Modelling and Simulation in Engineering

Volume 2017 (2017), Article ID 5246197, 19 pages

https://doi.org/10.1155/2017/5246197

## Modeling of Size Effects in Bending of Perforated Cosserat Plates

^{1}Department of Mathematics, University of Puerto Rico at Aguadilla, Aguadilla, PR 00604, USA^{2}Department of Mathematical Sciences, University of Puerto Rico at Mayagüez, Mayagüez, PR 00681, USA

Correspondence should be addressed to Lev Steinberg; ude.rpu@grebniets.vel

Received 27 August 2016; Revised 20 November 2016; Accepted 1 December 2016; Published 30 March 2017

Academic Editor: Chung-Souk Han

Copyright © 2017 Roman Kvasov and Lev Steinberg. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents the numerical study of Cosserat elastic plate deformation based on the parametric theory of Cosserat plates, recently developed by the authors. The numerical results are obtained using the Finite Element Method used to solve the parametric system of 9 kinematic equations. We discuss the existence and uniqueness of the weak solution and the convergence of the proposed FEM. The Finite Element analysis of clamped Cosserat plates of different shapes under different loads is provided. We present the numerical validation of the proposed FEM by estimating the order of convergence, when comparing the main kinematic variables with an analytical solution. We also consider the numerical analysis of plates with circular holes. We show that the stress concentration factor around the hole is less than the classical value, and smaller holes exhibit less stress concentration as would be expected on the basis of the classical elasticity.

#### 1. Introduction

A complete theory of asymmetric elasticity introduced by the Cosserat brothers [1] gave rise to a variety of beam, shell, and plate theories. The first theories of plates that take into account the microstructure of the material were developed in the 1960s. Eringen proposed a complete theory of plates in the framework of Cosserat (micropolar) elasticity [2], while independently Green et al. specialized their general theory of Cosserat surface to obtain the linear Cosserat plate [3]. Numerous plate theories were formulated afterwards; for the extensive review of the latest developments we recommend referring to [4].

The first theory of Cosserat elastic 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. 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 three-dimensional Cosserat elasticity for simply supported rectangular plate. The comparison of these solutions showed that the precision of the developed Cosserat plate theory is compatible with 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 develop 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 expected from the Reissner model. The modeling of Cosserat plates with simply supported rectangular holes is also provided.

The extension of the static model of Cosserat elastic plates to the dynamic problems is presented in [11]. The computations predict a new kind of natural frequencies associated with the material microstructure and were shown to be consistent with the size effect principle known from the Cosserat plate deformation reported in [10].

Since [10] was restricted only to the case of rectangular plates, the current article represents an extension of this work for the Finite Element modeling of the Cosserat plates for the different shapes, under different loads and different boundary conditions. We discuss the existence and uniqueness of the weak solution, convergence of the proposed FEM, and its numerical validation.

We also present the modeling of size effects in Cosserat plates with holes. We evaluate the stress concentration factor around the hole and show that the factor is smaller than the one based on the Reissner theory for simple elastic plates. The Finite Element comparison of the plates with holes confirms that smaller holes exhibit less stress concentration than the larger holes.

#### 2. Parametric Theory of Cosserat Plates

In this section we provide the main equations of the theory presented in [7].

Throughout this article Greek indices are assumed to range from 1 to 2, while the Latin indices range from 1 to 3 if not specified otherwise. We will also employ the Einstein summation convention according to which summation is implied for any repeated index.

Let us consider a thin plate of thickness and representing its middle plane. The sets and are the top and bottom surfaces contained in the planes and , respectively, and the curve is the boundary of the middle plane of the plate. The set of points forms the entire surface of the plate. is the part of the boundary where displacements and microrotations are prescribed, while is the part of the boundary where stress and couple stress are prescribed.

The equilibrium system of equations for Cosserat plate bending is given aswhere and are the bending moments, and are twisting moments, are shear forces, , are transverse shear forces, , , , are micropolar bending moments, , , , are micropolar twisting moments, and are micropolar couple moments, all defined per unit length. The initial pressure is represented here by the pressures and , where is the splitting parameter [7].

The constitutive formulas for Cosserat plate are given in the following form [7]:where and are symmetric constants and , , , and are the asymmetric Cosserat elasticity constants [5].

The vector of kinematic variables is given as where

The system of equilibrium equations is accompanied by the zero variation of the work density with respect to the splitting parameter . This allows us to split the bending pressure on the plate into two parts corresponding to different orders of stress approximation.whereand and are the Cosserat plate stress and strain sets

The optimal value of the splitting parameter is given as in [10]where .

The approximation of the components of the three-dimensional tensors and iswhere

The approximation of components of the three-dimensional strain and torsion tensors and is

The components of the three-dimensional displacements and microrotations [7] arewhere and as before.

We consider the vertical load and pure twisting momentum boundary conditions at the top and bottom of the plate

The expressions for the three-dimensional stress, strain, and displacement components satisfy the equilibrium Cosserat elasticity equationsthe strain-displacement and torsion-rotation relationsand the constitutive equationswhere is the Levi-Civita symbol.

In order to obtain the micropolar plate bending field equations in terms of the kinematic variables, the constitutive formulas in the reverse form are substituted into the bending system of (1). The obtained Cosserat plate bending field equations can be represented as an elliptic system of nine partial differential equations in terms of the kinematic variables [10]:where is a linear differential operator acting on the vector of kinematic variables (unknowns):and is the right-hand side vector that in general depends on :

The operators are given as follows:

The coefficients are given as

#### 3. Finite Element Method for Cosserat Plates

The right-hand side of the system (17) depends on the splitting parameter and so does the solution that we will formally be denoted as . Therefore the solution of the Cosserat elastic plate bending problem requires not only solving the system (17), but also an additional technique for the calculation of the value of the splitting parameter that corresponds to the unique solution. The use of the splitting parameter adds an additional degree of freedom to the modeling. The optimal value of this parameter provides the highest level of approximation to the original 3-dimensional problem. Considering that the elliptic systems of partial differential equations correspond to a state where the minimum of the energy is reached, the optimal value of the splitting parameter should minimize the elastic plate energy [12]. The minimization corresponds to the zero variation of the plate stress energy (5).

The Finite Element Method for Cosserat elastic plates is based on the algorithm for the optimal value of the splitting parameter. This algorithm requires solving the system of equations (17) for two different values of the splitting parameter , numerical calculation of stresses, strains, and the corresponding work densities. We will follow [10] in the description of our Finite Element Method algorithm.

(1) Use classical Galerkin FEM to solve two elliptic systems:for and , respectively.

(2) Calculate the optimal value of the splitting parameter using (8).

(3) Calculate the optimal solution of the Cosserat plate bending problem as a linear combination of and :

##### 3.1. Weak Formulation of the Clamped Cosserat Plate

Let us consider the following hard clamped boundary conditions similar to [13]:where and are the normal and the tangent vectors to the boundary. These conditions represent homogeneous Dirichlet type boundary conditions for the kinematic variables:

Let us denote by the standard space of square-integrable functions defined everywhere on and by the Hilbert space of functions that are square-integrable together with their first partial derivatives:

Let us denote the Hilbert space of functions from that vanish on the boundary as in [14]:

The space is equipped with the inner product:

Taking into account the fact that the boundary conditions for all variables are of the same homogeneous Dirichlet type, we look for the solution in the function space defined as

The space is equipped with the inner product :and relative to the metric induced by the norm , the space is a complete metric space and therefore is a Hilbert space [15].

Let us consider a dot product of both sides of the system of the field equations (17) and an arbitrary function : and then integrate both sides of the obtained scalar equation over the plate :

Let us introduce a bilinear form and a linear form defined as

The expression for is a summation over the terms of the formwhere , , and is a scalar differential operator.

There are 3 types of linear operators present in the field equations (17)—operators of order zero, one, and two, which are constant multiples of the following differential operators:where is a real matrix. For example, for the order two operator the corresponding matrix would be .

These operators act on the components of the vector and are multiplied by the components of the vector and the obtained expressions are then integrated over : where and .

The weak form of the second-order operator is obtained by performing the corresponding integration by parts and taking into account the fact that the test functions vanish on the boundary :

The expression for represents a summation over the terms of the form:

Taking into account the fact that the optimal solution of the field equations (17) minimizes the plate stress energy, we can give the weak formulation for the clamped Cosserat plate bending problem.

*Weak Formulation of the Clamped Cosserat Plate Bending Problem. Find all ** and ** that minimize the plate stress energy ** subject to*

##### 3.2. Construction of the Finite Element Spaces

Let us construct the Finite Element space, that is, finite dimensional subspace of the space , where we will be looking for an approximate Finite Element solution of the weak formulation (43).

Let us assume that the boundary is a polygonal curve. Let us make a triangulation of the domain by subdividing into nonoverlapping triangles with vertices : such that no vertex of the triangular element lies on the edge of another triangle (see Figure 1).