Minimizers with topological singularities in two dimensional elasticity

Jonathan Bevan; Xiaodong Yan

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

Abstract

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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 S1; the minimizer u is C1 and is such that det ∇ u vanishes at one point.


How to cite

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Bevan, Jonathan, and Yan, Xiaodong. "Minimizers with topological singularities in two dimensional elasticity." ESAIM: Control, Optimisation and Calculus of Variations 14.1 (2010): 192-209. <http://eudml.org/doc/90863>.

@article{Bevan2010,
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 S1; the minimizer u is C1 and is such that $\det\nabla u$ vanishes at one point.
},
author = {Bevan, Jonathan, Yan, Xiaodong},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Nonlinear elasticity; singular minimizer; stability; nonlinear elasticity},
language = {eng},
month = {3},
number = {1},
pages = {192-209},
publisher = {EDP Sciences},
title = {Minimizers with topological singularities in two dimensional elasticity},
url = {http://eudml.org/doc/90863},
volume = {14},
year = {2010},
}

TY - JOUR
AU - Bevan, Jonathan
AU - Yan, Xiaodong
TI - Minimizers with topological singularities in two dimensional elasticity
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
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 S1; the minimizer u is C1 and is such that $\det\nabla u$ vanishes at one point.

LA - eng
KW - Nonlinear elasticity; singular minimizer; stability; nonlinear elasticity
UR - http://eudml.org/doc/90863
ER -

References

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  1. E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal.86 (1984) 125–145.  
  2. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal.63 (1977) 337–403.  
  3. J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. Roy. Soc. London A306 (1982) 557–611.  
  4. J.M. Ball, Some open questions in elasticity. Geometry, mechanics, and dynamics. Springer, New York (2002) 3–59.  
  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.  
  6. J.M. Ball and F. Murat, W 1 , p -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58 (1984) 225–253.  
  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 Linéaire8 (1991) 119–157.  
  8. P. Bauman, N. Owen and D. Phillips, Maximal smoothness of solutions to certain Euler-Lagrange equations from noninear elasticity. Proc. Roy. Soc. Edinburgh119A (1991) 241–263.  
  9. R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and hardy spaces. J. Math. Pures. Appl.72 (1993) 247–286.  
  10. C. Horgan and D. Polignone, Cavitation in nonlinearly elastic solids; A review. Appl. Mech. Rev.48 (1995) 471–485.  
  11. R. Knops and C. Stuart, Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch. Rational Mech. Anal.86 (1984) 233–249.  
  12. S. Müller, Higher integrability of determinants and weak convergence in L 1 . J. reine angew. Math.412 (1990) 20–34.  
  13. R. Ogden, Large-deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids. Proc. Roy. Soc. Edinburgh328A (1972) 567–583.  
  14. P. Quintela-Estevez, Critical point in the energy of hyperelastic materials. RAIRO: Math. Modél. Numér. Anal.25 (1990) 103–132.  
  15. J. Sivaloganathan, The generalized Hamilton-Jacobi inequality and the stability of equilibria in nonlinear elasticity. Arch. Rational Mech. Anal.107 (1989) 347–369.  
  16. J. Sivaloganathan, Singular minimizers in the calculus of variations: a degenerate form of cavitation. Ann. Inst. H. Poincaré Anal. non linéaire9 (1992) 657–681.  
  17. J. Sivaloganathan, On the stability of cavitating equilibria. Q. Appl. Math.53 (1995) 301–313.  
  18. J. Spector, Linear deformations as global minimizers in nonlinear elasticity. Q. Appl. Math.52 (1994) 59–64.  
  19. V. Šverák, Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal.100 (1988) 105–127.  
  20. A. Taheri, Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations. Proc. Amer. Math. Soc.131 (2003) 3101–3107.  
  21. K. Zhang, Polyconvexity and stability of equilibria in nonlinear elasticity. Quart. J. Mech. appl. Math.43 (1990) 215–221.  
  22. K. Zhang, Energy minimizers in nonlinear elastostatics and the implicit function theorem. Arch. Rational Mech. Anal.114 (1991) 95–117.  

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