Acta Geotechnica

, Volume 7, Issue 2, pp 69–113 | Cite as

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

Research Paper


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 decomposing 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 alternating in the liquefaction process. This assumption sheds some light on the intricate transition from small pre-liquefaction 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 performance 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 centrifuge model tests. Finally, numerical simulation of liquefaction-related site response is performed for the Daikai subway station damaged during the 1995 Hyogoken-Nambu earthquake in Japan.


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

List of symbols

e, Dr

Void ratio and relative density


Atmospheric pressure


Simple shear stress

pe, ru

Excess pore water pressure and excess pore water pressure ratio

\({\sigma_{\text{c}}^{\prime } } \hbox{,} \)\({\sigma_{\text{m}}^{\prime } } \)

Initial effective consolidation stress and mean effective stress

p, q

Mean effective stress and deviatoric stress invariant

η, ηm

Shear stress ratio (η = q/p) and its maximum value in loading history


Total shear strain


Solid-like shear strain that occurs in non-zero effective confining stress state


Fluid-like shear strain that occurs in zero effective confining stress state


Preceding maximum cyclic shear strain

\( \dot{\gamma }_{\text{eff}} \)

Effective shear strain rate


Monotonic shear strain length


Reference shear strain length


Residual shear strain


Total volumetric strain

\( \varepsilon_{\text{v,recon}} \)

Reconsolidation volumetric strain


Volumetric strain component due to the change in p


Threshold volumetric strain to delimit whether the effective confining stress reaches zero, determined as εvc value at zero effective confining stress state


Threshold pressure for numerical calculation to delimit whether the effective confining stress reaches zero


Volumetric strain due to dilatancy


Irreversible dilatancy component

\( \varepsilon_{\text{vd,re}} \)

Reversible dilatancy component

\( {\varvec{\upsigma} } \)ij), s(sij)

Effective stress tensor and its deviatoric part

\( {\varvec{\upvarepsilon }}\)ij), e(eij)

Strain tensor and its deviatoric part


Deviatoric shear stress ratio tensor


Identity tensor of rank 2 (Kronecker delta)


Loading direction in stress ratio space


Flowing direction of plastic deviatoric strain increment


Projection center

\( \hat{f}(\hat{\varvec{\upsigma} }) \hbox{,} \, \bar{f}(\bar{\varvec{\upsigma} }) \)

Failure surface and maximum prestress memory surface serving as bounding surfaces


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, κ

Modulus parameters

Md,c, dre,1, dre,2

Reversible dilatancy parameters

\( d_{\text{ir}} ,\alpha ,\gamma_{\text{d,r}} \)

Irreversible dilatancy parameters


Lode angle

\( \rho ,\bar{\rho } \)

Mapping distances in stress ratio space



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.


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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute of Geotechnical Engineering, School of Civil Engineering/State Key Laboratory of Hydroscience and EngineeringTsinghua UniversityBeijingChina
  2. 2.Ertan Hydropower Development Company LimitedChengduChina

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