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Multi-body Spherically Symmetric Steady States of Newtonian Self-Gravitating Elastic Matter

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Abstract

We study the problem of static, spherically symmetric, self-gravitating elastic matter distributions in Newtonian gravity. To this purpose we first introduce a new definition of homogeneous, spherically symmetric (hyper)elastic body in Euler coordinates, i.e., in terms of matter fields defined on the current physical state of the body. We show that our definition is equivalent to the classical one existing in the literature and which is given in Lagrangian coordinates, i.e. in terms of the deformation of the body from a given reference state. After a number of well-known examples of constitutive functions of elastic bodies are re-defined in our new formulation, a detailed study of the Seth model is presented. For this type of material the existence of single and multi-body solutions is established.

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References

  1. Andréasson H., Calogero S.: Spherically symmetric steady states of John elastic bodies in general relativity. Class. Quantum Grav. 31, 165008 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  2. Ball J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. Lond. A 306, 557–611 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  3. Beig R., Schmidt B.G.: Static, self-gravitating elastic bodies. Proc. R. Soc. Lond. A 459, 109–115 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  4. Boulanger P., Hayes M.: Finite-amplitude waves in Mooney–Rivlin and Hadamard materials. In: Hayes, M., Saccomandi, G. (eds.) Topics in Finite Elasticity, pp. 131–167. Springer, Vienna (2001)

    Chapter  Google Scholar 

  5. Calogero S., Leonori T.: Ground states of self-gravitating elastic bodies. Calc. Var. Partial Differ. Equ. 54, 881–899 (2015)

    Article  MathSciNet  Google Scholar 

  6. Ciarlet P.G.: Mathematical Elasticity, vol. I. North-Holland, Amsterdam (1988)

    MATH  Google Scholar 

  7. Drozdov A.D.: Finite Elasticity and Viscoelasticity. World Scientific Publishing, singapore (1996)

    Book  Google Scholar 

  8. Heinzle J.M., Uggla C.: Newtonian stellar models. Ann. Phys. 308, 18–61 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  9. Heinzle J.M., Rendall A.D., Uggla C.: Theory of Newtonian self-gravitating stationary spherically symmetric systems. Math. Proc. Camb. Philos. Soc. 140, 177–192 (2006)

    Article  MathSciNet  Google Scholar 

  10. Kippenhahn R., Weigert A., Weiss A.: Stellar Structure and Evolution. Springer, Berlin (2012)

    Book  Google Scholar 

  11. Losert-Valiente Kroon, C.M.: Static elastic shells in Einsteinian and Newtonian gravity. arXiv:gr-qc/0603103

  12. Lemou M., Mehats F., Raphaël P.: Orbital stability of spherical gravitational systems. Invent. Math. 187, 145–194 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  13. Lichtenstein L.: Gleichgewichtsfiguren rotierender Flüssigkeiten. Springer, Berlin (1933)

    Book  Google Scholar 

  14. Lindblom L.: On the symmetries of equilibrium stellar models. Philos. Trans. R. Soc. Lond. A 340, 353–364 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  15. Love A.E.H.: A Treatise on the Mathematical Theory of Elasticity, vol. 4. Cambridge University Press, Cambridge (1927)

    MATH  Google Scholar 

  16. Lurie, A. I.: Non-Linear Theory of Elasticity. North-Holland Series in Applied Mathematics and Mechanics (2012)

  17. Müller, W., Weiss, W.: The State of Deformation in Earthlike Self-Gravitating Objects. Springer Briefs in Applied Sciences and Technology—Continuum Mechanics (2016)

    Book  Google Scholar 

  18. Ogden R.W.: Non-linear Elastic Deformations. Ellis Horwood Limited, Chicago (1984)

    MATH  Google Scholar 

  19. Park J.: Spherically symmetric static solutions of the Einstein equations with elastic matter source. Gen. Relativ. Gravit. 32, 235–252 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  20. Ramming T., Rein G.: Spherically symmetric equilibria for self-gravitating kinetic or fluid models in the nonrelativistic and relativistic case—a simple proof for finite extension. SIAM J. Math. Anal. 45, 900–914 (2013)

    Article  MathSciNet  Google Scholar 

  21. Rein G.: Static shells for the Vlasov–Poisson and Vlasov–Einstein systems. Indiana Univ. Math. J. 48, 335–346 (1999)

    Article  MathSciNet  Google Scholar 

  22. Rein G.: Stationary and static stellar dynamic models with axial symmetry. Nonlinear Anal. Theory Methods Appl. 41, 313–344 (2000)

    Article  MathSciNet  Google Scholar 

  23. Rein G.: Collisionless kinetic equations from astrophysics—the Vlasov–Poisson system. In: Dafermos, C.M., Feireisl, E. (eds.) Handbook of Differential Equations: Evolutionary Equations, vol. 3, Elsevier, Amsterdam (2007)

    Google Scholar 

  24. Rendall A.D, Schmidt B.G.: Existence and properties of spherically symmetric static fluid bodies with a given equation of state. Class. Quantum Gravity 8, 985–1000 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  25. Schulze A.: Existence and stability of static shells for the Vlasov–Poisson system. Analysis 26, 527–543 (2006)

    Article  MathSciNet  Google Scholar 

  26. Schulze A.: Existence of axially symmetric solutions to the Vlasov–Poisson system depending on Jacobi’s integral. Commun. Math. Sci. 6, 711–727 (2008)

    Article  MathSciNet  Google Scholar 

  27. Seth B.R.: Finite strain in elastic problems. Philos. Trans. R. Soc. Lond. 234, 231–264 (1935)

    Article  ADS  Google Scholar 

  28. Truesdell C., Noll W.: The Non-linear Field Theories of Mechanics. Springer, Berlin (2004)

    Book  Google Scholar 

Download references

Acknowledgements

A. A. is supported by the project (GPSEinstein) PTDC/MAT-ANA/1275/2014, by CAMGSD, Instituto Superior Técnico, through FCT/Portugal UID/MAT/04459/2013, and by the FCT Grant No. SFRH/BPD/85194/2012. Furthermore, A. A. thanks the Department of Mathematics at Chalmers University, Sweden, for the very kind hospitality.

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Authors and Affiliations

  1. Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    A. Alho

  2. Department of Mathematical Sciences, Chalmers University of Technology, University of Gothenburg, Gothenburg, Sweden

    S. Calogero

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  1. A. Alho
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  2. S. Calogero
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Correspondence to A. Alho.

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Communicated by P. Chrusciel

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Alho, A., Calogero, S. Multi-body Spherically Symmetric Steady States of Newtonian Self-Gravitating Elastic Matter. Commun. Math. Phys. 371, 975–1004 (2019). https://doi.org/10.1007/s00220-019-03380-0

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