Abstract

We analyze the concentration phenomena which occur in thin rods, solving the following optimization problem: a given fraction of elastic material must be distributed into a cylindrical design region with infinitesimal cross section in an optimal way, so that it maximizes the resistance to a given external load. For small volume fractions, the optimal configuration of material is described by a measure which concentrates on 2-rectifiable sets. For some choices of the external charging, the concentration phenomena turn out to be related to some new variants of the Cheeger problem of the cross section of the rod. The same study has already been carried out recently in the particular case of pure torsion regime by G. Bouchitté, I. Fragalà, I. Lucardesi and P. Seppecher ["Optimal thin torsion rods and Cheeger sets", SIAM J. Math. Anal. 44 (2012) 483-512]. Here we extend those results by enlarging the class of admissible loads.

Author information

Contact details are reproduced from the original publication and may be historical.

Ilaria Lucardesi

Laboratoire IMATH, Université du Sud Toulon-Var, Avenue de l'Université, 83957 La Garde, France

Keywords

  • Thin rods
  • optimization
  • compliance
  • duality
  • dimension reduction
  • Cheeger problem
  • generalized Cheeger sets

Mathematics Subject Classification

74K10, 49J35, 49Q10, 28A25

References

  1. G. Allaire: Shape Optimization by the Homogenization Method, Springer, New York (2002).https://doi.org/10.1007/978-1-4684-9286-6Google Scholar
  2. J. J. Alibert, G. Bouchitté, I. Fragalà, I. Lucardesi: A nonstandard free boundary problem arising in the shape optimization of thin torsion rods, Interfaces Free Bound. 15(1) (2013) 95–119.https://doi.org/10.4171/ifb/296Google Scholar
  3. F. Alter, V. Caselles: Uniqueness of the Cheeger set of a convex body, Nonlinear Anal. 70 (2009) 32–44.https://doi.org/10.1016/j.na.2007.11.032Google Scholar
  4. G. Bouchitté: Convex analysis and duality methods, in: Variational Techniques, Encyclopedia of Mathematical Physics, J.-P. Françoise, G. L. Naber, T. S. Tsun (eds.), Elsevier, Amsterdam (2006) 642–652.https://doi.org/10.1016/B0-12-512666-2/00487-9Google Scholar
  5. G. Bouchitté, I. Fragalà: Second-order energies on thin structures: variational theory and non-local effects, J. Funct. Anal. 204 (2003) 228–267.https://doi.org/10.1016/S0022-1236(02)00181-7Google Scholar
  6. G. Bouchitté, I. Fragalà, I. Lucardesi, P. Seppecher: Optimal thin torsion rods and Cheeger sets, SIAM J. Math. Anal. 44 (2012) 483–512.https://doi.org/10.1137/110828538Google Scholar
  7. G. Bouchitté, I. Fragalà, P. Seppecher: Structural optimization of thin plates: the three dimensional approach, Arch. Ration. Mech. Anal. 202 (2011) 829–874.https://doi.org/10.1007/s00205-011-0435-xGoogle Scholar
  8. A. Figalli, F. Maggi, A. Pratelli: A note on Cheeger sets, Proc. Amer. Math. Soc. 137 (2009) 2057–2062.https://doi.org/10.1090/S0002-9939-09-09795-0Google Scholar
  9. V. Fridman, B. Kawohl: Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Commentat. Math. Univ. Carol. 44 (2003) 659–667.Google Scholar
  10. I. R. Ionescu, T. Lachand-Robert: Generalized Cheeger sets related to landslides, Calc. Var. Partial Differ. Equ. 23 (2005) 227–249.https://doi.org/10.1007/s00526-004-0300-yGoogle Scholar
  11. B. Kawohl, T. Lachand Robert: Characterization of Cheeger sets for convex subsets of the plane, Pac. J. Math. 225 (2006) 103–118.https://doi.org/10.2140/pjm.2006.225.103Google Scholar
  12. S. Sorin: A First Course on Zero-Sum Repeated Games, Springer, Paris (2002).Google Scholar
  13. L. Trabucho, J. M. Viaño: Mathematical modelling of rods, in: Handbook of Numerical Analysis IV, North-Holland, Amsterdam (1996) 487–974.https://doi.org/10.1016/S1570-8659(96)80006-8Google Scholar

Suggested citation

I. Lucardesi. “Concentration Phenomena in the Optimal Design of Thin Rods.” Journal of Convex Analysis 22 (2015), No. 2, 303–338. https://doi.org/10.68381/jca22015

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2015. Rights now held by Banach Press.