One-dimensional rheological consolidation analysis of saturated clay using fractional order Kelvin’s model

LIU Zhong-yu1 YANG Qiang1

(1.School of Civil Engineering, Zhengzhou University, Zhengzhou, Henan Province, China 450001)

【Abstract】The Kelvin model is modified by the spring-pot element based on the Caputo fractional derivative to describe the one-dimensional rheological constitutive relation of saturated clay. The rheological consolidation equation is derived with the assumptions proposed in Terzaghi’s one-dimensional consolidation theory of saturated soil. The numerical analysis is performed by using the Laplace transform and its numerical inversion based on Fourier series expansions. To verify this method, the numerical solutions by the present method for the cases of the integer order derivative are compared with their analytical solutions. The applicability of the modified Kelvin model is verified by the simulation results of one-dimensional rheological consolidation test in the literature. The influences of the fractional order and the coefficient of viscosity of spring-pot component on the rheological consolidation process of soil are investigated. The calculated results show that the dissipation rate of pore water pressure is greater than that based on Terzaghi’s one-dimensional consolidation theory in a long period of consolidation and then less than the latter in the final stage of consolidation. Furthermore, the settlement rate of foundation is always less than that based on Terzaghi’s theory in the process. Overall, the settlement of ground lags the dissipation of pore water pressure. This phenomenon is obvious and it takes more time to achieve the final settlement with the decrease in the fractional order or the increase in the coefficient viscosity.

【Keywords】 rheological consolidation; fractional order derivative; Kelvin model; Laplace transform; pore water pressure; degree of consolidation;


【Funds】 National Natural Science Foundation of China (51578511)

Download this article


    [1] LIU Jun-xin, YANG Chun-he, XIE Qiang, et al. Settlement mechanism of unsaturated red layers embankment based on rheology and consolidation theories [J]. Rock and Soil Mechanics, 2015, 36(5): 1295–1305 (in Chinese).

    [2] TAYLOR D W, MERCHANT W. A theory of clay consolidation accounting for secondary compression [J]. Journal of Mathematics and Physics, 1940, 19(3): 167–185.

    [3] TAN Tjong-kie. One dimensional problems of consolidation and secondary time effects [J]. Chinese Journal of Civil Engineering, 1958, 5(1): 1–9 (in Chinese).

    [4] ZHAO Wei-bing. One-dimensional soil consolidation theory of saturated soil based on generalized Voigt model and its application [J]. Chinese Journal of Geotechnical Engineering, 1989, 11(5): 78–85 (in Chinese).

    [5] CHEN Xiao-ping, BAI Shi-wei. Research on creep-consolidation characteristics and calculating model of soft soil [J]. Chinese Journal of Rock Mechanics and Engineering, 2003, 22(5): 728–734 (in Chinese).

    [6] CAI Yuan-qiang, LIANG Xu, ZHENG Zhao-feng, et al. One-dimensional consolidation of viscoelastic soil layer with semi-permeable boundaries under cyclic loadings [J]. Chinese Journal of Civil Engineering, 2003, 36(8): 86–90 (in Chinese).

    [7] PAN Xiao-dong, CAI Yuan-qiang, XU Chang-jie, et al. One-dimensional consolidation of viscoelastic soil layer under cyclic loading [J]. Rock and Soil Mechanics, 2006, 27(2): 272–276 (in Chinese).

    [8] LI Xi-bin, XIE Kang-he, WANG Kui-hua, et al. Analytical solution of 1d visco-elastic consolidation of soils with impeded boundaries under cyclic loadings [J]. Engineering Mechanics, 2004, 21(5): 103–108 (in Chinese).

    [9] LI Xi-bin. Theoritical and experimental studies on rheological consolidation of soft soil [D]. Hangzhou: Zhejiang University, 2005 (in Chinese).

    [10] LIU Jia-cai, ZHAO Wei-bing, ZAI Jin-min, et al. One dimensional consolidation of viscoelastic soil layer [J]. Rock and Soil Mechanics, 2007, 28(4): 744–752 (in Chinese).

    [11] WANG Shao-mei, XIA Sen-wei, JIANG Jun. Study on one-dimensional consolidation behavior of saturated clay under cyclic loading [J]. Rock and Soil Mechanics, 2008, 29(2): 470–473 (in Chinese).

    [12] LIU Zhong-yu, YAN Fu-you, WANG Xi-jun. One-dimensional rheological consolidation analysis of saturated clay considering non-darcy flow [J]. Chinese Journal of Rock Mechanics and Engineering, 2013, 32(9): 1937–1944 (in Chinese).

    [13] ZHANG Chun-yuan. Viscoelastic Fracture Mechanics [M]. Beijing: Science Press, 2006.

    [14] BAGLEY R L, TORVIK P J. A theoretical basis for the application of fractional calculus to viscoelasticity [J]. Journal of Rheology, 1983, 27(3): 201–230.

    [15] BAGLEY R L, TORVIK P J. On the fractional calculus model of viscoelasticity behavior [J]. Journal of Rheology, 1986, 30(1): 133–155.

    [16] KOELLER R C. Applications of fractional calculus to the theory of viscoelasticity [J]. Journal of Applied Mechanics, 1984, 51(2): 299–307.

    [17] LIU Lin-chao, YAN Qi-fang, SUN Hai-zhong. Study on model of rheological property of soft clay [J]. Rock and Soil Mechanics, 2006, 27 (Supp. 1): 214–217 (in Chinese).

    [18] SUN Hai-zhong, ZHANG Wei. Analysis of soft soil with viscoelastic fractional derivative Kelvin model [J]. Rock and Soil Mechanics, 2007, 28(9): 1983–1986 (in Chinese).

    [19] YIN De-shun, REN Jun-juan, HE Cheng-liang, et al. New rheological model element for geomaterials [J]. Chinese Journal of Rock Mechanics and Engineering, 2007, 29(9): 1899–1903 (in Chinese).

    [20] HE Li-jun, KONG Ling-wei, WU Wen-jun, et al. Adescription of creep model for soft soil with fractional derivative [J]. Rock and Soil Mechanics, 2011, 32(Supp. 1): 239–249 (in Chinese).

    [21] LI Rui-duo, YUE Jin-chao. Nonlinear rheological constitute of soft soil based on fractional order derivative theory [J]. Journal of Basis Science and Engineering, 2014, 22(5): 856–863 (in Chinese).

    [22] YIN Jian-wu, KUANG Du-min, WANG Zhi-chao. Consolidation creep characteristics of compacted clay and its parameters analysis of rheological constitutive model based on fractional calculus [J]. Journal of Hunan University of Science & Technology (Nature Science Edition), 2015, 30(3): 46–51 (in Chinese).

    [23] DUBNER H, ABATE J. Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform [J]. Journal of Associating for Computing Machinery, 1968, 15(1): 115–123.

    [24] DUBINE F. Numerical inversion of Laplace transforms: an efficient improvement to Dubner & Abate’s method [J]. Compute Journal, 1974, 17(4): 371–376.

    [25] CRUMP K S. Numerical inversion of Laplace transforms using a Fourier series approximation [J]. Journal of Association for Computing Machinery, 1976, 23(1): 89–96.

This Article



Vol 38, No. 12, Pages 3680-3687+3697

December 2017


Article Outline


  • 1 Introduction
  • 2 Kelvin model based on Caputo fractional derivative
  • 3 Rheological consolidation equation based on modified Kelvin model and its solution
  • 4 Laplace transform and its numerical inversion
  • 5 Test verification of rheological consolidation model
  • 6 Parameter analysis
  • 7 Conclusions
  • References