Injective weak solutions in second-gradient nonlinear elasticity
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 863-871.

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.

DOI : https://doi.org/10.1051/cocv:2008050
Classification : 74B20,  49K20
Mots clés : gradient estimate, injective deformations, Euler-Lagrange equation, nonlinear elasticity
@article{COCV_2009__15_4_863_0,
author = {Healey, Timothy J. and Kr\"omer, Stefan},
title = {Injective weak solutions in second-gradient nonlinear elasticity},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {863--871},
publisher = {EDP-Sciences},
volume = {15},
number = {4},
year = {2009},
doi = {10.1051/cocv:2008050},
zbl = {1175.74017},
mrnumber = {2567249},
language = {en},
url = {http://www.numdam.org/item/COCV_2009__15_4_863_0/}
}
Healey, Timothy J.; Krömer, Stefan. Injective weak solutions in second-gradient nonlinear elasticity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 863-871. doi : 10.1051/cocv:2008050. http://www.numdam.org/item/COCV_2009__15_4_863_0/

[1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR 450957 | Zbl 0314.46030

[2] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337-403. | MR 475169 | Zbl 0368.73040

[3] J.M. Ball, Minimizers and Euler-Lagrange Equations, in Proceedings of I.S.I.M.M. Conf. Paris, Springer-Verlag (1983). | MR 755716 | Zbl 0547.73013

[4] 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. | MR 1919825 | Zbl 1054.74008

[5] P. Bauman, N.C. Owen and D. Phillips, 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. | Numdam | MR 1096601 | Zbl 0733.35015

[6] P. Bauman, D. Phillips and N.C. Owen, Maximal smoothness of solutions to certain Euler-Lagrange equations from nonlinear elasticity. Proc. Royal Soc. Edinburgh 119A (1991) 241-263. | MR 1135972 | Zbl 0744.49008

[7] P.G. Ciarlet, Mathematical Elasticity Volume I: Three-Dimensional Elasticity. Elsevier Science Publishers, Amsterdam (1988). | MR 936420 | Zbl 0648.73014

[8] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, New York (1989). | MR 990890 | Zbl 0703.49001

[9] G. Dal Maso, I. Fonseca, G. Leoni and M. Morini, Higher-order quasiconvexity reduces to quasiconvexity. Arch. Rational Mech. Anal. 171 (2004) 55-81. | MR 2029531 | Zbl 1082.49017

[10] E. Giusti, Direct Methods in the Calculus of Variations. World Scientific, New Jersey (2003). | MR 1962933 | Zbl 1028.49001

[11] E.L. Montes-Pizarro and P.V. Negron-Marrero, Local bifurcation analysis of a second gradient model for deformations of a rectangular slab. J. Elasticity 86 (2007) 173-204. | MR 2295040 | Zbl 1110.74025

[12] J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques. Masson, Paris (1967). | MR 227584

[13] X. Yan, Maximal smoothness for solutions to equilibrium equations in 2D nonlinear elasticity. Proc. Amer. Math. Soc. 135 (2007) 1717-1724. | MR 2286081 | Zbl 1108.35022