In the framework of nonlinear theory of Cosserat elasticity, also called micropolar elasticity, we provide the complete characterization of null Lagrangians for three dimensional bodies as well as for shells. Using the Gibb’s rotation vector for description of the microrotation, this task is possible by an application of a theorem stated by Olver and Sivaloganathan in ‘the structure of null Lagrangians’ (Nonlinearity 1:389–398, 1988). A set of necessary and sufficient conditions is also provided for the elasticity tensors to correspond to a null Lagrangian in linearized micropolar theory.

中文翻译：

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空Cosserat弹性中的拉格朗日

在Cosserat弹性（也称为微极弹性）非线性理论的框架中，我们提供了三维物体以及壳体的零拉格朗日数的完整表征。使用Gibb的旋转矢量来描述微旋转，通过应用Olver和Sivaloganathan在“零拉格朗日式结构”中定理的应用（非线性1：389-398，1988），可以完成此任务。还提供了一组必要和充分的条件，以使弹性张量对应于线性化微极理论中的零拉格朗日。