Interpenetration of matter in plate theories obtained as Γ-limits
ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 1, pp. 119-136.

We reconsider the derivation of plate theories as Γ-limits of 3-dimensional nonlinear elasticity and define a suitable notion for the interpenetration of matter in the limit configuration. This is done via the Brouwer degree. For the approximating maps, we adopt as definition of interpenetration of matter the notion of non-invertibility almost everywhere, see [J.M. Ball, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981) 315–328]. Given a limit map satisfying the former interpenetration property, we show that any recovery sequence (in the sense of Γ-convergence) has to consist of maps that satisfy the latter interpenetration property except for finitely many sequence elements. Then we explain how our result is applied in the context of the derivation of plate theories.

Received:
Accepted:
DOI: 10.1051/cocv/2015042
Classification: 73K10, 49J45
Keywords: Derivation of plate theories, Γ-convergence, nonlinear plate theory, interpenetration of matter
Olbermann, Heiner 1; Runa, Eris 2

1 Hausdorff Center for Mathematics & Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany.
2 Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany.
@article{COCV_2017__23_1_119_0,
     author = {Olbermann, Heiner and Runa, Eris},
     title = {Interpenetration of matter in plate theories obtained as $\Gamma{}$-limits},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {119--136},
     publisher = {EDP-Sciences},
     volume = {23},
     number = {1},
     year = {2017},
     doi = {10.1051/cocv/2015042},
     mrnumber = {3601018},
     zbl = {1364.49015},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2015042/}
}
TY  - JOUR
AU  - Olbermann, Heiner
AU  - Runa, Eris
TI  - Interpenetration of matter in plate theories obtained as $\Gamma{}$-limits
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2017
SP  - 119
EP  - 136
VL  - 23
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2015042/
DO  - 10.1051/cocv/2015042
LA  - en
ID  - COCV_2017__23_1_119_0
ER  - 
%0 Journal Article
%A Olbermann, Heiner
%A Runa, Eris
%T Interpenetration of matter in plate theories obtained as $\Gamma{}$-limits
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2017
%P 119-136
%V 23
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2015042/
%R 10.1051/cocv/2015042
%G en
%F COCV_2017__23_1_119_0
Olbermann, Heiner; Runa, Eris. Interpenetration of matter in plate theories obtained as $\Gamma{}$-limits. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 1, pp. 119-136. doi : 10.1051/cocv/2015042. http://archive.numdam.org/articles/10.1051/cocv/2015042/

J.M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics. In Vol. I. Nonlinear analysis and mechanics: Heriot-Watt Symposium, Edinburgh (1976). Res. Notes Math. Pitman, London (1977) 187–241. | MR | Zbl

J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1976/77) 337–403. | DOI | MR | Zbl

J.M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981) 315–328. | DOI | MR | Zbl

J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. Roy. Soc. London Ser. A 306 (1496) 557–611 (1982). | DOI | MR | Zbl

A.S. Besicovitch, Parametric surfaces. Bull. Amer. Math. Soc. 56 (1950) 288–296. | DOI | MR | Zbl

Ph.G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity. Arch. Ration. Mech. Anal. 97 (1987) 171–188. | DOI | MR | Zbl

S. Conti and F. Maggi, Confining thin elastic sheets and folding paper. Arch. Ration. Mech. Anal. 187 (2008) 1–48. | DOI | MR | Zbl

G. Dal Maso, An introduction to Γ-convergence. Vol. 8 of Progr. Nonlin. Differ. Eq. Appl. Birkhäuser Boston Inc., Boston, MA (1993). | MR | Zbl

K. Deimling, Nonlinear Functional Analysis. Springer-Verlag, Berlin (1985). | MR | Zbl

H. Federer, Geometric measure theory. Vol. 153 of Die Grundl. Math. Wiss. Springer-Verlag New York Inc., New York (1969). | MR | Zbl

G. Friesecke, R.D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55 (2002) 1461–1506. | DOI | MR | Zbl

G. Friesecke, R.D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. 180 (2006) 183–236. | DOI | MR | Zbl

D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics. Springer (2001). | MR | Zbl

H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549–578. | MR | Zbl

J. Malý and O. Martio, Lusin’s condition (N) and mappings of the class W 1,n . J. Reine Angew. Math. 458 (1995) 19–36. | MR | Zbl

S. Müller and S.J. Spector, An existence theory for nonlinear elasticity that allows for cavitation. Arch. Ration. Mech. Anal. 131 (1995) 1–66. | DOI | MR | Zbl

S. Müller, S.J. Spector and Q. Tang, Invertibility and a topological property of Sobolev maps. SIAM J. Math. Anal. 27 (1996) 959–976. | DOI | MR | Zbl

K. Zhang, A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 19 (1992) 313–326. | Numdam | MR | Zbl

W.P. Ziemer, Weakly differentiable functions, Sobolev spaces and functions of bounded variation. Vol. 120 of Grad. Texts Math. Springer-Verlag, New York (1989). | MR | Zbl

Cited by Sources: