Fractional modeling of Pasternak-type viscoelastic foundation

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.

[1]  A. Gemant,et al.  A Method of Analyzing Experimental Results Obtained from Elasto‐Viscous Bodies , 1936 .

[2]  G. W. Scott Blair,et al.  VI. An application of the theory of quasi-properties to the treatment of anomalous strain-stress relations , 1949 .

[3]  M. Hetényi A General Solution for the Bending of Beams on an Elastic Foundation of Arbitrary Continuity , 1950 .

[4]  P. L. Pasternak On a new method of analysis of an elastic foundation by means of two foundation constants , 1954 .

[5]  S. Timoshenko,et al.  THEORY OF PLATES AND SHELLS , 1959 .

[6]  K. Pister Viscoelastic Plate on a Viscoelastic Foundation , 1961 .

[7]  Arnold D. Kerr,et al.  Elastic and Viscoelastic Foundation Models , 1964 .

[8]  Peter J. Torvik,et al.  Fractional calculus-a di erent approach to the analysis of viscoelastically damped structures , 1983 .

[9]  Musharraf Zaman,et al.  Dynamic response of a thick plate on viscoelastic foundation to moving loads , 1991 .

[10]  Thomas L. Szabo,et al.  Time domain wave equations for lossy media obeying a frequency power law , 1994 .

[11]  Pol D. Spanos,et al.  Random Vibration of Systems with Frequency-Dependent Parameters or Fractional Derivatives , 1997 .

[12]  Andrei V. Metrikine,et al.  INSTABILITY OF VIBRATIONS OF A MASS MOVING UNIFORMLY ALONG AN AXIALLY COMPRESSED BEAM ON A VISCOELASTIC FOUNDATION , 1997 .

[13]  I. Podlubny Fractional differential equations , 1998 .

[14]  T. Szabo,et al.  A model for longitudinal and shear wave propagation in viscoelastic media , 2000, The Journal of the Acoustical Society of America.

[15]  Yung-Hsiang Chen,et al.  Dynamic stiffness of infinite Timoshenko beam on viscoelastic foundation in moving co-ordinate , 2000 .

[16]  Y.-H. Huang,et al.  RESPONSE OF AN INFINITE TIMOSHENKO BEAM ON A VISCOELASTIC FOUNDATION TO A HARMONIC MOVING LOAD , 2001 .

[17]  Lu Sun,et al.  A CLOSED-FORM SOLUTION OF A BERNOULLI-EULER BEAM ON A VISCOELASTIC FOUNDATION UNDER HARMONIC LINE LOADS , 2001 .

[18]  A. Oustaloup,et al.  Fractional Differentiation in Passive Vibration Control , 2002 .

[19]  S Holm,et al.  Modified Szabo's wave equation models for lossy media obeying frequency power law. , 2003, The Journal of the Acoustical Society of America.

[20]  M. Meerschaert,et al.  Finite difference methods for two-dimensional fractional dispersion equation , 2006 .

[21]  M. Wismer,et al.  Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation. , 2006, The Journal of the Acoustical Society of America.

[22]  Weihua Deng,et al.  Finite Element Method for the Space and Time Fractional Fokker-Planck Equation , 2008, SIAM J. Numer. Anal..

[23]  M. Di Paola,et al.  A generalized model of elastic foundation based on long-range interactions: Integral and fractional model , 2009 .

[24]  Y. Chen,et al.  Variable-order fractional differential operators in anomalous diffusion modeling , 2009 .

[25]  Hu Sheng,et al.  On mean square displacement behaviors of anomalous diffusions with variable and random orders , 2010 .

[26]  F. Mainardi Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .

[27]  Shuai Hu,et al.  Modal Analysis of Fractional Derivative Damping Model of Frequency-Dependent Viscoelastic Soft Matter , 2011 .

[28]  Bin Shi,et al.  Settlement analysis of viscoelastic foundation under vertical line load using a fractional Kelvin-Voigt model , 2012 .

[29]  Sverre Holm,et al.  On a fractional Zener elastic wave equation , 2012 .

[30]  Fanhai Zeng,et al.  The Finite Difference Methods for Fractional Ordinary Differential Equations , 2013 .

[31]  Deshun Yin,et al.  Fractional description of mechanical property evolution of soft soils during creep , 2013 .

[32]  Tieyuan Zhu,et al.  Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians , 2014 .

[33]  B T Cox,et al.  Modeling power law absorption and dispersion in viscoelastic solids using a split-field and the fractional Laplacian. , 2014, The Journal of the Acoustical Society of America.

[34]  Cheng-Cheng Zhang,et al.  Theoretical investigation of interaction between a rectangular plate and fractional viscoelastic foundation , 2014 .

[35]  Xiaomeng Duan,et al.  Time-based fractional longitudinal–transverse strain model for viscoelastic solids , 2014 .

[36]  Teodor M. Atanackovic,et al.  Vibrations of an elastic rod on a viscoelastic foundation of complex fractional Kelvin–Voigt type , 2015 .

[37]  Wen Chen,et al.  A causal fractional derivative model for acoustic wave propagation in lossy media , 2016 .