Asymptotic estimates of viscoelastic Green’s functions near the wavefront
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
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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
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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 (88i:26004)
- M. Caputo and F. Mainardi, New dissipation model based on memory mechanism, Pure Applied Geophysics 91 (1976), 134–147. 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 0149753 (26 \#7238)
- W. Desch and R. C. Grimmer, Initial-boundary value problems for integro-differential equations, Integro-differential evolution equations and applications (Trento, 1984), J. Integral Equations 10 (1985), no. 1-3, suppl., 73–97. MR 831236 (87f:45025)
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- 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 (91c:45003)
- 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 pp. MR 3232047
- 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.
- A. Hanyga and M. Seredyńska, Asymptotic wavefront expansions in hereditary media with singular memory kernels, Quart. Appl. Math. LX (2002), 213–244. MR 1900491 (2004f:35177)
- 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 (2008j:74010), DOI https://doi.org/10.1007/s10659-007-9112-6
- A. Hanyga and M. Seredyńska, Relaxation, dispersion, attenuation and finite propagation speed in viscoelastic media, J. Math. Phys. 51 (2010), no. 9, 092901–092916. MR 2742817 (2011j:74067)
- 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 (86j:45022)
- N. Jacob, Pseudo differential operators and Markov processes. Vol. I, Fourier analysis and semigroups, Imperial College Press, London, 2001. MR 1873235 (2003a:47104)
- 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 pamyatyu, Moskov. Gos. Univ., Moscow, 1982 (Russian). MR 676810 (84m:73031)
- F. Mainardi, Fractional calculus and waves in linear viscoelasticity, World-Scientific, 2010. MR 2676137 (2011e:74002)
- A. Molinari, Viscoélasticité linéaire et functions complètement monotones, J. Mécanique 12 (1975), 541–553. MR 0368558 (51 \#4799)
- S. P. Näsholm and S. Holm, On a fractional Zener elastic wave equation, Fract. Calc. Appl. Anal. 16 (2013), 26–50. MR 3016640
- J. Prüss, Positivity and regularity of hyperbolic Volterra equations in Banach spaces, Math. Ann. 279 (1987), 317–344. MR 0919509 (89h:45004)
- M. Renardy, W. J. Hrusa, and J. A. Nohel, Mathematical problems in viscoelasticity, Longman Scientific & Technical, Essex; John Wiley, New York, 1987. MR 0919738 (89b:35134)
- René L. Schilling, Renming Song, and Zoran Vondraček, Bernstein functions, Theory and applications, de Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter & Co., Berlin, 2010. MR 2598208 (2011d:60060)
- 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.
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Additional Information
Andrzej Hanyga
Affiliation:
ul. Bitwy Warszawskiej 1920r. 14/52, Warszawa, Poland
Email:
ajhbergen@yahoo.com
Keywords:
Viscoelasticity,
wavefront,
attenuation,
dispersion,
shock wave,
Bernstein function
Received by editor(s):
January 29, 2014
Published electronically:
September 14, 2015
Article copyright:
© Copyright 2015
Brown University