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Injective weak solutions in second-gradientnonlinear elasticity

Published online by Cambridge University Press:  19 July 2008

Timothy J. Healey  and
Stefan Krömer
Timothy J. Healey
Affiliation:
Cornell University, Ithaca, NY 14853, USA.
Stefan Krömer
Affiliation:
Universität Augsburg, 86135 Augsburg, Germany. stefan.kroemer@math.uni-augsburg.de
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Abstract

We consider a class of second-gradient elasticity models for which the internal potential energy is taken as the sum of a convex function of the second gradient of the deformation and a general function of the gradient. However, in consonance with classical nonlinear elasticity, the latter is assumed to grow unboundedly as the determinant of the gradient approaches zero. While the existence of a minimizer is routine, the existence of weak solutions is not, and we focus our efforts on that question here. In particular, we demonstrate that the determinant of the gradient of any admissible deformation with finite energy is strictly positive on the closure of the domain. With this in hand, Gâteaux differentiability of the potential energy at a minimizer is automatic, yielding the existence of a weak solution. We indicate how our results hold for a general class of boundary value problems, including “mixed” boundary conditions. For each of the two possible pure displacement formulations (in second-gradient problems), we show that the resulting deformation is an injective mapping, whenever the imposed placement on the boundary is itself the trace of an injective map.

Keywords

Gradient estimateinjective deformationsEuler-Lagrange equationnonlinear elasticity
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 4 , October 2009 , pp. 863 - 871
Copyright
© EDP Sciences, SMAI, 2008

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References

R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337–403. CrossRef
J.M. Ball, Minimizers and Euler-Lagrange Equations, in Proceedings of I.S.I.M.M. Conf. Paris, Springer-Verlag (1983).
J.M. Ball, Some open problems in elasticity, in Geometry, Mechanics and Dynamics, P. Newton, P. Holmes and A. Weinstein Eds., Springer-Verlag (2002) 3–59.
Bauman, P., Owen, N.C. and Phillips, D., Maximum principles and a priori estimates for a class of problems from nonlinear elasticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 119–157. CrossRef
Bauman, P., Phillips, D. and Owen, N.C., Maximal smoothness of solutions to certain Euler-Lagrange equations from nonlinear elasticity. Proc. Royal Soc. Edinburgh 119A (1991) 241–263. CrossRef
P.G. Ciarlet, Mathematical Elasticity Volume I: Three-Dimensional Elasticity. Elsevier Science Publishers, Amsterdam (1988).
B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, New York (1989).
Dal Maso, G., Fonseca, I., Leoni, G. and Morini, M., Higher-order quasiconvexity reduces to quasiconvexity. Arch. Rational Mech. Anal. 171 (2004) 55–81. CrossRef
E. Giusti, Direct Methods in the Calculus of Variations. World Scientific, New Jersey (2003).
Montes-Pizarro, E.L. and Negron-Marrero, P.V., Local bifurcation analysis of a second gradient model for deformations of a rectangular slab. J. Elasticity 86 (2007) 173–204. CrossRef
J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques. Masson, Paris (1967).
Yan, X., Maximal smoothness for solutions to equilibrium equations in 2D nonlinear elasticity. Proc. Amer. Math. Soc. 135 (2007) 1717–1724. CrossRef