Skip to main content
Springer Nature Link
Log in

Fractional modeling of Pasternak-type viscoelastic foundation

  • Published:
Mechanics of Time-Dependent Materials Aims and scope Submit manuscript

Abstract

In this paper, we propose a fractional Pasternak-type foundation model to characterize the time-dependent properties of the viscoelastic foundation. With varying fractional orders, the proposed model can govern the traditional Winkler model, Pasternak model, and viscoelastic model. We take the four-edge simply supported rectangular thin plate as an example to analyze the viscoelastic foundation reaction, and obtain the solution of the new governing equation. Theoretical results show that the fractional order has a dramatic influence on the deflection and bending moment. It can be further concluded that the softer foundation will become more time-dependent. Subsequently, the difference between fractional Pasternak-type and Winkler foundation model is presented in this work. The existence of constrained boundary is found to definitely affect deflection and bending moment. Such phenomenon, known as the wall effect, is deeply discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Atanackovic, T.M., Janev, M., Konjik, S., Pilipovic, S., Zorica, D.: Vibrations of an elastic rod on a viscoelastic foundation of complex fractional Kelvin–Voigt type. Meccanica 50, 1679–1692 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Bagley, R.L., Torvik, J.: Fractional calculus—a different approach to the analysis of viscoelastically damped structures. AIAA J. 21, 741–748 (1983)

    Article  MATH  Google Scholar 

  • Blair, G.S., Caffyn, J.E.: An application of the theory of quasi-properties to the treatment of anomalous strain–stress relations. Philos. Mag. 40, 80–94 (1949)

    Article  MATH  Google Scholar 

  • Chen, W., Holm, S.: Modified Szabo’s wave equation models for lossy media obeying frequency power law. J. Acoust. Soc. Am. 114, 2570–2574 (2003)

    Article  Google Scholar 

  • Chen, Y.H., Huang, Y.H.: Dynamic stiffness of infinite Timoshenko beam on viscoelastic foundation in moving coordinate. Int. J. Numer. Methods Eng. 48, 1–18 (2000)

    Article  MATH  Google Scholar 

  • Chen, Y.H., Huang, Y.H., Shih, C.T.: Response of an infinite Timoshenko beam on a viscoelastic foundation to a harmonic moving load. J. Sound Vib. 241, 809–824 (2001)

    Article  Google Scholar 

  • Chen, W., Hu, S., Cai, W.: A causal fractional derivative model for acoustic wave propagation in lossy media. Arch. Appl. Mech., 1–11 (2015)

  • Deng, W.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47, 204–226 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Di Paola, M., Marino, F., Zingales, M.: A generalized model of elastic foundation based on long-range interactions: integral and fractional model. Int. J. Solids Struct. 46, 3124–3137 (2009)

    Article  MATH  Google Scholar 

  • Filonenko-Borodich, M.M.: Some approximate theories of elastic foundations. Uchenie Zapiski Moskovskogo Gosudarstvennogo Universiteta. Mekhanica 46, 3–15 (1940)

    Google Scholar 

  • Gemant, A.: A method of analyzing experimental results obtained from elasto-viscous bodies. J. Appl. Phys. 7, 311–317 (1936)

    Google Scholar 

  • Hetényi, M.: A general solution for the bending of beams on an elastic foundation of arbitrary continuity. J. Appl. Phys. 21, 55–58 (1950)

    Article  MATH  Google Scholar 

  • Hu, S., Chen, W., Gou, X.: Modal analysis of fractional derivative damping model of frequency-dependent viscoelastic soft matter. Adv. Vib. Eng. 10, 187–196 (2011)

    Google Scholar 

  • Li, C., Zeng, F.: The finite difference methods for fractional ordinary differential equations. Numer. Funct. Anal. Optim. 34, 149–179 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific, Singapore (2010)

    Book  MATH  Google Scholar 

  • Näsholm, S.P., Holm, S.: On a fractional Zener elastic wave equation. Fract. Calc. Appl. Anal. 16, 26–50 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Meerschaert, M.M., Scheffler, H.P., Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211, 249–261 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Metrikine, A.V., Dieterman, H.A.: Instability of vibrations of a mass moving uniformly along an axially compressed beam on a viscoelastic foundation. J. Sound Vib. 201, 567–576 (1997)

    Article  MATH  Google Scholar 

  • Moreau, X., Ramus-Serment, C., Oustaloup, A.: Fractional differentiation in passive vibration control. Nonlinear Dyn. 29, 343–362 (2002)

    Article  MATH  Google Scholar 

  • Kerr, A.D.: Elastic and viscoelastic foundation models. J. Appl. Mech. 31, 491–498 (1964)

    Article  MATH  Google Scholar 

  • Pasternak, P.L.: On a New Method of Analysis of an Elastic Foundation by Means of Two Foundation Constants. Gosuderevstvennoe Izdatlesvo Literaturi po Stroitelstvu i Arkihitekture, Moscow, USSR (1954)

    Google Scholar 

  • Pister, K.S.: Viscoelastic plate on a viscoelastic foundation. J. Eng. Mech. Div. 87, 43–54 (1961)

    Google Scholar 

  • Podlubny, I.: Fractional Differential Equations. Academic Press, London (1999)

    MATH  Google Scholar 

  • Reissner, E.: A note on deflection of plates on a viscoelastic foundation. J. Appl. Phys. 80, 144–145 (1958)

    MATH  Google Scholar 

  • Spanos, P.D., Zeldin, B.A.: Random vibration of systems with frequency-dependent parameters or fractional derivatives. J. Eng. Mech. 123, 290–292 (1997)

    Article  Google Scholar 

  • Sun, L.: A closed-form solution of a Bernoulli–Euler beam on a viscoelastic foundation under harmonic line loads. J. Sound Vib. 242, 619–627 (2001)

    Article  Google Scholar 

  • Sun, H., Chen, W., Chen, Y.: Variable-order fractional differential operators in anomalous diffusion modeling. Physica A 388, 4586–4592 (2009)

    Article  Google Scholar 

  • Sun, H., Chen, W., Sheng, H., Chen, Y.: On mean square displacement behaviors of anomalous diffusions with variable and random orders. Phys. Lett. A 374, 906–910 (2010)

    Article  MATH  Google Scholar 

  • Szabo, T.L.: Time domain wave equations for lossy media obeying a frequency power law. J. Acoust. Soc. Am. 96, 491–500 (1994)

    Article  Google Scholar 

  • Szabo, T.L., Wu, J.: A model for longitudinal and shear wave propagation in viscoelastic media. J. Acoust. Soc. Am. 107, 2437–2446 (2000)

    Article  Google Scholar 

  • Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw-Hill, New York (1959)

    MATH  Google Scholar 

  • Treeby, B.E., Cox, B.: Modeling power law absorption and dispersion in viscoelastic solids using a split-field and the fractional Laplacian. J. Acoust. Soc. Am. 136, 1499–1510 (2014)

    Article  Google Scholar 

  • Vlasov, V.Z.: Beams, Plates and Shells on Elastic Foundations. Israel Program for Scientific Translations, Jerusalem (1966)

    MATH  Google Scholar 

  • Wismer, M.G.: Finite element analysis of broadband acoustic pulses through inhomogeneous media with power law attenuation. J. Acoust. Soc. Am. 120, 3493–3502 (2006)

    Article  Google Scholar 

  • Yin, D.S., Li, Y.Q., Wu, H., Duan, X.M.: Fractional description of mechanical property evolution of soft soils during creep. Water Sci. Eng. 6, 446–455 (2013)

    Google Scholar 

  • Yin, D.S., Duan, X., Zhou, X., Li, Y.: Time-based fractional longitudinal–transverse strain model for viscoelastic solids. Mech. Time-Depend. Mater. 18, 329–337 (2014)

    Article  Google Scholar 

  • Zhang, C., Zhu, H., Shi, B., Liu, L.: Theoretical investigation of interaction between a rectangular plate and fractional viscoelastic foundation. J. Rock Mech. Geotech. Eng. 6, 373–379 (2014)

    Article  Google Scholar 

  • Zaman, M., Taheri, M.R., Alvappillai, A.: Dynamic response of a thick plate on viscoelastic foundation to moving loads. Int. J. Numer. Anal. Methods 15, 627–647 (1991)

    Article  Google Scholar 

  • Zhu, H.H., Liu, L.C., Pei, H.F., Shi, B.: Settlement analysis of viscoelastic foundation under vertical line load using a fractional Kelvin–Voigt model. Geomech. Eng. 4, 67–78 (2012)

    Article  Google Scholar 

  • Zhu, T., Harris, J.M.: Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians. Geophysics 79, T105–T116 (2014)

    Article  Google Scholar 

Download references

Acknowledgements

The work described in this paper was supported by the National Natural Science Foundation of China (no. 11302069, 11372097, 11402075), the 111 project (Grant no. B12032).

Author information

Authors and Affiliations

  1. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, 210098, China

    Wei Cai, Wen Chen & Wenxiang Xu

Authors
  1. Wei Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wen Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenxiang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Wen Chen or Wenxiang Xu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cai, W., Chen, W. & Xu, W. Fractional modeling of Pasternak-type viscoelastic foundation. Mech Time-Depend Mater 21, 119–131 (2017). https://doi.org/10.1007/s11043-016-9321-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11043-016-9321-0

Keywords

Search

Navigation