Minimizers with topological singularities in two dimensional elasticity

Xiaodong Yan; Jonathan Bevan

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 1, page 192-209
  • ISSN: 1292-8119

Abstract

top
For a class of 2-D elastic energies we show that a radial equilibrium solution is the unique global minimizer in a subclass of all admissible maps. The boundary constraint is a double cover of S 1 ; the minimizer u is C 1 and is such that det ∇ u vanishes at one point.

How to cite

top

Yan, Xiaodong, and Bevan, Jonathan. "Minimizers with topological singularities in two dimensional elasticity." ESAIM: Control, Optimisation and Calculus of Variations 14.1 (2008): 192-209. <http://eudml.org/doc/245201>.

@article{Yan2008,
abstract = {For a class of 2-D elastic energies we show that a radial equilibrium solution is the unique global minimizer in a subclass of all admissible maps. The boundary constraint is a double cover of $S^\{1\}$; the minimizer $u$ is $C^\{1\}$ and is such that $\det \nabla u$ vanishes at one point.},
author = {Yan, Xiaodong, Bevan, Jonathan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {nonlinear elasticity; singular minimizer; stability},
language = {eng},
number = {1},
pages = {192-209},
publisher = {EDP-Sciences},
title = {Minimizers with topological singularities in two dimensional elasticity},
url = {http://eudml.org/doc/245201},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Yan, Xiaodong
AU - Bevan, Jonathan
TI - Minimizers with topological singularities in two dimensional elasticity
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2008
PB - EDP-Sciences
VL - 14
IS - 1
SP - 192
EP - 209
AB - For a class of 2-D elastic energies we show that a radial equilibrium solution is the unique global minimizer in a subclass of all admissible maps. The boundary constraint is a double cover of $S^{1}$; the minimizer $u$ is $C^{1}$ and is such that $\det \nabla u$ vanishes at one point.
LA - eng
KW - nonlinear elasticity; singular minimizer; stability
UR - http://eudml.org/doc/245201
ER -

References

top
  1. [1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125–145. Zbl0565.49010MR751305
  2. [2] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337–403. Zbl0368.73040MR475169
  3. [3] J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. Roy. Soc. London A 306 (1982) 557–611. Zbl0513.73020MR703623
  4. [4] J.M. Ball, Some open questions in elasticity. Geometry, mechanics, and dynamics. Springer, New York (2002) 3–59. Zbl1054.74008MR1919825
  5. [5] J.M. Ball, J. Currie and P. Olver, Null Lagrangians, weak continuity and variational problems of arbitrary order. J. Func. Anal. 41 (1981) 135–174. Zbl0459.35020MR615159
  6. [6] J.M. Ball and F. Murat, W 1 , p -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225–253. Zbl0549.46019MR759098
  7. [7] P. Bauman, N. Owen and D. Phillips, Maximum Principles and a priori estimates for a class of problems in nonlinear elasticity. Ann. Inst. H. Poincaré Anal. non Linéaire 8 (1991) 119–157. Zbl0733.35015MR1096601
  8. [8] P. Bauman, N. Owen and D. Phillips, Maximal smoothness of solutions to certain Euler-Lagrange equations from noninear elasticity. Proc. Roy. Soc. Edinburgh 119A (1991) 241–263. Zbl0744.49008MR1135972
  9. [9] R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and hardy spaces. J. Math. Pures. Appl. 72 (1993) 247–286. Zbl0864.42009MR1225511
  10. [10] C. Horgan and D. Polignone, Cavitation in nonlinearly elastic solids; A review. Appl. Mech. Rev. 48 (1995) 471–485. 
  11. [11] R. Knops and C. Stuart, Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch. Rational Mech. Anal. 86 (1984) 233–249. Zbl0589.73017MR751508
  12. [12] S. Müller, Higher integrability of determinants and weak convergence in L 1 . J. reine angew. Math. 412 (1990) 20–34. Zbl0713.49004MR1078998
  13. [13] R. Ogden, Large-deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids. Proc. Roy. Soc. Edinburgh 328A (1972) 567–583. Zbl0245.73032
  14. [14] P. Quintela-Estevez, Critical point in the energy of hyperelastic materials. RAIRO: Math. Modél. Numér. Anal. 25 (1990) 103–132. Zbl0692.73015MR1034901
  15. [15] J. Sivaloganathan, The generalized Hamilton-Jacobi inequality and the stability of equilibria in nonlinear elasticity. Arch. Rational Mech. Anal. 107 (1989) 347–369. Zbl0709.73014MR1004715
  16. [16] J. Sivaloganathan, Singular minimizers in the calculus of variations: a degenerate form of cavitation. Ann. Inst. H. Poincaré Anal. non linéaire 9 (1992) 657–681. Zbl0769.49030MR1198308
  17. [17] J. Sivaloganathan, On the stability of cavitating equilibria. Q. Appl. Math. 53 (1995) 301–313. Zbl0829.73079MR1330654
  18. [18] J. Spector, Linear deformations as global minimizers in nonlinear elasticity. Q. Appl. Math. 52 (1994) 59–64. Zbl0812.73014MR1262319
  19. [19] V. Šverák, Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 100 (1988) 105–127. Zbl0659.73038MR913960
  20. [20] A. Taheri, Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations. Proc. Amer. Math. Soc. 131 (2003) 3101–3107. Zbl1036.49024MR1993219
  21. [21] K. Zhang, Polyconvexity and stability of equilibria in nonlinear elasticity. Quart. J. Mech. appl. Math. 43 (1990) 215–221. Zbl0726.73016MR1059004
  22. [22] K. Zhang, Energy minimizers in nonlinear elastostatics and the implicit function theorem. Arch. Rational Mech. Anal. 114 (1991) 95–117. Zbl0734.73009MR1094432

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.