A two-surface plasticity model for cyclic behavior of saturated clay

  • Ren-Peng ChenEmail author
  • Shu Zhu
  • Peng-Yun Hong
  • Wei Cheng
  • Yu-Jun Cui
Research Paper


This paper presents a two-surface plasticity model for describing some important features of saturated clay under cyclic loading conditions, such as closed hysteresis loops, cyclic shakedown and degradation, and different stress–strain relations for two-way loading. The model, namely ACC-2-C, is based on the elastoplastic model ACC-2 (an adapted Modified Cam Clay model with two yield surfaces) developed by Hong et al. (Acta Geotech 11(4):871–885, 2015). The small-strain nonlinearity concept is adopted to achieve the nonlinear characteristics of clay during unloading–loading stage. The new hardening law related to accumulated deviatoric plastic strain is proposed for the inner surface to describe the cyclic shakedown and degradation. Following the advantages of the ACC-2 model, the constitutive equations are simply formulated based on the consistency condition for the inner yield surface. The model is conveniently implemented in a finite element code using a stress integration scheme similar to the Modified Cam Clay model. The simulation results are highly consistent with experimental data from drained and undrained isotropic cyclic triaxial tests in normally consolidated saturated clay under both one-way and two-way loadings.


Bounding surface plasticity Cyclic loading Saturated clay Small-strain nonlinearity 

List of symbols


Model parameter


Model parameter

De, Dep

Elastic and elastoplastic stiffness matrix, respectively

fB, fL

Bounding surface and loading yield surface, respectively

fBc, fBe

Bounding surface for compression and extension, respectively


Shear modulus

G1, GN

Secant modulus at cycles 1 and N, respectively

gB, gL

Plastic potentials associated with bounding surface and loading yield surface, respectively


Hardening modulus


Elastic bulk modulus


Material constant

kf, kg

Model parameters for yield surface and plastic potential, respectively

Mf, Mg

Stress ratio at the apex of the yield surface and critical state slope, respectively

Mfc, Mfe

Stress ratio at the apex of the yield surface in compression and extension, respectively


Material constant

p′, prev

Mean effective stress at the current stress point and at the stress reversal point, respectively

\(p_{c}^{{\prime }} ,\bar{p}_{c}^{{\prime }}\)

Actual loading yield stress and conventional yield stress on the normal consolidation line, respectively


Deviator stress


Positive scalar defining the size of the loading field surface


Model parameters

\(s_{{{\text{c}}0}} ,s_{{{\text{e}}0}}\)

Initial material constant under the first compression and extension cycle, respectively


Initial specific volume


Pore water pressure

X, Xs

Current stress state for mean effective stress and deviator stress, respectively



\(\varepsilon_{1} ,\varepsilon_{3}\)

Axial and lateral strains, respectively

\(\varepsilon_{\text{s}} ,\varepsilon_{\text{s}}^{\text{e}} ,\varepsilon_{\text{s}}^{\text{p}}\)

Shear strain, elastic shear strain and plastic deviator strain, respectively

\(\varepsilon_{\text{v}} ,\varepsilon_{\text{v}}^{\text{e}} ,\varepsilon_{\text{v}}^{\text{p}}\)

Volumetric strain, elastic volumetric strain and plastic volumetric strain, respectively


Total plastic strain

\(\varepsilon^{1} ,\varepsilon^{N}\)

Axial strains at cycles 1 and N, respectively

\(\eta ,\eta_{\text{rev}}\)

Normalized deviator stress at current stress point and at the stress reversal point, respectively

\(\kappa ,\kappa_{0}\)

Elastic slope and initial unloading slope in v − lnp space, respectively


Plastic multiplier


Slope of the normal consolidation compression line


Poisson’s ratio



\(\sigma_{1}^{{\prime }} ,\sigma_{3}^{{\prime }}\)

Axial and lateral effective stresses, respectively


Model parameter


Degradation index



The present work was carried out with the support of the National Key Research and Development Program of China (2016YFC0800207), National Natural Science Foundation of China (41472244, 51608188), the Provincial Key Research and Development Program of Hunan (0105679005), the Industrial Technology and Development Program of Zhongjian Tunnel Construction Co., Ltd. (17430102000417).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.MOE Key Laboratory of Building Safety and Energy EfficiencyChangshaChina
  2. 2.International Joint Research Center for Sustainable Building EnvironmentChangshaChina
  3. 3.College of Civil EngineeringHunan UniversityChangshaChina
  4. 4.Department of Civil EngineeringZhejiang UniversityHangzhouChina
  5. 5.Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTARUniversité Paris-EstMarne-la-ValléeFrance

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