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

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## 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 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.

### Keywords

Centrifuge tests Constitutive model Earthquake Liquefaction Large deformation Numerical analysis Site response### List of symbols

*e*,*D*_{r}Void ratio and relative density

*p*_{a}Atmospheric pressure

*τ*Simple shear stress

*p*_{e},*r*_{u}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*,*η*_{m}Shear stress ratio (

*η*=*q/p*) and its maximum value in loading history- γ
Total shear strain

- γ
_{d} Solid-like shear strain that occurs in non-zero effective confining stress state

- γ
_{o} Fluid-like shear strain that occurs in zero effective confining stress state

- γ
_{max} Preceding maximum cyclic shear strain

- \( \dot{\gamma }_{\text{eff}} \)
Effective shear strain rate

- γ
_{mono} Monotonic shear strain length

- γ
_{d,r} Reference shear strain length

- γ
_{r} Residual shear strain

- ε
_{v} Total volumetric strain

- \( \varepsilon_{\text{v,recon}} \)
Reconsolidation volumetric strain

- ε
_{vc} Volumetric strain component due to the change in

*p*- ε
_{vc,o} Threshold volumetric strain to delimit whether the effective confining stress reaches zero, determined as ε

_{vc}value at zero effective confining stress state*p*_{min}Threshold pressure for numerical calculation to delimit whether the effective confining stress reaches zero

- ε
_{vd} Volumetric strain due to dilatancy

- ε
_{vd,ir} Irreversible dilatancy component

- \( \varepsilon_{\text{vd,re}} \)
Reversible dilatancy component

- \( {\varvec{\upsigma} } \)(σ
_{ij}),**s**(*s*_{ij}) Effective stress tensor and its deviatoric part

- \( {\varvec{\upvarepsilon }}\)(ε
_{ij}),**e**(*e*_{ij}) Strain tensor and its deviatoric part

**r**(*r*_{ij})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

**α**Projection center

- \( \hat{f}(\hat{\varvec{\upsigma} }) \hbox{,} \, \bar{f}(\bar{\varvec{\upsigma} }) \)
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*,*D*_{ir},*D*_{re}Total, irreversible and reversible dilatancy rates

*D*_{re,gen},*D*_{re,rel}Reversible dilatancy rates in dilative and contractive phases

*M*_{f,c},*M*_{f,o}Failure stress ratios in triaxial compression stress state and torsional shear stress state

*G*_{o},*n*,*h*, κModulus parameters

*M*_{d,c},*d*_{re,1},*d*_{re,2}Reversible dilatancy parameters

- \( d_{\text{ir}} ,\alpha ,\gamma_{\text{d,r}} \)
Irreversible dilatancy parameters

- θ
_{σ} Lode angle

- \( \rho ,\bar{\rho } \)
Mapping distances in stress ratio space

## Notes

### 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.

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