Power-Law Wave Equations from Constitutive Equations

  • Sverre HolmEmail author


The exponential response has been the typical medium response in the first part of this book. Examples are the creep compliances of the Kelvin–Voigt and Zener models ( 3.30) and ( 3.30) as well as the relaxation responses of the Maxwell and Zener models (( 3.37) and ( 3.45)). But in many complex media this model is too simple, and power laws are observed instead, which may both be in time-domain responses and in the frequency-domain. As the Fourier transform of a temporal power law is itself a power law, these observations are equivalent. The first section therefore starts by reviewing empirical observations of power laws in the frequency and time domains.


  1. B. Alouache, D. Laux, A. Hamitouche, K. Bachari, T. Boutkedjirt, Ultrasonic characterization of edible oils using a generalized fractional model. Appl. Acoust. 131, 70–78 (2018)CrossRefGoogle Scholar
  2. R.L. Bagley, P.J. Torvik, Fractional calculus - a different approach to the analysis of viscoelastically damped structures. AIAA J. 21(5), 741–748 (1983)ADSzbMATHCrossRefGoogle Scholar
  3. R.L. Bagley, P.J. Torvik, On the fractional calculus model of viscoelastic behavior. J. Rheol. 30(1), 133–155 (1986)ADSzbMATHCrossRefGoogle Scholar
  4. G.S. Blair, M. Reiner, The rheological law underlying the Nutting equation. Appl. Sci. Res. 2(1), 225–234 (1951)zbMATHCrossRefGoogle Scholar
  5. M. Caputo, F. Mainardi, A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91(1), 134–147 (1971)ADSzbMATHCrossRefGoogle Scholar
  6. M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. Int. 13(5), 529–539 (1967)ADSCrossRefGoogle Scholar
  7. J.M. Carcione, F.J. Sanchez-Sesma, F. Luzón, J.J.P. Gavilán, Theory and simulation of time-fractional fluid diffusion in porous media. J.  Phys. A: Math. Theor. 46(34), 345501 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  8. E.L. Carstensen, K. Li, H.P. Schwan, Determination of the acoustic properties of blood and its components. JASA 25(2), 286–289 (1953)CrossRefGoogle Scholar
  9. K.S. Cole, Permeability and impermeability of cell membranes for ions, in Cold Spring Harbor Symposia on Quantitative Biology, vol. 8 (Cold Spring Harbor Laboratory Press, 1940), pp. 110–122Google Scholar
  10. K.S. Cole, R.H. Cole, Dispersion and absorption in dielectrics I. Alternating current characteristics. J. Chem. Phys. 9(4), 341–351 (1941)Google Scholar
  11. C. Coussot, S. Kalyanam, R. Yapp, M. Insana, Fractional derivative models for ultrasonic characterization of polymer and breast tissue viscoelasticity. IEEE Trans. Ultrason. Ferroelectr., Freq. Control 56(4), 715–725 (2009)CrossRefGoogle Scholar
  12. D.O. Craiem, F.J. Rojo, J.M. Atienza, G.V. Guinea, R.L. Armentano, Fractional calculus applied to model arterial viscoelasticity. Latin. Am. Appl. Res. 38(2), 141–145 (2008)Google Scholar
  13. G.B. Davis, M. Kohandel, S. Sivaloganathan, G. Tenti, The constitutive properties of the brain paraenchyma. Part 2. Fractional derivative approach. Med. Eng. Phys. 28(5), 455–459 (2006)CrossRefGoogle Scholar
  14. L. Deng, X. Trepat, J.P. Butler, E. Millet, K.G. Morgan, D.A. Weitz, J.J. Fredberg, Fast and slow dynamics of the cytoskeleton. Nat. Mat. 5(8), 636 (2006)CrossRefGoogle Scholar
  15. V.D. Djordjević, J. Jarić, B. Fabry, J.J. Fredberg, D. Stamenović, Fractional derivatives embody essential features of cell rheological behavior. Ann. Biomed. Eng. 31, 692–699 (2003)CrossRefGoogle Scholar
  16. F.A. Duck, Physical Properties of Tissues: a Comprehensive Reference Book (Academic, New Jersey, 2012)Google Scholar
  17. B. Fabry, G.N. Maksym, J.P. Butler, M. Glogauer, D. Navajas, J.J. Fredberg, Scaling the microrheology of living cells. Phys. Rev. Lett. 87(14), 148102 (2001)ADSCrossRefGoogle Scholar
  18. R. Garrappa, F. Mainardi, G. Maione, Models of dielectric relaxation based on completely monotone functions. Fract. Calc. Appl. Anal. 19(5), 1105–1160 (2016)Google Scholar
  19. A. Gemant, A method of analyzing experimental results obtained from elasto-viscous bodies. Physics 7(8), 311–317 (1936)ADSCrossRefGoogle Scholar
  20. W.G. Glöckle, T.F. Nonnenmacher, Fractional integral operators and Fox functions in the theory of viscoelasticity. Macromolecules 24(24), 6426–6434 (1991)ADSCrossRefGoogle Scholar
  21. N.M. Grahovac, M. Zigic, Modelling of the hamstring muscle group by use of fractional derivatives. Comput. Math. Appl. 59(5), 1695–1700 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  22. S. Grimnes, Ø.G. Martinsen, Cole electrical impedance model-a critique and an alternative. IEEE Trans. Biomed. Eng. 52(1), 132–135 (2005)CrossRefGoogle Scholar
  23. R. Hill, A. Jonscher, The dielectric behaviour of condensed matter and its many-body interpretation. Contemp. Phys. 24(1), 75–110 (1983)ADSCrossRefGoogle Scholar
  24. B.D. Hoffman, G. Massiera, K.M. Van Citters, J.C. Crocker, The consensus mechanics of cultured mammalian cells. Proc. Natl. Acad. Sci. 103(27), 10259–10264 (2006)ADSCrossRefGoogle Scholar
  25. S. Holm, S.P. Näsholm, A causal and fractional all-frequency wave equation for lossy media. J. Acoust. Soc. Am. 130(4), 2195–2202 (2011)ADSCrossRefGoogle Scholar
  26. S. Holm, R. Sinkus, A unifying fractional wave equation for compressional and shear waves. J. Acoust. Soc. Am. 127, 542–548 (2010)ADSCrossRefGoogle Scholar
  27. J.G. Liu, M.Y. Xu, Higher-order fractional constitutive equations of viscoelastic materials involving three different parameters and their relaxation and creep functions. Mech. Time-Depend. Mat. 10, 263–279 (2006)ADSCrossRefGoogle Scholar
  28. K. Jüttner, Electrochemical impedance spectroscopy (EIS) of corrosion processes on inhomogeneous surfaces. Electrochim. Acta 35(10), 1501–1508 (1990)CrossRefGoogle Scholar
  29. D. Klatt, U. Hamhaber, P. Asbach, J. Braun, I. Sack, Noninvasive assessment of the rheological behavior of human organs using multifrequency MR elastography: A study of brain and liver viscoelasticity. Phys. Med. Biol. 52(24), 7281–7294 (2007)CrossRefGoogle Scholar
  30. R. Kohlrausch, Theorie des elektrischen Rückstandes in der Leidener Flasche (Theory of the electric residue of a Leidner bottle). Ann. Phys. 167(2), 179–214 (1854)ADSCrossRefGoogle Scholar
  31. M. Kohandel, S. Sivaloganathan, G. Tenti, K. Darvish, Frequency dependence of complex moduli of brain tissue using a fractional Zener model. Phys. Med. Biol. 50(12), 2799–2805 (2005)CrossRefGoogle Scholar
  32. S. Konjik, L. Oparnica, D. Zorica, Waves in fractional Zener type viscoelastic media. J. Math. Anal. Appl. 365(1), 259–268 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  33. Y.-C. Lin, N.Y. Yao, C.P. Broedersz, H. Herrmann, F.C. MacKintosh, D.A. Weitz, Origins of elasticity in intermediate filament networks. Phys. Rev. Lett. 104(5), 058101 (2010)ADSCrossRefGoogle Scholar
  34. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelesticity: An Introduction to Mathematical Models (Imperial College Press, London, UK, 2010)zbMATHCrossRefGoogle Scholar
  35. T. Meidav, Viscoelastic properties of the standard linear solid. Geophys. Prospect. 12(1), 1365–2478 (1964)CrossRefGoogle Scholar
  36. F. Meral, T. Royston, R. Magin, Fractional calculus in viscoelasticity: An experimental study. Commun. Nonlinear Sci. 15(4), 939–945 (2010)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  37. R. Metzler, T.F. Nonnenmacher, Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials. Int. J. Plast. 19(7), 941–959 (2003)zbMATHCrossRefGoogle Scholar
  38. D. Mizuno, C. Tardin, C.F. Schmidt, F.C. MacKintosh, Nonequilibrium mechanics of active cytoskeletal networks. Science 315(5810), 370–373 (2007)ADSCrossRefGoogle Scholar
  39. S.P. Näsholm, S. Holm, On a fractional Zener elastic wave equation. Fract. Calc. Appl. Anal. 16, 26–50 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  40. R. Nigmatullin, Y.E. Ryabov, Cole-Davidson dielectric relaxation as a self-similar relaxation process. Phys. Solid State 39(1), 87–90 (1997)ADSCrossRefGoogle Scholar
  41. P. Nutting, A new general law of deformation. J. Franklin. Inst. 191(5), 679–685 (1921)CrossRefGoogle Scholar
  42. J. Ormachea, R.J. Lavarello, S.A. McAleavey, K.J. Parker, B. Castaneda, Shear wave speed measurements using crawling wave sonoelastography and single tracking location shear wave elasticity imaging for tissue characterization. IEEE Trans. Ultrason. Ferroelectr., Freq. Control 63(9), 1351–1360 (2016)CrossRefGoogle Scholar
  43. V. Pandey, S.P. Näsholm, S. Holm, Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity. Fract. Calc. Appl. Analysis. 19(2), 498–515 (2016)Google Scholar
  44. L.M. Petrovic, D.T. Spasic, T.M. Atanacković, On a mathematical model of a human root dentin. Dent. Mater. 21(2), 125–128 (2005)CrossRefGoogle Scholar
  45. H. Pollard, The completely monotonic character of the Mittag-Leffler function \(E_{-\alpha }(-x)\). Bull. Am. Math. Soc. 54(12), 1115–1116 (1948)zbMATHGoogle Scholar
  46. O. Posnansky, J. Guo, S. Hirsch, S. Papazoglou, J. Braun, I. Sack, Fractal network dimension and viscoelastic powerlaw behavior: I. A modeling approach based on a coarse-graining procedure combined with shear oscillatory rheometry. Phys. Med. Biol. 57(12), 4023–4040 (2012)ADSCrossRefGoogle Scholar
  47. T. Pritz, Analysis of four-parameter fractional derivative model of real solid materials. J. Sound. Vib. 195(1), 103–115 (1996)ADSzbMATHCrossRefGoogle Scholar
  48. T. Pritz, Loss factor peak of viscoelastic materials: Magnitude to width relations. J. Sound. Vib. 246(2), 265–280 (2001)ADSCrossRefGoogle Scholar
  49. T. Pritz, Five-parameter fractional derivative model for polymeric damping materials. J. Sound. Vib. 265(5), 935–952 (2003)ADSCrossRefGoogle Scholar
  50. S. Rogosin, F. Mainardi, George William Scott Blair-the pioneer of fractional calculus in rheology (Commun. Appl, Ind, Math, 2014)zbMATHCrossRefGoogle Scholar
  51. Y.A. Rossikhin, Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids. Appl. Mech. Rev. 63(1), 010701–1–010701–12 (2010)ADSCrossRefGoogle Scholar
  52. Y.A. Rossikhin, M.V. Shitikova, Analysis of rheological equations involving more than one fractional parameters by the use of the simplest mechanical systems based on these equations. Mech. Time-Depend. Mat. 5(2), 131–175 (2001)ADSCrossRefGoogle Scholar
  53. Y.A. Rossikhin, M.V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results. Appl. Mech. Rev. 63, 010801–1–25 (2010)ADSCrossRefGoogle Scholar
  54. I. Sack, B. Beierbach, J. Wuerfel, D. Klatt, U. Hamhaber, S. Papazoglou, P. Martus, J. Braun, The impact of aging and gender on brain viscoelasticity. NeuroImage 46(3), 652–657 (2009)CrossRefGoogle Scholar
  55. M. Sasso, G. Palmieri, D. Amodio, Application of fractional derivative models in linear viscoelastic problems. Mech. Time-Depend. Mat. 15, 367–387 (2011)ADSCrossRefGoogle Scholar
  56. R. Sinkus, S. Lambert, K.Z. Abd-Elmoniem, C. Morse, T. Heller, C. Guenthner, A.M. Ghanem, S. Holm, A.M. Gharib, Rheological determinants for simultaneous staging of hepatic fibrosis and inflammation in patients with chronic liver disease. NMR Biomed. e3956, 1–10Google Scholar
  57. D. Stamenović, Rheological behavior of mammalian cells. Cell Mol. Life Sci. 65(22), 3592–3605 (2008)CrossRefGoogle Scholar
  58. T.L. Szabo, Causal theories and data for acoustic attenuation obeying a frequency power law. J. Acoust. Soc. Am. 97, 14–24 (1995)ADSCrossRefGoogle Scholar
  59. T.L. Szabo, Diagnostic Ultrasound Imaging: Inside Out, 2nd edn. (Academic, New Jersey, 2014)Google Scholar
  60. T.L. Szabo, J. Wu, A model for longitudinal and shear wave propagation in viscoelastic media. J. Acoust. Soc. Am. 107, 2437–2446 (2000)ADSCrossRefGoogle Scholar
  61. X. Wang, D.R. Bauer, R. Witte, H. Xin, Microwave-induced thermoacoustic imaging model for potential breast cancer detection. IEEE Trans. Biomed. Eng. 59(10), 2782–2791 (2012)CrossRefGoogle Scholar
  62. B.J. West, Colloquium: Fractional calculus view of complexity: A tutorial. Rev. Mod. Phys. 86(4), 1169 (2014)ADSCrossRefGoogle Scholar
  63. S. Westerlund, Dead matter has memory!. Phys Scripta 43(2), 174 (1991)ADSMathSciNetCrossRefGoogle Scholar
  64. S. Westerlund, L. Ekstam, Capacitor theory. IEEE Trans. Dielectr. Electr. Ins. 1(5), 826–839 (1994)CrossRefGoogle Scholar
  65. M.G. Wismer, Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation. J. Acoust. Soc. Am. 120, 3493–3502 (2006)ADSCrossRefGoogle Scholar
  66. W. Zhang, S. Holm, Estimation of shear modulus in media with power law characteristics. Ultrason. 64, 170–176 (2016)CrossRefGoogle Scholar
  67. M. Zhang, B. Castaneda, Z. Wu, P. Nigwekar, J.V. Joseph, D.J. Rubens, K.J. Parker, Congruence of imaging estimators and mechanical measurements of viscoelastic properties of soft tissues. Ultrasound Med. Biol. 33(10), 1617–1631 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of InformaticsUniversity of OsloOsloNorway

Personalised recommendations