Acta Geotechnica

, Volume 14, Issue 2, pp 329–345 | Cite as

A critical state constitutive model for clean and silty sand

  • X. Wei
  • J. YangEmail author
Research Paper


The mechanical behavior of silty sand is highly dependent on the percentage of fines in addition to the packing density and confining pressure. Properly modeling the diverse behavior of silty sand remains an area of difficulty and uncertainty. This paper presents an attempt to formulate a critical state-based constitutive model for sand with varying fines content based on several new laboratory findings. A marked feature of the model is a unified description of the state-dependent elastic modulus as well as a unified description of plastic hardening modulus such that only one set of elastic and hardening parameters is required for sand with different fines contents. The model is calibrated and validated using the results from a structured experimental program. It shows that the model can produce reasonably good predictions for undrained shear responses of sand specimens under a range of void ratios, confining stresses and fines contents. In particular, it successfully predicts the laboratory observation that under otherwise similar conditions, the presence of non-plastic fines increases the liquefaction susceptibility of sand.


Constitutive modeling Critical state Liquefaction Silty sands State parameter 

List of symbols


Fitting parameter of G using F(e)


Fitting parameter of G in F(e)


Fitting parameter of G using F(ψ)


Fitting parameter of G in F(ψ)


Model parameter in Cr


Reduction factor for elastic shear modulus




Dilatancy parameter


Deviatoric strain increment


Elastic deviatoric strain increment


Plastic deviatoric strain increment


Volumetric strain increment


Elastic volumetric strain increment


Plastic volumetric strain increment


Void ratio


Initial void ratio prior to shearing (i.e., post-consolidation void ratio ec)


Intercept of critical state line (CSL) in the e − (p′/Pa)ξ plane


Void ratio function

f(X1, X2, X3…)

Function of X1, X2, X3


State parameter function


Fines content (%)


Fines content in decimal


Elastic shear modulus

h, h1, h2

Hardening parameters


Elastic bulk modulus


Pressure exponent of modulus


Model parameter in Cr


Plastic hardening modulus


Loading index


Dilatancy parameter


Stress ratio (η) at critical state


Hardening parameter


Mean effective stress


Post-consolidation pressure (i.e., initial mean effective stress)


Reference stress equaling to 1 atm


Particle size distribution


Phase transformation state


Deviatoric stress


Roundness of sand particle


Combined roundness


Undrained instability state


Model parameter in Kp


Deviatoric strain


Elastic deviatoric strain


Plastic deviatoric strain


Volumetric strain


Elastic volumetric strain


Plastic volumetric strain


Accumulated plastic deviatoric strain


Stress ratio q/p′


Stress ratio (η) at peak state


Stress ratio (η) at phase transformation state


Magnitude of the slope of CSL


Poisson’s ratio


Pressure exponent of CSL formulation


Critical state friction angle


State parameter


Initial state parameter prior to shearing



This work was supported by the Research Grants Council of Hong Kong through the General Research Fund (17250316, 17205717). This support is gratefully acknowledged.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil EngineeringThe University of Hong KongHong KongChina

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