Cosserat Plate Theory

This book presents the foundation and validation of the Cosserat Plate Theory, numerical experiments of deformation and vibration, and the unique properties of the Cosserat plates. Our approach incorporates the high accuracy assumptions of the Cosserat plate deformation consistent with the Cosserat Elasticity equilibrium equations, constitutive formulas, strain-displacement and torsion-microrotation relations. The Cosserat Plate Theory is parametric, where the ＂splitting parameter＂ minimizes the Cosserat plate energy. The validation of the theory is based on the comparison with the three-dimensional Cosserat Elastostatics and Elastodynamics. The numerical results are obtained using the Finite Element Method (FEM) specifically developed to solve the parametric system of equations. The analysis of deformation of a variety of Cosserat plates shows the stress concentration reduction, higher stiffness of Cosserat plates, and the size-effect related to the microstructure. The analysis of vibration of Cosserat plates predicts size-related properties of the plate vibration, the existence of the additional so-called Cosserat plate resonances, and the dynamic anisotropy, related to the dependency of the resonances on the microelement’s shapes and orientations.

Roman Kvasov is a professor of Mathematics at the University of Puerto Rico at Aguadilla, USA. He is an expert in Scientific Computing and the Finite Element Method and an active researcher in the area of Cosserat Plates and Mathematical Modeling. His research has been published in such prestigious journals as Applied Mathematics, Thin-Walled Structures, Modelling and Simulation in Engineering and the chapter in the book ＂Dynamical Systems Theory＂.

Dr. Lev Steinberg is a professor of Mathematics at the University of Puerto Rico at Mayagüez. His academic, scientific, and industrial background includes advising of graduate students, research in Mathematical Modelling, experience at Sandia National Laboratories and US Naval Warfare Centers. Most of his work is related to the Continuum Mechanics, Solids with Complex Microstructures, and Inverse Problems. He is the author and co-author of dozens of patents and research publications in Applied Differential Equation and Applied Mechanics.