We rigorously derive a Kirchhoff plate theory, via Γ-convergence, from a three-dimensional model that describes the finite elasticity of an elastically heterogeneous, thin sheet. The heterogeneity in the elastic properties of the material results in a spontaneous strain that depends on both the thickness and the plane variables x′. At the same time, the spontaneous strain is h-close to the identity, where h is the small parameter quantifying the thickness. The 2D Kirchhoff limiting model is constrained to the set of isometric immersions of the mid-plane of the plate into ℝ3, with a corresponding energy that penalizes deviations of the curvature tensor associated with a deformation from an x′-dependent target curvature tensor. A discussion on the 2D minimizers is provided in the case where the target curvature tensor is piecewise constant. Finally, we apply the derived plate theory to the modeling of swelling-induced shape changes in heterogeneous thin gel sheets.
Accepté le :
DOI : 10.1051/cocv/2018046
Keywords: Dimension reduction, Γ-convergence, Kirchhoff plate theory, incompatible tensor fields, polymer gels, geometry of energy minimizers
Agostiniani, Virginia 1 ; Lucantonio, Alessandro 1 ; Lučić, Danka 1
@article{COCV_2019__25__A24_0,
author = {Agostiniani, Virginia and Lucantonio, Alessandro and Lu\v{c}i\'c, Danka},
title = {Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2019},
publisher = {EDP Sciences},
volume = {25},
doi = {10.1051/cocv/2018046},
mrnumber = {3986363},
zbl = {1434.74077},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2018046/}
}
TY - JOUR AU - Agostiniani, Virginia AU - Lucantonio, Alessandro AU - Lučić, Danka TI - Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2018046/ DO - 10.1051/cocv/2018046 LA - en ID - COCV_2019__25__A24_0 ER -
%0 Journal Article %A Agostiniani, Virginia %A Lucantonio, Alessandro %A Lučić, Danka %T Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2018046/ %R 10.1051/cocv/2018046 %G en %F COCV_2019__25__A24_0
Agostiniani, Virginia; Lucantonio, Alessandro; Lučić, Danka. Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 24. doi: 10.1051/cocv/2018046
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