预应变不协调的板
Plates with incompatible prestrain
摘要：
本文研究了预应变不协调板的有效弹性行为，这里的预应变与厚度无关，并且在整个厚度上是均匀的。我们将这种板模拟为具有规定的逐点无应力状态的三维弹性体，其特征在于具有上述性质的黎曼度量$G$，并寻求厚度趋于零时的极限行为。所得结果推广了M. Lewicka，M. R. Pakzad ESAIM控制优化计算值变种17（2011），第4号。我们首先建立了$Γ$ -极限是Kirchhoff型弯曲。此外，我们表明，最小能量配置包含非平凡基尔霍夫型弯曲-即，三维能量的标度是板厚度的立方的数量级--当且仅当$G$的黎曼曲率$R^3_{112}, R^3_{221}$和$ R_{1212}$不相同地为零。我们通过例子证明了一个新的区域的存在，其中$G$的上述三个曲率消失（而板的中平面可能是平的，也可能不是平的），但极限构型仍然具有Föppl - von Kármán板的能量。最后，我们将这些结果应用到一个模型的玻璃，包括表征的条件时，度量是浸入式的，为$G=\mbox{Id}_3 +γ\vec n\otimes \vec n$给出的不均匀的单位导演场分布$\vec n\in\mathbb{R}^3$。
英文摘要：
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 $Γ$-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 Föppl - von Kármán plates. Finally, we apply these results to a model of nematic glass, including a characterization of the condition when the metric is immersible, for $G=\mbox{Id}_3 +γ\vec n\otimes \vec n$ given in terms of the inhomogeneous unit director field distribution $\vec n\in\mathbb{R}^3$.
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