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Fractional plasticity for over-consolidated soft soil


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.


  1. 1.

    Singh RP, Nimbalkar S, Singh S, Choudhury D (2019) Field assessment of railway ballast degradation and mitigation using geotextile. Geotext Geomembr.

    Article  Google Scholar 

  2. 2.

    El Jirari S, Wong H, Deleruyelle F, Branque D, Berthoz N, Leo C (2020) Analytical modelling of a tunnel accounting for elastoplastic unloading and reloading with reverse yielding and plastic flow. Comput Geotech 121:103441.

    Article  Google Scholar 

  3. 3.

    Wu Y, Zhou X, Gao Y, Zhang L, Yang J (2019) Effect of soil variability on bearing capacity accounting for non-stationary characteristics of undrained shear strength. Comput Geotech 110:199–210.

    Article  Google Scholar 

  4. 4.

    Yin ZY, Chang CS (2009) Microstructural modelling of stress-dependent behaviour of clay. Int J Solids Struct 46(6):1373–1388.

    Article  MATH  Google Scholar 

  5. 5.

    Nakai T, Matsuoka H, Okuno N, Tsuzuki K (1986) True triaxial tests on normally consolidated clay and analysis of the observed shear behaviour using elastoplastic constitutive models. Soils Found 26(4):67–78.

    Article  Google Scholar 

  6. 6.

    Whittle AJ (1993) Evaluation of a constitutive model for overconsolidated clays. Géotechnique 43:289–313.

    Article  Google Scholar 

  7. 7.

    Liu MD, Carter JP (2002) A structured Cam Clay model. Can Geotech J 39(6):1313–1332.

    Article  Google Scholar 

  8. 8.

    Pestana JM, Whittle AJ, Gens A (2002) Evaluation of a constitutive model for clays and sands: part II—clay behaviour. Int J Numer Anal Meth Geomech 26(11):1123–1146.

    Article  MATH  Google Scholar 

  9. 9.

    Roscoe K, Schofield A, Thurairajah A (1963) Yielding of clays in states wetter than critical. Géotechnique 13(3):211–240

    Article  Google Scholar 

  10. 10.

    Yao Y, Liu L, Luo T, Tian Y, Zhang JM (2019) Unified hardening (UH) model for clays and sands. Comput Geotech 110:326–343.

    Article  Google Scholar 

  11. 11.

    Yao Y, Hou W, Zhou AN (2009) UH model: three-dimensional unified hardening model for overconsolidated clays. Géotechnique 59(5):451–469.

    Article  Google Scholar 

  12. 12.

    Matsuoka H, Yao Y, Sun D (1999) The cam-clay models revised by the SMP criterion. Soils Found 39(1):81–95.

    Article  Google Scholar 

  13. 13.

    Nakai T, Hinokio M (2004) A simple elastoplastic model for normally and overconsolidated soils with unified material parameters. Soils Found 44(2):53–70.

    Article  Google Scholar 

  14. 14.

    Shi XS, Nie J, Zhao J, Gao Y (2020) A homogenization equation for the small strain stiffness of gap-graded granular materials. Comput Geotech 121:103440.

    Article  Google Scholar 

  15. 15.

    Shi XS, Zhao J (2020) Practical estimation of compression behavior of clayey/silty sands using equivalent void-ratio concept. J Geotech Geoenviron Eng 146(6):04020046.

    Article  Google Scholar 

  16. 16.

    Jocković S, Vukićević M (2017) Bounding surface model for overconsolidated clays with new state parameter formulation of hardening rule. Comput Geotech 83:16–29.

    Article  Google Scholar 

  17. 17.

    Lu D, Liang J, Du X, Ma C, Gao Z (2019) Fractional elastoplastic constitutive model for soils based on a novel 3D fractional plastic flow rule. Comput Geotech 105:277–290.

    Article  Google Scholar 

  18. 18.

    Wichtmann T, Triantafyllidis T (2017) Monotonic and cyclic tests on kaolin: a database for the development, calibration and verification of constitutive models for cohesive soils with focus to cyclic loading. Acta Geotech.

    Article  Google Scholar 

  19. 19.

    Ye GL, Ye B, Zhang F (2014) Strength and dilatancy of overconsolidated clays in drained true triaxial tests. J Geotech Geoenviron Eng.

    Article  Google Scholar 

  20. 20.

    Yang ZX, Xu TT, Li XS (2019) J2-deformation type model coupled with state dependent dilatancy. Comput Geotech 105:129–141.

    Article  Google Scholar 

  21. 21.

    Yin ZY, Xu Q, Hicher PY (2013) A simple critical-state-based double-yield-surface model for clay behavior under complex loading. Acta Geotech 8(5):509–523.

    Article  Google Scholar 

  22. 22.

    Xiao Y, Desai CS (2019) Constitutive modeling for overconsolidated clays based on disturbed state concept. I: theory. Int J Geomech.

    Article  Google Scholar 

  23. 23.

    Zhang S, Ye G, Wang J (2018) Elastoplastic model for overconsolidated clays with focus on volume change under general loading conditions. Int J Geomech 18(3):04018005.

    Article  Google Scholar 

  24. 24.

    Been K, Jefferies MG (1985) A state parameter for sands. Géotechnique 35(2):99–112.

    Article  Google Scholar 

  25. 25.

    Sun Y, Gao Y, Shen Y (2019) Mathematical aspect of the state-dependent stress-dilatancy of granular soil under triaxial loading. Géotechnique 69(2):158–165.

    Article  Google Scholar 

  26. 26.

    Sun Y, Sumelka W (2019) State-dependent fractional plasticity model for the true triaxial behaviour of granular soil. Arch Mech 71(1):23–47.

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Sumelka W, Nowak M (2018) On a general numerical scheme for the fractional plastic flow rule. Mech Mater 116:120–129.

    Article  Google Scholar 

  28. 28.

    Sumelka W (2014) A note on non-associated Drucker–Prager plastic flow in terms of fractional calculus. J Theor Appl Mech 52(2):571–574.

    Article  Google Scholar 

  29. 29.

    Sumelka W (2014) Fractional viscoplasticity. Mech Res Commun 56:31–36.

    Article  Google Scholar 

  30. 30.

    Sumelka W, Nowak M (2016) Non-normality and induced plastic anisotropy under fractional plastic flow rule: a numerical study. Int J Numer Anal Methods Geomech 40(5):651–675.

    Article  Google Scholar 

  31. 31.

    Liang J, Lu D, Zhou X, Du X, Wu W (2019) Non-orthogonal elastoplastic constitutive model with the critical state for clay. Comput Geotech 116:103200.

    Article  Google Scholar 

  32. 32.

    Liang J, Lu D, Du X, Wu W, Ma C (2020) Non-orthogonal elastoplastic constitutive model for sand with dilatancy. Comput Geotech 118:103329.

    Article  Google Scholar 

  33. 33.

    Sun Y, Sumelka W, Gao Y (2020) Advantages and limitations of an α-plasticity model for sand. Acta Geotech 15:1423–1437.

    Article  Google Scholar 

  34. 34.

    Sun Y, Wichtmann T, Sumelka W, Kan M (2020) Karlsruhe fine sand under monotonic and cyclic loads: modelling and validation. Soil Dyn Earthq Eng 133:106119.

    Article  Google Scholar 

  35. 35.

    Hardin BO, Blandford GE (1989) Elasticity of particulate materials. J Geotech Eng 115(6):788–805.

    Article  Google Scholar 

  36. 36.

    Pastor M, Zienkiewicz OC, Chan AHC (1990) Generalized plasticity and the modelling of soil behaviour. Int J Numer Anal Methods Geomech 14(3):151–190.

    Article  MATH  Google Scholar 

  37. 37.

    Sun Y, Nimbalkar S (2019) Stress-fractional soil model with reduced elastic region. Soils Found 59(6):2007–2023.

    Article  Google Scholar 

  38. 38.

    Sun Y, Zheng C (2019) Fractional-order modelling of state-dependent non-associated behaviour of soil without using state variable and plastic potential. Adv Differ Equ 2019(83):1–18.

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Schofield A, Wroth P (1968) Critical state soil mechanics. McGraw-Hill London, New York

    Google Scholar 

  40. 40.

    Wang ZL, Dafalias YF, Shen CK (1990) Bounding surface hypoplasticity model for sand. J Eng Mech 116(5):983–1001

    Article  Google Scholar 

  41. 41.

    Tafili M (2020) Triantafyllidis T state-dependent dilatancy of soils: experimental evidence and constitutive modeling. In: Triantafyllidis T (ed) Recent developments of soil mechanics and geotechnics in theory and practice. Springer, Cham, pp 54–84

    Chapter  Google Scholar 

  42. 42.

    Yu HS, Khong CD, Wang J, Zhang G (2005) Experimental evaluation and extension of a simple critical state model for sand. Granul Matter 7(4):213–225.

    Article  Google Scholar 

  43. 43.

    Gens A (1982) Stress–strain and strength of a low plasticity clay. Imperial College, London University, London

    Google Scholar 

  44. 44.

    Ladd CC, Varallyay J (1965) The influence of the stress system on the behaviour of saturated clays during undrained shear (trans: Engineering DoC). MIT, Cambridge

    Google Scholar 

  45. 45.

    Biarez J, Hicher PY (1994) Elementary mechanics of soil behaviour. Balkema, Rotterdam

    Google Scholar 

  46. 46.

    Zervoyannis C (1982) Étude synthétique des propriétés mécaniques des argiles saturées et des sables sur chemin oedométrique et triaxial de révolution. École centrale Paris, Paris

    Google Scholar 

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

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

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:

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

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).

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