We reconsider the derivation of plate theories as $\Gamma $-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 $\Gamma $-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.

Accepted:

DOI: 10.1051/cocv/2015042

Keywords: Derivation of plate theories, Γ-convergence, nonlinear plate theory, interpenetration of matter

^{1}; Runa, Eris

^{2}

@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/

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