Variational Asymptotic Modeling of Cosserat Elastic Plates

One of the most important branches of applied mechanics is the theory of plates-defined to be plane structural elements whose thickness is very small when compared to the two planar dimensions. There is an abundance of plate models in the literature owing to advantages such as reduced computational effort and a simpler, yet elegant, resulting mathematical formulation. Recently, there has been a steady growth of interest in modeling materials with microstructure that exhibit length-scale dependent behavior, generally known as Cosserat elastic materials. Traditional plate models derived from classical elasticity theory, such as the Reissner-Mindlin type, are incapable of accounting for such length-scale effects, which can only be predicted when one starts from a higher-order elasticity theory. The objective of this work is the formulation of a theory of Cosserat elastic plates. The mathematical foundation of the approach used is the variational asymptotic method, a powerful tool used to construct asymptotically correct reduced-dimensional models. Unlike existing Cosserat plate models in the literature, the variational asymptotic method allows for a plate formulation that is free of ad hoc assumptions regarding the kinematics. The result is a systematic derivation of the two-dimensional constitutive relations and a set of geometrically exact, fully intrinsic equations governing the motion of a plate. An important consequence is the extraction of the so-called drilling stiffness associated with the drilling degree of freedom. This stiffness cannot be extracted from classical elasticity theory and is therefore only associated with higher-order elasticity theories. The present approach connects the two-dimensional Cosserat plate theory with a three-dimensional elasticity theory, in stark contrast to the usual approach of regarding the Cosserat plate theory as phenomenological and thus disconnected from the three-dimensional world.