Volume 25, 2019
|Number of page(s)||32|
|Published online||30 July 2019|
Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets
Università di Verona,
Strada le Grazie 15,
2 SISSA, via Bonomea 265, 34136 Trieste, Italy.
* Corresponding author: firstname.lastname@example.org
Accepted: 6 September 2018
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.
Mathematics Subject Classification: 49J45 / 74B20 / 74K20 / 74F10
Key words: Dimension reduction / Γ-convergence / Kirchhoff plate theory / incompatible tensor fields / polymer gels / geometry of energy minimizers
© EDP Sciences, SMAI 2019
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