Poincaré’s Equations for Cosserat Media: Application to Shells

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• 作者：Frederic Boyer ; Federico Renda
• 关键词：Cosserat media ; Euler ; Poincaré reduction ; Geometrically exact shells
• 刊名：Journal of Nonlinear Science
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：27
• 期：1
• 页码：1-44
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis; Theoretical, Mathematical and Computational Physics; Classical Mechanics; Appl.Mathematics/Computational Methods of Engineering; Economic Theory/Quantitative Economics/Mathematical Methods;
• 出版者：Springer US
• ISSN：1432-1467
• 卷排序：27

文摘

In 1901, Henri Poincaré discovered a new set of equations for mechanics. These equations are a generalization of Lagrange’s equations for a system whose configuration space is a Lie group which is not necessarily commutative. Since then, this result has been extensively refined through the Lagrangian reduction theory. In the present contribution, we apply an extended version of these equations to continuous Cosserat media, i.e. media in which the usual point particles are replaced by small rigid bodies, called microstructures. In particular, we will see how the shell balance equations used in nonlinear structural dynamics can be easily deduced from this extension of the Poincaré’s result. In future, these results will be used as foundations for the study of squid locomotion, which is an emerging topic relevant to soft robotics.