Zeitschrift für angewandte Mathematik und Physik

, Volume 65, Issue 6, pp 1261–1288 | Cite as

Variational analysis of the coupling between a geometrically exact Cosserat rod and an elastic continuum

Article

Abstract

We formulate the static mechanical coupling of a geometrically exact Cosserat rod to a nonlinearly elastic continuum. In this setting, appropriate coupling conditions have to connect a one-dimensional model with director variables to a three-dimensional model without directors. Two alternative coupling conditions are proposed, which correspond to two different configuration trace spaces. For both, we show existence of solutions of the coupled problems, using the direct method of the calculus of variations. From the first-order optimality conditions, we also derive the corresponding conditions for the dual variables. These are then interpreted in mechanical terms.

Mathematics Subject Classification (2000)

74K10 74B20 49K20 

Keywords

Coupling conditions Energy minimization Cosserat rod Hyperelastic material 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Antman S.S.: Nonlinear Problems of Elasticity, volume 107 of Applied Mathematical Sciences. Springer, Berlin (1991)Google Scholar
  2. 2.
    Ball J.M.: Some open problems in elasticity. In: Newton, P., Holmes, P., Weinstein, A. (eds) Geometry, Mechanics, and Dynamics, pp. 3–59. Springer, Berlin (2002)CrossRefGoogle Scholar
  3. 3.
    Bethuel F.: The approximation problem for Sobolev maps between two manifolds. Acta Math. 167, 153–206 (1991)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Blanco P., Feijóo R., Urquiza S.: A variational approach for coupling kinematically incompatible structural models. Comput. Methods Appl. Mech. Eng. 197, 1577–1602 (2008)CrossRefMATHGoogle Scholar
  5. 5.
    Blanco P.J., Discacciati M., Quarteroni A.: Modeling dimensionally-heterogeneous problems: analysis, approximation and applications. Numer. Math. 119(2), 299–335 (2011)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Brezis H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2011)MATHGoogle Scholar
  7. 7.
    Chouaïeb, N.: Kirchhoff’s problem of helical solutions of uniform rods and their stability properties. Ph.D. thesis, Ecole Polytechnique Fédérale Lausanne (2003)Google Scholar
  8. 8.
    Ciarlet P., LeDret H., Nzengwa R.: Junctions between three-dimensional and two-dimensional linearly elastic structures. J. Math. Pures Appl. 68, 261–295 (1989)MATHMathSciNetGoogle Scholar
  9. 9.
    Ciarlet, P.G.: Mathematical Elasticity, vol. I: Three-Dimensional Elasticity. North-Holland, Amsteram (1988)Google Scholar
  10. 10.
    Ekeland I., Temam R.: Convex Analysis and Variational Problems. SIAM, Philadelphia, PA (1999)CrossRefMATHGoogle Scholar
  11. 11.
    Ern A., Guermond J.-L.: Theory and Practice of Finite Elements. Springer, Berlin (2004)CrossRefMATHGoogle Scholar
  12. 12.
    Formaggia L., Gerbeau J., Nobile F., Quarteroni A.: On the coupling of 3D and 1D Navier–Stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Eng. 191, 561–582 (2001)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Karcher H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541 (1977)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Kehrbaum, S.: Hamiltonian formulations of the equilibrium conditions governing elastic rods: qualitative analysis and effective properties. Ph.D. thesis, University of Maryland (1997)Google Scholar
  15. 15.
    Lagnese J., Leugering G., Schmidt E.: Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures. Birkhäuser, Basel (1994)CrossRefMATHGoogle Scholar
  16. 16.
    Markou, G.: Detailed three-dimensional nonlinear hybrid simulation for the analysis of large scale reinforced concrete structures. Ph.D. thesis, National Technical University of Athens (2011)Google Scholar
  17. 17.
    Mielke A.: Hamiltonian and Lagrangian flows on center manifolds with applications to elliptic variational problems, volume 1489 of Lecture Notes in Mathematics. Springer, Berlin (1991)Google Scholar
  18. 18.
    Mielke A., Holmes P.: Spatially complex equilibria of buckled rods. Arch. Ration. Mech. Anal. 101, 319–348 (1988)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Monaghan, D.J., Doherty, I.W., Court, D.M., Armstrong, C.G.: Coupling 1D beams to 3D bodies. In: Proceedings of 7th International Meshing Roundtable. Sandia National Laboratories (1998)Google Scholar
  20. 20.
    Neff P.: A geometrically exact Cosserat shell-model including size effects, avoiding degeneracy in the thin shell limit. Part I: Formal dimensional reduction for elastic plates and existence of minimizers for positive Cosserat couple modulus. Continuum Mech. Thermodyn. 16(6), 577–628 (2004)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Neff P.: A geometrically exact Cosserat shell-model including size effects, avoiding degeneracy in the thin shell limit. Existence of minimizers for zero Cosserat couple modulus. Math. Models Methods Appl. Sci. 17(3), 363–392 (2007)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Neff, P., Lankeit, J., Madeo, A.: On Grioli’s minimum property and its relation to Cauchy’s polar decomposition. Int. J. Eng. Sci. arXiv:1310.7826 (submitted)Google Scholar
  23. 23.
    Palais R.S.: Morse theory on Hilbert manifolds. Topology 2, 299–340 (1963)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Sander, O.: Multidimensional coupling in a human knee model. Ph.D. thesis, Freie Universität Berlin (2008)Google Scholar
  25. 25.
    Sander, O.: Coupling geometrically exact Cosserat rods and linear elastic continua. In: Proceedings of DD20 (to appear)Google Scholar
  26. 26.
    Seidman T., Wolfe P.: Equilibrium states of an elastic conducting rod in a magnetic field. Arch. Ration. Mech. Anal. 102(4), 307–329 (1988)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Wloka J.: Partielle Differentialgleichungen. Teubner-Verlag, Wiesbaden (1982)CrossRefMATHGoogle Scholar
  28. 28.
    Zeidler E.: Nonlinear Functional Analysis and its Applications, vol. I. Springer, Berlin (1986)CrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.RWTH Aachen, Institut für Geometrie und Praktische, MathematikAachenGermany
  2. 2.Hamburg University of Technology, Institute of MathematicsHamburgGermany

Personalised recommendations