Distinctive Characteristics of Cosserat Plate Free Vibrations

Lev Steinberg and Roman Kvasov

## Abstract

In this chapter, we present the theoretical analysis of the distinctive characteristics of Cosserat plate vibrations. This analysis is based on the dynamic model of the Cosserat plates, which we developed as an extension of the Reissner plate theory. Primarily, we describe the validation of the model, which is based on the comparison with three-dimensional exact solutions. We present the results of the computer simulations, which allow us to identify different characteristics of the plate vibrations. Particularly, we illustrate and discuss the detection and the classification of the additional high resonance frequencies of a plate depending on the shape and orientation of microelements incorporated into the Cosserat plates.

Keywords: variational principle, Cosserat plate vibrations, frequencies of micro-vibrations

## 1. Introduction

The theory of asymmetric elasticity introduced in 1909 by the Cosserat brothers [1] gave rise to a variety of Cosserat plate theories. In 1960s, Green and Naghdi specialized their general theory of Cosserat surface to obtain the linear Cosserat plate [2], while independently Eringen proposed a complete theory of plates in the framework of Cosserat elasticity [3]. Numerous plate theories were formulated afterwards; for the review of the latest developments in the area of Cosserat plates we recommend to turn to [4].

The first theory of Cosserat plates based on the Reissner plate theory was developed in [5] and its finite element modeling is provided in [6]. The parametric theory of Cosserat plate, presented by the authors in [7], includes some additional assumptions leading to the introduction of the splitting parameter. This provided the highest level of approximation to the original three-dimensional problem. The theory provides the equilibrium equations and constitutive relations, and the optimal value of the minimization of the elastic energy of the Cosserat plate. The paper [7] also provides the analytical solutions of the presented plate theory and the threedimensional Cosserat elasticity for simply supported rectangular plate. The comparison of these solutions showed that the precision of the developed Cosserat plate theory is similar to the precision of the classical plate theory developed by Reissner [8, 9].

The numerical modeling of bending of simply supported rectangular plates is given in [10]. We developed the Cosserat plate field equations and a rigorous formula for the optimal value of the splitting parameter. The solution of the

Cosserat plate was shown to converge to the Reissner plate as the elastic asymmetric parameters tend to zero. The Cosserat plate theory demonstrates the agreement with the size effect, confirming that the plates of smaller thickness are more rigid than is expected from the Reissner model. The modeling of Cosserat plates with simply supported rectangular holes is also provided. The finite element analysis of the perforated Cosserat plates is given in [11].

The extension of the static model of Cosserat elastic plates to the dynamic problems is presented in [12]. The computations predict a new kind of natural frequencies associated with the material microstructure and were shown to be compatible with the size effect principle reported in [10] for the Cosserat plate bending.

This chapter represents an extension of the paper [12] for different shapes and orientations of micro-elements incorporated into the Cosserat plates. It is based on the generalized variational principle for elastodynamics and includes a nondiagonal rotatory inertia tensor. The numerical computations of the plate free vibrations showed the existence of some additional high frequencies of microvibrations depending on the orientation of micro-elements. The comparison with three-dimensional Cosserat elastodynamics shows a high agreement with the exact values of the eigenvalue frequencies.
