Wave-number-independent theology in a sphere

Michele Caputo

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1987)

  • Volume: 81, Issue: 2, page 175-207
  • ISSN: 1120-6330

Abstract

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We determine the displacement vector and the relaxation caused by time varying surface tractions on a spherical Earth model whose anelastic properties are described by a class of stress strain relations representing a wide variety of rheologies including the Maxwell model and the type which approximates polycrystalline halite and granite. It is seen that this class of rheologies is almost wave number independent, and that once a solution is found for a boundary condition whose time variation does not depend on the wave number, then an approximate solution is found for any other boundary condition with the same geometry. We then study the particular case of the Burger solid and see that its relaxation is described by three different exponentials, one of which is rigorously wave number independent, whereas the other two are weakly wave number dependent. We also study the general solution of the equations of elasticity for a spherical Earth model with assigned surface tractions when the stress-strain relations are defined using derivatives of fractional order and find a subclass of models having the same relaxation time; we see that the relaxation time is not sufficient to describe the rheology of a medium but that the relaxation of the medium needs a much more detailed description in the time domain. This rheology causes a splitting of the free modes in a set of very close lines in the frequency domain. We discuss the effect of successive glacial loads on the Earth for this subclass of models and find it is possible to still see today the effect of more than one glaciation and that the effect of the last glaciation may be masked by the quasi-fossil effect of the previous ones. A relation between migration of isotherms and rheology is also considered. We show that the reciprocity theorem of Betti is valid also with the stress strain relations of the generalized Maxwell models. We finally find a general solution of the equations which govern the deformation of an elastic sphere with Maxwell rheology.

How to cite

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Caputo, Michele. "Wave-number-independent theology in a sphere." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 81.2 (1987): 175-207. <http://eudml.org/doc/287421>.

@article{Caputo1987,
abstract = {We determine the displacement vector and the relaxation caused by time varying surface tractions on a spherical Earth model whose anelastic properties are described by a class of stress strain relations representing a wide variety of rheologies including the Maxwell model and the type which approximates polycrystalline halite and granite. It is seen that this class of rheologies is almost wave number independent, and that once a solution is found for a boundary condition whose time variation does not depend on the wave number, then an approximate solution is found for any other boundary condition with the same geometry. We then study the particular case of the Burger solid and see that its relaxation is described by three different exponentials, one of which is rigorously wave number independent, whereas the other two are weakly wave number dependent. We also study the general solution of the equations of elasticity for a spherical Earth model with assigned surface tractions when the stress-strain relations are defined using derivatives of fractional order and find a subclass of models having the same relaxation time; we see that the relaxation time is not sufficient to describe the rheology of a medium but that the relaxation of the medium needs a much more detailed description in the time domain. This rheology causes a splitting of the free modes in a set of very close lines in the frequency domain. We discuss the effect of successive glacial loads on the Earth for this subclass of models and find it is possible to still see today the effect of more than one glaciation and that the effect of the last glaciation may be masked by the quasi-fossil effect of the previous ones. A relation between migration of isotherms and rheology is also considered. We show that the reciprocity theorem of Betti is valid also with the stress strain relations of the generalized Maxwell models. We finally find a general solution of the equations which govern the deformation of an elastic sphere with Maxwell rheology.},
author = {Caputo, Michele},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Rheology; wave number; isostasy; glaciations; migration of isotherms; reciprocity; Apennines},
language = {eng},
month = {6},
number = {2},
pages = {175-207},
publisher = {Accademia Nazionale dei Lincei},
title = {Wave-number-independent theology in a sphere},
url = {http://eudml.org/doc/287421},
volume = {81},
year = {1987},
}

TY - JOUR
AU - Caputo, Michele
TI - Wave-number-independent theology in a sphere
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1987/6//
PB - Accademia Nazionale dei Lincei
VL - 81
IS - 2
SP - 175
EP - 207
AB - We determine the displacement vector and the relaxation caused by time varying surface tractions on a spherical Earth model whose anelastic properties are described by a class of stress strain relations representing a wide variety of rheologies including the Maxwell model and the type which approximates polycrystalline halite and granite. It is seen that this class of rheologies is almost wave number independent, and that once a solution is found for a boundary condition whose time variation does not depend on the wave number, then an approximate solution is found for any other boundary condition with the same geometry. We then study the particular case of the Burger solid and see that its relaxation is described by three different exponentials, one of which is rigorously wave number independent, whereas the other two are weakly wave number dependent. We also study the general solution of the equations of elasticity for a spherical Earth model with assigned surface tractions when the stress-strain relations are defined using derivatives of fractional order and find a subclass of models having the same relaxation time; we see that the relaxation time is not sufficient to describe the rheology of a medium but that the relaxation of the medium needs a much more detailed description in the time domain. This rheology causes a splitting of the free modes in a set of very close lines in the frequency domain. We discuss the effect of successive glacial loads on the Earth for this subclass of models and find it is possible to still see today the effect of more than one glaciation and that the effect of the last glaciation may be masked by the quasi-fossil effect of the previous ones. A relation between migration of isotherms and rheology is also considered. We show that the reciprocity theorem of Betti is valid also with the stress strain relations of the generalized Maxwell models. We finally find a general solution of the equations which govern the deformation of an elastic sphere with Maxwell rheology.
LA - eng
KW - Rheology; wave number; isostasy; glaciations; migration of isotherms; reciprocity; Apennines
UR - http://eudml.org/doc/287421
ER -

References

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  1. AQUARO, G. (1949) - Sul calcolo delle deformazioni di uno strato sferico elastico. «Atti Naz. Lincei, Rend. Cl. Sc. Fis. mat. e nat.», 7, 289-296. Zbl0037.10801MR36143
  2. ARCA, S. e BERETTA, G.P. (1985) - Prima sintesi geodetico-geologica sui movimenti verticali del suolo nell'Italia Settentrionale (1897-1957): «Boll. Geod. e Sc. Affini», 2, 125-156. 
  3. BARBARELLA, M. (1973) - Un Modello per le deformazioni di Pozzuoli, «Riv. Ital. Geof.», 22, 8-12. 
  4. CALDWELL, J.G. and TURCOTTE, D.L. (1979) - Dependence of the thickness of the elastic oceanic lithosphère on age, «J. Geophys. Res.», 84, 7572-7576. 
  5. CAPUTO, M. (1961) - Elastostatica di una sfera stratificata e sue deformazioni causate da masse superficiali, «Annali di Geofisica», 14 (4), 363-378. 
  6. CAPUTO, M. (1963) - Free modes of layered oblate planets. «J. Geophys. Res.», 68, 497-503. 
  7. CAPUTO, M. (1966) - Linear models of dissipation whose Q is almost frequency independent. «Annali di Geofisica», XXI, (4), 383-393. MR2379269
  8. CAPUTO, M. (1967) - Linear Models of dissipation whose Q is almost frequency independent. «Geophys. J.R. astr. Soc.», 13, 529-539. 
  9. CAPUTO, M. (1979) - Which is the correct stress-strain relation for the interior of the Earth? «Geophys. J.R. astr. Soc.», 59, 227-230. 
  10. CAPUTO, M. (1981) - Elastic radiation from a source in a medium with an almost frequency independent Q, «J. Phys. Earth.», 29, 487-497. 
  11. CAPUTO, M. (1983) - Determination of creep, fatigue and activation energy from constant strain rate experiments, «Teconophysics», 91, 157-164. Zbl0541.73039MR728103
  12. CAPUTO, M. (1984a) - Relaxation and free modes of a self gravitating planet, Geophys. «J.R. Astr. Soc.», 77 (3), 789-808. 
  13. CAPUTO, M. (1984b) - Spectral rheology in a sphere, Proc. Symp. Space Techn. for Geodynamics, Sopron (Hungary), July 9-13. 
  14. CAPUTO, M. (1985) - Generalized rheology and geophysical consequences, «Tectonophysics», 116, 163-172. 
  15. CAPUTO, M. (1986) - Linear and nonlinear inverse rheologies of rocks, «Tectonophysics», 122, 53-71. 
  16. CAPUTO, M., MILANA, G. and RAYHORN, J. (1984) - Topography and its isostatic compensation as a cause of the seismicity of the Appennines, «Tectonophysics», 79, 73-83. 
  17. CAPUTO, M., MANZETTI, V. and NICELLI, R. (1985) - Topography and its isostatic compensation as a cause of seismicity, a revision, «Tectonophysics», 111, 25-39. 
  18. CARTER, N.L. and HEARD, H.C. (1970) - Temperature and rate dependent deformation of halite, «Am. J. Sc.», 269, 193-249. 
  19. CARTER, N.L. and HANSEN, F.D. (1983) - Creep of rock salt, «Tectonophysics», 92, 275-333. 
  20. FICHERA, G. (1949) - Sul calcolo delle deformazioni dotate di simmetria assiale, di uno strato sferico elastico. «Atti Acc. Naz. Lincei Rend. Cl. Sc. Fis. Nat. Mat.», 6, 582-589, 1949. Zbl0034.40902MR35169
  21. GRAFFI, D. (1939) - Sui teoremi di reciprocità nei fenomeni dipendenti dal tempo, «Annali di matematica pura e applicata», IV, XVIII, 173-200. Zbl0022.14703MR1553317JFM65.1025.03
  22. HEARD, H.C. and RAYLEIGH, C.B. (1972) - Steady state flow in marble at 500 C to 800 C, «Geol. Soc. Am. Bull.», 83, 935-956. 
  23. HEARD, H.C. (1972) - Steady state flow in polycrystaline halite at pressure of 2 Kbar. «Geophys. Monogr. A.G.U.», 16, 191-203. 
  24. PICONE, M. (1936) - Nuovi indirizzi di ricerca nella teoria e nel calcolo delle soluzioni di talune equazioni lineari delle derivate parziali della Fisica Matematica, «Ann. Scuola Norm. Sup. di Pisa», 5, 19-37. JFM62.0564.04
  25. SABADINI, R., YUEN, D.A. and GASPERINI, P. (1985) - The effect of transient rehology on the interpretation of lower mantle viscosity, «Geophys. Res. Lett.», 12 (6): 361-364. 
  26. SLICHTER, L.B. and CAPUTO, M. (1960) - Deformation of an Earth Model by Surfaces Pressures, «Journal of Geoph. Research.», 12, December, 4151-4156. 

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