Gaussian Curvature as an Identifier of Shell Rigidity
Gaussian Curvature as an Identifier of Shell Rigidity
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高斯曲率作为壳刚度的标识符
DOI：
10.1007/s00205-017-1143-y
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发表时间：
2016
影响因子：
2.5
通讯作者：
D. Harutyunyan
中科院分区：
文献类型：
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作者：
D. Harutyunyan
本文讨论了高斯曲率为非零的壳。我们推导出尖锐的科恩的第一（线性几何刚度估计）和第二不等式，这种壳为零或周期Dirichlet，Neumann和Robin型边界条件。我们证明了如果高斯曲率为正，那么第一个Korn不等式中的最佳常数的尺度为h，如果高斯曲率为负，那么Korn常数的尺度为h4/3，其中h是壳的厚度。这些结果在连续介质力学，特别是壳理论中具有经典的味道。Korn第一不等式是Friesecke等人著名的几何刚性估计的线性版本。对于Arch Ration Mech Anal 180（2）：183-236，2006中的板（其中他们表明非线性Korn第一不等式中的Korn常数如h2），扩展到具有非零曲率的壳。我们还恢复了Lewicka和Müller在Annales de l 'Institute Henri Poincare（C）Non Linear Anal 28（3）：443-469，2011中针对“无边界”壳证明的一致Korn-Poincaré不等式。新的估计也可以应用到寻找的比例律的临界屈曲载荷下的壳面内载荷，以及推导出能量比例律在预屈曲制度。指数1和4/3在本工作中出现的第一次在任何尖锐的几何刚度估计。
In the paper we deal with shells with non-zero Gaussian curvature. We derive sharp Korn’s first (linear geometric rigidity estimate) and second inequalities on that kind of shell for zero or periodic Dirichlet, Neumann, and Robin type boundary conditions. We prove that if the Gaussian curvature is positive, then the optimal constant in the first Korn inequality scales like h, and if the Gaussian curvature is negative, then the Korn constant scales like h4/3, where h is the thickness of the shell. These results have a classical flavour in continuum mechanics, in particular shell theory. The Korn first inequalities are the linear version of the famous geometric rigidity estimate by Friesecke et al. for plates in Arch Ration Mech Anal 180(2):183–236, 2006 (where they show that the Korn constant in the nonlinear Korn’s first inequality scales like h2), extended to shells with nonzero curvature. We also recover the uniform Korn–Poincaré inequality proven for “boundary-less” shells by Lewicka and Müller in Annales de l’Institute Henri Poincare (C) Non Linear Anal 28(3):443–469, 2011 in the setting of our problem. The new estimates can also be applied to find the scaling law for the critical buckling load of the shell under in-plane loads as well as to derive energy scaling laws in the pre-buckled regime. The exponents 1 and 4/3 in the present work appear for the first time in any sharp geometric rigidity estimate.
高斯曲率为零的壳的 Korn 不等式
DOI：
10.1016/j.anihpc.2017.04.004
发表时间：
2018
期刊：
Annales de l'Institut Henri Poincare (C
影响因子：
--
作者：
Grabovsky, Yury;Harutyunyan, Davit
通讯作者：
Harutyunyan, Davit
本质凸W 2.2面的正则性及凸壳均匀弯曲理论的推导
DOI：
10.1016/j.matpur.2018.04.008
发表时间：
2018
期刊：
Journal de Mathématiques Pures et Appliquées
影响因子：
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作者：
Peter Hornung;Igor Velčić
通讯作者：
Igor Velčić