Plates with Incompatible Prestrain
Plates with Incompatible Prestrain
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DOI：
10.1007/s00205-015-0958-7
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发表时间：
2016-07-01
影响因子：
2.5
通讯作者：
Schaeffner, Mathias
中科院分区：
文献类型：
--
作者：
Bhattacharya, Kaushik;Lewicka, Marta;Schaeffner, Mathias
本文研究了预应变不协调薄板的有效弹性行为，其中预应变与板厚无关，且在整个板厚h上是均匀的。我们将这种板模拟为三维弹性体，具有由黎曼度量G表征的规定的逐点无应力状态，并寻求极限行为。我们首先建立，当单位体积的能量作为h的二次方缩放时，所得到的极限是基尔霍夫型弯曲理论。然后，我们展示了一个有点令人惊讶的结果，即存在不可浸入的度量G，其下确界能量（每体积）小于h（2）。这意味着，最小化序列的变形进行非平凡的剩余三维能量，但它有零弯曲能量，从极限基尔霍夫理论的角度来看。另一个含义是，其他渐近情景是有效的，在适当的较小尺度制度的能量。我们刻画了度量G的上述性质，证明了当且仅当G的Riemann曲率R（1213），R（1223）和R（1212）为零时，在Kirchhoff极限中弯曲能为零.我们用例子说明我们的研究结果;特别感兴趣的是一个例子，其中，G的二维限制，是平坦的，但板仍然表现出能量标度的Foppl-von Karman型。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，为给定的不均匀的单位导演场分布。
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 . We first establish that when the energy per volume scales as the second power of h, the resulting -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 , the two-dimensional restriction of G, is flat but the plate still exhibits the energy scaling of the Foppl-von Karman type. Finally, we apply these results to a model of nematic glass, including a characterization of the condition when the metric is immersible, for given in terms of the inhomogeneous unit director field distribution .