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Plates with Incompatible Prestrain

Bhattacharya, Kaushik and Lewicka, Marta and Schäffner, Mathias (2016) Plates with Incompatible Prestrain. Archive for Rational Mechanics and Analysis, 221 (1). pp. 143-181. ISSN 0003-9527. https://resolver.caltech.edu/CaltechAUTHORS:20160509-105130410

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Abstract

We study effective elastic behavior of the incompatibly prestrained thin plates, where the prestrain is independent of thickness and uniform through the plate’s thickness h. We model such plates as three-dimensional elastic bodies with a prescribed pointwise stress-free state characterized by a Riemannian metric G, and seek the limiting behavior as h→0. We first establish that when the energy per volume scales as the second power of h, the resulting Γ-limit is a Kirchhoff-type bending theory. We then show the somewhat surprising result that there exist non-immersible metrics G for whom the infimum energy (per volume) scales smaller than h^2. This implies that the minimizing sequence of deformations carries nontrivial residual three-dimensional energy but it has zero bending energy as seen from the limit Kirchhoff theory perspective. Another implication is that other asymptotic scenarios are valid in appropriate smaller scaling regimes of energy. We characterize the metrics G with the above property, showing that the zero bending energy in the Kirchhoff limit occurs if and only if the Riemann curvatures R_(1213), R_(1223) and R_(1212) of G vanish identically. We illustrate our findings with examples; of particular interest is an example where G_(2×2), the two-dimensional restriction of G, is flat but the plate still exhibits the energy scaling of the Föppl–von Kármán type. Finally, we apply these results to a model of nematic glass, including a characterization of the condition when the metric is immersible, for G=Id_3+γn⊗n given in terms of the inhomogeneous unit director field distribution n∈R^3.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s00205-015-0958-7DOIArticle
http://link.springer.com/article/10.1007%2Fs00205-015-0958-7PublisherArticle
https://arxiv.org/abs/1401.1609arXivDiscussion Paper
http://rdcu.be/tteiPublisherFree ReadCube access
ORCID:
AuthorORCID
Bhattacharya, Kaushik0000-0003-2908-5469
Additional Information:© 2016 Springer-Verlag Berlin Heidelberg. Received: 27 August 2014; Accepted: 21 December 2015; First online: 13 January 2016. Marta Lewicka was partially supported by the NSF Career grant DMS-0846996 and by the NSF Grant DMS-1406730. Kaushik Bhattacharya was partially supported by the NSF PIRE Grant OISE-0967140.
Funders:
Funding AgencyGrant Number
NSFDMS-0846996
NSFDMS-1406730
NSFOISE-0967140
Issue or Number:1
Record Number:CaltechAUTHORS:20160509-105130410
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160509-105130410
Official Citation:Bhattacharya, K., Lewicka, M. & Schäffner, M. Arch Rational Mech Anal (2016) 221: 143. doi:10.1007/s00205-015-0958-7
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:66747
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:09 May 2016 17:58
Last Modified:03 Oct 2019 09:59

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