Modeling permanent strains of granular soil under cyclic loading with variable confining pressure


Previous experimental studies showed that variable confining pressure (VCP) promoted the development of permanent strains of granular soils in most cases and indicated VCP should not be ignored. However, existing empirical formulas do not fully account for the effect of VCP on permanent strains. Therefore, three series of cyclic triaxial tests with constant confining pressure (CCP) and VCP are performed on saturated sand at three relative densities. These tests aim at investigating the influential factors which could reflect the effect of VCP on permanent strains and should be considered in establishing the empirical formula. Test results show that an increasing initial void ratio has great effect on the permanent strains of sand. The permanent strains increase with an increase in the stress ratio of maximum deviatoric stress to average confining pressure \({{q^{\hbox{max} } } \mathord{\left/ {\vphantom {{q^{\hbox{max} } } {\sigma_{3}^{\text{av}} }}} \right. \kern-0pt} {\sigma_{3}^{\text{av}} }}\) and the length of stress path L in both CCP and VCP tests. Based on that, a four-parameter equation considering the initial void ratio, stress ratio, length of stress path, and number of loading cycles is proposed to predict the permanent strains under cyclic loading with VCP. Comparison between the measured and predicted permanent strains shows that the equation can well capture the permanent strain response in tests with CCP and VCP.

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Constant confining pressure

Dr c :

Relative density after consolidation


Function of void ratio

L :

Length of stress path

N :

Number of loading cycles

p a :

Reference stress

p ampl :

Amplitude of mean principal stress

p av :

Average mean principal stress

p max :

Maximum mean principal stress

p min :

Minimum mean principal stress

q ampl :

Amplitude of deviatoric stress

q av :

Average deviatoric stress

q cyc :

Cyclic deviatoric stress

q max :

Maximum deviatoric stress

q min :

Minimum deviatoric stress


Variable confining pressure

α1, α2, α3, and α4 :

Parameters of the equation

\(\varepsilon_{\text{a}}^{\text{p}}\) :

Axial permanent strain

η ampl :

Inclination of stress path

\(\sigma_{3}^{\text{ampl}}\) :

Amplitude of confining pressure

\(\sigma_{3}^{\text{av}}\) :

Average confining pressure

\(\sigma_{3}^{\text{cyc}}\) :

Cyclic confining pressure


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The work presented in this paper was supported by the National Natural Science Foundation of China under Grant Nos. 51608394, 51508417, and the Natural Science Foundation of Zhejiang Province Nos. LY18E080030 and LY19E080021. These financial supports are gratefully acknowledged.

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Correspondence to Quanyang Dong.

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Sun, Q., Dong, Q., Cai, Y. et al. Modeling permanent strains of granular soil under cyclic loading with variable confining pressure. Acta Geotech. 15, 1409–1421 (2020).

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  • Cyclic loading
  • Granular soil
  • Permanent strain
  • Variable confining pressure