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Minimizers with topological singularitiesin two dimensional elasticity

Published online by Cambridge University Press:  21 September 2007

Jonathan Bevan  and
Xiaodong Yan
Jonathan Bevan
Affiliation:
Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK; j.bevan@surrey.ac.uk
Xiaodong Yan
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA; xiayan@math.msu.edu
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Abstract

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


Keywords

Nonlinear elasticitysingular minimizerstability
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 1 , January 2008 , pp. 192 - 209
Copyright
© EDP Sciences, SMAI, 2007

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References

Acerbi, E. and Fusco, N., Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125–145. CrossRef
Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337–403. CrossRef
Ball, J.M., Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. Roy. Soc. London A 306 (1982) 557–611. CrossRef
J.M. Ball, Some open questions in elasticity. Geometry, mechanics, and dynamics. Springer, New York (2002) 3–59.
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.
Ball, J.M. and Murat, F., $W^{1,p}$ -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225–253. CrossRef
Bauman, P., Owen, N. and Phillips, D., 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. CrossRef
Bauman, P., Owen, N. and Phillips, D., Maximal smoothness of solutions to certain Euler-Lagrange equations from noninear elasticity. Proc. Roy. Soc. Edinburgh 119A (1991) 241–263. CrossRef
Coifman, R., Lions, P. L., Meyer, Y. and Semmes, S., Compensated compactness and hardy spaces. J. Math. Pures. Appl. 72 (1993) 247–286.
C. Horgan and D. Polignone, Cavitation in nonlinearly elastic solids; A review. Appl. Mech. Rev. 48 (1995) 471–485. CrossRef
Knops, R. and Stuart, C., Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch. Rational Mech. Anal. 86 (1984) 233–249. CrossRef
S. Müller, Higher integrability of determinants and weak convergence in $L^{1}.$ J. reine angew. Math. 412 (1990) 20–34.
Large-deformation, R. Ogden isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids. Proc. Roy. Soc. Edinburgh 328A (1972) 567–583.
Quintela-Estevez, P., Critical point in the energy of hyperelastic materials. RAIRO: Math. Modél. Numér. Anal. 25 (1990) 103–132.
Sivaloganathan, J., The generalized Hamilton-Jacobi inequality and the stability of equilibria in nonlinear elasticity. Arch. Rational Mech. Anal. 107 (1989) 347–369. CrossRef
Sivaloganathan, J., Singular minimizers in the calculus of variations: a degenerate form of cavitation. Ann. Inst. H. Poincaré Anal. non linéaire 9 (1992) 657–681. CrossRef
Sivaloganathan, J., On the stability of cavitating equilibria. Q. Appl. Math. 53 (1995) 301–313. CrossRef
Spector, J., Linear deformations as global minimizers in nonlinear elasticity. Q. Appl. Math. 52 (1994) 59–64. CrossRef
Šverák, V., Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 100 (1988) 105–127. CrossRef
Taheri, A., Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations. Proc. Amer. Math. Soc. 131 (2003) 3101–3107. CrossRef
Zhang, K., Polyconvexity and stability of equilibria in nonlinear elasticity. Quart. J. Mech. appl. Math. 43 (1990) 215–221.
Zhang, K., Energy minimizers in nonlinear elastostatics and the implicit function theorem. Arch. Rational Mech. Anal. 114 (1991) 95–117. CrossRef