Plates with Incompatible Prestrain

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• 作者：Kaushik Bhattacharya ; Marta Lewicka…
• 刊名：Archive for Rational Mechanics and Analysis
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：221
• 期：1
• 页码：143-181
• 全文大小：722 KB
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• 作者单位：Kaushik Bhattacharya (1)
  Marta Lewicka (2)
  Mathias Schäffner (3)
  
  1. Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA, 91125, USA
  2. Department of Mathematics, University of Pittsburgh, 139 University Place, Pittsburgh, PA, 15260, USA
  3. Institute for Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074, Würzburg, Germany
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mechanics
  Electromagnetism, Optics and Lasers
  Mathematical and Computational Physics
  Complexity
  Fluids
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0673

文摘

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 \({h \to 0}\). We first establish that when the energy per volume scales as the second power of h, the resulting \({\Gamma}\) -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 \({G_{2 \times 2}}\), 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 \({G = Id_{3} + \gamma n \otimes n}\) given in terms of the inhomogeneous unit director field distribution \({ n \in \mathbb{R}^3}\).