Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Fractal Viscoelastic Models

  • Wen Chen
  • Wei Cai
  • Hongguang Sun
  • Yingjie Liang
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_78-1

Synonyms

Definition

Introduction of the Hausdorff derivative, a new kind of methodology, for modeling viscoelastic behaviors.

Introduction

Viscoelastic Behaviors of Materials

A wide range of materials, such as rubber, soft soil, blood, colloid, and polymer in the real world, are observed to simultaneously exhibit both the elastic and viscous behaviors. It is well known that the stress, respectively, in purely elastic and viscous process is linear to the strain and the rate of strain. In contrast, the viscoelastic stress response of viscoelastic materials is dependent on time and strain rate, leading to complex behaviors of creep and relaxation in the case of certain stress or constraint. Such power-law responses are usually considered as memorial behaviors, in other words, history-dependent process.

In addition, some viscoelastic materials present certain abnormal properties in the frequency domain, such as frequency-dependent damping and energy...

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

References

  1. Bagley RL (1989) Power law and fractional calculus model of viscoelasticity. AIAA J 27:1412–1417CrossRefGoogle Scholar
  2. Balankin AS, Elizarraraz BE (2012) Map of fluid flow in fractal porous medium into fractal continuum flow. Phys Rev E 85:056314CrossRefGoogle Scholar
  3. Blair GS, Caffyn J (1949) VI. An application of the theory of quasi-properties to the treatment of anomalous strain-stress relations. Lond Edinb Dublin Philos Mag J Sci 40:80–94CrossRefGoogle Scholar
  4. Bland DR (1960) The theory of linear viscoelasticity, international series of monographs on pure and applied mathematics, vol 10. Pergamon, OxfordGoogle Scholar
  5. Cai W, Chen W, Xu W (2016) Characterizing the creep of viscoelastic materials by fractal derivative models. Int J Nonlin Mech 87:58–63CrossRefGoogle Scholar
  6. Cai W, Chen W, Wang F (2018) Three-dimensional Hausdorff derivative diffusion model for isotropic/anisotropic fractal porous media. Therm Sci 22:S1–S6CrossRefGoogle Scholar
  7. Chen W (2006) Time–space fabric underlying anomalous diffusion. Chaos Soliton Fract 28:923–929CrossRefGoogle Scholar
  8. Chen W (2016) Non-power-function metric: a generalized fractal. viXra Preprint. http://viXra.org/abs/1612.0409
  9. Chen W (2017) Fractal geometric theory of Hausdorff calculus and fractional calculus models. Comput Aided E 27:1–6 (in Chinese)Google Scholar
  10. Chen W, Liang Y (2017) New methodologies in fractional and fractal derivatives modeling. Chaos Soliton Fract 102:72–77MathSciNetCrossRefGoogle Scholar
  11. Chen W, Sun H, Zhang X, Korošak D (2010) Anomalous diffusion modeling by fractal and fractional derivatives. Comput Math Appl 59:1754–1758MathSciNetCrossRefGoogle Scholar
  12. Chen W, Liang Y, Hei X (2016) Structural derivative based on inverse Mittag-Leffler function for modeling ultraslow diffusion. Fract Calc Appl Anal 19:1250MathSciNetzbMATHGoogle Scholar
  13. Christensen RM (1971) Theory of viscoelasticity. Academic, New YorkGoogle Scholar
  14. Garas VY, Kurtis KE, Kahn LF (2012) Creep of UHPC in tension and compression: effect of thermal treatment. Cem Concr Compos 34:493–502CrossRefGoogle Scholar
  15. Garra R, Mainardi F, Spada G (2017) A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus. Chaos Soliton Fract 102:333–338MathSciNetCrossRefGoogle Scholar
  16. Heki K, Miyazaki S, Tsuji H (1997) Silent fault slip following an interplate thrust earthquake at the Japan trench. Nature 386:595–598CrossRefGoogle Scholar
  17. Jaishankar A, McKinley GH (2012) Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations. Proc Roy Soc A-Math Phys 469(2149):20120284MathSciNetCrossRefGoogle Scholar
  18. Jumarie G (2005) On the representation of fractional Brownian motion as an integral with respect to. Appl Math Lett 18:739–748MathSciNetCrossRefGoogle Scholar
  19. Li J, Ostoja-Starzewski M (2009) Fractal solids, product measures and fractional wave equations. Proc Roy Soc A-Math Phys 465:2521–2536MathSciNetCrossRefGoogle Scholar
  20. Li J, Ostoja-Starzewski M (2013) Comment on “hydrodynamics of fractal continuum flow” and “map of fluid flow in fractal porous medium into fractal continuum flow”. Phys Rev E 88:057001CrossRefGoogle Scholar
  21. Liang Y, Allen QY, Chen W, Gatto RG, Colon-Perez L, Mareci TH, Magin RL (2016) A fractal derivative model for the characterization of anomalous diffusion in magnetic resonance imaging. Commun Nonlinear Sci 39:529–537MathSciNetCrossRefGoogle Scholar
  22. Lin G (2016) Instantaneous signal attenuation method for analysis of PFG fractional diffusions. J Magn Reson 269:36–49CrossRefGoogle Scholar
  23. Lin G (2017) Analyzing signal attenuation in PFG anomalous diffusion via a modified Gaussian phase distribution approximation based on fractal derivative model. Physica A 467:277–288MathSciNetCrossRefGoogle Scholar
  24. Liu X, Sun HG, Lazarević MP, Fu Z (2017) A variable-order fractal derivative model for anomalous diffusion. Therm Sci 21:51–59CrossRefGoogle Scholar
  25. Mainardi F (1996) Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Soliton Fract 7:1461–1477MathSciNetCrossRefGoogle Scholar
  26. Mainardi F (2010) Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. Imperial College Press, LondonGoogle Scholar
  27. Mandare P, Winter HH (2006) Ultraslow dynamics in asymmetric block copolymers with nanospherical domains. Colloid Polym Sci 284:1203–1210CrossRefGoogle Scholar
  28. Martinez-Garcia JC, Rzoska SJ, Drozd-Rzoska A, Martinez-Garcia J (2013) A universal description of ultraslow glass dynamics. Nat Commun 4:1823CrossRefGoogle Scholar
  29. Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77MathSciNetCrossRefGoogle Scholar
  30. Milovanov AV, Rypdal K, Rasmussen JJ (2007) Stretched exponential relaxation and ac universality in disordered dielectrics. Phys Rev B 76:104201CrossRefGoogle Scholar
  31. Paulsen JD, Nagel SR (2017) A model for approximately stretched-exponential relaxation with continuously varying stretching exponents. J Stat Phys 167:1–14MathSciNetCrossRefGoogle Scholar
  32. Reyes-Marambio J, Moser F, Gana F, Severino B, Calderón-Muñoz WR, Palma-Behnke R, Estevez PA, Orchard M, Cortés M (2016) A fractal time thermal model for predicting the surface temperature of air-cooled cylindrical Li-ion cells based on experimental measurements. J Power Sources 306:636–645CrossRefGoogle Scholar
  33. Rossikhin YA (2010) Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids. Appl Mech Rev 63:010701CrossRefGoogle Scholar
  34. Sun H, Meerschaert MM, Zhang Y, Zhu J, Chen W (2013) A fractal Richards’ equation to capture the non-Boltzmann scaling of water transport in unsaturated media. Adv Water Resour 52:292–295CrossRefGoogle Scholar
  35. Tarasov VE (2014) Anisotropic fractal media by vector calculus in non-integer dimensional space. J Math Phys 55:083510MathSciNetCrossRefGoogle Scholar
  36. Zähle M (1997) Fractional differentiation in the self-affine case. V-the local degree of differentiability. Math Nachr 185:279–306MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Wen Chen
    • 1
  • Wei Cai
    • 2
  • Hongguang Sun
    • 1
  • Yingjie Liang
    • 1
  1. 1.State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Institute of Soft Matter Mechanics, College of Mechanics and MaterialsHohai UniversityNanjingChina
  2. 2.College of Mechanical and Electrical EngineeringHohai UniversityChangzhouChina

Section editors and affiliations

  • Yury A. Rossikhin (deceased)
    • 1
  • Marina V. Shitikova
    • 2
  1. 1.Research Center on Dynamics of Solids and StructuresVoronezh State Technical UniversityVoronezhRussia
  2. 2.Research Center on Dynamics of Solids and StructuresVoronezh State Technical UniversityVoronezhRussia