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Fractional plasticity for over-consolidated soft soil

Abstract

The stress–strain response of over-consolidated soft soil, e.g., clay, is dependent on its pre-consolidation history and material state. In this study, a state-dependent constitutive model for over-consolidated soft soils is proposed by extending the fractional plasticity originally developed for granular soil. The state-dependent stress-dilatancy and peak failure behaviour of over-consolidated soft soil are analytically captured through stress-fractional gradient of the current yielding surface. In addition, a reference yielding surface describing the pre-consolidation history of soft soil is proposed. A combined hardening rule expressed as a function of both the incremental plastic volumetric and shear strains is suggested. To validate the proposed model, a series of drained and undrained test results of different soft soils with a wide range of over-consolidation ratios are simulated and compared. It is found that without using additional plastic potentials and state parameters, the developed fractional model can capture the state-dependent hardening and softening responses of soft soils subjected to different over-consolidation states.

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Availability of data and material

All the data used in this study can be found in the relevant literatures, which has been cited properly besides each dataset.

Code availability

Codes related to this study can be provided upon proper request.

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Funding

The first author would like to thank Prof. Wen Chen for the invaluable inspiration. The financial support from the National Natural Science Foundation of China (Grant No. 41630638), the National Science Centre, Poland (Grant No. 2017/27/B/ST8/00351) and the Alexander Von Humboldt Research Foundation are appreciated.

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Correspondence to Yifei Sun.

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Appendix

Appendix

Following Matsuoka et al. [12], the critical-state stress ratio, \(M\left( \theta \right)\), is defined as:

$$ M\left( \theta \right) = \frac{{\sqrt 3 M_{c} \left( {\sqrt {8 + {\text{sin}}^{2} \varphi_{0} } - {\text{sin}}\varphi_{0} } \right)}}{{4\sqrt {2 + {\text{sin}}^{2} \varphi_{0} } {\text{cos}}\Psi \left( \theta \right)}} $$
(34)

where Mc is the slope of the CSL in the \(p^{\prime} - q\) plane and \(\Psi\) is a function of the Lode’s angle (θ), which can be defined as:

$$ M_{c} = \frac{{6{\text{sin}}\phi_{c} }}{{3 - {\text{sin}}\phi_{c} }} $$
(35)
$$ \Psi \left( \theta \right) = \frac{1}{3}{\text{arccos}}\left[ { - \left( {\frac{3}{{2 + {\text{sin}}^{2} \varphi_{0} }}} \right)^{3/2} {\text{sin}}\varphi_{0} {\text{cos}}3\theta } \right] $$
(36)
$$ \varphi_{0} = {\text{arctan}}\left( {\frac{2\sqrt 2 }{3}{\text{tan}}\phi_{c} } \right) $$
(37)
$$ \theta = \frac{1}{3}{\text{arccos}}\left\{ {\frac{9}{2}\frac{{{\text{s}}_{ij} {\text{s}}_{jk} {\text{s}}_{ki} }}{{\left( {3/2{\text{s}}_{ct} {\text{s}}_{ct} } \right)^{3/2} }}} \right\} $$
(38)

where \(\phi_{c}\) is the critical-state friction angle obtained under triaxial compression.

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Sun, Y., Sumelka, W. & Gao, Y. Fractional plasticity for over-consolidated soft soil. Meccanica (2021). https://doi.org/10.1007/s11012-021-01343-1

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Keywords

  • Fractional derivative
  • State dependence
  • Over-consolidation
  • Soft soil