Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Time Derivatives in Material and Spatial Description—What Are the Differences and Why Do They Concern Us? Read chapter Read first chapter

2016 | OriginalPaper | Chapter

A Closed-Form Solution for a Linear Viscoelastic Self-gravitating Sphere

Authors : Wolfgang H. Müller, Elena N. Vilchevskaya

Published in: Advanced Methods of Continuum Mechanics for Materials and Structures

Publisher: Springer Singapore

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Following up on the classical solutions by Love for a linear-elastic self-gravitating sphere, this paper presents the corresponding extension to a linear viscoelastic body of the Kelvin–Voigt type. The solution is expressed in closed form by making use of Laplace transforms. Applications to the genesis of terrestrial planets are sought and the evolution of the Love radius and possible extensions to large deformations are discussed. As a new result, it turns out that in the early days of planet formation there is no Love radius and that it takes time for the Love radius to develop.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter The Influence of Distributed Dislocations on Large Deformations of an Elastic Sphere
next chapter Constitutive Modelling of the Glass Transition and Related Phenomena: Relaxation of Shear Stress Under Pressure
Appendix
Available only for authorised users
Footnotes
1
It is said that also the gas giants initially need a rocky core of sufficient size which is then able to attract gas, if available in the region of its formation.
 
2
Some readers may want to consult Lakes (2009), pg. 4 or Müller and Müller (2009), pg. 370 for more information.
 
3
The label \(\tau = 0\) in Figs. 3, 4 is to be understood in the sense \(\tau \approx 0\) (i.e., very small but not equal to zero). A more detailed discussion of this degenerated case can be found in Müller and Weiss (2016).
 
4
The interested reader is referred to Müller and Weiss (2016).
 
5
The interested reader may want to consult Müller and Weiss (2016) for further information.
 
Literature
Bose, S.C., Chattarji, P.P.: A note on the finite deformation in the interior of the Earth. Bull. Calcutta Math. Soc. 55(1), 11–18 (1963)
Campbell, D.L.: The loading problem for a linear viscoelastic Earth: I. Compressible, non-gravitating models. Pageoph 112, 997–1010 (1974)CrossRef
Chattarji, P.P.: Finite deformation in the interior of the Earth. Bull. Calcutta Math. Soc. 45, 113–118 (1953)MathSciNet
Frelova, X.: A viscoelastic model of the state of deformation of the eye. Master thesis, St Petersburg Polytechnical State University supervised by E N Vilchevskaya (2016)
Gad-el Hak, M., Bandyopadhyay, P.R.: Questions in fluid mechanics. J. Fluids Eng. 117, 3–5 (1995)CrossRef
Kenyon, B.B.C.S.J.: Terrestrial planet formation. I. The transition from oligarchic growth to chaotic growth. Astron. J. 131(3), 1837–1850 (2006)MathSciNetCrossRef
Kienzler, R., Schröder, R.: Einführung in die Höhere Festigkeitslehre. Springer, Dordrecht (2009)CrossRef
Lissauer, J.J.: Planet formation. Annu. Rev. Astron. Astrophys. 31, 129–174 (1993)CrossRef
Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, vol. 1. Cambridge University Press, Cambridge (1892)MATH
Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, Second edn. Cambridge University Press, Cambridge (1906)MATH
Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, Fourth edn. Cambridge University Press, Cambridge (1927)MATH
Müller, I.: Thermodynamik—Grundlagen der Materialtheorie. Bertelsmann-Universitäts verlag, Düsseldorf (1973)
Müller, I., Müller, W.H.: Fundamentals of Thermodynamics and Applications: With Historical Annotations and Many Citations from Avogadro to Zermelo. Springer Science and Business Media, Heidelberg (2009)MATH
Müller, W.H., Weiss, W.: The State of Deformation in Earthlike Self-Gravitating Objects. Springer, Singapore (2016)CrossRef
Ragazzo, C., Ruiz, L.S.: Dynamics of an isolated, viscoelastic, self-gravitating body. Celest. Mech. Dyn. Astron. 122, 303–332 (2015)MathSciNetCrossRefMATH
Sokolnikoff, I.S.: Mathematical Theory of Elasticity, 4th edn. McGraw-Hill Book Company, Inc., New York (1956)MATH
Tanaka, Y., Klemann, V., Fleming, K., Martinec, Z.: Spectral finite element approach to postseismic deformation in a viscoelastic self-gravitating spherical Earth. Geophys. J. Int. 176(3), 715–739 (2009)CrossRef
Timoshenko, S., Goodier, J.N.: Theory of Elasticity, 4th edn. McGraw-Hill Book Company, Inc., New York (1951)MATH
Truesdell, C., Toupin, R.: Handbook of Physics: The Classical Field Theories. Springer, Berlin (1960)
Wang, C.C., Truesdell, C.: Introduction to Rational Elasticity, vol. 1. Springer Science and Business Media, Heidelberg (1973)MATH
Wetherill, G.W.: Formation of the Earth. Annu. Rev. Earth Planet. Sci. 18, 205–256 (1990)CrossRef
Metadata
Title
A Closed-Form Solution for a Linear Viscoelastic Self-gravitating Sphere
Authors
Wolfgang H. Müller
Elena N. Vilchevskaya
Copyright Year
2016
Publisher
Springer Singapore
Book
Advanced Methods of Continuum Mechanics for Materials and Structures

Print ISBN: 978-981-10-0958-7

Electronic ISBN: 978-981-10-0959-4

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-981-10-0959-4_5

Premium Partners

    Image Credits