# Modeling of Size Effects in Bending of Perforated Cosserat Plates

Authors: Roman Kvasov; Lev Steinberg
Journal: Modelling and Simulation in Engineering
 (2017)

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## 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.


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## Body

## 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 plateP of thickness h and x3=0 representing its middle plane. The sets T and B are the top and bottom surfaces contained in the planes x3=h/2 and x3=-h/2, respectively, and the curve Γ is the boundary of the middle plane of the plate. The set of points P=Γ×[-h/2,h/2]∪T∪B forms the entire surface of the plate. Γu×[-h/2,h/2] is the part of the boundary where displacements and microrotations are prescribed, while Γσ×-h/2,h/2 is the part of the boundary where stress and couple stress are prescribed.The equilibrium system of equations for Cosserat plate bending is given as(1)Mαβ,α-Qβ=0,Qα,α∗+p^1=0,Rαβ,α+ε3βγQγ∗-Qγ=0,ε3βγMβγ+Sα,α∗=0,Q^α,α+p^2=0,Rαβ,α∗+ε3βγQ^γ=0,where M11 and M22 are the bending moments, M12 and M21 are twisting moments, Qα are shear forces, Qα∗, Q^α are transverse shear forces, R11, R22, R11∗, R22∗ are micropolar bending moments, R12, R21, R12∗, R21∗ are micropolar twisting moments, and Sα∗ are micropolar couple moments, all defined per unit length. The initial pressure p is represented here by the pressures p^1=ηoptp and p^2=(2/3)1-ηoptp, where ηopt is the splitting parameter [7].The constitutive formulas for Cosserat plate are given in the following form [7]:(2)eαα=12λ+μh3μ3λ+2μMαα-εαβ36λh3μ3λ+2μMββ-λ3p1+5p25hμ3λ+2μ,eαβ=3α+μh3αμMαβ+3α-μh3αμMβα,α≠β,ωα=3α-μ10hαμQα∗+3α+μ10hαμQα+3α-μ8hαμQ^α,ωα∗=3α-μ10hαμQα+3α+μ10hαμQα∗+3α+μ8hαμQ^α,ω^α∗=3α-μ8hαμQα+3α+μ8hαμQα∗+9α+μ16hαμQ^α,ταα0=6β+γ5hγ3β+2γRαα-εαβ33β5hγ3β+2γRββ∗+3β+γ2hγ3β+2γRαα∗-εαβ33β4hγ3β+2γRββ,ταβ0=3ϵ+γ10hγϵRαβ-3γ-ϵ10hγϵRβα+3ϵ+γ8hγϵRαβ∗-3γ-ϵ8hγϵRβα∗,α≠β,ταα∗=3β+γ2hγ3β+2γRαα-εαβ39β8hγ3β+2γRββ∗+3β+γ4hγ3β+2γRαα∗εαβ33β4hγ3β+2γRββ,ταβ∗=3ϵ+γ8hγϵRαβ-3γ-ϵ8hγϵRβα+3ϵ+γ16hγϵRαβ∗-3γ-ϵ16hγϵRβα∗,α≠β,τ3α=3γ+ϵh3γϵSα∗,where λ and μ are symmetric constants and α, β, γ, and ϵ are the asymmetric Cosserat elasticity constants [5].The vector of kinematic variables is given as(3)v=Ψα,W,Ω3,Ωα0,W∗,Ωα0T, where(4)eαβ=Ψβ,α+ε3αβΩ3,ωα=Ψα+ε3αβΩβ0+Ω^β,ωα∗=W,α+45W,α∗+ε3αβΩβ0+Ω^β,ω^α=32W,α+W,α∗+ε3αβ54Ωβ0+32Ω^β,τ3α=Ω3,α,ταβ=Ωβ,α0+Ω^β,α,ταβ∗=54Ωβ,α0+32Ω^β,α.The system of equilibrium equations is accompanied by the zero variation of the work density with respect to the splitting parameterη. This allows us to split the bending pressure on the plate p(x1,x2) into two parts corresponding to different orders of stress approximation.(5)δWη=0,where(6)Wη=Sη·Eη=Mαβeαβ+Qαωα+Q3α∗ωα∗+Q^αω^α+Rαβταβ+Rαβ∗ταβ∗+Sα∗τ3α,and S(η) and E(η) are the Cosserat plate stress and strain sets(7)Sη=Mαβ,Qα,Qα∗,Q^α,Rαβ,Rαβ∗,Sβ∗,Eη=eαβ,ωβ,ωα∗,ω^α,τ3α,ταβ,ταβ∗.The optimal valueηopt of the splitting parameter is given as in [10](8)ηopt=2W00-W10-W012W11+W00-W10-W01,where W(ij)=S(i)·E(j).The approximation of the components of the three-dimensional tensorsσji and μji is(9)σαβ=6h2ζMαβx1,x2,σ3β=32h1-ζ2Qβx1,x2,σβ3=32h1-ζ2Qβ∗x1,x2+32hQ^βx1,x2,σ33=-3413ζ3-ζp1x1,x2+ζp2x1,x2+σ0x1,x2,μαβ=32h1-ζ2Rαβx1,x2+32hRαβ∗x1,x2,μβ3=6h2ζSβ∗x1,x2,μ3β=0,μ33=ζvx1,x2+tx1,x2,where(10)ζ=2hx3,px1,x2=σtx1,x2-σbx1,x2,σ0x1,x2=12σtx1,x2+σbx1,x2,vx1,x2=12μtx1,x2-μbx1,x2,tx1,x2=12μtx1,x2+μbx1,x2,p1x1,x2=ηoptpx1,x2,p2x1,x2=1-ηopt2px1,x2.The approximation of components of the three-dimensional strain and torsion tensorsγji and χji is(11)γαβ=6h2ζeαβx1,x2,γ3β=32h1-ζ2ωβx1,x2,γβ3=32h1-ζ2ωβ∗x1,x2+32hω^βx1,x2,χαβ=32h1-ζ2ταβx1,x2+32hταβ∗x1,x2,χ3β=6h2ζτβ∗x1,x2.The components of the three-dimensional displacementsui and microrotations ϕi [7] are(12)uα=h2ζΨαx1,x2,u3=Wx1,x2+1-ζ2W∗x1,x2,ϕα=Ωα0x1,x21-ζ2+Ω^αx1,x2,ϕ3=ζΩ3x1,x2,where ζ=2x3/h and α,β∈{1,2} as before.We consider the vertical load and pure twisting momentum boundary conditions at the top and bottom of the plate(13)σ33x1,x2,h2=σtx1,x2,σ33x1,x2,-h2=σbx1,x2,σ3βx1,x2,±h2=0,μ33x1,x2,h2=μtx1,x2,μ33x1,x2,-h2=μbx1,x2,μ3βx1,x2,±h2=0.The expressions for the three-dimensional stress, strain, and displacement components satisfy the equilibrium Cosserat elasticity equations(14)σji,j=0,εijkσjk+μji,j=0,the strain-displacement and torsion-rotation relations(15)γji=ui,j+εijkϕk,χji=ϕi,j,and the constitutive equations(16)σji=μ+αγji+μ-αγij+λγkkδij,μji=γ+ϵχji+γ-ϵχij+βχkkδij,where εijk 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]:(17)Lv=fη,where L is a linear differential operator acting on the vector of kinematic variables v (unknowns):(18)L=L11L12L13L140L16k1L130L16L12L22L23L24L160k1L23L160-L13-L23L330L35L36k1L77L38L39L41L420L44000000-L16-L380L55L56-k1L35L580L160-L390L56L66-k1L360L58-L13-L14L730L35L36k1L77L78L790-L16-L780L85L56-k1L35k1L88k1L56L160-L790L56L55-k1L36k1L56k1L99,and fη is the right-hand side vector that in general depends on η:(19)fη=-h2λ3p1,1+5p2,130λ+2μ-h2λ3p1,2+5p2,230λ+2μ-p1000h23p1+4p22400T.The operatorsLij are given as follows:(20)L11=c1∂2∂x12+c2∂2∂x22-c3,L12=c1-c2∂2∂x1x2,L13=c11∂∂x1,L14=c12∂∂x2,L16=c13,L17=k1c11∂∂x1,L22=c2∂2∂x12+c1∂2∂x22-c3,L23=c11∂∂x2,L24=-c12∂∂x1,L33=c3∂2∂x12+∂2∂x22,L35=-c13∂∂x2,L36=c13∂∂x1,L38=-c10∂∂x2,L39=c10∂∂x1,L41=-c12∂∂x2,L42=c12∂∂x1,L44=c6∂2∂x12+∂2∂x22-2c12,L55=c7∂2∂x12+c8∂2∂x22-2c13,L56=c7-c8∂2∂x1x2,L58=-c9,L66=c8∂2∂x12+c7∂2∂x22-2c13,L73=c5∂2∂x12+∂2∂x22,L77=c4∂2∂x12+∂2∂x22,L78=-c14∂∂x2,L79=c14∂∂x1,L85=c7∂2∂x12+c8∂2∂x22-2c13,L88=c7∂2∂x12+c8∂2∂x22-c15,L99=c8∂2∂x12+c7∂2∂x22-c15.The coefficientsci are given as (21)c1=h3μλ+μ3λ+2μ,c2=h3α+μ12,c3=5hα+μ6,c4=5hα-μ26α+μ,c5=h5α2+6αμ+5μ26α+μ,c6=h3γϵ3γ+ϵ,c7=10hγβ+γ3β+2γ,c8=5hγ+ϵ6,c9=10hα23α+μ,c10=5hαα-μ3α+μ,c11=5hα-μ6,c12=h3α6,c13=5hα3,c14=hα5α+3μ3α+μ,c15=2hα5α+4μ3α+μ.


## 3. Finite Element Method for Cosserat Plates

The right-hand side of the system (17) depends on the splitting parameter η and so does the solution v that we will formally be denoted as vη. 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 η, 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:(22)Lv0=f0,Lv1=f1,for v0 and v1, respectively.(2) Calculate the optimal value of the splitting parameterηopt using (8).(3) Calculate the optimal solutionvη0 of the Cosserat plate bending problem as a linear combination of v0 and v1:(23)vη0=1-ηoptv0+ηoptv1.


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

Let us consider the following hard clamped boundary conditions similar to [13]:(24)W=0,W∗=0,Ψ·s^=0,Ψ·n^=0,Ω3=0,Ω0·s^=0,Ω0·n^=0,Ω^·s^=0,Ω^·n^=0,where n^ and s^ are the normal and the tangent vectors to the boundary. These conditions represent homogeneous Dirichlet type boundary conditions for the kinematic variables:(25)W=0,W∗=0,Ψ1=0,Ψ2=0,Ω3=0,Ω^10=0,Ω^20=0,Ω^10=0,Ω^20=0.Let us denote byL2B0 the standard space of square-integrable functions defined everywhere on B0(26)L2B0=v:∫B0v2ds<∞ and by H1B0 the Hilbert space of functions that are square-integrable together with their first partial derivatives:(27)H1B0=v:v∈L2B0,∂iv∈L2B0.Let us denote the Hilbert space of functions fromH1B0 that vanish on the boundary as in [14]:(28)H01B0=v∈H1B0,v=0on∂B0.The spaceH01B0 is equipped with the inner product:(29)u,vH01=∫B0uv+∂iu∂ivdsforu,v∈H01B0.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 spaceHB0 defined as(30)H=H01B09.The spaceH is equipped with the inner product u,vH:(31)u,vH=∑i=19ui,viH01foru,v∈Hand relative to the metric(32)du,v=u-vHforu,v∈H, induced by the norm x=x,xH, the space H 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 v∈H:(33)v·Lu=v·fη and then integrate both sides of the obtained scalar equation over the plate B0:(34)∫B0v·Luds=∫B0v·fηds.Let us introduce a bilinear formau,v:H×H→R and a linear form b(η)v:H→R defined as(35)au,v=∫B0v·Luds,bηv=∫B0v·fηds.The expression forau,v(36)av,u=∫B0viLijujds is a summation over the terms of the form(37)aijvm,un=∫B0vmL^unds,where vm∈Hm, un∈Hn, and L^ 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:(38)L0=1,L1=∂∂xα,L2=-∇·A∇,where A is a 2×2 real matrix. For example, for the order two operator k1(∂2/∂x12)+k2(∂2/∂x22) the corresponding matrix A would be A=k100k2.These operators act on the components of the vectoru and are multiplied by the components of the vector v and the obtained expressions are then integrated over B0:(39)∫B0vmL0unds=∫B0vmunds,∫B0vmL1unds=∫B0vm∂un∂xαds,∫B0vmL2unds=-∫B0vm∇·A∇unds, where vm∈Hm and un∈Hn.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 functionsvm vanish on the boundary ∂B0:(40)∫B0vmL2unds=-∫B0vm∇·A∇unds=-∫∂B0A∇un·nvmdτ+∫B0A∇un·∇vmds,av,u=∫B0A∇un·∇vmds.The expression forb(η)v(41)bηv=∫B0vifiηdsrepresents a summation over the terms of the form:(42)∫B0vmf^ηds.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
u
∈
H
and
η
∈
R
that minimize the plate stress energy
U
K
S
u
,
η
subject to
(43)
a
v
,
u
=
b
η
v
∀
v
∈
H
.


### 3.2. Construction of the Finite Element Spaces

Let us construct the Finite Element space, that is, finite dimensional subspaceHh of the space H, where we will be looking for an approximate Finite Element solution of the weak formulation (43).Let us assume that the boundary∂B0 is a polygonal curve. Let us make a triangulation of the domain B0 by subdividing B0 into l nonoverlapping triangles Ki with m vertices Nj:(44)B0=⋃i=1lKi=K1∪K2∪⋯∪Kl such that no vertex of the triangular element lies on the edge of another triangle (see Figure 1).Figure 1
Example of the Finite Element triangulation of the domainB0.Let us introduce the mesh parameterh as the greatest diameter among the elements Ki:(45)h=maxi=1,l¯dKi, which for the triangular elements corresponds to the length of the longest side of the triangle.We now define the finite dimensional spaceH^h as a space of all continuous functions that are linear on each element Kj and vanish on the boundary:(46)H^h=Hih=v:v∈CB0,vis  linear  on  everyKj,v=0on∂B0.By definition Hih⊂Hi, and the Finite Element space Hh is then defined as(47)Hh=H^h9.The approximate weak solutionuh can be found from the Galerkin formulation of the clamped Cosserat plate bending problem [16].Galerkin Formulation of the Clamped Cosserat Plate. Find all
u
h
∈
H
h
and
η
∈
R
that minimize the stress plate energy
U
K
S
u
h
,
η
subject to
(48)
a
v
h
,
u
h
=
b
η
v
h
∀
v
h
∈
H
h
.The description of the functionvih∈Hih is provided by the values vihNk at the nodes Nk (k=1,m¯).Let us define the set of basis functionsϕ1,ϕ2,…,ϕm of each space Hih as(49)ϕjNk=δjk,j,k=1,m¯ excluding the points Nk on the boundary ∂B0.Therefore(50)Hih=span⁡ϕ1,ϕ2,…,ϕm=v:v=∑j=1mαjiϕj and the functions ϕj are nonzero only at the node Nj and those that belong to the specified boundary and the support of ϕj consists of all triangles Ki with the common node Nj (see Figure 2).Figure 2
Example of the Finite Element basis function.Since the spacesHih are identical they will also have identical sets of basis functions ϕj (j=1,m¯). 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 Hih are ϕji. 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(51)avh,uh=aijvih,ujh=∫B0vihLijujhds.Sinceujh∈Hjh then there exist such constants αp(j)∈R that(52)ujh=αpjϕpi.Since (51) is satisfied for all vih∈Hih then it is also satisfied for all basis functions ϕk(i) (k=1,m¯):(53)aijvih,ujh=aijϕki,αpjϕpj=αpjaijϕki,ϕpj,where(54)aijv,u=∫B0vLijuds.We define the block stiffness matricesKij (i,j=1,9¯):(55)Kij=aijϕ1i,ϕ1j⋯aijϕ1i,ϕmj⋮⋱⋮aijϕmi,ϕ1j⋯aijϕmi,ϕmj.For computational purposes the superscripts of the basis functions can be dropped and the block stiffness matricesKij can be calculated as(56)Kij=aijϕ1,ϕ1⋯aijϕ1,ϕm⋮⋱⋮aijϕm,ϕ1⋯aijϕm,ϕm.Let us define the block load vectorsFi(η) (i=1,9¯)(57)Fiη=bηiϕ1⋮bηiϕmand the solution block vectors αi corresponding to the variable uih (i=1,9¯):(58)αi=α1i⋮αmi.Equation (48) of the Galerkin formulation can be rewritten as(59)Kijαi=Fjη.The global stiffness matrix consists of 81 block stiffness matricesKij, the global load vector consists of 9 block load vectors Fi(η), and the global displacement vector is represented by the 9 blocks of coefficients αi. The entries of the block matrices Kij and the block vectors Fi(η) can be calculated as(60)Kmnij=∫B0ϕmLijϕnds,Fmiη=∫B0ϕmfiηds.The block matrix form of equation (48) is given as (61)K11K12⋯K19K21K22⋯K29⋮⋮⋱⋮K91K92⋯K99α1α2⋮α9=F1ηF2η⋮F9η.


### 3.4. Existence and Convergence Remarks

We will follow [17, 18], where the analysis of the analytic regularity for linear elliptic systems and their general treatment are recently presented.Employing integration by parts for the second-order operatorsLij the bilinear form a(·,·) can be rewritten in the following form:(62)au,v=∑β,γ≤1∫B0cij∂βvi∂γujds, where β and γ are multi-indices.Since coefficientscij are constant and therefore bounded on B0, the bilinear form a(·,·) is continuous over H [17]; that is, there exists a constant C>0 such that(63)av,u≤CvHuH∀u,v∈H.The strong ellipticity of the operatorL was shown in [10]. Since the operator L is strongly elliptic on B0 the bilinear form a(·,·) is V-elliptic on H [12, 17]; that is, there exists a constant α>0 such that(64)au,u≥αuH2∀v∈H. 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 a(·,·) restricted on Hh obviously remains bilinear, continuous, and V-elliptic [12]. 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 H=H01B09 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 Hn⊂Hn+1 for every n∈N and the sequence of spaces Hn is dense in H [22], and thus(65)⋃n=1∞Hn¯=H and un converges to u as n→∞ [12, 23].It was shown that there exists a sequence of triangulations that ensures optimal rates of convergence inH1-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 m [24]. For the details on different norms and rates of convergence of the FEM we refer the reader to [20].


## 3.1. Weak Formulation of the Clamped Cosserat Plate

Let us consider the following hard clamped boundary conditions similar to [13]:(24)W=0,W∗=0,Ψ·s^=0,Ψ·n^=0,Ω3=0,Ω0·s^=0,Ω0·n^=0,Ω^·s^=0,Ω^·n^=0,where n^ and s^ are the normal and the tangent vectors to the boundary. These conditions represent homogeneous Dirichlet type boundary conditions for the kinematic variables:(25)W=0,W∗=0,Ψ1=0,Ψ2=0,Ω3=0,Ω^10=0,Ω^20=0,Ω^10=0,Ω^20=0.Let us denote byL2B0 the standard space of square-integrable functions defined everywhere on B0(26)L2B0=v:∫B0v2ds<∞ and by H1B0 the Hilbert space of functions that are square-integrable together with their first partial derivatives:(27)H1B0=v:v∈L2B0,∂iv∈L2B0.Let us denote the Hilbert space of functions fromH1B0 that vanish on the boundary as in [14]:(28)H01B0=v∈H1B0,v=0on∂B0.The spaceH01B0 is equipped with the inner product:(29)u,vH01=∫B0uv+∂iu∂ivdsforu,v∈H01B0.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 spaceHB0 defined as(30)H=H01B09.The spaceH is equipped with the inner product u,vH:(31)u,vH=∑i=19ui,viH01foru,v∈Hand relative to the metric(32)du,v=u-vHforu,v∈H, induced by the norm x=x,xH, the space H 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 v∈H:(33)v·Lu=v·fη and then integrate both sides of the obtained scalar equation over the plate B0:(34)∫B0v·Luds=∫B0v·fηds.Let us introduce a bilinear formau,v:H×H→R and a linear form b(η)v:H→R defined as(35)au,v=∫B0v·Luds,bηv=∫B0v·fηds.The expression forau,v(36)av,u=∫B0viLijujds is a summation over the terms of the form(37)aijvm,un=∫B0vmL^unds,where vm∈Hm, un∈Hn, and L^ 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:(38)L0=1,L1=∂∂xα,L2=-∇·A∇,where A is a 2×2 real matrix. For example, for the order two operator k1(∂2/∂x12)+k2(∂2/∂x22) the corresponding matrix A would be A=k100k2.These operators act on the components of the vectoru and are multiplied by the components of the vector v and the obtained expressions are then integrated over B0:(39)∫B0vmL0unds=∫B0vmunds,∫B0vmL1unds=∫B0vm∂un∂xαds,∫B0vmL2unds=-∫B0vm∇·A∇unds, where vm∈Hm and un∈Hn.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 functionsvm vanish on the boundary ∂B0:(40)∫B0vmL2unds=-∫B0vm∇·A∇unds=-∫∂B0A∇un·nvmdτ+∫B0A∇un·∇vmds,av,u=∫B0A∇un·∇vmds.The expression forb(η)v(41)bηv=∫B0vifiηdsrepresents a summation over the terms of the form:(42)∫B0vmf^ηds.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
u
∈
H
and
η
∈
R
that minimize the plate stress energy
U
K
S
u
,
η
subject to
(43)
a
v
,
u
=
b
η
v
∀
v
∈
H
.


## 3.2. Construction of the Finite Element Spaces

Let us construct the Finite Element space, that is, finite dimensional subspaceHh of the space H, where we will be looking for an approximate Finite Element solution of the weak formulation (43).Let us assume that the boundary∂B0 is a polygonal curve. Let us make a triangulation of the domain B0 by subdividing B0 into l nonoverlapping triangles Ki with m vertices Nj:(44)B0=⋃i=1lKi=K1∪K2∪⋯∪Kl such that no vertex of the triangular element lies on the edge of another triangle (see Figure 1).Figure 1
Example of the Finite Element triangulation of the domainB0.Let us introduce the mesh parameterh as the greatest diameter among the elements Ki:(45)h=maxi=1,l¯dKi, which for the triangular elements corresponds to the length of the longest side of the triangle.We now define the finite dimensional spaceH^h as a space of all continuous functions that are linear on each element Kj and vanish on the boundary:(46)H^h=Hih=v:v∈CB0,vis  linear  on  everyKj,v=0on∂B0.By definition Hih⊂Hi, and the Finite Element space Hh is then defined as(47)Hh=H^h9.The approximate weak solutionuh can be found from the Galerkin formulation of the clamped Cosserat plate bending problem [16].Galerkin Formulation of the Clamped Cosserat Plate. Find all
u
h
∈
H
h
and
η
∈
R
that minimize the stress plate energy
U
K
S
u
h
,
η
subject to
(48)
a
v
h
,
u
h
=
b
η
v
h
∀
v
h
∈
H
h
.The description of the functionvih∈Hih is provided by the values vihNk at the nodes Nk (k=1,m¯).Let us define the set of basis functionsϕ1,ϕ2,…,ϕm of each space Hih as(49)ϕjNk=δjk,j,k=1,m¯ excluding the points Nk on the boundary ∂B0.Therefore(50)Hih=span⁡ϕ1,ϕ2,…,ϕm=v:v=∑j=1mαjiϕj and the functions ϕj are nonzero only at the node Nj and those that belong to the specified boundary and the support of ϕj consists of all triangles Ki with the common node Nj (see Figure 2).Figure 2
Example of the Finite Element basis function.Since the spacesHih are identical they will also have identical sets of basis functions ϕj (j=1,m¯). 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 Hih are ϕji. 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(51)avh,uh=aijvih,ujh=∫B0vihLijujhds.Sinceujh∈Hjh then there exist such constants αp(j)∈R that(52)ujh=αpjϕpi.Since (51) is satisfied for all vih∈Hih then it is also satisfied for all basis functions ϕk(i) (k=1,m¯):(53)aijvih,ujh=aijϕki,αpjϕpj=αpjaijϕki,ϕpj,where(54)aijv,u=∫B0vLijuds.We define the block stiffness matricesKij (i,j=1,9¯):(55)Kij=aijϕ1i,ϕ1j⋯aijϕ1i,ϕmj⋮⋱⋮aijϕmi,ϕ1j⋯aijϕmi,ϕmj.For computational purposes the superscripts of the basis functions can be dropped and the block stiffness matricesKij can be calculated as(56)Kij=aijϕ1,ϕ1⋯aijϕ1,ϕm⋮⋱⋮aijϕm,ϕ1⋯aijϕm,ϕm.Let us define the block load vectorsFi(η) (i=1,9¯)(57)Fiη=bηiϕ1⋮bηiϕmand the solution block vectors αi corresponding to the variable uih (i=1,9¯):(58)αi=α1i⋮αmi.Equation (48) of the Galerkin formulation can be rewritten as(59)Kijαi=Fjη.The global stiffness matrix consists of 81 block stiffness matricesKij, the global load vector consists of 9 block load vectors Fi(η), and the global displacement vector is represented by the 9 blocks of coefficients αi. The entries of the block matrices Kij and the block vectors Fi(η) can be calculated as(60)Kmnij=∫B0ϕmLijϕnds,Fmiη=∫B0ϕmfiηds.The block matrix form of equation (48) is given as (61)K11K12⋯K19K21K22⋯K29⋮⋮⋱⋮K91K92⋯K99α1α2⋮α9=F1ηF2η⋮F9η.


## 3.4. Existence and Convergence Remarks

We will follow [17, 18], where the analysis of the analytic regularity for linear elliptic systems and their general treatment are recently presented.Employing integration by parts for the second-order operatorsLij the bilinear form a(·,·) can be rewritten in the following form:(62)au,v=∑β,γ≤1∫B0cij∂βvi∂γujds, where β and γ are multi-indices.Since coefficientscij are constant and therefore bounded on B0, the bilinear form a(·,·) is continuous over H [17]; that is, there exists a constant C>0 such that(63)av,u≤CvHuH∀u,v∈H.The strong ellipticity of the operatorL was shown in [10]. Since the operator L is strongly elliptic on B0 the bilinear form a(·,·) is V-elliptic on H [12, 17]; that is, there exists a constant α>0 such that(64)au,u≥αuH2∀v∈H. 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 a(·,·) restricted on Hh obviously remains bilinear, continuous, and V-elliptic [12]. 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 H=H01B09 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 Hn⊂Hn+1 for every n∈N and the sequence of spaces Hn is dense in H [22], and thus(65)⋃n=1∞Hn¯=H and un converges to u as n→∞ [12, 23].It was shown that there exists a sequence of triangulations that ensures optimal rates of convergence inH1-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 m [24]. For the details on different norms and rates of convergence of the FEM we refer the reader to [20].


## 4. Validation of the FEM for Different Boundary Conditions

Let us consider the plateB0 to be a square plate of size [0,a]×[0,a] with the boundary G=G1∪G2∪G3∪G4 and the hard simply supported boundary conditions similar to [13], written in terms of the kinematic variables in the mixed Dirichlet-Neumann type:(66)G1∪G2:W=0,W∗=0,Ψ2=0,Ω10=0,Ω^10=0,∂Ω3∂n=0,∂Ψ1∂n=0,∂Ω20∂n=0,∂Ω^20∂n=0,G3∪G4:W=0,W∗=0,Ψ1=0,Ω20=0,Ω^20=0,∂Ω3∂n=0,∂Ψ2∂n=0,∂Ω10∂n=0,∂Ω^10∂n=0,where(67)G1=x1,x2:x1=0,x2∈0,a,G2=x1,x2:x1=a,x2∈0,a,G3=x1,x2:x1∈0,a,x2=0,G4=x1,x2:x1∈0,a,x2=a.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 [24]. For the case of piecewise linear polynomials the optimal rate of convergence in H1-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 inH1-norm and L2-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 [25]: (68)E=299.5MPa,ν=0.44,lt=0.62mm,lb=0.327,N2=0.04.Taking into account the fact that the ratioβ/γ is equal to 1 for bending [25], these values of the technical constants correspond to the following values of symmetric and asymmetric parameters: (69)λ=762.616MPa,μ=103.993MPa,α=4.333MPa,β=39.975MPa,γ=39.975MPa,ϵ=4.505MPa.Let us consider homogeneous Dirichlet boundary conditions. We will assume the analytical solutionui of the form(70)ui=Uisin⁡πx1asin⁡πx2a,Ui∈R,i=1,9¯,which 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 f. The results of the error estimation of the FEM approximation in H1 and L2 norms performed for the elastic parameters corresponding to the polyurethane foam are given in Tables 1 and 2, respectively.Table 1
Order of convergence inH1-norm for homogeneous Dirichlet BC.
Refinements
Number of nodes
Diameter
Error inH1-norm
Convergence rate
0
177
0.302456
1.620369
1
663
0.151228
0.711098
1.19
2
2565
0.075614
0.322016
1.14
3
10089
0.037807
0.150149
1.10
4
40017
0.018903
0.073481
1.03
5
159393
0.009451
0.036512
1.01Table 2
Order of convergence inL2-norm for homogeneous Dirichlet BC.
Refinements
Number of nodes
Diameter
Error inL2-norm
Convergence rate
0
177
0.302456
0.279484
1
663
0.151228
0.069632
2.00
2
2565
0.075614
0.018175
1.94
3
10089
0.037807
0.004598
1.98
4
40017
0.018903
0.001153
2.00
5
159393
0.009451
0.000288
2.00Let 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 [7] for some fixed value of the parameter η. The results of the error estimation of the FEM approximation in H1 and L2 norms performed for the elastic parameters corresponding to the polyurethane foam are given in Tables 3 and 4, respectively.Table 3
Order of convergence inH1-norm for mixed Neumann-Dirichlet BC.
Refinements
Number of nodes
Diameter
Error inH1-norm
Convergence rate
0
177
0.302456
0.236791
1
663
0.151228
0.115809
1.03
2
2565
0.075614
0.054195
1.09
3
10089
0.037807
0.026233
1.05
4
40017
0.018903
0.012986
1.01
5
159393
0.009451
0.006475
1.00Table 4
Order of convergence inL2-norm for mixed Neumann-Dirichlet BC.
Refinements
Number of nodes
Diameter
Error inL2-norm
Convergence rate
0
177
0.302456
6.214
×
1
0
-
2
1
663
0.151228
1.638
×
1
0
-
2
1.92
2
2565
0.075614
4.219
×
1
0
-
3
1.96
3
10089
0.037807
1.065
×
1
0
-
3
1.99
4
40017
0.018903
2.678
×
1
0
-
4
1.99
5
159393
0.009451
6.772
×
1
0
-
5
1.98The boundary condition for the variableΩ3 is a Neumann type boundary condition:(71)∂Ω3∂n=0onG and thus we will look for Ω3 in the space H1Δ,B0, where(72)H1Δ,B0=u∈H1B0:Δu∈L2B0.The boundary condition for the variablesW and W∗ is a Dirichlet type boundary condition:(73)W=0onG,W∗=0onG and thus we will look for W and W∗ in the space H01B0 defined as [14](74)H01B0=v∈H1B0,v=0onG.The boundary condition for the variablesΨ1, Ω20, and Ω^20 is of mixed Dirichlet-Neumann type:(75)∂Ψ1∂n=0,∂Ω20∂n=0,∂Ω^20∂n=0,onG1∪G2,Ψ1=0,Ω20=0,Ω^20=0,onG3∪G4and thus we will look for Ψ1, Ω20, and Ω^20 in the following space [14]:(76)HV1=v∈H1Δ,B0,v=0onG3∪G4.The boundary condition for the variablesΨ2, Ω10, and Ω^10 is of mixed Dirichlet-Neumann type:(77)Ψ2=0,Ω10=0,Ω^10=0,onG1∪G2,∂Ψ2∂n=0,∂Ω10∂n=0,∂Ω^10∂n=0,onG3∪G4 and thus we will look for Ψ2, Ω10, and Ω^10 in the following space [14]:(78)HH1=v∈H1Δ,B0,v=0onG1∪G2.Therefore we will look for the solution(79)Ψ1,Ψ2,W,Ω3,Ω10,Ω20,W∗,Ω^1,Ω^2Tof the Cosserat plate field equations (17) in the space H defined as(80)H=H1×H2×H3×H4×H5×H6×H7×H8×H9,where(81)H1=H6=H9=HV1B0,H2=H5=H8=HH1B0,H3=H7=H01B0,H4=H1Δ,B0.The spaceH is a Hilbert space equipped with the inner product u,vH defined on H as follows:(82)u,vH=∑i=19ui,viHiforu,v∈H, where u,vHi is an inner product defined on the Hilbert space Hi, respectively.Taking into account the essential boundary conditions we define the Finite Element spacesHih as follows:(83)H1h=H6h=H9h=v:v∈CB0,vis  linear  on  everyKj,v=0onG1∪G2,(84)H2h=H5h=H8h=v:v∈CB0,vis  linear  on  everyKj,v=0onG3∪G4,(85)H3h=H7h=v:v∈CB0,vis  linear  on  everyKj,v=0onG,(86)H4h=v:v∈CB0,vis  linear  on  everyKj.The finite dimensional spaceHh is then defined as(87)Hh=H1h×H2h×H3h×H4h×H5h×H6h×H7h×H8h×H9h.We solve the field equations using described Finite Element Method and compare the obtained results with the analytical solution for the square plate made of polyurethane foam derived in [7].The initial distribution of the pressure is assumed to be sinusoidal:(88)px1,x2=sin⁡πx1asin⁡πx2a.The estimation of the error inH1 norms shows that the order of the error is optimal (linear) in H1-norm for the piecewise linear elements for the simply supported plate. The results of the error estimation of the FEM approximation in H1 and L2 norms performed for the elastic parameters corresponding to the polyurethane foam are given in Tables 5 and 6, respectively.Table 5
Order of convergence inH1-norm for simply supported plate.
Refinements
Number of nodes
Diameter
Error inH1-norm
Convergence rate
0
177
0.302456
0.256965
1
663
0.151228
0.119234
1.11
2
2565
0.075614
0.054701
1.12
3
10089
0.037807
0.026301
1.05
4
40017
0.018903
0.012994
1.01
5
159393
0.009451
0.006476
1.00Table 6
Order of convergence inL2-norm for simply supported plate.
Refinements
Number of nodes
Diameter
Error inL2-norm
Convergence rate
0
177
0.302456
8.253
×
1
0
-
2
1
663
0.151228
2.260
×
1
0
-
2
1.87
2
2565
0.075614
5.860
×
1
0
-
3
1.95
3
10089
0.037807
1.482
×
1
0
-
3
1.98
4
40017
0.018903
3.720
×
1
0
-
4
1.99
5
159393
0.009451
9.355
×
1
0
-
5
1.99The comparison of the maximum of the displacementsui and microrotations ϕi calculated using Finite Element Method with 320 thousand elements and the analytical solution for the micropolar plate theory is provided in Table 7. The relative error of the approximation of the optimal value of the splitting parameter is 0.09%.Table 7
Relative error of the maximum values of the displacement and microrotations.
Optimalη
u
1
u
2
u
3
ϕ
1
ϕ
2
Finite element solution
0.040760
-
0.014891
-
0.014891
0.307641
0.046767
-
0.046767
Analytical solution
0.040799
-
0.014892
-
0.014892
0.307674
0.046770
-
0.046770
Relative error (%)
0.09
0.03
0.03
0.04
0.03
0.03


## 5. Numerical Results

In this section we study the Cosserat elastic plates with different shapes and boundary conditions. We investigate the size effect of circular clamped plate and of plate with holes. We also consider the plates with holes and calculate the stress concentration factor around the hole.


### 5.1. Clamped Cosserat Plate

Figure3 represents the Finite Element modeling of the bending of the simply supported square plate made of polyurethane foam. The comparison of the distribution of the vertical deflection of the clamped and simply supported plates is given in Figure 4. The bending of a clamped circular plate under the uniform load is given in Figure 5. Figures 6–8 represent the Finite Element modeling of the bending of the clamped polyurethane foam plates of different shapes under different loads including plates with holes.Figure 3
Hard simply supported square plate 3.0 m× 3.0 m × 0.1 m made of polyurethane foam: the initial mesh and the isometric view of the resulting vertical deflection of the plate.Figure 4
The cross section that contains the center of the micropolar square plate 2.0 m× 2.0 m × 0.1 m made of polyurethane foam under the sinusoidal load: Cosserat clamped plate, solid blue line, Cosserat simply supported plate, solid orange line, and initial sinusoidal load, dashed black line.Figure 5
The circular Cosserat clamped plate of radiusR=1.0m and thickness h=0.1m made of polyurethane foam under the uniform load: the initial mesh and the isometric view of the resulting vertical deflection of the plate.Figure 6
The circular Cosserat clamped plate of radiusR=1.0m and thickness h=0.1m made of polyurethane foam with circular clamped holes under the uniform load: the initial mesh and the isometric view of the resulting vertical deflection of the plate.Figure 7
The clamped plate of size 10.0 m× 6.0 m × 0.1 m made of polyurethane foam under the uniform load: the initial mesh and the isometric view of the resulting vertical deflection of the plate.Figure 8
The clamped polyurethane gasket under the uniform load: the initial mesh and the isometric view of the resulting vertical deflection of the plate.The size effect of the Cosserat plate is illustrated in Figures9 and 10. As in [10] we perform computations for different levels of the asymmetric microstructure by reducing the values of the elastic asymmetric parameters. For example, 1% of the microstructure means that 1% of the original values of the asymmetric parameters were used. The influence of the microstructure in thinner plates results in a more rigid plate than would be expected on the basis of the classical elasticity.Figure 9
Comparison of the maximum vertical deflections of the circular Cosserat plateWC with Reissner plate WR for different values of width to thickness ratio a/h, which illustrates the influence of the thickness of the Cosserat plate.Figure 10
Comparison of the maximum vertical deflections of the circular plate with 4 holes,WC (Cosserat) with WR (Reissner) for different values of width to thickness ratio a/h, which illustrates the influence of the thickness of the Cosserat plate.


### 5.2. Stress Concentration

In this section we will consider the plates with holes and calculate the stress concentration factor around the hole as a ratio of the highest stressσmax to a reference stress σref of the gross cross section: (89)K=σmax⁡σref.Let us consider a square plate of size[0,a]×[0,a] for a=10.0 of thickness h=1.0m and a hole of diameter d varying from 1.0 m to 5.0 m located in the center of the plate (note that in the notation of Figure 11, the diameter d defined as the largest distance in the interior of the hole is equal to 4r).Figure 11
Square plate of sizea×a and thickness h with a hole located in the center of the plate.The exterior boundary of the plate isG=G1∪G2∪G3∪G4 and the boundary of the hole is H=H1∪H2∪H3∪H4. The plate is loaded under the uniform load and the following mixed hard simply supported and clamped boundary conditions written in terms of the kinematic variables:(90)G1∪G2:W=0,W∗=0,Ψ2=0,Ω10=0,Ω^10=0,∂Ω3∂n=0,∂Ψ1∂n=0,∂Ω20∂n=0,∂Ω^20∂n=0,G3∪G4:W=0,W∗=0,Ψ1=0,Ψ2=0,Ω10=0,Ω20=0,Ω^10=0,Ω^20=0,Ω30=0,H1∪H2:W=0,W∗=0,Ψ1=0,Ψ2=0,Ω10=0,Ω20=0,Ω^10=0,Ω^20=0,Ω30=0,H3∪H4:W=0,W∗=0,Ψ2=0,Ω10=0,Ω^10=0,∂Ω3∂n=0,∂Ψ1∂n=0,∂Ω20∂n=0,∂Ω^20∂n=0, where(91)G1=x1,x2:x1=0,x2∈0,a,G2=x1,x2:x1=a,x2∈0,a,G3=x1,x2:x1∈0,a,x2=0,G4=x1,x2:x1∈0,a,x2=a,H1=x1,x2:x1=a2-r-r2-x22,x2∈-r,r,H2=x1,x2:x1=a2-r+r2-x22,x2∈-r,r,H3=x1,x2:x1∈0,a,x2=0,H4=x1,x2:x1∈0,a,x2=a.The stress distribution for the polyurethane foam plate and the stress concentration around the hole is provided in Figure12.Figure 12
The distribution of the stressσ11 for the square polyurethane foam plate 10.0 m × 10.0 m × 1.0 m demonstrates the stress concentration around the hole (160,000 elements used).The Finite Element calculations of the stress concentration factors around the hole show that the stress concentration factor for the Cosserat plate is always smaller than the classical value [26]. The comparison of the stress concentration factors for Cosserat and Reissner plates is given in Figure 13.Figure 13
Comparison of the stress concentration factors of the Cosserat plate with Reissner plate for different ratiosd/a: Cosserat stress concentration factor is smaller than the classical value.The ratios of the Cosserat plate stress concentration factor to the Reissner plate concentration factor are plotted for different values ofd/a and given in Figure 14. The presence of the length scale for Cosserat materials results in the predicted size effect. This graph confirms that the small holes exhibit less stress concentration than the larger holes as it is expected on the basis of the classical theory [26] and results in the enhanced toughness [27]. Similar size effect results were obtained earlier for the vertical deflection of Cosserat plates for the case of bending and for the main eigenfrequencies in the case of free vibration [10, 11].Figure 14
Comparison of the ratios of the stress concentration factor of the Cosserat plateK with the stress concentration factor of the Reissner plate KR for different hole sizes. Smaller holes exhibit less stress concentration than what would be expected on the basis of Reissner theory.


## 5.1. Clamped Cosserat Plate

Figure3 represents the Finite Element modeling of the bending of the simply supported square plate made of polyurethane foam. The comparison of the distribution of the vertical deflection of the clamped and simply supported plates is given in Figure 4. The bending of a clamped circular plate under the uniform load is given in Figure 5. Figures 6–8 represent the Finite Element modeling of the bending of the clamped polyurethane foam plates of different shapes under different loads including plates with holes.Figure 3
Hard simply supported square plate 3.0 m× 3.0 m × 0.1 m made of polyurethane foam: the initial mesh and the isometric view of the resulting vertical deflection of the plate.Figure 4
The cross section that contains the center of the micropolar square plate 2.0 m× 2.0 m × 0.1 m made of polyurethane foam under the sinusoidal load: Cosserat clamped plate, solid blue line, Cosserat simply supported plate, solid orange line, and initial sinusoidal load, dashed black line.Figure 5
The circular Cosserat clamped plate of radiusR=1.0m and thickness h=0.1m made of polyurethane foam under the uniform load: the initial mesh and the isometric view of the resulting vertical deflection of the plate.Figure 6
The circular Cosserat clamped plate of radiusR=1.0m and thickness h=0.1m made of polyurethane foam with circular clamped holes under the uniform load: the initial mesh and the isometric view of the resulting vertical deflection of the plate.Figure 7
The clamped plate of size 10.0 m× 6.0 m × 0.1 m made of polyurethane foam under the uniform load: the initial mesh and the isometric view of the resulting vertical deflection of the plate.Figure 8
The clamped polyurethane gasket under the uniform load: the initial mesh and the isometric view of the resulting vertical deflection of the plate.The size effect of the Cosserat plate is illustrated in Figures9 and 10. As in [10] we perform computations for different levels of the asymmetric microstructure by reducing the values of the elastic asymmetric parameters. For example, 1% of the microstructure means that 1% of the original values of the asymmetric parameters were used. The influence of the microstructure in thinner plates results in a more rigid plate than would be expected on the basis of the classical elasticity.Figure 9
Comparison of the maximum vertical deflections of the circular Cosserat plateWC with Reissner plate WR for different values of width to thickness ratio a/h, which illustrates the influence of the thickness of the Cosserat plate.Figure 10
Comparison of the maximum vertical deflections of the circular plate with 4 holes,WC (Cosserat) with WR (Reissner) for different values of width to thickness ratio a/h, which illustrates the influence of the thickness of the Cosserat plate.


## 5.2. Stress Concentration

In this section we will consider the plates with holes and calculate the stress concentration factor around the hole as a ratio of the highest stressσmax to a reference stress σref of the gross cross section: (89)K=σmax⁡σref.Let us consider a square plate of size[0,a]×[0,a] for a=10.0 of thickness h=1.0m and a hole of diameter d varying from 1.0 m to 5.0 m located in the center of the plate (note that in the notation of Figure 11, the diameter d defined as the largest distance in the interior of the hole is equal to 4r).Figure 11
Square plate of sizea×a and thickness h with a hole located in the center of the plate.The exterior boundary of the plate isG=G1∪G2∪G3∪G4 and the boundary of the hole is H=H1∪H2∪H3∪H4. The plate is loaded under the uniform load and the following mixed hard simply supported and clamped boundary conditions written in terms of the kinematic variables:(90)G1∪G2:W=0,W∗=0,Ψ2=0,Ω10=0,Ω^10=0,∂Ω3∂n=0,∂Ψ1∂n=0,∂Ω20∂n=0,∂Ω^20∂n=0,G3∪G4:W=0,W∗=0,Ψ1=0,Ψ2=0,Ω10=0,Ω20=0,Ω^10=0,Ω^20=0,Ω30=0,H1∪H2:W=0,W∗=0,Ψ1=0,Ψ2=0,Ω10=0,Ω20=0,Ω^10=0,Ω^20=0,Ω30=0,H3∪H4:W=0,W∗=0,Ψ2=0,Ω10=0,Ω^10=0,∂Ω3∂n=0,∂Ψ1∂n=0,∂Ω20∂n=0,∂Ω^20∂n=0, where(91)G1=x1,x2:x1=0,x2∈0,a,G2=x1,x2:x1=a,x2∈0,a,G3=x1,x2:x1∈0,a,x2=0,G4=x1,x2:x1∈0,a,x2=a,H1=x1,x2:x1=a2-r-r2-x22,x2∈-r,r,H2=x1,x2:x1=a2-r+r2-x22,x2∈-r,r,H3=x1,x2:x1∈0,a,x2=0,H4=x1,x2:x1∈0,a,x2=a.The stress distribution for the polyurethane foam plate and the stress concentration around the hole is provided in Figure12.Figure 12
The distribution of the stressσ11 for the square polyurethane foam plate 10.0 m × 10.0 m × 1.0 m demonstrates the stress concentration around the hole (160,000 elements used).The Finite Element calculations of the stress concentration factors around the hole show that the stress concentration factor for the Cosserat plate is always smaller than the classical value [26]. The comparison of the stress concentration factors for Cosserat and Reissner plates is given in Figure 13.Figure 13
Comparison of the stress concentration factors of the Cosserat plate with Reissner plate for different ratiosd/a: Cosserat stress concentration factor is smaller than the classical value.The ratios of the Cosserat plate stress concentration factor to the Reissner plate concentration factor are plotted for different values ofd/a and given in Figure 14. The presence of the length scale for Cosserat materials results in the predicted size effect. This graph confirms that the small holes exhibit less stress concentration than the larger holes as it is expected on the basis of the classical theory [26] and results in the enhanced toughness [27]. Similar size effect results were obtained earlier for the vertical deflection of Cosserat plates for the case of bending and for the main eigenfrequencies in the case of free vibration [10, 11].Figure 14
Comparison of the ratios of the stress concentration factor of the Cosserat plateK with the stress concentration factor of the Reissner plate KR for different hole sizes. Smaller holes exhibit less stress concentration than what would be expected on the basis of Reissner theory.


## 6. Conclusion

The article provides numerical results obtained using the Finite Element Method for solving the parametric system of 9 kinematic equations. The Finite Element modeling of the clamped Cosserat plates of different shapes under different loads is provided. The article also presents the numerical validation of the proposed FEM by estimating the order of convergence, when comparing the main kinematic variables with the analytical solution. Numerical simulation of size effects of Cosserat plates has been presented. It was shown that the stress concentration factor around the hole in the plates is less than the classical value, and smaller holes exhibit less stress concentration than would be expected on the basis of the classical elasticity.



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*Source: 5246197-2017-03-30.xml*