Rheologies quasi wave number independent in a sphere and splitting the spectral line

Michele Caputo

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1988)

  • Volume: 82, Issue: 3, page 507-526
  • ISSN: 1120-6330

Abstract

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The solution of the equations which govern the slow motions (for which the inertia forces are negligible) in an elastic sphere is studied for a great variety of rheological models and surface tractions with rotational symmetry (Caputo 1984a). The solution is expressed in terms of spherical harmonics and it is shown that its time dependent component is dependent on the order of the harmonic. The dependence of the time component of the solution on the order of the harmonic number is studied. The problem of causality is then discussed showing that the rheological models defined by strees-strain relations of the generalized Maxwell type (Caputo 1984b), which contain derivatives of real order, are causal. It is also seen that the rheological model based on stress strain relations of the generalized Maxwell type multiplies the number of spectral lines of the free modes of a spherical shell. The same applies also to the rheologies of Voigt, Maxwell and of the standard linear solid.

How to cite

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Caputo, Michele. "Rheologies quasi wave number independent in a sphere and splitting the spectral line." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 82.3 (1988): 507-526. <http://eudml.org/doc/287248>.

@article{Caputo1988,
abstract = {The solution of the equations which govern the slow motions (for which the inertia forces are negligible) in an elastic sphere is studied for a great variety of rheological models and surface tractions with rotational symmetry (Caputo 1984a). The solution is expressed in terms of spherical harmonics and it is shown that its time dependent component is dependent on the order of the harmonic. The dependence of the time component of the solution on the order of the harmonic number is studied. The problem of causality is then discussed showing that the rheological models defined by strees-strain relations of the generalized Maxwell type (Caputo 1984b), which contain derivatives of real order, are causal. It is also seen that the rheological model based on stress strain relations of the generalized Maxwell type multiplies the number of spectral lines of the free modes of a spherical shell. The same applies also to the rheologies of Voigt, Maxwell and of the standard linear solid.},
author = {Caputo, Michele},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Splitting; Eigenvalues; Rheology; Wavenumber; Quasistatic; surface tractions; rotational symmetry; spherical harmonics; dependence; time component of the solution; order of the harmonic number; stress- strain relations; generalized Maxwell type},
language = {eng},
month = {9},
number = {3},
pages = {507-526},
publisher = {Accademia Nazionale dei Lincei},
title = {Rheologies quasi wave number independent in a sphere and splitting the spectral line},
url = {http://eudml.org/doc/287248},
volume = {82},
year = {1988},
}

TY - JOUR
AU - Caputo, Michele
TI - Rheologies quasi wave number independent in a sphere and splitting the spectral line
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1988/9//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 3
SP - 507
EP - 526
AB - The solution of the equations which govern the slow motions (for which the inertia forces are negligible) in an elastic sphere is studied for a great variety of rheological models and surface tractions with rotational symmetry (Caputo 1984a). The solution is expressed in terms of spherical harmonics and it is shown that its time dependent component is dependent on the order of the harmonic. The dependence of the time component of the solution on the order of the harmonic number is studied. The problem of causality is then discussed showing that the rheological models defined by strees-strain relations of the generalized Maxwell type (Caputo 1984b), which contain derivatives of real order, are causal. It is also seen that the rheological model based on stress strain relations of the generalized Maxwell type multiplies the number of spectral lines of the free modes of a spherical shell. The same applies also to the rheologies of Voigt, Maxwell and of the standard linear solid.
LA - eng
KW - Splitting; Eigenvalues; Rheology; Wavenumber; Quasistatic; surface tractions; rotational symmetry; spherical harmonics; dependence; time component of the solution; order of the harmonic number; stress- strain relations; generalized Maxwell type
UR - http://eudml.org/doc/287248
ER -

References

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  1. CAPUTO, M., Elastodinamica ed elastoplastica di un modello della Terra e sue auto-oscillazioni toroidali, Boll. Geof. Teor. Appl., 3, 10, 1-20, 1961. 
  2. CAPUTO, M., Generalized rheologies and geophysical consequences, Tecnophysics, 116, 163-172, 1985. 
  3. CAPUTO, M., Linear and non-linear independent rheologies of rocks, Tectonophysics, 122, 53-71, 1986. 
  4. CAPUTO, M., Deformation, creep, fatigue and activation energy from constant strain rate experiments, Tectonophysics, 91, 157-164, 1983. MR728103
  5. CAPUTO, M., Elasticità e dissipazione, ZanichelliBologna, 1969. 
  6. CAPUTO, M., Wave number independent rheology in a sphere, Atti Acc. Lincei Rend, fis., (8) LXXXI, 175-207, 1987. MR999430
  7. CAPUTO, M., Spectral rheology in a sphere, Proc. Int., Symp. Space Techniques for Geodynamics, Somogi and Reigberg Eds. Sopron, Hungary, 1984a. 
  8. CAPUTO, M., Relaxation and free modes of a self gravitating planet, Geophys. J.R. astr. Soc., 77, 789-808, 1984b. 
  9. CAPUTO, M., Free modes of layered oblate planets, J. Geophys. Res, 68, 497-503, 1964,. 
  10. BOZZI ZADRO, and CAPUTO, M., Spectral and byspectral analysis of the free modes of the Earth, Atti IV sympopsium on Theory and Computers, Supplemento al Nuovo Cimento, VI, 1, 67-81, 1968. 
  11. MARUSSI, A., I primi risultati ottenuti nella stazione per lo studio delle maree della verticale della Grotta Gigante, Bol. Geodesia e Scienze Affini, 19, 4, 1960. 
  12. STACEY, F.D., Physics of the Earth, J. Wiley & Sons, New York, Sidney, 1977. 
  13. CAPUTO, M. and MARCUCCI, S., The identification of the S 1 2 mode of the spheroidal oscillations of the Earth, (Unpublished), 1982. 

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