Research Article | Open Access
Modeling of Size Effects in Bending of Perforated Cosserat Plates
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.
A complete theory of asymmetric elasticity introduced by the Cosserat brothers  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 , while independently Green et al. specialized their general theory of Cosserat surface to obtain the linear Cosserat plate . Numerous plate theories were formulated afterwards; for the extensive review of the latest developments we recommend referring to .
The first theory of Cosserat elastic plates based on the Reissner plate theory was developed in  and its Finite Element modeling is provided in . The parametric theory of Cosserat plate, presented by the authors in , 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  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 . 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 . 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 .
Since  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 .
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 .
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 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  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 :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 . 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  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 :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 :
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 .
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).
Let us introduce the mesh parameter as the greatest diameter among the elements : which for the triangular elements corresponds to the length of the longest side of the triangle.
We now define the finite dimensional space as a space of all continuous functions that are linear on each element and vanish on the boundary:By definition , and the Finite Element space is then defined as
The approximate weak solution can be found from the Galerkin formulation of the clamped Cosserat plate bending problem .
Galerkin Formulation of the Clamped Cosserat Plate. Find all and that minimize the stress plate energy subject to
The description of the function is provided by the values at the nodes ().
Let us define the set of basis functions of each space as excluding the points on the boundary .
Therefore and the functions are nonzero only at the node and those that belong to the specified boundary and the support of consists of all triangles with the common node (see Figure 2).
Since the spaces are identical they will also have identical sets of basis functions (). Sometimes we will need to distinguish between the basis functions of different spaces assigning the superscript of the functions space to the basis function; that is, the basis functions for the space are . For computational purposes these superscripts will be dropped.
3.3. Calculation of the Stiffness Matrix and the Load Vector
The bilinear form of the Galerkin formulation (48) is given as
Since then there exist such constants that
Since (51) is satisfied for all then it is also satisfied for all basis functions ():where
We define the block stiffness matrices ():
For computational purposes the superscripts of the basis functions can be dropped and the block stiffness matrices can be calculated as
Let us define the block load vectors ()and the solution block vectors corresponding to the variable ():
Equation (48) of the Galerkin formulation can be rewritten as
The global stiffness matrix consists of 81 block stiffness matrices , the global load vector consists of 9 block load vectors , and the global displacement vector is represented by the 9 blocks of coefficients . The entries of the block matrices and the block vectors can be calculated as
The block matrix form of equation (48) is given as
3.4. Existence and Convergence Remarks
Employing integration by parts for the second-order operators the bilinear form can be rewritten in the following form: where and are multi-indices.
Since coefficients are constant and therefore bounded on , the bilinear form is continuous over ; that is, there exists a constant such that
The strong ellipticity of the operator was shown in . Since the operator is strongly elliptic on the bilinear form is -elliptic on [12, 17]; that is, there exists a constant such that The existence of the solution of the weak form (43) and its uniqueness are the consequences of the Lax-Milgram Theorem [19, 20]. Note that the existence and uniqueness of the Galerkin weak problem (48) are also a consequence of the Lax-Milgram theorem, since the bilinear form restricted on obviously remains bilinear, continuous, and -elliptic . Lax-Milgram theorem also states that the solution is bounded by the right-hand side which represents the stability condition for the Galerkin method.
The convergence of the Galerkin approximation follows from Céa’s lemma and an additional convergence theorem [12, 21]. On the polygonal domains the sequence of subspaces of can be obtained by the successive uniform refinement of the initial mesh using the midpoints as new nodes thus subdividing every triangle into 4 congruent triangles. Therefore for every and the sequence of spaces is dense in , and thus and converges to as [12, 23].
It was shown that there exists a sequence of triangulations that ensures optimal rates of convergence in -norm for the FEM approximation of the second-order strongly elliptic system with zero Dirichlet boundary condition on polyhedron domain with continuous, piecewise polynomials of degree . For the details on different norms and rates of convergence of the FEM we refer the reader to .
4. Validation of the FEM for Different Boundary Conditions
Let us consider the plate to be a square plate of size with the boundary and the hard simply supported boundary conditions similar to , written in terms of the kinematic variables in the mixed Dirichlet-Neumann type:where
The existence of a sequence of triangulations that ensures the optimal rates of convergence for the Finite Element approximation of the solution of a second-order strongly elliptic system with homogeneous Dirichlet boundary condition on polyhedron domain with continuous piecewise polynomials was shown in . For the case of piecewise linear polynomials the optimal rate of convergence in -norm is linear.
We propose using the uniform refinement to form the sequence of triangulations and estimate the order of the error of approximation of the proposed FEM in -norm and -norm.
In our computations we consider plates made of polyurethane foam—a material reported in the literature to behave Cosserat-like and the values of the technical elastic parameters are presented in :
Taking into account the fact that the ratio is equal to 1 for bending , these values of the technical constants correspond to the following values of symmetric and asymmetric parameters:
Let us consider homogeneous Dirichlet boundary conditions. We will assume the analytical solution of the formwhich automatically satisfies homogeneous Dirichlet boundary conditions. Substituting the solution (70) into the system of field equations (17) we can find the corresponding right-hand side function . The results of the error estimation of the FEM approximation in and norms performed for the elastic parameters corresponding to the polyurethane foam are given in Tables 1 and 2, respectively.
Let us consider mixed Neumann-Dirichlet boundary conditions. Simply supported boundary conditions represent this type of boundary conditions and therefore the FEM approximation can be compared with the analytical solution developed in  for some fixed value of the parameter . The results of the error estimation of the FEM approximation in and norms performed for the elastic parameters corresponding to the polyurethane foam are given in Tables 3 and 4, respectively.
The boundary condition for the variable is a Neumann type boundary condition: and thus we will look for in the space , where
The boundary condition for the variables and is a Dirichlet type boundary condition: and thus we will look for and in the space defined as 
The boundary condition for the variables , , and is of mixed Dirichlet-Neumann type:and thus we will look for , , and in the following space :
The boundary condition for the variables , , and is of mixed Dirichlet-Neumann type: and thus we will look for , , and in the following space :
Therefore we will look for the solutionof the Cosserat plate field equations (17) in the space defined aswhere
The space is a Hilbert space equipped with the inner product defined on as follows: where is an inner product defined on the Hilbert space , respectively.
Taking into account the essential boundary conditions we define the Finite Element spaces as follows: