epvd;ir

_cp¼

dir exp �aepvd;ir

�

1þ cmono

�cd;r

� �2ð57Þ

The performance of Eq. 57 is shown in Fig. 27 where

dir = 0.2, a = 50 and cd,r = 5%. It can be seen that two

decrease tendencies of the irreversible dilatancy rate for

cyclic shearing are well captured.

5.4 Numerical algorithm

5.4.1 Model operation after initial liquefaction

In the present model, no other assumptions and parameters

are needed in the calculation of the post-liquefaction shear

strain, but the model operation in zero effective confining0.0 0.5 1.0 1.5 2.0 2.5-3

0

3

Number of cycles, N

γ (%

)

(a)

-0.8 -0.4 0.0 0.4

0.00

-0.04

-0.08

-0.12

-0.4 0.0 0.4 0.8

1th cycle

(b)

ε vd,

re(%

)

(c)2nd cycle

τ/σc ' τ/σc '

Fig. 26 Performance of the formulas for reversible dilatancy

component

-3 -2 -1 0 1 22.5

2.0

1.5

1.0

0.5

0.0

2 4 6 8 10 12

ε vd,

ir(%

)

γ (%)

(a) (b)

Number of cycles, N0

Fig. 27 Performance of the formulas for irreversible dilatancy

component

Acta Geotechnica (2012) 7:69–113 89

123

stress state must be carefully dealt with. Since sand is a

pressure-dependent material, both its modulus and strength

depend on the current mean effective stress. In order to

avoid numerical difficulties at vanishing effective stress, a

small positive value is assigned to the mean effective

stress, denoted as pmin, i.e.,

p [ pmin and evc [ evc;o non-zero effective stress state

p ¼ pmin and evc� evc;o zero effective stress state

(

ð58Þ

Obviously, pmin is an important threshold parameter for

numerical calculations to delimit whether the effective

confining stress reaches zero, thereby defined as the

threshold pressure.

According to our model at zero effective confining stress

(evc� evc:o), the mean effective stress p always remains

equal to the threshold pressure pmin, independent of any

given value of evc. Under this restraint condition, the shear

strain c, the reversible dilatancy component epvd;re and the

irreversible dilatancy component epvd;ir are calculated based

on plasticity theory. As a result, evc is determined as

evc ¼ ev � epvd;re� ep

vd;ir. evc can be regarded as a state vari-

able. Whether the effective confining stress reaches zero

can be judged by comparing evc and the threshold volu-

metric strain evc;o. If evc is greater than evc;o, the effective

confining stress is different from zero.

The model response in a typical post-liquefaction

loading cycle is shown in Fig. 28, where the threshold

pressure pmin is exaggerated for explanation. In the figure,

phase b–c and e–f both are in state of zero effective con-

fining stress.

5.4.2 Stress integration algorithm

In a state of zero effective confining stress, evc will change

due to the change in evd;re and evd;ir. The mean effective

stress p always remains at the threshold pressure pmin when

evc is less than evc;o. For the case of evc\evc;o, common

integration techniques, for classical plasticity ([88, 120]) or

bounding surface models [70], cannot be directly applied to

the present model. A new stress integration algorithm is

developed based on the backward Euler method.

Assume that stress, strain and internal variables

(rn; en;wn) at the beginning of the step n are known and a

strain increment Denþ1 occurred in the step n ? 1 is also

given, we proceed to obtain the stress/strain and internal

variables (rnþ1; enþ1;wnþ1) at the end of the step n ? 1.

Based on the decomposition of strain rate, the stress at the

end of the step n ? 1 can be written as

rnþ1 ¼ rn þ E : Denþ1 � Depnþ1

� �¼ rtrial

nþ1 � Drcornþ1 ð59Þ

in which E is elastic modulus tensor of the fourth order,

rtrialnþ1 ¼ rn þ E : Denþ1 is the elastic trial stress, Dep

nþ1 is the

plastic strain increment and Drcornþ1 ¼ E : Dep

nþ1 is plastic

corrector.

The procedure of the stress integration algorithm is

presented in ‘‘Appendix’’ and can be readily implemented

into a finite element code. The main features of the algo-

rithm include the following: (1) The mean effective stress

must be calculated before the shear stress, in order to deal

with zero mean effective stress state. (2) The convergence

of plastic consistent condition as well as the relationship

between p and evc is checked in the iterative process.

For simplicity, the relationship between p and evc is

represented by a function p ¼ f evcð Þ. The following

expressions can be obtained by integrating Eqs. 48 and 58

as

p ¼ f evcð Þ

¼ papo

pa

� 1�n

þ 1� nð Þ 1þek evc

� � 11�n

; for evc [ evc;o

pmin; for evc� evc;o

8<

:

ð60Þ

wherein po is the initial mean effective stress. This non-

linear elastic relationship between evc and p can greatly

simplify the stress integration algorithm. That is the reason

why the incremental plastic response with change in p

(known as compression mechanism) was ignored in the

presented model as described in Sect. 5.2.

Obviously, the threshold volumetric strain evc;o can be

obtained by substituting p = pmin into the first equation of

Eq. 60 to give

-4 -2 0 2 4

(d)

g f

e

dc

b

a

γ (%)

-2

-1

0

1

2 (a)

pmin

f

e

d

c

bτ(k

Pa)

(b)

g

f

e

d

c

b

aag

Mf pmin

-Mf pmin

Mf

-0.6 -0.4 -0.2 0

(c)

gf

e

dc

b

τ /σ

εvc (%)

a

-4 -2 0 2 4

γ (%)0 1 2 3 4

(kPa)p

0

0.5

1.0

-0.5

-1.0

εvc,o

Fig. 28 Schematic of model response in a typical post-liquefaction

shearing cycle

90 Acta Geotechnica (2012) 7:69–113

123

evc;o ¼ �j

1þ eð Þ 1� nð Þpo

pa

� �1�n

� pmin

pa

� �1�n !

ð61Þ

in which the threshold pressure pmin is a parameter for

numerical calculations, independent of the material prop-

erty. The effect of pmin on model performance is shown in

Fig. 29, where the stress–strain curves and effective stress

paths are given for three cases with pmin equal to 0.1 po,

0.01 po and 0.0001 po, respectively. As seen from this

figure, the threshold pressure pmin has no influence on the

model behavior in the pre-liquefaction regime. In the post-

liquefaction regime, however, the fluid-like shear strain co

increases slightly with pmin. This is because evc;o increases

with pmin as defined by Eq. 61. This in turn gives rise to an

increase in evd,re for the sake of Eq. 6. As a result, the fluid-

like shear strain co increases to enable the phase transfor-

mation from the fluid-like to solid-like state. The model

performance is not sensitive to pmin. In fact, a variation of

pmin by a factor of 100 from 0.01 po to 0.0001 po results in

only small difference in the stress–strain response. More-

over, no instability is observed in the calculation. There-

fore, the threshold pressure pmin can be assumed in the

range of 0.01 po to 0.0001 po.

5.5 Model parameters

There are eleven material parameters in the presented

model as listed in Table 1. They can be classified into four

categories: (1) failure parameters, (2) modulus parameters,

(3) reversible dilatancy parameters and (4) irreversible

dilatancy parameters. These groups of parameters are dis-

cussed in the following.

Failure parameters: The failure surface is defined by

Mf,c, which is the failure stress ratio in triaxial compres-

sion. This parameter can be easily determined either by the

stress at failure or from the critical stress state line (CSL)

for undrained condition. The slope of CSL from undrained

cyclic tests remains constant [161, 162]. This seems to be

at odd with well-known fact that the slope of CSL depends

on the stress level. The reason lies in the fact that the initial

effective confining pressure varies in a relatively small

range of less than 300 kPa for most liquefaction problems

in practice. Instead, the parameter Mf,c can be obtained by

first determining the failure friction angle at triaxial com-

pression /f;c and then calculating it using Eq. 40.

Modulus parameters: The elastic parameters Go and n

can be determined by fitting the relationship between the

initial stiffness of shear stress–strain curve and the effec-

tive confining pressure. Alternatively, these parameters can

also be determined by small strain tests, such as resonant

column tests or bender element tests. The parameter h can

be obtained by fitting shear stress–shear strain (c� q) curve

as described by Wang et al. [139] and Li and Dafalias [63].

The parameter j can be determined from the rebound

curves of e over logp.

Reversible dilatancy parameters: The parameters Md,c

and dre,1 among the three reversible dilatancy parameters

are important for the reversible dilatancy rate. Many

experimental observations ([123, 94, 111]) show that a

unique stress-dilatancy relationship exists for both mono-

tonic and cyclic loading irrespective of void ratio and stress

-40

-20

0

20

40(a)S

hear

str

ess

τ (k

Pa)

Shear strain γ (%)

pmin

= 0.1po

pmin

= 0.01po

po =100kPa

-40

-20

0

20

40

She

ar s

tres

s τ

(kP

a)

Shear strain γ (%)

pmin

= 0.01po

pmin

= 0.0001po

(b)

po =100kPa

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10 -8 -6 -4 -2 0 2 4 6 8 10

0 20 40 60 80 100-40

-20

0

20

40

She

ar s

tres

s τ

(kP

a)

Mean effective stress p (kPa)

(c)

Fig. 29 Influence of pmin on calculated stress–strain response

Table 1 List of model parameters

Category Symbol Reference value

Strength Mf,c 1.4–1.8

Modulus Go 100–200

j 0.001–0.01

n 0.5–1.0

h 0.7–1.2

Reversible dilatancy Md,c 0.3–1.0

dre,1 0.4

dre,2 1,000–1,500

Irreversible dilatancy dir –

a –

cd,r 0.05

Acta Geotechnica (2012) 7:69–113 91

123

level. In addition, Zhang [166] showed that the irreversible

dilatancy has a significant impact on the stress-dilatancy

behavior only in the range immediately after the stress

reversal. Along with further increasing shear strain, how-

ever, the irreversible dilatancy diminishes and the revers-

ible dilatancy becomes dominant. If the initial stage after

stress reversal is excluded from the test data, the stress-

dilatancy relationship of g over devd=dc becomes linear.

Therefore, the parameters Md,c and dre,1 can be obtained by

fitting the linear part of the stress-dilatancy data.

Another reversible dilatancy parameter dre,2 can be

assigned a relatively large value in the range of

1,000–1,500 so as to ensure that the generated reversible

dilatancy is completely released after a relatively small

shear strain.

Irreversible dilatancy parameters: Determining the

three irreversible dilatancy parameters dir, a and cd,r is the

most important but difficult part. These three parameters

control both the pore pressure in the pre-liquefaction

regime and the shear strain in the post-liquefaction regime.

Because of its strong dependence on both the initial density

and confining pressure, the irreversible dilatancy rate under

undrained condition is considerably different from that

under drained condition. Therefore, the three parameters

cannot be determined by a single drained cyclic test.

Moreover, because the irreversible dilatancy cannot be

separated from undrained cyclic test data, it is also difficult

to determine the three parameters through an undrained

cyclic test.

In order to examine the influence of the three parameters

on model performance, some parametric study is carried

out. Some typical results for Toyoura sand at Dr = 70%

are shown in Fig. 30. The numerical results in Fig. 30 are

obtained with a shear stress amplitude of 20 kPa for four

combinations of the irreversible dilatancy parameters with

the same parameters of Mf,c = 1.79, Go = 125, n = 0.5,

j = 0.005, Md,c = 0.5, dre,1 = 0.4, dre,2 = 1,200. A sen-

sitivity analysis of the results in Fig. 30 can be made with

the help of the following two criteria: (1) Number of

loading cycles required to reach the initial liquefaction,

denoted as NL, and (2) Number of loading cycles required

to develop the specified single amplitude shear strain of 6%

after the initial liquefaction, denoted as N6%. It is seen from

Fig. 30 that the larger dir and cd,r are, the smaller a is, the

less NL and N6% become. Particularly, the parameter dir is

found to be the most significant parameter governing both

the pore pressure in the pre-liquefaction regime and the

shear strain in the post-liquefaction regime.

As stated above, the three irreversible dilatancy

parameters are ‘‘hidden parameters’’ in the sense that they

cannot be explicitly determined, neither by drained nor by

undrained test. However, these three parameters can be

identified with the help of NL and N6%. This presents an

optimization problem, which can be easily solved with a

standard computer code. An example is shown in Fig. 31

with the following procedure: (1) Determine the eight

parameters Mf,c, Go, n, j, Md,c, dre,1 and dre,2; (2) Set the

value of po and pmin as well as the cyclic stress amplitude;

(3) Determine NL and N6% according to the test data; (4)

Vary the three parameters dir, cd,r and a until the values of

NL and N6% from model and test are close enough.

5.6 Model performance

5.6.1 Simulation of torsional tests on Toyoura sand

A series of drained and undrained torsional tests were

performed on hollow cylindrical specimens of Toyoura

sand with different relative densities. The test conditions

are described in Sect. 2.1. The model parameters for

Toyoura sand with four different relative densities are lis-

ted in Table 2. It can be seen that the model parameters are

nearly the same for different relative densities except the

strength and irreversible dilatancy parameters.

Although the model is proposed mainly for undrained

cyclic behavior of sand and especially for large post-liq-

uefaction deformation, the drained behavior of sand can

also be simulated and compared quantitatively with

experimental data. As shown in Fig. 32, both the hysteresis

of the shear strain–shear stress curves and the accumulation

of the volumetric strain are well captured by our model.

-20

-10

0

10

20

τ( k

Pa)

(b)

-20

-10

0

10

20 (c)

-20

-10

0

10

20 (a)

-6 -4 -2 0 2 4 6

-20

-10

0

10

20

0 50 100

(d)

p (kPa)γ (%)

dir=0.2α=80γd,r=0.05

dir=0.2a=20γd,r=0.05

dir=0.2a=80γd,r=0.10

dir=0.1a=80γd,r=0.05

τ (k

Pa)

τ(k

Pa)

τ (k

Pa)

Fig. 30 Effect of irreversible dilatancy parameters on calculated

stress–strain response and effective stress path

92 Acta Geotechnica (2012) 7:69–113

123

Figure 33 shows the calculated and tested time histories

of shear stress, excess pore water pressure and shear strain for

Toyoura sand with Dr = 70%. By following closely the

loading program, the evolution of both excess pore water

pressure and shear strain is well reproduced. The predicted

evolution of the two dilatancy components is shown in

Fig. 34. The corresponding stress–strain relation and effec-

tive stress path in test and in simulation are given in Fig. 35.

Similar comparison shown in Figs. 36 and 37 is made for two

specimens with Dr = 60% and 48%, respectively.

The fairly good agreement between test and simulation

as shown in Figs. 35, 36 and 37 brings out the potential of

our model in simulating the complex behavior in the post-

liquefaction regime. In particular, excellent agreement in

the stress–strain curves can be observed. A perusal of

Figs. 35, 36 and 37 reveals, however, that the effective

stress paths before the initial liquefaction are not well

reproduced. This is because the pre-liquefaction stress–

strain response is governed by many factors, some of which

are not taken into account in the present model. Since our

focus is on the large deformation in the post-liquefaction

regime, the model performance beyond the initial lique-

faction is far more important than before the initial

liquefaction.

Shown in Fig. 38 are the predicted non-symmetrical

stress–strain relations and stress paths with three different

Fig. 31 Identification of model parameters using a visual computer on Microsoft Windows by C ?? Builder

Table 2 Model parameters for Toyoura sand at different relative densities

Dr (%) Mf,c Go j n h Md,c dre,1 dre,2 dir a cd,r

72 1.79 125 0.005 0.6 0.9 0.5 0.4 1,200 0.25 170 0.05

70 1.79 125 0.005 0.6 0.9 0.5 0.4 1,200 0.26 80 0.05

60 1.72 125 0.006 0.6 1.0 0.5 0.4 1,200 0.25 40 0.05

48 1.61 125 0.006 0.6 1.0 0.5 0.4 1,200 0.25 30 0.05

Acta Geotechnica (2012) 7:69–113 93

123

static driving shear stresses for Toyoura sand at Dr = 70%.

It is seen that influence of the static driving shear stress on

the stress–strain and stress path response can be well

reproduced by our model.

The other one merit of our model is that it can be also

used to predict the reconsolidation volumetric strain after

the initial liquefaction. According to Eq. 8 in Sect. 4.4, the

reconsolidation volumetric strain ev;recon is mainly com-

posed of the irreversible dilatancy component evd;ir and the

residual reversible dilatancy evd;re

� �residu

. As shown in

Fig. 34, the irreversible dilatancy component evd;re increa-

ses with the number of loading cycles and the increase rate

after the initial liquefaction is much larger than that before

the initial liquefaction. Our model can better predict the

developments of two dilatancy components evd;re and evd;ir

and shear strain c as well as their residual values. The

residual reversible dilatancy component evd;re

� �residu

can be

determined by the calculated residual shear strain cc, while

the reconsolidation volumetric strain ev;recon can be

-30

-20

-10

0

10

20

30

Drained torsional test

σ'c=30 kPa

Dr=72%Toyoura sand

Tested Calculated

She

ar s

tres

s τ

(kP

a)

Shear strain γ (%)

(a)

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

Tested Calculated

Vol

umet

ric

stra

in ε

v(%

)

Shear strain γ (%)

(b)

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

Tested Calculated

Vol

umet

ric

stra

in ε

v(%

)

Shear stress τ (kPa)

(c)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-30 -20 -10 0 10 20 30

0 1 2 3 4 5 6 7 8 9 10

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

Drained torsional test

σ'c=30 kPa

Dr=72%Toyoura sand

Tested CalculatedV

olum

etri

c st

rain

εv(%

)

Number of cycles, N

(d)

Fig. 32 Simulation of drained torsional test result for Toyoura sand at

Dr = 72%

-80

-40

0

40

80

σ'c=100kPaDr =70%

(a)

τ (k P

a)

0

50

100

0 1 2 3 4 5 6 7 8 9 10 11 12-10-5

0

5

Number of cycles, N

Tested

(b)

(c)

Calculatedp e(k

Pa)

γ(%

)

Fig. 33 A comparison between calculated and tested results of the

undrained torsional test for Toyoura sand at Dr = 70%, (a) shear

stress, (b) excess pore pressure, (c) shear strain

0

2

4

0 1 2 3 4 5 6 7 8 9 10 11 12-4

-2

0

Pre-

(a)

(b)

ε vd ,

re

Number of cycles, N

Post-liquefaction

ε vd ,

ir

Fig. 34 Calculated accumulation process of two dilatancy compo-

nents in the undrained torsional test for Toyoura sand at Dr = 70%

94 Acta Geotechnica (2012) 7:69–113

123

obtained as sum of evd;ir and evd;re

� �residu

. As a result, cor-

relation between cc and ev;recon can be also evaluated. This

implies that our model is capable of revealing the interplay

between the residual shear strain cc and the reconsolidation

volumetric strain ev;recon. This phenomenon, however,

cannot be evaluated by most existing models, e.g., Parra

[91], Yang et al. [146], Wang and Dafalias [140] and

Zhang and Wang [167].

5.6.2 Simulation of cyclic tests on Nevada sand

In addition to the simulation of Toyoura sand, we have also

simulated some tests in the literature. The well-docu-

mented tests on Nevada sand with different relative den-

sities [5] are considered. Several undrained cyclic triaxial

and simple shear tests on specimens with relative densities

of 40 and 60% were modeled. These cyclic tests were

stress-controlled with a sinusoidal loading frequency of

approximately 1 Hz. The model parameters for Nevada

sand at the two densities are listed in Table 3.

Figures 39 and 40 shows the undrained cyclic simple

shear test result and model simulation with the same rela-

tive density of 60% at two different cyclic stress ratios,

while Fig. 41 compares test and simulation with the rela-

tive density of 40%. In Fig. 42, the numerical simulation of

an anisotropically consolidated undrained cyclic triaxial

test with relative density of 40% is compared with test

result. Good performance of our model is shown in simu-

lating the cyclic stress–strain development after the initial

liquefaction. Especially as shown in Fig. 42, the cyclic

shear strain accumulation cycle by cycle toward a direction

is well simulated when there exists an application of initial

driving shear stress. It should be pointed out that a rela-

tively high loading frequency of about 1 Hz was adopted in

these tests. As a result, when the stress passes through the

zero effective confining stress state, high shear strain rate

leads to shear resistance due to the viscosity of specimen

and equipment system. The ensuing rate-dependent

behavior gives rise to uncertainty of the test results. This

issue was discussed by Zhang [161] and Zhang et al. [164].

The shear resistance was not taken into consideration in the

present model, which is rate independent.

6 Numerical modeling of centrifuge model tests

The constitutive model presented in Sect. 5 was imple-

mented in the finite element code DIANA- SWANDYNE

II developed by Chan [21]. This two-dimensional finite

element code is capable of solving static and dynamic

-10 -5 0 5-40

-20

0

20

40

-5 0 5 10

Toyoura sand

Dr=70%

(a)

Tested Calculated

±10

Shear strain γ (%)

She

ar s

tres

s τ

(kP

a)

0 20 40 60 80-40

-20

0

20

40

20 40 60 80 100

(b)

Tested Calculated

0

Mean effective stress p (kPa)

She

ar s

tres

s τ

(kP

a )

Fig. 35 A comparison between the tested and calculated results of

the undrained cyclic torsional test for Toyoura sand at Dr = 70%,

(a) stress–strain relation, (b) effective stress path

0 20 40 60 80-30

-20

-10

0

10

20

30

20 40 60 80 100

(b)

0

Mean effective stress p (kPa)

She

ar s

tres

s (k

Pa)

τS

hear

str

ess

(kP

a)τ

Tested Calculated

-10 -5 0 5-30

-20

-10

0

10

20

30

-5 0 5 10

(a)

±10

Toyoura sand

Dr=60%

Tested Calculated

Shear strain γ (%)

Fig. 36 A comparison between the tested and calculated results of

the undrained torsional test for Toyoura sand at Dr = 60%, (a) stress–

strain relation, (b) effective stress path

Acta Geotechnica (2012) 7:69–113 95

123

problems under both drained and undrained conditions. A

detailed description of the numerical method used is

available from Chan [20] and Zienkiewicz et al. [177]. An

extensive program of centrifuge model tests was reported

by Arulanandan and Scott [4]. The tests were conducted in

the VELACS project sponsored by the U.S. National Sci-

ence Foundation. These tests are often used for verification

of numerical models concerning liquefaction problems. In

this section, two typical model tests (VELACS Models 1

and 2) are simulated using DIANA-SWANDYNE II

together with our constitutive model. To comply with

conventions in numerical analysis, matrices are used

instead of vectors and tensors in line with the notations in

Chan [20] and Zienkiewicz et al. [177].

6.1 Numerical method

Following the Biot formulation governing the behavior of

porous media [11] and neglecting fluid acceleration relative

to solid, the following equation of motion can be readily

written out for the solid phase [176]:

M½ €uf g þZ

XB½ T r0f gdX� Q½ pef g ¼ f uf g ð62Þ

in which the displacement of the soil skeleton u and pore

pressure pe are unknowns, [M] is the global mass matrix,

[B] is the strain–displacement matrix, r0f g is the effective

stress vector, [Q] is s discrete gradient operator coupling

the solid and fluid phases, f uf g is a force vector for the

solid phase and €uf g and pef g are the acceleration and pore

pressure matrices, respectively.

For the fluid phase, similar equation can be given

G½ €uf g þ Q½ T _uf g þ S½ _pef g þ H½ pef g ¼ f pf g ð63Þ

-10 -5 0 5-30

-20

-10

0

10

20

30

-5 0 5 10

(a)

±10

Toyoura sand

Dr=48%

Shear strain γ (%)

She

ar s

tres

s τ

(kP

a)

Tested Calculated

0 20 40 60 80-30

-20

-10

0

10

20

30

20 40 60 80 100

(b)

0

Mean effective stress p (kPa)

She

ar s

tres

s τ

(kP

a)

Tested Calculated

Fig. 37 A comparison between the tested and calculated results of

the undrained torsional test for Toyoura sand at Dr = 48%, (a) stress–

strain relation, (b) effective stress path

-6 -4 -2 0 2 4 6 8

-20

-10

0

10

20

30

0 50 100

Predicted

(a)

Mean effective stress p (kPa)Shear strain γ (%)

She

ar s

tres

s τ

(kP

a)

-2 0 2 4 6 8 10 12-20

-10

0

10

20

30

40

0 50 100

Predicted

(b)

Mean effective stress p (kPa)Shear strain γ (%)

She

ar s

tres

s τ

(kP

a)

0 5 10-10

0

10

20

30

40

0 50 100

Predicted

(c)

Mean effective stress p (kPa)

She

ar s

tres

s τ

(kP

a )

Shear strain γ (%)

Fig. 38 Predicted stress–strain responses and stress paths with three

different initial driving shear stresses for Toyoura sand at Dr = 70%

Table 3 Model parameters for Nevada sand at Dr = 60% and 40%

Dr (%) Mf,c Go j n h Md,c dre,1 dre,2 dir a cd,r

60 1.52 125 0.005 0.5 0.9 0.3 0.4 1,200 0.35 20 0.05

40 1.46 125 0.006 0.5 0.9 0.3 0.4 1,200 0.45 20 0.05

96 Acta Geotechnica (2012) 7:69–113

123

where [G] is the dynamic seepage force matrix, [S] is the

compressibility matrix, [H] is the permeability matrix and

f pf g is a force vector for the fluid phase. A superposed dot

denotes time derivative in these two equations.

Equations 62 and 63 are solved using a finite difference

technique called the generalized Newmark time integration

scheme [57].

6.2 Model configuration and FE model

The test configuration of the VELACS centrifuge model

tests Models 1 and 2 is depicted in Fig. 43. VELACS

Model 1 represents an infinite level site, and VELACS

Model 2 represents a mildly infinite slope with an incli-

nation angle of 2�. The ground consists of a 20 cm high,

uniform Nevada sand layer, which was placed in a 1-D

laminated box at a relative density of about 40%. The

centrifugal experiments were performed at a gravitational

acceleration of 50 g throughout. Under this gravity field,

the centrifuge models correspond to a prototype soil layer

of 10 m depth with infinite lateral extent.

Due to the large lateral extent, two models can be ide-

alized as a 1-D soil column and analyzed with a single

column of elements. This one-dimensional representation

is equivalent to assuming that the stresses and strains in the

centrifuge model are uniform across any lateral plane.

Figure 44 shows the finite element mesh used, where an

8-4 node (8 nodes for the solid phase and 4 nodes in cor-

ners for the fluid phase) element is employed. The

boundary conditions in the FE model are that (1) nodes at

the same depth are tied together to reproduce the shear

beam effect, (2) the base and lateral boundaries are

assumed impervious and (3) the surface is traction free

with zero prescribed pore pressure.

The actual relative density of the sand in both tests was

about Dr = 45% (e = 0.724), somewhat larger than

Dr = 40% originally specified. According to the model

parameters for Nevada sand with Dr = 40% and the

agreement of the numerical results with the experiments,

the model parameters for Nevada sand with Dr = 45%

were identified as shown in Table 4. The permeability of

water is about 6.6E-5 m/s for Nevada sand at Dr = 45%.

For the FE Model in prototype scale, the permeability is

upscaled by 50 times and equal to 3.3E-3 m/s for the

analysis. A static stress field of gravity was applied prior to

seismic excitation. The resulting fluid hydrostatic pressure

and soil stress states along the soil column served as initial

conditions for the subsequent dynamic analysis.

-20 -10 0 10 20-20

-10

0

10

20

-20 -10 0 10 20

Tested

(a)Cyclic undrainedsimple shear

Dr=60%

Calculated

Nevada sand

Shear strain γ (%)

-20 0 20 40 60-20

-10

0

10

20

0 20 40 60 80

(b)

Mean effective stress p (kPa)

She

ar s

tres

s τ

(kP

a)

Tested Calculated

Fig. 39 A comparison between the tested and calculated results of

the undrained simple shear test for Nevada sand at Dr = 60%,

(a) stress–strain relation, effective stress path (test data from [5]

-20 -10 0 10 20-40

-20

0

20

40

-20 -10 0 10 20

(a)

Tested

Cyclic undrainedsimple shear

Dr=60%

Calculated

Nevada sand

Shear strain γ (%)

She

ar s

tres

s τ

(kP

a)

-20 0 20 40 60-40

-20

0

20

40

0 20 40 60 80

(b)

Mean effective stress p (kPa)

She

ar s

tres

s τ

(kP

a)Tested Calculated

Fig. 40 A comparison between the tested and calculated results of

the undrained simple shear test for Nevada sand at Dr = 60%,

(a) stress–strain relation, (b) effective stress path (test data from [5])

Acta Geotechnica (2012) 7:69–113 97

123

6.3 Results and comparisons

In this section, the numerical results are presented and

compared with model tests. It should be noted that all the

results are given in prototype scale.

6.3.1 Model 1: level site

The numerical simulation and experimental instrumenta-

tion include lateral acceleration, excess pore pressure, lat-

eral displacement and surface settlement at certain depths.

The evolutions of lateral acceleration and excess pore

water pressure are shown in Figs. 45 and 46. The test

results are obtained from the centrifuge experiments by

Taboada and Dobry [125]. Excellent agreement between

test and simulation can be observed in terms of acceleration

and excess pore pressure. Due to the liquefaction of sub-

jacent sand, the horizontal acceleration response at the

surface (AH3) becomes almost zero after about 4 s, and

similarly, the acceleration response at 2.5 m depth also

approaches zero after about 6 s. Fig. 46 shows the evolu-

tion of excess pore pressure together with the excess pore

pressure ratio ru ¼ pe

�rv;o, where pe is the excess pore

pressure and rv;o is the initial vertical effective stress. The

excess pore pressure ratio reaches the critical value of 1.0

from surface down to a depth of 5 m. This means that the

sand in the range from surface down to 5 m depth enters

into the post-liquefaction regime. The liquefaction state

(ru = 1.0) persists in the upper part after the shaking event

stops, which is due to the upward diffusion of pore water

from the subjacent sand. As shown in Fig. 46, the lique-

faction state lasts about 21 s and 19 s, respectively, at

1.25 m depth (P5) and 2.5 m depth (P6), although the

shaking event has already ceased at 13 s.

-15 -10 -5 0 5 10-30

-20

-10

0

10

20

30

-10 -5 0 5 10 15

(a)

±15

Tested

Cyclic undrainedsimple shear

Dr=40%

Calculated

Nevada sand

Shear strain γ (%)

She

ar s

tres

s τ

(kP

a)

0 40 80 120 160-30

-20

-10

0

10

20

30

0 40 80 120 160

(b)

Mean effective stress p (kPa)

She

ar s

tres

s τ

(kP

a)

Tested Calculated

Fig. 41 A comparison between the tested and calculated results of

the undrained simple shear test for Nevada sand at Dr = 40%,

(a) stress–strain relation, effective stress path (test data from [5])

-5 0 5 10-40

-20

0

20

40

60

80

0 5 10 15

Dr=40%

Tested

(a)Cyclic triaxial test

Caluclated

Dev

iato

ric s

tres

s σ 1

-σ3

(kP

a)

Axial strain ε1 (%)

Nevada sand

0 40 80 120-40

-20

0

20

40

60

80

0 40 80 120 160

(b)

160

Dev

iato

ric s

tres

s σ 1

- σ3

(kP

a)

Mean effective stress p (kPa)

Tested Caluclated

Fig. 42 A comparison between the tested and calculated results of

the undrained simple shear test for Nevada sand at Dr = 40%,

(a) stress–strain relation, (b) effective stress path (test data from [5])

α

α =0o for Model 1; =2o for Model 2.

Accelerometer (Horizontal) PWP (Pore Water Pressure Transducer)

LVDT (Linear Variable Differential Transducer)

P5P6

P7

P8

AH3

AH4

AH5

AH1

LVDT1LVDT2

LVDT3

LVDT4

LVDT5

LVDT6

22.86m

2.5m

10.0

m

Nevada Sand (Dr=40%)

2.5m

2.5m

2.5m

Laminar Box

Fig. 43 Configuration for RPI Models 1 and 2

98 Acta Geotechnica (2012) 7:69–113

123

The lateral displacements are relatively small with a

maximum amplitude of about 5 cm near the surface as

illustrated in Fig. 47. The residual displacement is close to

zero at the end of shaking, which is also expected for the

model with horizontal surface. The surface settlements are

plotted in Fig. 48 where the calculated result is obviously

smaller than observed in the experiment. Particularly, the

main part of the observed settlement was induced during the

shaking, but the pore pressure ratio still remains about 1.0

and no significant diffusion and dissipation of the pore

pressure took place at the same time. This test result seems

to be at odds with the existing recognition that the earth-

quake-induced surface settlements for saturated sandy level

ground are mainly produced in the post-earthquake recon-

solidation regime and necessarily accompanied with the

upward seepage of the pore water. This paradox may be

attributed to the following three main aspects. (1) The

actual seepage capacity of the pore water during shaking in

the gravity field may be too underestimated. (2) Some larger

deviations exist in the identification of the model parame-

ters. (3) Since the model was shaken with higher frequen-

cies under a centrifugal field of 50 g, some dynamic effects

such as the pore water acceleration relative to the soil

skeleton should be considered in the numerical simulation,

but they are neglected in the adopted u–p formulation.

These explanations need to be further investigated theo-

retically and experimentally.

Figure 49 provides the calculated cyclic shear stress–

shear strain curves and stress path at different depths,

which shows typical pattern of stiffness degradation and

even loss due to excess pore pressure build-up.

6.3.2 Model 2: sloping site

The performance of numerical model for problems with

inclined ground surface is shown by comparing the tested

and calculated results of Model 2 including lateral accel-

eration, excess pore pressure, lateral displacement and

surface settlement. The test results are from the database of

node both with freedom of displacement u and pore pressure p

node only with freedom of displacement u

10.0

m

2.5m

Specified input motion

Impervious base

Impe

rvio

us

Impe

rvio

us

zero pore pressurep=0

8-4 node coupled element

Y

X

gcosα

gsinα

α =0o for Model 1;=2o for Model 2.

Fig. 44 Finite Element discretization and boundary conditions for

Models 1 and 2

Table 4 Model parameters for Nevada sand at Dr = 45%

Dr (%) Mf,c Go j n h Md,c dre,1 dre,2 dir a cd,r

45 1.46 125 0.006 0.5 0.9 0.3 0.4 1,200 0.4 20 0.05

-2024

-202

-202

0 2 4 6 8 10 12 14 16-4-202

(a)

AH3(surface)

(b)

AH4(-2.0m)

Late

ral a

ccer

atio

n (m

/ s2 )

(c)

AH5(-5.0m)

(d)

AH1(Input)

Time (s)

Tested Calculated

Fig. 45 Comparison of tested and calculated lateral accelerations of

Model 1

0

10

20

0

10

20

0

20

40

0 5 10 15 20 25 30 35 400

204060

(a)

P5(-1.25m)

(b)

P6(-2.5m)

Exc

ess

pore

pre

ssur

e (k

Pa)

(c)

P7(-5.0m)

(d)

P8(-7.5m)

Time (s)

Tested Calculated

End of shaking events at 13s

ru=1.0

ru=1.0

ru=1.0

ru=1.0

Fig. 46 Comparison of tested and calculated excess pore pressures of

Model 1

Acta Geotechnica (2012) 7:69–113 99

123

centrifuge tests by Taboada and Dobry [126]. Due to the 2�surface inclination, a static driving shear stress exists in

Model 2 prior to shaking. As a consequence, the responses

of Model 2 are different from Model 1. The tested and

simulated lateral accelerations at different depths from the

input at the base to the response at the surface are com-

pared in Fig. 50. The recorded and calculated excess pore

pressure histories are shown in Fig. 51, where a number of

instantaneous sharp drops of pore pressure after the initial

liquefaction can be observed. Figure 52 illustrates the lat-

eral deformation, which accumulates cycle by cycle toward

the downslope direction. However, the test results show

larger fluctuations than the numerical simulation. Figure 53

provides a comparison of the calculated and tested surface

settlements. The former is significantly larger than the

latter, similar to the results of Model 1. This may be due to

the same reasons with those of Model 1.

The calculated shear stress–shear strain curve and

effective stress path at different depths are shown in

Fig. 54. It is clearly seen that the effective confining stress

begins to increase when the shear strain reaches certain

value in each loading cycle, and the increase in the effec-

tive confining stress leads to the regain of stiffness and

shear strength. This mechanism ensures that the static

driving shear stress can only cause limited lateral defor-

mation in each loading cycle. The cycle-by-cycle accu-

mulation of the shear strains at different depths is

-0.050.000.05

-0.050.000.05

-0.050.000.05

0 2 4 6 8 10 12 14 16-0.10-0.050.000.05

(a)

LVDT3(surface)

(b)

LVDT4(-2.5m)

Late

ral d

ispl

acem

ent (

m)

(c)

LVDT5(-5.0m)

(d)

LVDT6(-7.5m)

Time (s)

Tested Calculated

Fig. 47 Comparison of tested and calculated lateral displacements of

Model 1

0 5 10 15 20 25 30-0.20

-0.15

-0.10

-0.05

0.00

0.05

LVDT1(Tested)

LVDT2(Tested)

Calculated

Sur

face

set

telm

ent (

m)

End of shaking events at 13s

Fig. 48 Comparison of tested and calculated surface settlements of

Model 1

-0.50.00.51.0

-505

-100

10

-0.4 -0.2 0.0 0.2 0.4-30-15

015

(a)

(b)

She

ar s

tres

s τ x

y(k

Pa)

(c)

(d)

Shear strain γxy (%)0 20 40 60 80

-0.94m

-3.44m

-6.44m

-8.44m

Effective vertical stress σv (kPa)

Fig. 49 Calculated shear stress–shear strain curve and stress path at

different depths for Model 1

-2024

-202

-202

0 2 4 6 8 10 12 14 16-4-202

(a)

AH3(surface)

(b)

AH4(-2.0m)

(c)

AH5(-5.0m)

(d)

AH1(Input)

Tested Calculated

Late

ral A

ccel

erat

ion

(m/s

2 )

Time (s)

Fig. 50 Comparison of tested and calculated lateral accelerations of

Model 2

0

10

20

0

10

20

0

20

40

0 5 10 15 20 25 30 35 400

204060

(a)

P5(-1.25m)

(b)

P6(-2.5m)

(c)

P7(-5.0m)

(d)

P8(-7.5m)

Time (s)

Tested Calculated

Exc

ess

pore

pre

ssur

e (k

Pa)

End of shaking events at 13s

ru=1.0

ru=1.0

ru=1.0

ru=1.0

Fig. 51 Comparison of tested and calculated excess pore pressures of

Model 2

100 Acta Geotechnica (2012) 7:69–113

123

responsible for the large lateral deformation. The fairly

good agreement between the calculated and recorded lat-

eral displacements along the soil column gives some con-

fidence in the constitutive model and numerical scheme in

simulating large post-liquefaction deformation.

7 Seismic response analysis of Daikai subway station

In this section, the seismic response of the Daikai subway

station subjected to the 1995 Hyogoken-Nambu earthquake

in Japan is investigated in order to validate our numerical

model in soil–structure dynamic response analysis.

7.1 Earthquake-induced damage to Daikai station

During the 1995 Hyogoken-Nambu earthquake, some

subway stations and tunnels suffered heavy damage, which

had not been observed before. This event caused wide

safety concerns and research interest, because underground

structures were considered relatively safe in earthquakes in

comparison with aboveground structures.

There were some 18 subway stations in Kobe City,

among which 5 stations experienced damage to some

extent. The damage to the Daikai station was the most

severe. Figures 55 and 56 show the plan and cross section

of the Daikai station. The station is a two-story under-

ground structure of reinforced concrete. The B2 floor

consists of platforms and rail lines, and the B1 floor is a

concourse with a ticket barrier. The thickness of the

overburden soil is about 4.8 m at the B2 floor. As illus-

trated in Fig. 55, the main part of the B2 floor is a frame

structure with columns at the center, which is 17 m wide

and 7.17 m high in the outside dimension and 120 m long

in the longitudinal direction. The center column is 3.82 m

high and has cross section of 0.4 m*1.0 m, and the distance

between adjacent columns is about 3.5 m. Figure 56 shows

the typical damaged profile of the B2 floor.

According to the damage pattern, Iida et al. [46] pointed

out that a strong horizontal force was probably imposed on

the structure in the transversal direction, which caused

deformation of the box frame structure. The following was

regarded as an explanation. Due to the strong horizontal

force, large relative displacement occurred between the

ceiling level and base level of the station, which led to the

collapse of the center columns under a combination of

bending and shearing. Then, the box was compressed to

failure under the gravity of the overburden soil as illus-

trated in Fig. 56. During the reconstruction process, dis-

continuity of about 3 cm was found in a ventilation tower

0.00.20.40.60.8

0.00.20.40.6

0.00.20.40.6

0 2 4 6 8 10 12 14 160.00.20.40.6

(a)

LVDT3(surface)

(b)

LVDT4(-2.5m)

(c)LVDT5(-5.0m)

(d)LVDT6(-7.5m)

Time (s)

Late

ral d

ispl

acem

ent (

m)

Tested Calculated

Fig. 52 Comparison of tested and calculated lateral displacements of

Model 2

0 5 10 15 20 25 30-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

Sur

face

set

telm

ent

( m)

End of shaking events at 13s

LVDT1(Tested)

LVDT2(Tested)

Calculated

Fig. 53 Comparison of tested and calculated surface settlements of

Model 2

0 20 40 60 80

-0.94m

-3.44m

-6.44m

-8.44m

Effective vertical stress σv (kPa)

-0.50.00.5

-505

-100

1020

-2 0 2 4 6 8 10 12-20

020

(a)

(b)

(c)

(d)

Shear strain γxy (%)

She

ar s

tres

s τ x

y(k

Pa)

Fig. 54 Calculated shear stress–shear strain curve and stress path at

different depths for Model 2

B-1

B-3

D-1

B-2

B-4

Boreholes

PlatformNagata

Platform

Switchingstaion room

Electric facility room

B2 floor

Mountain

Shinkaichi

(Unit: mm)

Sea

120 000

1

12

2

Fig. 55 Plan view of Daikai station (after [46])

Acta Geotechnica (2012) 7:69–113 101

123

for the subway at a depth of about 4 m with the upper part

moving toward the mountain side [46]. These observations

indicate that there was large shearing horizontal displace-

ment in the surrounding subsoils.

Many analyses have performed on the Daikai station,

such as An et al. [1], Takewaki et al. [127], Yamato et al.

[144] and Nishimura et al. [85]. In these studies, however,

the emphasis was placed on the structure, and the behavior

of surrounding subsoils was over-simplified. The equiva-

lent linear models or direct nonlinear models were used to

describe the behavior of subsoils during the earthquake.

The generation, diffusion and dissipation of excess pore

pressures of saturated sand strata during shaking were

ignored. Geological investigation shows that the ground

water table was rather shallow. Water leakage from the

cracks of the base slab and lateral wall was observed after

the earthquake. In order to obtain sufficiently large defor-

mation to reproduce the observed collapse, the calculated

response of the soil and structure are usually too large in

these previous studies. Therefore, the damage mechanisms

of the Daikai station from these studies are still doubtful.

Unlike aboveground structures, underground structures

are surrounded by subsoils, which are characterized by

strong nonlinear soil–structure interaction. The deforma-

tion behavior of the subsoils is especially important to

evaluate the dynamic response of underground structures.

In this section, a fully coupled effective stress analysis is

carried out for the seismic response of the Daikai station.

Emphasis is placed on the displacements in the subsoils

and their effect on the structure performance.

7.2 Finite element model

7.2.1 Geological conditions and FE mesh

The Dakai station is located west of Kobe Port, as shown in

Fig. 57, and is about 15 km northeast of the epicenter of

the Hyogoken-Nambu earthquake. It is situated in a low-

lying area with a ground surface elevation of about 5 m.

The geology of the site is essentially soft sandy silt of back

marsh origin overlying medium to course granitic sands

containing gravel.

Soil investigations were made both before the con-

struction of the subway and after the earthquake. The

investigation results after the earthquake were shown in

Fig. 58 along with the soil profiles. The plan locations of

the boreholes in Fig. 58 were marked in Fig. 55. As

illustrated in Fig. 58, the Daikai station is buried in a non-

uniformly layered soil stratum. The properties of soil layers

from up to down were backfill, clay, sand, clay, gravel,

clay and gravel in turn. The depth of water table was

between 6 and 8 meters after the earthquake, which is

lower than that before the earthquake by about 3 m.

17.00

1.72 2.19

7.17

Unit: m

Fig. 56 Typical section and damage patterns of Daikai station B2

floor

Fig. 57 Geomorphological map of the Kobe city and the location of Daikai station (After [130])

102 Acta Geotechnica (2012) 7:69–113

123

The transverse section of B2 floor between borehole D1

and B3 is studied by a 2D FEM analysis. This section is in

the heaviest damaged region of the station. The boreholes

D1 and B3 were located on either side of the section,

respectively (Fig. 55). Referencing from An et al. [1] and

Fig. 58 the properties of soil layers at this site can be

estimated, and the results are listed in Table 5. The

underground water table is assumed to be 5.6 m below

ground surface before the earthquake. The FE mesh of the

transverse section and the soil layers are given in Fig. 59.

The shear wave velocity of the seventh layer is about

453 m/s, which is higher than the overburden soil. The

stiffness of this layer is large enough to assume this layer as

base rock, and thus, the input acceleration wave is applied

on the bottom of the sixth layer. The horizontal extent of

the soil layer is about 50 m beyond the station lateral wall

in the FE mesh. The nodes at the lateral boundary of the

model of the same depth are tied together to reproduce the

far field response.

The structure and soil layers above the underground

water level are discretized with 8-nodes isoparametric

displacement elements. The soil layers below underground

water level are modeled with 8-4 nodes displacement–pore

water pressure coupled isoparametric elements, among

which 8 nodes are used for the solid phase and 4 nodes for

the fluid phase. The base and lateral boundaries are

assumed impervious. The nodes at the underground water

level are traction free with zero prescribed pore pressure.

7.2.2 Material parameters and input motion

The Daikai station is a reinforced concrete (RC) under-

ground structure. An elastic model is adopted for consti-

tutive description of RC. Elastic modulus and Poisson ratio

of RC are assumed to be 25 GPa and 0.167, respectively.

The soil layers at the site of the Daikai station can be

classified into three categories: sand, clay and gravel. A

simplified version of the present model is used to model the

stress–strain behavior of clay and gravel. Only the non-

linear shear stress–shear strain responses for clay and

gravel are considered, but their shear-dilatancy features are

ignored. The simplified version of the model is achieved by

Fig. 58 Soil profile based on the borehole investigation after the earthquake (after [46])

Table 5 Properties of soil layers at the site of Daikai station

Layer

No.

Soil

type

Layer

height

(m)

SPT-N Vs

(m/s)

G(100 kPa)

q(g/cm3)

Layer 1 Back fill 2.2 8 188 633 1.8

Layer 2 Clay 1 8 199 633 1.6

Layer 3 Sand 5.8 8 183 633 1.8

Layer 4 Clay 1.1 9 197 696 1.8

Layer 5 Gravel 2.4 18 240 1,212 2.1

Layer 6 Clay 4.75 13 228 934 1.8

Layer 7 Gravel [10 90 453 4,391 2.1

Acta Geotechnica (2012) 7:69–113 103

123

setting the shear-dilatancy parameters to zero. There are

only four parameters in the simplified version of the model:

elastic modulus G and K, plastic modulus parameter h, and

failure parameter Mf.c. According to the properties of soil

layers listed in Table 5, the parameters of the simplified

model for clay and gravel are estimated and listed in

Table 6.

In this study, special attention is paid to the influence of

liquefaction of saturated sand layer on the dynamic

response of the station. Only the SPT-N value and shear

velocity of the sand layer are known, and no shear-dilat-

ancy characteristics under drained conditions neither pore

water pressure generation characteristics under undrained

conditions are available. Three sets of parameters, named

as Param 1, Param 2 and Param 3, respectively, are

assumed for the saturated sand layer. As listed in Table 7,

the only difference between these three sets of parameters

lies in the irreversible dilatancy parameter dir. The larger

dir is, the more susceptible the sand becomes. By using

these three sets of parameters in the analysis, we can

compare effects of different liquefaction extent of the

saturated sand layer on the dynamic response of the station

and surrounding subsoils. The void ratios are all set to be

0.6 for the three sets of parameters.

Seismic records at the Daikai station during the 1995

Hyogoken-Nambu earthquake are not available. However,

a vertical array of geophones had been installed at depths

of GL-83 m, -32 m, -16 m and 0 m from the ground sur-

face in the Port island, which is located about 4 km to the

east of the Daikai station [54]. The south–north component

of the recorded acceleration is adopted as the input motion.

The time history of the acceleration is given in Fig. 60. The

peak acceleration is about 6.78 m/s2 at 3.4 s, and the

accelerations at other times are smaller than 3.5 m/s2.

The Daikai station was constructed with the open-cut-

method, and the excavation depth is about 12 m. A static

analysis of simulating the excavation and construction

procedure is performed to obtain the initial condition for

the dynamic analysis, including the initial effective stress

and pore water pressure distribution.

0 10-10

(a)

(b)

-20-30-40-50-60 20 30 40 50 60

0

-5

-10

-15

-20

GL-2.2m Back Fill

ClayGL-3.2m

GL-9.0mSand

GL-11.1mClay

GL-11.97m

GL-17.25m

Gravel

Clay

8 nodes displacement element

8-4 nodes coupled element

GL-5.6m

Input motion

Whole mesh

Local mesh, soil profile and location of acceleration and pore pressure output points

GL0.0m

AH2

AH1

Acceleration output point

AH4

AH3

P3

P1 P2

P4

Excess pore pressure output point

Fig. 59 Simplified soil profile and FE mesh of Daikai station B2 floor

Table 6 Simplified model parameters for clay and gravel

Soil type G/100 (kPa) K/100 (kPa) h Mf,c

Layer 1: Backfill 633 1,055 1.0 1.3

Layer 2: Clay 633 1,055 1.0 1.3

Layer 4: Clay 696 1,160 1.0 1.3

Layer 5: Gravel 1,212 2,020 1.0 2.0

Layer 6: Clay 934 1,555 1.0 1.4

104 Acta Geotechnica (2012) 7:69–113

123

7.3 Results and analysis

7.3.1 Acceleration response

Figure 61 shows the acceleration responses calculated at

four typical points whose locations are depicted in Fig. 59.

There is little difference in the acceleration response before

the 4th second among the results of the three parameter

sets. The surface peak acceleration at far field is 4.56 m/s2,

and the surface peak acceleration above the station is

4.77 m/s2. The response peak acceleration takes place

nearly at the same time of the input peak acceleration and

is smaller than the input peak acceleration. The attenuation

of the horizontal acceleration is due to nonlinearity of soils.

This phenomenon is corroborated by the observation in the

Port Island, where the recorded peak horizontal accelera-

tion at the ground surface is the minimum and increases

with depth.

7.3.2 Excess pore pressure response

Figure 62 shows time histories of the excess pore pressure

response at the four typical points, whose locations are

illustrated in Fig. 59. Because the irreversible dilatancy

parameter dir in Param 1 is zero, the excess pore pressure is

negligible in this case, and will not be presented here. The

maximum excess pore pressure ratio is about 0.6 in the

results of Param 2, and it reaches 1.0 for Param 3. This means

that the saturated sand layer (layer 3) experiences zero

effective confining stress state or reaches the post-liquefac-

tion state for Param 3. Being closer to the structure, the

excess pore pressure responses at P2 and P4 near the station

show stronger fluctuations than P1 and P3, which are further

away from the station. Figure 63 shows the distribution of

excess pore pressure and effective vertical stress in the

domain at the 8th second, when the model parameters of

Param 3 are adopted. It can be seen that the excess pore

pressure is generated mainly in layer 3, because the layer

below is clay, which prohibits diffusion of excess pore water

to the underlying layer. At this time, the effective vertical

stress is about 10 kPa and is far smaller than the initial

effective vertical stress of about 120 kPa.

7.3.3 Stress–strain response of saturated sand

Plotted in Fig. 64 are the shear stress–shear strain curves

and effective stress paths of a typical element in the

Table 7 Model parameters of sand

Content Symbols Param 1 Param 2 Param 3

Failure parameters Mf,c 1.72 1.72 1.72

Modulus Go 200 200 200

j 0.002 0.002 0.002

n 0.5 0.5 0.5

h 1.2 1.2 1.2

Reversible dilatancy

parameters

Md,c 0.5 0.5 0.5

dre,1 0.4 0.4 0.4

dre,2 1,000 1,000 1,000

Irreversible dilatancy

parameters

dir 0 0.07 0.15

a 50 50 50

cd, r 0.05 0.05 0.05

5 10 15 20-8

-4

0

4

8

Acc

eler

atio

n (m

/s2 )

Time (s)0

amax=6.79m/s2

Fig. 60 Input motion for numerical analysis

-3036

-303

-303

0 5 10 15 20-6-303

Param 1 Param 2 Param 3(a)

AH4

(b)

AH3

Hor

izon

tal a

ccel

erat

ion

(m/s

2 )

(c)

AH2

(d)

AH1

Time (s)

amax = -3.98m/s2

amax = -4.56m/s2

amax = -4.77m/s2

amax = -3.43m/s2

Fig. 61 Time series of calculated acceleration response

0

50

100

150

0

50

100

0

50

100

0 5 10 15 200

50

100

(a) P4

Param 2Param 3

(b) P3

Exc

ess

pore

pre

ssur

e (k

Pa)

(c) P2

(d) P1

Time (s)

ru=1.0

ru=1.0

ru=1.0

ru=1.0

Fig. 62 Time histories of calculated excess pore pressure

Acta Geotechnica (2012) 7:69–113 105

123

saturated sand layer. Because the irreversible dilatancy

parameter dir in Param 1 is zero, the effective vertical stress

is almost equal to the initial effective vertical stress, and

the shear strain–shear stress curve is similar to that of

constant mean effective stress. When the model parameters

of layer 3 are set to Param 2, the effective vertical stress in

layer 3 decreases gradually without reaching zero. Mean-

while, the shear modulus reduces along with increasing

shear strain due to the reduction of the effective vertical

stress. The maximum shear strain is about 0.5%. When the

model parameters of layer 3 are set to Param 3, the

effective vertical stress in layer 3 reaches zero. During

shearing after initial liquefaction, the cyclic shear strain

increased rapidly, and the maximum shear strain is about

2.5%. The stress–strain curve shows characteristics of

saturated sand during cyclic mobility.

7.3.4 Deformation of the station and surrounding subsoils

Figure 65 shows time histories of the relative displacement

between the celling slab and the base of the station. For

comparison, the relative displacements of the far field at

0-10-20-30 10 20 30

0

-10

-20

50

5010

50100

150

150 100

100100

100

150 150

10050

1050

5050

0

50100120100

0

0 0 50100

120100

0

-10-20-30 10 20 30

0

-10

-20

(a) Distribution of excess pore pressure (kPa)

(b) Distribution of effective vertical stress (kPa)

Fig. 63 Distribution of excess pore pressure and effective vertical stress (Param 3 at 8 s)

-30

0

30

60

-30

0

30

-3 -2 -1 0 1 2 3-60

-30

0

30

Param 1(a)

(b) Param 2

(c) Param 3

Shea

r st

ress

τ xy

( kPa

)

Shear strain γxy (%)0 50 100 150

Effective vertical stress σv (kPa)

Fig. 64 Stress–strain curves of typical sand element

-10

-5

0

5

10

0 5 10 15 20-10

-5

0

5

10Param 1Param 2Param 3

Rel

ativ

e di

spla

cem

ent (

Cm

)

(a) Relative displacement of far field soil

(b) Relative displacement of structure

Time (s)

Fig. 65 Time histories of relative displacement of structure and free

field soil

106 Acta Geotechnica (2012) 7:69–113

123

these two depths are also plotted in the same figure.

Figure 66 provides the deformed mesh of the station and

the surrounding soil at the time when the maximum relative

displacement occurs. Figure 67 illustrates the deformed

profile of the station structure. The maximum relative

displacements of the station and of the far field are listed in

Table 8. It can be seen that the relative displacement of the

structure and soil increases largely, when the pore pressure

of saturated sand increases. When the model parameters of

saturated sand are set to Param 3, the relative movement of

far field soil is about 8.1 cm, which can be compared with

the relative movement of station of about 4.1 cm.

7.3.5 Collapse mechanism of Daikai station

Many studies have been made to reveal the collapse

mechanism of the Daikai station qualitatively. The main

results of these studies can be summarized as follows. A

strong horizontal load was transmitted to the transverse

section of the station during earthquake, which caused the

deformation of the box frame structure. Moreover, the

center columns collapsed first due to a combination of

bending and shears resulting from the deformation of the

box frame. Nishimura et al. [85] performed an analysis of

the Daikai station using response displacement method. At

first, the relative displacement of the subsoils between the

top and the bottom of station of 3–4 cm was calculated by

seismic response analysis of free field. Then, this relative

displacement was applied to the structure, resulting in the

collapse of the structure. Provided the analysis of structure

by Nishimura et al. [85] is correct, we can draw the con-

clusion that a relative displacement of the structure

between the celling slab and the base of 3–4 cm would lead

to structure collapse. According to our analysis, however, if

(c) Param 3 (amplified 50 times)

2.1cm

2.6cm

8.1cm

(b) Param 2 (amplified 50 times)

(a) Param 1 (amplified 50 times)

Fig. 66 Deformed mesh at maximum displacement

Param1 Param2 Param3

Fig. 67 Deformed profile of Daikai station (amplified by 200 times)

Table 8 Comparison of the relative displacement of structure and far

field soils

Content Param

1

Param

2

Param

3

Relative displacement of far field soils

(cm)

2.1 2.6 8.1

Relative displacement of structure (cm) 1.9 2.1 4.1

Time that the maximum relative

displacement happened (s)

6.28 6.28 8.02

Acta Geotechnica (2012) 7:69–113 107

123

the excess pore water pressure in the saturated sand layer is

relatively high, the relative displacement of soil between

the celling slab level and the base slab can be larger than

3 cm. This means that liquefaction in the saturated sand

layer may cause structure collapse of the station. Therefore,

the relative displacement of the soil due to liquefaction of

the saturated sand layer is thought to be the main reason for

the large deformation of the box frame structure and the

collapse of the station structure. This type of displacement

may be negligible for small structures, but it can be crucial

for large structures in transverse section such as the Daikai

station. The nonlinear behavior of the surrounding subsoil

plays a very predominant rule in the dynamic analysis of

large underground structures.

8 Conclusions

Experimental observations, physical explanations and the-

oretical descriptions are made to establish a coherent the-

oretical framework for evaluating large post-liquefaction

deformation of saturated sand. Special attention is paid to

the mechanical laws, physical mechanism, constitutive

model and numerical algorithm as well as practical appli-

cation. The main achievements are summarized as follows.

1. The post-liquefaction shear strain for saturated sand

subjected to undrained cyclic shear is decomposed into

a solid-like shear strain cd that occurs in non-zero

effective confining stress states, which depends only

on current effective stress, and a fluid-like shear strain

co that is triggered in zero effective confining stress

state, which depends on strain history.

2. The physical background of large post-liquefaction

deformation in saturated sand is revealed by the

experimental observation that the volumetric strain due

to dilatancy is composed of a reversible and an

irreversible dilatancy component, which are intimately

related to three physical states of sand particles after

the initial liquefaction. An intrinsic relationship is

found to exist between the shear deformation and the

reversible and irreversible dilatancy in the post-lique-

faction regime, and this relationship is governed by the

evolution laws of three volumetric strain components,

i.e., two dilatancy components due to shearing and a

component due to the change in mean effective stress.

The above mechanism provides a rational approach to

establish both the triggering condition for unstable

flow deformation and the method of determining large

post-liquefaction reconsolidation volumetric strains.

3. Based on the mechanism in the post-liquefaction

regime, a general approach is proposed to predict the

solid-like and fluid-like shear strains during undrained

cyclic shearing. The description of the two dilatancy

components is the most crucial issue.

4. A constitutive model within the frame of bounding

surface plasticity is developed following the above

approach. The present model captures small to large

shear strain in both pre- and post-liquefaction regimes.

Moreover, the model is also capable of simulating the

accumulation process of large volumetric strain

induced during post-liquefaction reconsolidation based

on intrinsic relationship between residual volumetric

strain and shear strain. In addition, the model calibra-

tion procedures are outlined, and a large set of

experimental data is used to identify the parameters

in the constitutive model.

5. The constitutive model is implemented into a fully

coupled finite element code. A robust numerical

algorithm using the constitutive model is developed

to deal with instability in the vicinity of zero effective

confining stress, which appears repeatedly in the post-

liquefaction regime. The numerical model is validated

by simulating two dynamic centrifuge model tests.

6. Finally, the seismic response of the Daikai subway

station during the 1995 Hyogoken-Nambu earthquake

is investigated. The large displacements induced by a

liquefied sand layer are shown to be the main reason

for the serious damage in the underground structure

during the earthquake. Our numerical model provides

a rational approach to boundary-value problems with

large post-liquefaction deformation.

Acknowledgments The present study is financially supported by

the National Natural Science Foundation of China (No. 51038007,

No. 51079074, and No. 50979046). The authors wish to express their

sincere appreciation to Professor W. Wu for his valuable comments

during preparation of this paper.

Appendix: Local stress integration algorithm

1. Obtain the strains and stress quantities as well as

volumetric strain due to change in mean effective

stress at the beginning of the step evcð Þn and other

history variables. Moreover, get strain increment by

geometric update from a global converged state.

Denþ1 ¼ enþ1 � en ð64Þ

2. Elastic predictor: Assume that all the strain increments

are elastic, the trial stress state can be predicted as

evcð Þtrnþ1¼ evcð Þnþ Devð Þnþ1

ptrnþ1 ¼ f evcð Þtrnþ1

� �

108 Acta Geotechnica (2012) 7:69–113

123

strnþ1 ¼ sn þ 2Gnþ1Denþ1

3. Check for plastic yielding: is Df tr ¼ Dstrnþ1 :

nL � Dptrnþ1rtr

nþ1 : nL [ 0?

NO: Update strains and stress quantities and EXIT

YES: Perform plastic correction (Step 4).

4. Plastic corrector for yielding states: by Newton–

Raphson iterative procedure. Initialization of iterative

variables:

k ¼ 0;Depð0Þnþ1 ¼ 0;Deeð0Þ

nþ1 ¼ Denþ1;DLð0Þ ¼ 0;Dkð0Þ

¼ 0;Df ð0Þ ¼ Df tr

wherein the variable in the superscript embraced by

brackets indicates iterative number.

Iterative procedure on k

dkðkÞ ¼ Df ðkÞ

H þ 3G� KD r : nLð Þ

Dkðkþ1Þ ¼ DkðkÞ þ dkðkÞ;DLðkþ1Þ ¼ DLðkÞ þ HdkðkÞ

Depv

� �ðkþ1Þnþ1¼ Dkðkþ1ÞD; Depð Þðkþ1Þ

nþ1 ¼ Dkðkþ1ÞnL;

Deev

� �ðkþ1Þnþ1¼ Devð Þnþ1� Dep

v

� �ðkþ1Þnþ1

;

Deeð Þðkþ1Þnþ1 ¼ Denþ1 � Depð Þðkþ1Þ

nþ1 ;

evcð Þðkþ1Þnþ1 ¼ evcð Þnþ Dee

v

� �ðkþ1Þnþ1

pðkþ1Þnþ1 ¼ f evcð Þðkþ1Þ

nþ1

� ;

snþ1 ¼ sn þ 2Gðkþ1Þnþ1 Deeð Þðkþ1Þ

nþ1

where H is plastic modulus, G and K are elastic

moduli, D is the total shear-dilatancy rate, DL is the

increment of loading index.

5. Convergence check: Calculate the residual value of

yielding function

Df ðkþ1Þ ¼ Dsðkþ1Þnþ1 : nL � Dp

ðkþ1Þnþ1 r

ðkþ1Þnþ1 : nL � DLðkþ1Þ

Is Df ðkþ1Þ � TOL1 and pðkþ1Þnþ1 � p

ðkÞnþ1

� TOL2?

NO: k k þ 1; continue iteration

YES: Update strains, stress and internal variables, then

EXIT

snþ1 ¼ sðkþ1Þnþ1 ; pnþ1 ¼ p

ðkþ1Þnþ1 ; evcð Þnþ1¼ evcð Þðkþ1Þ

nþ1 :

References

1. An XH, Shawky AA, Maekawa K (1997) The collapse mecha-

nism of a subway station during the Great Hanshin Earthquake.

Cem Concr Compos 19:241–257

2. Asaoka A, Nahano M, Noda T (1998) Superloading yield con-

cept for the saturated structured soils. In: Proceedings, fourth

European conference on numerical methods in geotechnical

engineering, pp 232–242

3. Aubry, D, Hujeux, JG, Lassoudiere, F, Moimen Y (1982) A

double memory model with multiple mechanisms for cyclic soil

behavior. In: Proceedings, International symposium on numer-

ical models in geomechanics, Zurich, pp 3–13

4. Arulanandan K, Scott RF (1993) Proceedings of verification of

numerical procedures for the analysis of soil liquefaction

problems, vol 1. A. A. Balkema Publishers, Netherlands

5. Arulmoli K, Muraleetharan KK, Hossain MM, Fruth LS (1992)

VELACS: Verification of liquefaction analysis by centrifuge

studies, laboratory testing program, soil data report. Project No.

90-0562, The Earth Technology Corporation, Irvine, California

6. Armstrong PJ, Frederick OC (1966) A mathematical represen-

tation of the multiaxial Bauschinger effect. C.E.G.B Report

RD/B/N731, Central Electricity Generating Board

7. Bardet JP (1986) Boundary surface plasticity model for sands.

J Eng Mech ASCE 112(12):1198–1217

8. Bardet JP, Tobita T, Mace N, Hu J (2002) Regional modeling of

liquefaction-induced ground deformation. Earthq Spectra

18(1):19–46

9. Bartlett SF, Youd TL (1992) Empirical analysis of horizontal

ground displacement generated by liquefaction-induced lateral

spreading. In: National center for earthquake engineering

research technical report NCEER-92-0021, 114

10. Bartlett SF, Youd TL (1995) Empirical prediction of liquefac-

tion-induced lateral spread. J Geotech Eng ASCE

121(4):316–329

11. Biot MA (1956) Theory of propagation of elastic waves in

a fluid-saturated porous solid. Part I: Low frequency range.

J Acoust Soc Am 28(2):168–178

12. Biot MA (1956) Theory of propagation of elastic waves in

a fluid-saturated porous solid. Part I: Low frequency range.

J Acoust Soc Am 28(2):179–191

13. Borja RI, Chao HY, Montans F, Lin CH (1999) Nonlinear

ground response at Lotung LSST site. J Geotech Geoenviron

Eng ASCE 125(3):187–197

14. Borja RI, Chao HY, Montans F, Lin CH (1999) SSI Effects on

ground motion at Lotung LSST site. J Geotech Geoenviron Eng

ASCE 125(9):760–770

15. Boulanger RW, Idriss IM, Mejia LH (1995) Investigation and

evaluation of liquefaction related ground displacements at Moss

Landing during the 1989 Loma Prieta earthquake, Repot No.

UCD/CGM-95/02, Department of Civil and Environmental

Engineering, UC-Davis

16. Castro G (1975) Liquefaction and cyclic mobility of saturated

sands. J Geotech Eng Div ASCE 101(GT6): 551–569

17. Castro G, Poulos SJ (1977) Factors affecting liquefaction and

cyclic mobility. J Geotech Eng Div ASCE 103(GT6): 501–516

18. Chaboche PJ (1977) Viscoplastic constitutive equations for the

description of cyclic and anisotropic behavior of metals. Bull

Acad Polon Sci Ser Tech 25:33–42

19. Chaboche PJ, Rousselier G (1983) On the plastic and visco-

plastic constitutive equations. J Pressure Vessel Tech ASME

105:153–164

20. Chan AHC (1988) A unified finite element solution to static and

dynamic geomechanics problem. PhD dissertation, University

College of Swansea, Wales, 1988

21. Chan AHC (1993) User’s manual for DIANA-SWANDYNE II.

School of Civil Engineering. University of Birmingham,

Birmingham

22. Dafalias YF, Popov EP (1975) A model of nonlinearly hard-

ening materials for complex loading. Acta Mech 21(3):173–192

Acta Geotechnica (2012) 7:69–113 109

123

23. Dafalias YF, Popov EP (1977) Cyclic loading for materials with

a vanishing elastic domain. Nucl Eng Des 41:293–302

24. Dafalias YF (1979) A model for soil behavior under monotonic

and cyclic loading conditions. In: 5th international conference

on structural mechanics and reactor technology, West Berlin,

German K: K1/8

25. Dafalias YF, Herrmann LR (1982) Bounding surface formula-

tion of soil plasticity. Soil Mechanics. In: Pande GN, Zie-

nkiewicz OC (eds) Transient and Cyclic Loads. pp 253–282

26. Dafalias YF (1986) Bounding surface plasticity. I: Mathematical

foundation and hypoplasticity. J Eng Mech ASCE

112(9):966–987

27. Davis RO, Berrill JB (1982) Energy dissipation and seismic

liquefaction in sands. Earthq Eng Struct Dyn 10(1):59–68

28. Doanh T, Dubujet Ph, Touron G (2010) Exploring the undrained

induced anisotropy of Hostun RF loose sand. Acta Geotech

5(4):239–256

29. Drucker DC, Gibson RE, Henkel DJ (1957) Soil mechanics and

working hardening theories of plasticity. Trans ASCE

122:338–346

30. Elgamal A, Yang ZH, Parra E (2002) Computational modeling

of cyclic mobility and postliquefaction site response. Soil Dyn

Earthq Eng 22(4):259–271

31. Elgamal A, Yang ZH, Parra E (2003) Modeling of cyclic

mobility in saturated cohesionless soils. Int J Plast 19:883–903

32. Finn WDL, Lee KW, Martin GR (1977) An effective stress

model for liquefaction. J Geotech Eng Div ACSE 103(GT6):

517–533

33. Finn WDL, Bhata SK (1981) Prediction of seismic pore-water

pressure. In: Proceedings, tenth international conference on soil

mechanics and foundation engineering, vol 3, pp 201–206

34. Finn WDL (1982) Soil liquefaction studies in the People’s

Republic of China. In: Soil mechanics. In: Pande & Zienkiewicz

(eds) Transient and cyclic loads. Wiley, pp 609–626

35. Finn WDL (2000) State-of-the-art of geotechnical earthquake

engineering practice. Soil Dyn Earthq Eng 20:1–15

36. Hamada M, Yasuda S, Isoyama R, Emoto K (1986) Study on

liquefaction induced permanent ground displacements. Associ-

ation for the Development of Earthquake Prediction pp 1–

87

37. Hamada M, O’Rourke TD (1992) Case studies of liquefaction

and lifeline performance during past earthquakes. Volume one:

Japanese case studies. Technical report NCEER-92–0001.

National center for earthquake engineering research, Buffalo

38. Hardin BO, Drnevich VP (1972) Shear modulus and damping in

soils: design equations and curves. J Soil Mech Found Div

ASCE 98(SM7): 667–692

39. Hashiguchi K, Ueno M (1977) Elastoplastic constitutive laws of

granular material. In: Proceedings ninth international conference

on soil mechanics and foundation engineering, Special session 9,

pp 73–82

40. Hashiguchi K (1980) Constitutive equations of elastoplastic

materials with elasto-plastic transition. J Appl Mech ASME

47(2):266–272

41. Hashiguchi K (1989) Subloading surface model in unconven-

tional plasticity. Int J Solids Struct 25:917–945

42. Hashiguchi K, Chen ZP (1998) Elastoplastic constitutive equa-

tion of soils with the subloading surface and the rotation hard-

ening. Int J Num Anal Methods Geomech 22:197–227

43. Huang WX (1959) Study on liquefaction of saturated sand

foundation and slopes. Water Resour Hydropower Eng 15:1–8

(in Chinese)

44. Huang WX (1961) Investigation of stability of saturated sand

foundation and slopes against liquefaction. In: Proceedings, 5th

international conference on soil mechanics and foundation

engineering, vol 2. Paris, France pp 629–631

45. Iai S, Matsunaga Y, Kameoka T (1992) Strain space plasticity

model for cyclic mobility. Soils Found 32(2):1–15

46. Iida H, Hiroto T, Yoshida N, Iwafuji M (1996) Damage to

Daikai subway station. Soils and foundations, special issue on

geotechnical aspects of the January 17, 1995 Hyogoken-Nambu

Earthquake pp 283–300

47. Idriss IM, Dobry R, Singh RD (1978) Nonlinear behavior of soft

clays during cyclic loading. J Geotech Eng Div ACSE, 104

(GT12): 1427–1447

48. Idriss IM, Boulanger RW (2008) Soil liquefaction during

earthquake, Earthquake Engineering Institute, EERI Publication

MNO-12

49. Ishihara K (1985) Stability of natural deposits during earth-

quakes. In: Proceedings eleventh international conference on

soil mechanics and foundation engineering, Rotterdam,

Netherlands

50. Ishihara K (1986) Evaluation of soil properties for use in

earthquake response analysis. In: Dungar R, Studer JA (eds)

Geomechanical modeling in engineering practice. A. A. Balk-

ema, Rotterdam, pp 241–275

51. Ishihara K, Tatsuoka F, Yasuda S (1975) Undrained deformation

and liquefaction of sand under cyclic stresses. Soils Found

15(1):29–44

52. Ishihara K, Yoshimine M (1992) Evaluation of settlements in

sand deposits following liquefaction during earthquakes. Soils

Found 32(1):173–188

53. Iwan WD (1967) On a class of models for the yielding behavior

of continuous and composite systems. J Appl Mech ASME

34:612–617

54. Iwasaki Y, Tai M (1996) Strong motion records at Kobe Port

Island. Soils and foundations, special issue on geotechnical

aspects of the January 17 1995 Hyogoken- Nambu Earthquake

vol 38, pp 29–40

55. Kabilamany K, Ishihara K (1990) Stress dilatancy and hardening

laws for rigid granular model of sand. Soil Dyn Earthq Eng

9(2):66–77

56. Katada T, Suemasa N, Satoh H, Hamada M (1993) A study of

cyclic behavior on liquefied sand. In: Proceedings of 29th Japan

national conference on soil mechanics and foundation engi-

neering, vol 2, pp 831–832 (in Japanese)

57. Katona MG, Zienkiewicz OC (1985) A unified set of single step

algorithms Part 3: the Beta-m method, a generalization of the

Newmark scheme. Int J Num Methods Eng 21:1345–1359

58. Kondner RL, Zelasko JS (1963) A hyperbolic stress-strain for-

mulation of sands. In: Proceedings, 2nd Pan American confer-

ence on soil mechanics and foundation engineering, Sao Paulo,

Brazil, pp 289–324

59. Krieg RD (1975) A practical two surface plasticity theory.

J Appl Mech ASME 42(3):641–646

60. Ladd CC, Foott R, Ishihara K, Schlosser F, Poulos HG (1977)

Stress-deformation and strength characteristics. In: Proceedings

9th international conference on soil mechanics and foundation

engineering, Tokyo, 2: 421–494

61. Law KT, Cao YL, He GN (1990) An energy approach for

assessing seismic liquefaction potential. Can Geotech J

27(3):320–329

62. Lee KL, Albaisa A (1974) Earthquake induced settlements in

saturated sands. J Geotech Eng Div ASCE 100(GT4): 387–406

63. Li XS, Dafalias YF (2000) Dilatancy for cohesionless soils.

Geotechnique 50(4):449–460

64. Li XS (2002) A sand model with state-dependent dilatancy.

Geotechnique 52(3):173–186

65. Ling HI, Yang S (2006) Unified sand model based on the critical

state and generalized plasticity. J Eng Mech 132(12):1380–1391

66. Liou CP, Streeter VL, Richart FE (1977) Numerical model for

liquefaction. J Geotech Eng Div ASCE 103(6):589–606

110 Acta Geotechnica (2012) 7:69–113

123

67. Loukidis D, Salgado R (2009) Modeling sand response using

two-surface plasticity. Comput Geotech 36(1–2):166–186

68. Lu J, He L, Yang Z, Abdoun T, Elgamal A (2004a) Three-

dimensional finite element analysis of dynamic pile behavior in

liquefied ground. In: Doolin D, Kammerer A, Nogami T, Seed

RB (eds) Proceedings, the 11th international conference on soil

dynamics and earthquake engineering. I.T., Berkeley, CA, Jan-

uary 7–9, 1, pp 144–148

69. Lu J, Peng J, Elgamal A, Yang Z, Law KH (2004) Parallel finite

element modeling of earthquake liquefaction response. Int J

Earthq Eng Eng Vib 3(1):23–37

70. Manzari MT, Nour MA (1997) On implicit integration of

bounding surface plasticity models. Comput Struct 63(3):385–

395

71. Manzari MT, Dafalias YF (1997) A critical state two-surface

plasticity model for sands. Geotechnique 47(2):255–272

72. Martin GR, Finn WDL, Seed HB (1975) Fundamentals of liq-

uefaction under cyclic loading. J Geotech Eng Div ASCE

101(GT5): 423–438

73. Masing G (1926) Eigenspannungen und verfestigung beim

messing. In: Proceedings, second international congress for

applied mechanics, Zurich, Switzerland, pp 332–335 (in

German)

74. Matsuoka H (1974) A microscopic study on shear mechanism of

granular materials. Soils Found 14(1):29–43

75. Mogami T, Kubo K (1953) The behavior of soil during vibra-

tion. In: Proceedings, third international conference on soil

mechanics and foundation engineering vol 1, pp 152–155

76. Mroz Z (1967) On the description of anisotropic workhardening.

J Mech Phy Solids 15:163–175

77. Mroz Z (1969) An attempt to describe the behavior of metals

under cyclic loads using a more general working hardening

model. Acta Mech 7:199–212

78. Mroz Z, Norris VA, Zienkiewicz OC (1978) An anisotropic

hardening model for soils and its application to cyclic loading.

Int J Num Anal Meth Geomech 2(3):203–221

79. Mroz Z, Norris VA, Zienkiewicz OC (1979) Application of an

anisotropic hardening model in the analysis of elasto-plastic

deformation of soils. Geotechnique 29(1):1–34

80. Mroz Z, Norris VA, Zienkiewicz OC (1981) An anisotropic,

critical state model for soils subjected to cyclic loading. Geo-

technique 31(4):451–469

81. Mroz Z, Zienkiewicz OC (1984) Uniform formulation of con-

stitutive equations for clays and sands. In: Desai CS, Gallagher

RH (eds) Mechanics of Engineering Materials, pp 415–449

82. Nagase H, Ishihara K (1988) Liquefaction-induced compaction

and settlement of sand during earthquake. Soils Found 28:66–76

83. Nemat-Nasser S, Shokooh A (1979) A unified approach to

densification and liquefaction of cohesionless sand in cyclic

shearing. Can Geotech J 16:659–678

84. Nemat-Nasser S, Tobita Y (1982) Influence of fabric on lique-

faction and densification potential of cohesionless sand. Mech

Mat 1(1):43–62

85. Nishimura A, Sawada R, Danamura S (1996) Damage analyses

of railway open-cut-method-tunnels in Hyogoken-nambu

earthquake. In: Proceedings, 31th conference of Japan geo-

technical society, pp 2245–2246 (in Japanese)

86. Nishimura S, Towhata I (2004) A three-dimensional stress-strain

model of sand undergoing cyclic rotation of principal stress

axes. Soils Found 44(2):103–116

87. Nova R, Wood DM (1979) A constitutive model for sand in

triaxial compression. Int. J Num Anal Meth Geomech

3(3):255–278

88. Ortiz M, Simo JC (1986) An analysis of a new class of inte-

gration algorithms for elastoplastic constitutive relations. Int J

Num Meth Eng 23(3):353–366

89. Pan H, Chen GX, Liu HL, Wang BH (2011) Behavior of large

post-liquefaction deformation in saturated Nanjing fine sand.

Earthq Eng Eng Vib 10(2):187–193

90. Papadimitriou AG, Bouckovalas GD, Dafalias YF (2001) Plas-

ticity model for sand under small and large cyclic strains.

J Geotech Geoenviron Eng ASCE 127(11):973–983

91. Parra E (1996) Numerical modeling of liquefaction and lateral

ground deformation including cyclic mobility and dilation

response in soil systems. Ph.D. thesis, Department of Civil

Engineering, Rensselaer Polytechnic Institute, Troy, New York

92. Pastor M, Zienkiewicz OC, Chan AC (1990) Theme/feature

paper: Generalized plasticity and modeling of soil behavior. Int J

Anal Mech Geomech 14:151–190

93. Poorooshasb HB, Pietruszczak S (1986) A generalized flow

theory for sand. Soils Found 26(2):1–15

94. Pradhan TBS, Tatsuoka F, Sato Y (1989) Experimental stress-

dilatancy relations of sand subjected to cyclic loading. Soils

Found 29(1):45–64

95. Prevost JH (1978) Plasticity theory for soil stress-strain behav-

ior. J Eng Mech Div ASCE 104(5):1177–1194

96. Prevost JH (1985) A simple plasticity theory for frictional

cohesionless soils. Int J Soil Dyn Earthq Eng 4(1):9–17

97. Prevost JH, Keane CM (1990) Shear stress-strain curve gener-

ation from simple material parameter. J Geotech Eng ASCE

116(8):1255–1263

98. Ramamurthy T, Rawat PC (1973) Shear strength of sand under

general stress system. In: Proceedings, 8th international con-

ference on soil mechanics and foundation engineering. Moscow,

1: 339–342

99. Ramberg W, Osgood WR (1943) Description of stress-strain

curves by three parameters. National Advisory Committee for

Aeronautics, Washington, DC Technical Note No. 902, 1943-

07

100. Rauch AF, Martin JR II (2000) EPOLLS model for predicting

average displacements on lateral spreads. J Geotech Geoenviron

Eng ASCE 126(4):360–371

101. Richart FE (1975) Some effects of dynamic soil properties on

soil-structure interaction. J Geotech Eng Div ACSE 101(GT12):

1197–1240

102. Roscoe KH, Schofield AN, Wroth CP (1958) On the yield of

soils. Geotechnique 8(1):22–52

103. Roscoe KH, Burland JB (1968) On the generalised stress/strain

behavior of ‘‘wet’’ clay. In: Hayman J, Lockheed FA (eds)

Engineering plasticity. Cambridge University Press, Cambridge,

pp 535–609

104. Rowe PW (1962) The stress-dilatancy relation for static equi-

librium on an assembly of particles in contact. Proc R Soc Lond

A 269:500–527

105. Salciarini D, Tamagnini C (2009) A hypoplastic macroelement

model for shallow foundations under monotonic and cyclic

loads. Acta Geotech 4(3):163–176

106. Seed HB, Lee KL (1966) Liquefaction of saturated sands during

cyclic loading. J Soil Mech Found Eng Div ASCE 92 (SM6):

105–134

107. Seed HB, Lee KL, Idriss IM, Makdisi F (1975) The slides in the

San Fernando dams during the earthquake of February 9, 1971.

J Geotech Eng Div ASCE 101(GT7): 651–688

108. Seed HB, Martin PP, Lysmer J (1976) Pore-pressure changes

during soil liquefaction. J Geotech Eng Div ASCE 102(GT4):

323–346

109. Seed HB (1979) Soil liquefaction and cyclic mobility evaluation

for level ground during earthquakes. J Geotech Eng Div ASCE

105(GT2): 201–255

110. Sfriso A, Weber G (2010) Formulation and validation of a

constitutive model for sands in monotonic shear. Acta Geotech

5(4):257–272

Acta Geotechnica (2012) 7:69–113 111

123

111. Shahnazari H, Towhata I (2002) Torsional shear tests on cyclic

stress-dilatancy relationship of sand. Soils Found 42(1):105–

119

112. Shamoto Y, Zhang JM, Goto S, Tokimatsu K (1996) A new

approach to predict post-liquefaction permanent deformation in

saturated sand. In: Proceedings, 11th WCEE, Acapulco, Mexico

113. Shamoto Y, Sato M, Zhang JM (1996) Simplified estimation of

earthquake-induced settlements in saturated sand deposits. Soils

Found 36(1):39–50

114. Shamoto Y, Zhang JM, Goto S (1997) Mechanism of large post-

liquefaction deformation in saturated sands. Soils Found

37(2):71–80

115. Shamoto Y, Zhang JM (1998) Evaluation of seismic settlement

potential of saturated sandy ground based on concept of relative

compression. Soils Found 38(S2):57–68

116. Shamoto Y, Zhang JM, Tokimatsu K (1998) New methods for

evaluating large residual post-liquefaction ground settlement

and horizontal displacement. Soils Found 38(S2):69–84

117. Shamoto Y, Zhang JM, Tokimatsu K (1998) New charts for

predicting large residual post-liquefaction ground deformation.

Soil Dyn Earthq Eng 17(7):427–438

118. Shamoto Y, Zhang JM, Tokimatsu K (1998) Horizontal residual

post-liquefaction deformation of level. ASCE Geotech Special

Publ 1(75):373–384

119. Shen ZJ (1980) Analysis of dynamically coupled percolation

and deformation problems of saturated sands. Chin J Hydraul

Eng 2:14–22 (in Chinese)

120. Simo JC, Meschke G (1993) A new class of algorithms for

classical plasticity extended to finite strains. Application to

geomaterials. Comput Mech 11(4):253–278

121. Sturm H (2009) Numerical investigation of the stabilisation

behaviour of shallow foundations under alternate loading. Acta

Geotech 4(4):283–292

122. Sutherland HB, Mesday MS (1969) The influence of the inter-

mediate principal stress on the strength of sand. In: Proceedings

of 7th international conference on soil mechanics and founda-

tion engineering. Mexico City, vol 1, pp 391–399

123. Tatsuoka F (1978) On the theoretical studies on deformation

behavior of granular material I, Tsuchi-to-Kiso. JSSMFE

26(6):82–89

124. Tatsuoka F, Sasaki T, Yamada S (1984) Settlement in saturated

sand induced by cyclic undrained simple shear. In: Proceedings,

8th world conference on earthquake engineering. San Francisco,

California, vol 3, pp 95–102

125. Taboada VM, Dobry R (1993a) Experimental results of model

No 1 at RPI. In: Arulanandan K, Scott RF (eds) Proceedings of

verification of numerical procedures for the analysis of soil

liquefaction problems, vol. 1. A.A. Balkema Publishers, Neth-

erlands, pp 3–17

126. Taboada VM, Dobry R (1993b) Experimental results of model

No 2 at RPI. In: Arulanandan K, Scott RF (eds) Proceedings,

verification of numerical procedures for the analysis of soil

liquefaction problems, vol 1. A.A. Balkema Publishers, Neth-

erlands, pp 277–294

127. Takewaki N, Ohtsuki A, Ohnishi Y (1996) An examination on

failure mechanism of a subway station due to the 1995 Hy-

ogoken-Nanbu Earthquake. In: Proceedings of Technical Con-

ference on the Great Hanshin-Awaji Earthquake, JSCE,

pp 221–226. (in Japanese)

128. Tejchman J, Gorski J (2010) Finite element study of patterns of

shear zones in granular bodies during plane strain compression.

Acta Geotech 5(2):95–112

129. Tokimatsu K, Seed HB (1987) Evaluation of settlements in

sands due to earthquake shaking. J Geotech Eng 113(8):861–878

130. Tokimatsu K, Mizuno H, Kakurai M (1996) Building damage

associated with geotechnical problems. Soils and foundations,

special issue on geotechnical aspects of the January 17 1995

Hyogoken-Nambu Earthquake, 36: 219–234

131. Tokimatsu K, Asaka Y (1998) Effects of liquefaction-induced

ground displacements on pile performance in the 1995 Hyogo-

ken-Nambu earthquake. Soils and Foundations, Special issue

No. 2 on geotechnical aspects of the January 17, 1995 Hyogo-

ken-Nambu earthquake, pp 163–178

132. Tokimatsu K, Taya Y, Zhang JM (2001) Effect of pore water

redistribution on post-liquefaction deformation of sands. In:

Proceedings, 15th international conference on soil mechanics

and geotechnical engineering, Balkema, Rotterdam, The Neth-

erlands, 1:289–292

133. Vaid YP, Thomas J (1995) Liquefaction and post liquefaction

behavior of sand. J Geotech Eng ASCE 121(2):163–173

134. Wang WS (1962) Study on pore water pressures of saturated

sands due to vibration. J Hydraul Eng 2:25–31 (in Chinese)

135. Wang WS (1964) Generation, diffusion and dissipation of

excess pore water pressure in saturated sand under cyclic

loading. In: Proceedings, first China national conference on soil

mechanics and foundation engineering, China civil engineering

society. Tianjin pp 224–235 (in Chinese)

136. Wang WS (1979/1980) Some findings in soil liquefaction,

research report of China water resources and hydropower

research institute, 1979: 1–17 (in English); Chin J Geotech Eng

2(3): 55–63 (in Chinese)

137. Wang WS (1980) Strength, liquefaction and failure of saturated

sands under cyclic loading. J Hydraul Eng 1: 14–27(in Chinese)

138. Wang WS (1981) Liquefaction mechanism of soil. J Hydraul

Eng 5:22–34 (in Chinese)

139. Wang ZL, Dafalias YF, Shen CK (1990) Bounding surface

hypoplasticity model for sand. J Eng Mech ASCE 116(5):

983–1001

140. Wang ZL, Dafalias YF (2002) Simulation of post-liquefaction

deformation of sand. In: Proceedings, ASCE 15th engineering

mechanics conference. Columbia University, New York, NY

141. Xie DY, Zhang JM, Shi HY (1989) A new approach to predict

seismic pore pressure, shear stress and acceleration in a satu-

rated sand foundation. A research report of Department of Civil

Engineering, Rensselaer Polytechnic Institute, pp 1–45

142. Xie DY, Zhang JM (1990) Transient change mechanism of pore

water pressure in saturated sand under cyclic loading. China Civ

Eng J 23(2):51–60 (in Chinese)

143. Xu ZY, Shen ZJ (1981) 2D dynamic analysis of effective stress

of seismic liquefaction. J East China Coll Hydraul Eng 2:1–14

(in Chinese)

144. Yamato T, Umehara T, Aoki H, Nakamura S, Ezaki J, Suetomi I

(1996) Damage to Daikai subway station, Kobe rapid transit

system and estimation of its reason. In: Proceedings, technical

conference on the great Hanshin-Awaji Earthquake, JSCE,

Tokyo, pp 247–254. (in Japanese)

145. Yang ZH (2000) Numerical modeling of earthquake site

response including dilation and liquefaction. Ph.D. thesis,

Department of Civil Engineering and Engineering Mechanics,

Columbia University, New York

146. Yang ZH, Elgamal A, Parra E (2003) Computational model for

cyclic mobility and associated shear deformation. J Geotech

Geoenviron Eng ASCE 129(12):1119–1127

147. Yang ZH, Elgamal A (2008) Multi-surface cyclic plasticity sand

model with Lode angle effect. Geotech Geol Eng 26(3):335–348

148. Yasuda S, Nagase H, Kiku H, Uchida Y (1992) The mechanism

and a simplified procedure for the analysis of permanent ground

displacement due to liquefaction. Soils Found 32(1):149–160

149. Yasuda S, Masuda T, Yoshida N, Nagase H, Kiku H, Itafuji S,

Mine K, Sata K (1994) Torsional shear and triaxial compression

tests on deformation characters of sands before and after

liquefaction. In: Proceedings, 5th US-Japan workshop on

112 Acta Geotechnica (2012) 7:69–113

123

earthquake resistant design of lifelines and countermeasures

against soil liquefaction, pp 249–265

150. Yi F (2010a) Procedures to evaluate liquefaction-induced set-

tlements based on shear wave velocity. In: Proceedings, Ninth

US National and 10th Canadian conference on earthquake

engineering, Paper No. 291

151. Yi F (2010b) Procedures to evaluate liquefaction-induced lateral

spreading based on shear wave velocity. In: Proceedings, fifth

international conference on recent advances in geotechnical

earthquake engineering and soil dynamics, Paper No. 1.57a

152. Yoshida N, Yasuda S, Kiku M, Masuda T, Finn WDL (1994)

Behavior of sand after liquefaction. In: Proceedings, 5th U.S.-

Japan workshop on earthquake resistant design of lifelines and

countermeasures against soil liquefaction, pp 181–198

153. Yoshimi Y, Kuwabara F, Tokimatsu K (1975) One-dimensional

volume change characteristics of sands under very low confining

stresses. Soils Found 15(3):51–60

154. Yoshimi Y (1994) Evaluation of in situ liquefaction resistance

of sands based on high-quality undisturbed specimens. In: Pro-

ceedings of symposium on developments in geotechnical engi-

neering, Bangkok

155. Yoshimine M, Nishizaki H, Amano K, Hosono Y (2006) Flow

deformation of liquefied sand under constant shear loading and

its application to analysis of flow slide of infinite slope. Soil Dyn

Earthq Eng 26(2–4):253–264

156. Youd TL, Perkins M (1978) Mapping liquefaction-induced

ground failure potential. J Geotech Eng Div ASCE 104(4):

433–446

157. Youd TL, Hansen CM, Bartlett SF (2002) Revised multi-linear

regression equations for lateral spread displacement. J Geotech

Geoenviron Eng ASCE 128(12):1007–1017

158. Young YL, White JA, Xiao H, Borja RI (2009) Liquefaction

potential of coastal slopes induced by solitary waves. Acta

Geotech 4(1):17–34

159. Zhang G, Robertson PK, Brachman RWI (2002) Estimating

liquefaction-induced ground settlements from CPT for level

ground. Can Geotech J 39(5):1168–1180

160. Zhang G, Robertson PK, Brachman RWI (2004) Estimating

liquefaction-induced lateral displacements using the standard

penetration test or Cone Penetration test. J Geotech Geoenviron

Eng ASCE 130(8):861–871

161. Zhang JM (1994) Transient shear strength of saturated sand

under cyclic loading considering strain-rate effect. Soils Found

34(4):51–65

162. Zhang JM (1997) Cyclic critical stress state theory of sand with

its application to geotechnical problems. Ph D thesis, Tokyo

Institute of Technology, Tokyo, pp 1–269

163. Zhang JM, Shamoto Y, Tokimatsu K (1997) Moving critical and

phase-transformation stress state lines of saturated sand under

undrained cyclic loading. Soils Found 37(2):51–59

164. Zhang JM, Shamoto Y, Tokimatsu K (1998) Cyclic critical

stress states of sand with non-frictional effects. J Eng Mech

ASCE 125(10):1106–1114

165. Zhang JM, Tokimatsu K, Taya Y (1999) Effect of water

absorption during post-liquefaction shearing in saturated sand.

Chin J Geotech Eng 21(4):398–402 (in Chinese)

166. Zhang JM (2000) Reversible and irreversible dilatancy of sand.

Chin J Geotech Eng 22(1):12–17 (in Chinese)

167. Zhang JM, Wang G (2004) A constitutive model for evaluating

small to large cyclic strains of saturated sand during liquefaction

process. Chin J Geotech Eng 26(4):546–552

168. Zhang JM, Wang G (2005) Fundamentals and constitutive

model of large post-liquefaction deformation in saturated sand.

In: Invited lecture, proceedings, second china-japan geotechni-

cal symposium, Shanghai, China pp 43–67

169. Zhang JM, Wang G (2006) Mechanism of large post-liquefac-

tion deformation of saturated sand. Chin J Geotech Eng

28(7):835–840 (in Chinese)

170. Zhang JM, Yang ZY, Gao JZ, Tong ZX (2010) Lessons from

damages to high embankment dams in the May 12, 2008

Wenchuan earthquake. ASCE Geotech Special Pub 201:1–31

171. Zhang JM, Tong ZX, Luo G (2011) A cyclic constitutive model

for granular materials considering reversible and irreversible

dilatancy. Research report of Tsinghua University. Tsinghua

University, Beijing, pp 1–40

172. Zhang F, Ye B, Jin YJ, Nakai T (2011a) Cyclic mobility of sand

and its simulation in boundary value problems. Bifurcations,

Instabilities and Degradations in Geomaterials 0: 113–132

173. Zhang F, Ye B, GL Y (2011) United description of sand

behavior. Front Archit Civ Eng China 5(2):121–150

174. Zhou J, Xu ZY (1981) 3D dynamic analysis of effective stress of

earth dams. Earthq Eng Eng Vib 4(3):60–70 (in Chinese)

175. Zienkiewicz OC, Mroz Z (1984) Generalized plasticity formu-

lation and application to geomechanics. In: Desai CS, Gallagher

RH (eds) Mech Eng Materials. John Wiley & Sons, pp 655–679

176. Zienkiewicz OC, Shiomi T (1984) Dynamic behavior of satu-

rated porous media: the generalized Biot formulation and its

numerical solution. Int J Num Anal Methods Geomech 8:71–96

177. Zienkiewicz OC, Chan AHC, Schrefler BA, Shiomi T (1999)

Computational geomechanics with special reference to earth-

quake engineering. Wiley, pp 335–369

Acta Geotechnica (2012) 7:69–113 113

123