Plates with incompatible prestrain

Abstract

We study the effective elastic behavior of incompatibly prestrained plates, where the prestrain is independent of thickness as well as uniform through the thickness. We model such plates as three-dimensional elastic bodies with a prescribed pointwise stress-free state characterized by a Riemannian metric $G$ with the above properties, and seek the limiting behavior as the thickness goes to zero. Our results extand the prior analysis in M. Lewicka, M. R. Pakzad ESAIM Control Optim. Calc. Var. 17 (2011), no. 4. We first establish that the $\Gamma$-limit is a Kirchhoff type bending. Further, we show that the minimum energy configuration contains non-trivial Kirchhoff type bending -- i.e., the scaling of the three-dimensional energy is of the order of the cube of the plate thickness -- if and only if the Riemann curvatures $R^3_{112}, R^3_{221}$ and $ R_{1212}$ of $G$ do not identically vanish. We demonstrate through examples, the existence of a new regime where the three above curvatures of $G$ vanish (while the mid-plane of the plate may or may not be flat), but the limiting configuration still has energy that is of the order of F\"oppl - von K\'arm\'an plates. Finally, we apply these results to a model of nematic glass, including a characterization of the condition when the metric is immersible, for $G=\mbox{Id}_3 +\gamma\vec n\otimes \vec n$ given in terms of the inhomogeneous unit director field distribution $\vec n\in\mathbb{R}^3$.