Asymptotic estimates of viscoelastic Green’s functions near the wavefront

Hanyga, Andrzej

doi:10.1090/qam/1400

Author: Andrzej Hanyga
Journal: Quart. Appl. Math. 73 (2015), 679-692
MSC (2010): Primary 74D05
DOI: https://doi.org/10.1090/qam/1400
Published electronically: September 14, 2015
MathSciNet review: 3432278
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Asymptotic behavior of viscoelastic Green’s functions near the wavefront is expressed in terms of a causal function $g(t)$ defined in Hanyga and Seredyńska (2012) in connection with the Kramers-Kronig dispersion relations. Viscoelastic Green’s functions exhibit a discontinuity at the wavefront if $g(0) < \infty$. Estimates of continuous and discontinuous viscoelastic Green’s functions near the wavefront are obtained.

References

N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871
M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Fract. Calc. Appl. Anal. 10 (2007), no. 3, 309–324. Reprinted from Pure Appl. Geophys. 91 (1971), no. 1, 134–147; With an editorial note by Virginia Kiryakova. MR 2382783

J. M. Carcione, F. Cavallini, F. Mainardi, and A. Hanyga, Time-domain seismic modeling of constant-Q wave propagation using fractional derivatives, Pure Appl. Geophys. 159 (2002), 1714–1736.

R. M. Christensen, Theory of viscoelasticity: An introduction, Academic Press, New York, 1971.

Boa-Teh Chu, Stress waves in isotropic linear viscoelastic materials. I, J. Mécanique 1 (1962), 439–462. MR 149753
W. Desch and R. C. Grimmer, Initial-boundary value problems for integro-differential equations, J. Integral Equations 10 (1985), no. 1-3, suppl., 73–97. Integro-differential evolution equations and applications (Trento, 1984). MR 831236
W. Desch and R. Grimmer, Smoothing properties of linear Volterra integro-differential equations, SIAM J. Math. Anal. 20 (1989), no. 1, 116–132. MR 977492, DOI https://doi.org/10.1137/0520009

J. M. Greenberg, The existence of steady shock waves for a class of nonlinear dissipative materials with memory, Quart. Appl. Math. 26 (1968), 27–34.

G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990. MR 1050319
Andrzej Hanyga, Wave propagation in linear viscoelastic media with completely monotonic relaxation moduli, Wave Motion 50 (2013), no. 5, 909–928. MR 3070949, DOI https://doi.org/10.1016/j.wavemoti.2013.03.002
A. Hanyga, Dispersion and attenuation for an acoustic wave equation consistent with viscoelasticity, J. Comput. Acoust. 22 (2014), no. 3, 1450006, 22. MR 3232047, DOI https://doi.org/10.1142/S0218396X14500064

A. Hanyga, Attenuation and shock waves in linear hereditary viscoelastic media. Strick-Mainardi and Jeffreys-Lomnitz-Strick creep compliances., arXiv:1401.3094 [math-ph] (2014).

A. Hanyga and M. Seredyńska, Some effects of the memory kernel singularity on wave propagation and inversion in poroelastic media, I: Forward modeling, Geophys. J. Int. 137 (1999), 319–335.

Andrzej Hanyga and M. Seredyńska, Asymptotic and exact fundamental solutions in hereditary media with singular memory kernels, Quart. Appl. Math. 60 (2002), no. 2, 213–244. MR 1900491, DOI https://doi.org/10.1090/qam/1900491
Andrzej Hanyga and Małgorzata Seredyńska, Relations between relaxation modulus and creep compliance in anisotropic linear viscoelasticity, J. Elasticity 88 (2007), no. 1, 41–61. MR 2337270, DOI https://doi.org/10.1007/s10659-007-9112-6
M. Seredyńska and Andrzej Hanyga, Relaxation, dispersion, attenuation, and finite propagation speed in viscoelastic media, J. Math. Phys. 51 (2010), no. 9, 092901, 16. MR 2742817, DOI https://doi.org/10.1063/1.3478299

S. Havriliak and S. Negami, A complex plane analysis of alpha-dispersions in some polymer systems, J. Polym. Sci. 14 (1966), 99–117.

W. J. Hrusa and M. Renardy, On wave propagation in linear viscoelasticity, Quart. Appl. Math. 43 (1985), no. 2, 237–254. MR 793532, DOI https://doi.org/10.1090/S0033-569X-1985-0793532-0
N. Jacob, Pseudo differential operators and Markov processes. Vol. I, Imperial College Press, London, 2001. Fourier analysis and semigroups. MR 1873235

E. Kjartansson, Constant Q-wave propagation and attenuation, J. Geophys. Res. 84 (1979), 4737–4748.

A. A. Lokshin and Yu. V. Suvorova, Matematicheskaya teoriya rasprostraneniya voln v sredakh s pamyat′yu, Moskov. Gos. Univ., Moscow, 1982 (Russian). MR 676810
Francesco Mainardi, Fractional calculus and waves in linear viscoelasticity, Imperial College Press, London, 2010. An introduction to mathematical models. MR 2676137
Alain Molinari, Viscoélasticité linéaire et fonctions complètement monotones, J. Mécanique 12 (1973), 541–553 (French, with English summary). MR 368558
Sven Peter Näsholm and Sverre Holm, On a fractional Zener elastic wave equation, Fract. Calc. Appl. Anal. 16 (2013), no. 1, 26–50. MR 3016640, DOI https://doi.org/10.2478/s13540-013-0003-1
Jan Prüss, Positivity and regularity of hyperbolic Volterra equations in Banach spaces, Math. Ann. 279 (1987), no. 2, 317–344. MR 919509, DOI https://doi.org/10.1007/BF01461726
Michael Renardy, William J. Hrusa, and John A. Nohel, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR 919738
René L. Schilling, Renming Song, and Zoran Vondraček, Bernstein functions, De Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter & Co., Berlin, 2010. Theory and applications. MR 2598208
Peter Straka, Mark M. Meerschaert, Robert J. McGough, and Yuzhen Zhou, Fractional wave equations with attenuation, Fract. Calc. Appl. Anal. 16 (2013), no. 1, 262–272. MR 3016653, DOI https://doi.org/10.2478/s13540-013-0016-9

E. Strick, A predicted pedestal effect for a pulse propagating in constant Q solids, Geophysics 35 (1970), 387–403.

E. Strick, An explanation of observed time discrepancies between continuous and conventional well velocity surveys, Geophysics 36 (1971), 285–295.

P. J. Torvik and R. L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mechanics 51 (1983), 294–298.