Open access peer-reviewed chapter - ONLINE FIRST

Distinctive Characteristics of Cosserat Plate Free Vibrations

By Lev Steinberg and Roman Kvasov

Submitted: May 1st 2019Reviewed: May 24th 2019Published: July 10th 2019

DOI: 10.5772/intechopen.87044

Downloaded: 34

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 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 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 non-diagonal rotatory inertia tensor. The numerical computations of the plate free vibrations showed the existence of some additional high frequencies of micro-vibrations 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.

2. Cosserat linear elastodynamics

2.1 Fundamental equations

The Cosserat linear elasticity balance laws are

σji,j=pit,E1
εijkσjk+μji,j=qit,E2

where the σjiis the stress tensor, μjithe couple stress tensor, pi=ρuitand qi=Jjiϕjtare the linear and angular momenta, ρand Jjiare the material density and the rotatory inertia characteristics, εijkis the Levi-Civita tensor.

We will also consider the constitutive equations as in [13]:

σji=μ+αγji+μαγij+λγkkδij,E3
μji=γ+εχji+γεχij+βχkkδij,E4

and the kinematic relations in the form

γji=ui,j+εijkϕkandχji=ϕi,j,E5

Here uiand ϕirepresent the displacement and microrotation vectors, γjiand χjirepresent the strain and bend-twist tensors, μ, λare the Lamé parameters and α, β, γ, εare the Cosserat elasticity parameters.

The constitutive Eqs. (3)(4) can be written in the reverse form [5].

γji=μ+ασji+μασij+λσkk,E6
χji=γ+εμji+γεμij+βμkk,E7

where μ=14μ, α=14α, γ=14γ, ε=14ε, λ=λ6μλ+2μ3and β=β6μβ+2γ3.

We will consider the boundary conditions given in [12].

ui=ui0,ϕi=ϕi0,onG1t=B0\Bσ×t0t,E8
σjini=σj0,μjini=μj0onG2t=Bσ×t0t,E9

and initial conditions

uix0=Ui0,ϕix0=Φi0,inB0,E10
u̇ix0=U̇i0,ϕ̇ix0=Φ̇i0,inB0,E11

where ui0and ϕi0are prescribed on G1, σj0and μj0on G2, and niis the unit vector normal to the boundary B0of the elastic body B0.

2.2 Cosserat elastic energy

The strain stored energy UCof the body B0is defined by the integral [13]:

UC=B0Wγχdv,E12

where

Wγχ=μ+α2γijγij+μα2γijγji+λ2γkkγnn+γ+ε2χijχij+γε2χijχji+β2χkkχnn,E13

is non-negative. The relations Eqs. (3)(4) can be written in the form [12]:

σ=γWandμ=χW.E14

The stress energy is given as

UK=B0Φσμdv,E15

where

Φσμ=μ+α2σijσij+μα2σijσji+λ2σkkσnn+γ+ε2μijμij+γε2μijμji+β2μkkμnn,E16

and the relations Eqs. (6)(7) can be written as [12].

γ=∂Φσ,andχ=∂Φμ.E17

We consider the work done by the stresses σand μover the strains γand χas in [13].

U=B0σγ+μχdvE18

and

U=UK=UCE19

Here σγ=σjiγjiand μχ=μjiχji.

The stored kinetic energy TCis defined as

TC=B0ϒCdv=12B0ρut2+Jϕt2dv,E20

The kinetic energy TKis given as

TK=B0ϒKpqdv=12B0p2ρ1+q2J1dv,E21

where

p=ϒCu̇=ρutandq=ϒCϕ̇=Jϕt,E22

and

ut=ϒKp=pρ1andϕt=ϒKq=qJ1,E23

The work TWdone by the inertia forces over displacement and microrotation is given as in [12].

TW=B0ϒWdv=B0ptu+qtϕdvE24

Keeping in mind that the variation of pu, q, ϕ, δu, and δϕis zero at t0and tkwe can integrate by parts

t0tkTKdt=t0tkTIdt=12B0pu+qϕdvt0tkt0tkTWdtE25
δt0tkTK=δt0tkTWE26

or

δTC=δTK=δTWE27

and therefore

t0tkB0pδut+qδϕtdvdt=t0tkB0ptδu+qtδϕdvdtE28

2.3 Variational principle for elastodynamics

We modify the HPR principle [14] for the case of Cosserat elastodynamics in the following way: for any set Av of all admissible states s=uϕγχσμthat satisfy the strain-displacement and torsion-rotation relations Eq. (5), the zero variation

δΘs=0

of the functional

Θs=t0tkUK+TCB0σγ+μχ+put+qϕtdvdt+t0tG1σnuu0+μnϕϕ0dadt+t0tG2σ0u+μ0ϕdadtE29

at sAis equivalent of sto be a solution of the system of equilibrium Eqs. (1)(2), constitutive relations Eqs. (6)(7), which satisfies the mixed boundary conditions Eqs. (8)(9).

Proof of the variational principle for elastodynamics

Let us consider the variation of the functional Θs:

δΘs=t0tkδUK+δTCdtt0tkB0δσγ+σδγ+δμχ+μδχ+utδp+pδut+δqϕt+qδϕtdvdt+t0tG1δσnuu0+σnδu+δμnϕϕ0+μnδϕdadt+t0tG2σ0δu+μ0δϕdadt

Taking into account Eq. (5) we can perform the integration by parts

B0σδγdv=B0σnδudaB0δudivσdv+B0εσδϕdvB0μδχdv=B0μnδϕdaB0δϕdivμdv

and based on Eqs. (17)(23)

δΦ=∂Φσδσ+∂Φμδμ,δϒC=ϒCutδut+ϒCϕtδϕt.

Then keeping in mind that δTK=δTand Eq. (28) we can rewrite the expression for the variation of the functional δΘsin the following form

δΘs=t0tB0∂Φσγδσdvdt+t0tB0∂Φμχδμdvdt+t0tB0ρutpδutdvdt+t0tB0Jφtqδφtdvdt+t0tB0divσptδudvdt+t0tB0divμ+εσqtδudvdt+t0tG1uu0δσndadt+t0tG1ϕϕ0δμndadt+t0tG2σnσ0δudadt+t0tG2μnμ0δϕdadt

3. Dynamic Cosserat plate theory

In this section we review our stress, couple stress and kinematic assumptions of the Cosserat plate [7]. We consider the thin plate P, where his the thickness of the plate and x3=0represents its middle plane. The sets Tand Bare the top and bottom surfaces contained in the planes x3=h/2, x3=h/2respectively and the curve Γis the boundary of the middle plane of the plate.

The set of points P=Γ×[h2h2]TBforms the entire surface of the plate and Γu×[h2,h2]is the lateral part of the boundary where displacements and microrotations are prescribed. The notation Γσ=Γ\Γuof the remainder we use to describe the lateral part of the boundary edge Γσ×[h2,h2]where stress and couple stress are prescribed. We also use notation P0for the middle plane internal domain of the plate.

In our case we consider the vertical load and pure twisting momentum boundary conditions at the top and bottom of the plate, which can be written in the form:

σ33x1x2h/2t=σtx1x2t,σ33x1x2h/2t=σbx1x2t,E30
σ3βx1x2±h/2t=0,E31
μ33x1x2h/2t=μtx1x2t,μ33x1x2h/2t=μbx1x2t,E32
μ3βx1x2±h/2t=0,E33

where x1x2P0.

We will also consider the rotatory inertia Jin the form

J=J11J120J12J22000J33

Let Adenote the set of all admissible states that satisfy the Cosserat plate strain-displacement relation Eq. (5) and let Θbe a functional on Adefined by

Θsη=UKS+TCSP0SE+PUtP̂W+vΩ30da+ΓσSoUUods+ΓuSnUds,E34

for every s=UESA.Here P̂=p̂1p̂2and W=WW, p̂1=ηpand p̂2=231ηp

Here the plate stress and kinetic energy density by the formulas

UKS=P0ΦSda,TKS=P0ϒCUtdaE35

where P0is the internal domain of the middle plane of the plate.

ΦS=3λMααMββh3μ3λ+2μ+3α+μMαβ22h3αμ+3α+μ160h3αμ8Q̂αQ̂α+15QαQ̂α+20Q̂αQα+8QαQα+3αμMαβ22h3αμ+αμ280h3αμ21Qα5Q̂α+4Qαγε160hγε24Rαα2+45Rαα+60RαβRαβ+48R12R21+3γ+εSαSα2h3γε+γ+ε160h3γε8Rαβ2+15RαβRαβ+20RαβRαβ3β803β+2γ8RααRββ15RααRββ20RααRααβ4γ3β+2γ2Rαα+3RααthV2+T2+λ5603λ+2μ5+3η1+ηpMαα+λ+μh840μ3λ+2μ140+168η+51η241+η2p2+λ+μh2μ3λ+2μσ02+εh12hγε3T2+V2E36

and

ϒCUt=2Wt2+415Wt2+23WtWt+h3ρ24Ψαt2+4hJαβ15Ωα0tΩβ0t+hJαβ2Ω̂α0tΩ̂β0t+2hJαβ3Ωα0tΩ̂β0t+hJ336Ω3t2.

S, Uand Eare the Cosserat plate stress, displacement and strain sets

S=MαβQαQαQ̂αRαβRαβSβ,E37
Sn=MˇαQˇQˇα̂RˇαRˇαSˇ,E38
So=ΠΠo3Πo3MMMo3,E39
U=ΨαWΩ3Ωα0WΩα0,E40
E=eαβωβωαω̂ατ3αταβταβ,E41

where Mαβnβ=Π, Rαβnβ=M, Qαnα=Πo3, Sαnα=Mo3, Q̂αnα=Πo3, Rαβnβ=M, Mˇα=Mαβnβ, Qˇ=Qβnβ, Rˇα=Rαβnβ, Sˇ=Sβnβ, Qˇ̂=Qˇβ̂nβ, Rˇα=Rˇαβnβ. (nβis the outward unit normal vector to Γu).

The plate characteristics provide the approximation of the components of the three-dimensional tensors σjiand μji

σαβ=6h2ζMαβx1x2t,E42
σ3β=32h1ζ2Qβx1x2t,E43
σβ3=32h1ζ2Qβx1x2t+32hQ̂βx1x2t,E44
σ33=3413ζ3ζp1x1x2t+ζp2x1x2t+σ0x1x2t,E45
μαβ=32h1ζ2Rαβx1x2t+32hRαβx1x2t,E46
μβ3=6h2ζSβx1x2t,E47
μ3β=0,E48
μ33=ζVx1x2t+Tx1x2t,E49

where

px1x2t=σtx1x2tσbx1x2t,E50
σ0x1x2t=12σtx1x2t+σb(x1x2t),E51
Vx1x2t=12μtx1x2tμb(x1x2t),E52
Tx1x2t=12μtx1x2t+μb(x1x2t).E53

The pressures p1and p2are chosen in the form

p1x1x2t=ηpx1x2t,E54
p2x1x2t=1η2px1x2t.E55

and ηRis called splitting parameter.

The three-dimensional displacements uiand microrotations ϕi

uα=h2ζΨαx1x2t,E56
u3=Wx1x2t+1ζ2Wx1x2t,E57
ϕα=Ωα0x1x2t1ζ2+Ω̂αx1x2t,E58
ϕ3=ζΩ3x1x2t,E59

and the three-dimensional strain and torsion tensors γjiand χji

γαβ=6h2ζeαβx1x2t,E60
γ3β=32h1ζ2ωβx1x2t,E61
γβ3=32h1ζ2ωβx1x2t+32hω̂βx1x2t,E62
χαβ=32h1ζ2ταβx1x2t+32hταβx1x2t,E63
χ3β=6h2ζτβx1x2t,E64

where ζ=2x3h.

Then zero variation of the functional

δΘsη=0

is equivalent to the plate bending system of equations (A) and constitutive formulas (B) mixed problems.

A. The bending equilibrium system of equations:

Mαβ,αQβ=I12Ψβt2,E65
Qα,α+p̂1=I22Wt2,E66
Rαβ,α+ε3βγQγQγ=Iαβ2Ωα0t2,E67
ε3βγMβγ+Sα,α=I32Ω3t2,E68
Q̂α,α+p̂2=I22Wt2,E69
Rαβ,α+ε3βγQ̂γ=Iαβ02Ω̂α0t2,E70

where I1=h312ρ, I2=2h3ρ, Iαβ=5h6Jαβ, I3=h26J33, Iαβ0=2h3Jαβ, p̂1=ηoptp, and p̂2=231ηoptp, with the resultant traction boundary conditions:

Mαβnβ=Π,Rαβnβ=M,E71
Qαnα=Πo3,Sαnα=ϒo3,E72

at the part Γσand the resultant displacement boundary conditions

Ψα=Ψ,W=Wo,Ωα0=Ω0,Ω3=Ωo3,E73

at the part Γu.

B. Constitutive formulas in the reverse form:1

Mαα=μλ+μh33λ+2μΨα,α+λμh36λ+2μΨβ,β+3p1+5p2λh230λ+2μ,E74
Mβα=μαh312Ψα,β+μ+αh312Ψβ,α+1α'αh36Ω3,E75
Rβα=5γεh6Ωβ,α0+5γ+εh6Ωα,β0,E76
Rαα=10β+γ3β+2γΩα,α0+5hβγ3β+2γΩβ,β0,E77
Rβα=2γεh3Ω̂β,α+2γ+εh3Ω̂α,β,E78
Rαα=8γγ+βh3β+2γΩ̂α,α+4γβh3β+2γΩ̂β,β,E79
Qα=5μ+αh6Ψα+5μαh6W,α+2μαh3W,α+1β53Ωβ0+1β53Ω̂β,E80
Qα=5μαh6Ψα+5μα2h6μ+αW,α+2μ+αh3W,α+1α53Ωβ0+μαμ+αΩ̂β,E81
Q̂α=8αμh3μ+αW,α+1α8αμh3μ+αΩ̂β,E82
Sα=5γεh33γ+εΩ3,α,E83

and the optimal value ηoptof the splitting parameter is given as in [10]

ηopt=2W00W10W012W11+W00W10W01.E84

where

Wij=Sη=iEη=j.

We also assume that the initial condition can be presented in the form

Ux1x20=U0x1x2,Utx1x20=V0x1x2

4. Cosserat plate dynamic field equations

The Cosserat plate field equations are obtained by substituting the relations Eqs. (74)(83) into the system of Eqs. (65)(70) similar to [10]:

LU=K2Ut2+Fη,E85

where

L=L11L12L13L140L16kL130L16L12L22L23L24L160kL23L160L13L23L330L35L36L77L38L39L41L420L44000000L16L380L55L56kL35L580L160L390L56L66kL360L58L13L14L730L35L36L77L78L790L16L780L85L56kL35L88kL56L160L790L56L55kL36kL56L99,
K=h312ρ000000000h312ρ0000000002h3ρ000000000h26J330000000005h6J115h6J1200000005h6J125h6J220000000002h3ρ0000000002h3J112h3J1200000002h3J122h3J22,
U=Ψ1Ψ2WΩ3Ω10Ω20WΩ10Ω20T,
Fη=3h2λ3p1,1+5p2,130λ+2μ,3h2λ3p1,2+5p2,230λ+2μ,p1,0,0,0,h23p1+4p224,0,0T,
p1=ηp,p2=1η2p

The operators Lijare given as follows

L11=c12x12+c22x22c3,L12=c1c22x1x2,L13=c11x1,L14=c12x2,L16=c13,L17=k1c11x1,L22=c22x12+c12x22c3,L23=c11x2,L24=c12x1,L33=c32x12+2x22,L35=c13x2,L36=c13x1,L38=c10x2,L39=c10x1,L41=c12x2,L42=c12x1,L44=c62x12+2x222c12,L55=c72x12+c82x222c13,L56=c7c82x1x2,L58=c9,L66=c82x12+c72x222c13,L73=c52x12+2x22,L77=c42x12+2x22,L78=c14x2,L79=c14x1,L85=c72x12+c82x222c13,L88=c72x12+c82x22c15,L99=c82x12+c72x22c15.

The coefficients ciare given as

c1=h3μλ+μ3λ+2μ,c2=h3α+μ12,c3=5hα+μ6,c4=5hαμ26α+μ,c5=h5α2+6αμ+5μ26α+μ,c6=h3γε3γ+ε,c7=10β+γ3β+2γ,c8=5hγ+ε6,c9=10hα23α+μ,c10=5αμ3α+μ,c11=5hαμ6,c12=h3α6,c13=53,c14=5α+3μ3α+μ,c15=25α+4μ3α+μ.

5. Numerical validation

For the validation purposes we provide the algorithm and computation results for the three-dimensional Cosserat elastodynamics. We also present the analysis of the numerical results based on the plate theory for the microelements of different shapes and orientations incorporated into the Cosserat plate.

5.1 Analysis of Cosserat plate vibrations based on the three-dimensional theory

In our computations we consider the plates made of polyurethane foam—a material reported in the literature to behave Cosserat like—and the values of the technical elastic parameters presented in [15]: E=299.5MPa, ν=0.44, lt=0.62mm, lb=0.327mm, N2=0.04. Taking into account that the ratio β/γis equal to 1 for bending [15], these values of the technical constants correspond to the following values of Lamé and Cosserat parameters: λ=762.616MPa, μ=103.993MPa, α=4.333MPa, β=39.975MPa, γ=39.975MPa, ε=4.505MPa. We consider a low-density rigid foam usually characterized by the densities of 24–50 kg/m3 [16]. In all further numerical computations we used the density value ρ=34 kg/m3 and different values the rotatory inertia J.

Let us consider the plate B0being a rectangular cuboid 0a×0,a]×h2h2. Let the sets Tand Bbe the top and the bottom surfaces contained in the planes x3=h2and x3=h2respectively, and the curve Γ=Γ1Γ2be the lateral part of the boundary:

Γ1=x1x2x3:x10ax20ax3h2h2,Γ2=x1x2x3:x10ax20ax3h2h2,

We solve the three-dimensional Cosserat equilibrium Eqs. (1)(2) accompanied by the constitutive Eqs. (3)(4) and strain-displacement and torsion-rotation relations Eq. (5) complemented by the following boundary conditions:

Γ1:u2=0,u3=0,φ1=0,σ11=0,μ12=0,μ13=0;E86
Γ2:u1=0,u3=0,φ2=0,σ22=0,μ21=0,μ23=0;E87
T:σ33=px1x2,μ33=0;E88
B:σ33=0,μ33=0.E89

where the initial distribution of the pressure is given as p=sinπx1asinπx2asinωtand the rotatory inertia tensor Jis assumed to have a diagonal form

J=Jx000Jy000Jz.E90

Using the method of separation of variables and taking into account the boundary conditions Eqs. (86)(87), we express the kinematic variables in the form:

u1=cosπx1asinπx2az1x3sinωt,E91
u2=sinπx1acosπx2az2x3sinωt,E92
u3=sinπx1asinπx2az3x3sinωt,E93
ϕ1=sinπx1acosπx2az4x3sinωt,E94
ϕ2=cosπx1asinπx2az5x3sinωt,E95
ϕ3=cosπx1acosπx2az6x3sinωt,E96

where the functions zix3represent the transverse variations of the kinematic variables.

If we substitute the expressions Eqs. (91)(96) into Eqs. (3)(4) and then into Eqs. (1)(2), we will obtain the following eigenvalue problem

Bz=ω2AzE97

where

B=b1L2+b2L0b3L0b4L10b5L1b6L0b3L0b1L2+b2L0b4L1b5L10b6L0b4L1b4L1b7L2b6L0b6L000b5L1b6L0b9L2+b10L0b11L0b12L1b5L10b6L0b11L0b9L2+b10L0b12L1b6L0b6L00b12L1b12L1b13L2+b2L14,E98
A=a2ρ000000a2ρ000000a2ρ000000a2Jx000000a2Jy000000a2Jz,E99
z=z1,z2,z3,z4,z5,z6T,E100

and the differential operators Liare defined as

L0=I,L1=ddx3,L2=d2dx32

and the coefficients biare defined as

b1=a2μ+α,b2=π2α+λ+3μ,b3=π2λ+μα,b4=λ+μα,b5=2a2α,b6=2aπα,b7=a22μ+λ,b8=2π2α+μ,b9=a2γ+ε,b10=π2β+ε+3γ,b11=π2β+γε,b12=β+γε,b13=a2β+2γ,b14=2π2γ+ε4a2α

The system of differential Eq. (97) is complemented by the following boundary conditions Dz=D0for x3=h2and Dz=0for x3=h2.

D=d1L10d2L00d3L000d1L1d2L0d3L000d4L0d4L0d5L1000000d6L10d7L00000d6L1d7L0000d8L0d8L0d9L1,E101
D0=0,0,a,0,0,0T,E102

and the coefficients diare defined as

d1=aμ+α,d2=πμα,d3=2,d4=aλ+2μ,d5=πλ,d6=aγ+ε,d7=aγε,d8=πβ,d9=aβ+2γ.

The idea for the solution of the eigenvalue problem Eq. (97) is based on the following algorithm:

Step 1.Fix certain frequency value.

We fix certain value of the frequency ωand force the Cosserat body to vibrate at this frequency.

Step 2.Solve the three-dimensional Cosserat system of equations.

Mathematically, fixing certain value of ωimplies that three-dimensional system of Eq. (97) has a constant right-hand side and therefore can be solved for the kinematic variables as a static system of equations. We solve the system Eq. (97) using the high-precision Runge-Kutta method incorporated in Mathematica software similar to how it was done in [7].

Step 3.Find large amplitudes of the kinematic variables.

We run ωthrough an interval of positive real values and take note where the solution changes its sign and the amplitude of the solutions starts to grow indefinitely. This corresponds to the oscillation of the Cosserat body at its resonant frequency. Thus, when the frequency ωcoincides with the natural frequency of the plate the resonance will occur and the large amplitude linear vibrations can be observed (Figure 1).

Figure 1.

Large amplitude linear vibrations of the Cosserat body forced to vibrate close to its natural frequency ω1.

The comparison of the eigenfrequencies of the Cosserat plate with the eigenfrequencies of the three-dimensional Cosserat elasticity is given in the Table 1. The rotatory inertia principle moments used are Jx=0.001, Jy=0.001, Jz=0.001, which represent a ball-shaped microelement (Figure 2). The relative error of the natural macro frequencies associated with the rotation of the middle plane and the flexural motion is less than 1%.

ω1, ω2ω3, ω7ω4ω5, ω8ω6, ω9
Plate theory0.31017.881501.13205.62338.95
D Cosserat elasticity0.30917.763530.82211.98317.87

Table 1.

Comparison of the eigenfrequencies ωi(Hz) with the exact values of the 3D Cosserat elasticity.

Figure 2.

Ball-shaped micro-elements: Jx=0.001, Jy=0.001, Jz=0.001 (left) and horizontally stretched ellipsoid micro-elements: Jx=0.002, Jy=0.001, Jz=0.0001right.

5.2 Analysis of Cosserat plate vibrations based on the plate theory

We consider a plate a×aof thickness hwith the boundary G=G1G2

G1=x1x2:x10ax20a
G2=x1x2:x20ax10a

and the following hard simply supported boundary conditions [7]:

G1:W=0,W=0,Ψ2=0,Ω10=0,Ω̂10=0,Ω3=0,Ψ1n=0,Ω20n=0,Ω̂20n=0;
G2:W=0,W=0,Ψ1=0,Ω20=0,Ω̂20=0,Ω3=0,Ψ2n=0,Ω10n=0,Ω̂10n=0.

Similar to [12] we apply the method of separation of variables for the eigenvalue problem Eq. (85) to solve for the kinematic variables Ψα, W, Ω3, Ωα0, Wand Ωα0. The kinematic variables can be further expressed in the following form

Ψ1nm=A1cosx1asinx2asinωt+B1sinx1acosx2asinωt,Ψ2nm=A2sinx1acosx2asinωt+B2cosx1asinx2asinωt,Wnm=A3sinx1asinx2asinωt+B3cosx1acosx2asinωt,Ω3nm=A4cosx1acosx2asinωt+B4sinx1asinx2asinωt,Ω10,nm=A5sinx1acosx2asinωt+B5cosx1asinx2asinωt,Ω20,nm=A6cosx1asinx2asinωt+B6sinx1acosx2asinωt,W,nm=A7sinx1asinx2asinωt+B7cosx1acosx2asinωt,Ω̂10,nm=A8sinx1acosx2asinωt+B8cosx1asinx2asinωt,Ω̂20,nm=A9cosx1asinx2asinωt+B9sinx1acosx2asinωt,

where Aiand Biare constants.

We solve an eigenvalue problem by substituting these expressions into the system of Eq. (85). The obtained nine sequences of positive eigenfrequencies ωinmare associated with the rotation of the middle plane (ω1nmand ω2nm), flexural motion and its transverse variation (ω3nmand ω7nm), micro rotatory inertia (ω4nm, ω5nmand ω6nm) and its transverse variation (ω8nmand ω9nm) [12].

We perform all our numerical simulations for a=3.0 m and h=0.1 m. We consider different forms of micro elements: ball-shaped elements, horizontally and vertically stretched ellipsoids (see Figure 2). For simplicity we will use the notation ωifor the first elements ωi11of the sequences ωinm. The results of the computations are given in the Table 2. The shape of the micro-elements does not effect the natural macro frequencies ω1and ω2associated with the rotation of the middle plane and ω3and ω7associated with the flexural motion and its transverse variation. The ellipsoid elements have higher micro frequencies associated with the micro rotatory inertia (ω4, ω5and ω6) and its transverse variation (ω8and ω9), than the ball-shaped elements.

ω1, ω2ω3, ω7ω4ω5, ω8ω6, ω9
ShapeJxJyJzω1, ω2ω3, ω7ω4ω5, ω8ω6, ω9
Ball0.0010.0010.00117.880.31501.13205.62338.95
Vertical ellipsoid0.0010.0010.000117.880.31501.13650.22338.95
Horizontal ellipsoid0.00010.0010.00117.880.311363.01205.62394.08

Table 2.

Eigenfrequencies ωi11(Hz) for different shapes of micro-elements.

Let Jx, Jyand Jzbe the principal moments of inertia of the microelements corresponding to the principal axes of their rotation. We assume that the quantities Jx, Jyand Jzare constant throughout the plate B0. If the microelements are rotated around the z-axis by the angle θthe rotatory inertia tensor Jcan be expressed as

J=Jxcos2θ+Jysin2θJxJysin2θ0JxJysin2θJxsin2θ+Jycos2θ000JzE103

The eigenfrequencies for different angles of microrotation of the microelements are given in the Table 3 and the Figure 3. The rotatory inertia principle moments used are Jx=0.002, Jy=0.001, Jz=0.0001, which represent a horizontally stretched ellipsoid microelement. The case when the microelements are not aligned with the edges of the plate the model predicts some additional natural frequencies related with the microstructure of the material.

Angle θω1ω2ω3ω7ω4ω5ω8ω6ω9
017.8817.880.310.31650.221265.37265.37450.61450.61
1017.8817.880.310.31650.221255.59279.40429.89469.93
2017.8817.880.310.31650.221247.75295.33406.70484.79
3017.8817.880.310.31650.221242.57313.65382.94495.14
4017.8817.880.310.31650.221239.99333.10360.57500.46
4517.8817.880.310.31650.221239.68338.95354.35501.13
5017.8817.880.310.31650.221239.99333.10360.57500.46
6017.8817.880.310.31650.221242.57313.65382.94495.14
7017.8817.880.310.31650.221247.75295.33406.70484.79
8017.8817.880.310.31650.221255.59279.40429.89469.93
9017.8817.880.310.31650.221265.37265.37450.61450.61

Table 3.

Eigenfrequencies ωi11(Hz) for different angles of rotation of horizontal ellipsoid micro-elements.

Figure 3.

Micro frequencies ω4, ω5, ω8, ω6 and ω9.

6. Conclusions

In this chapter, we presented a mathematical model of Cosserat plate vibrations. The dynamic model of the plates has been developed as a dynamic extension of the Reissner plate theory. The equations has been presented in both tensorial and the matrix forms. We also described the validation of the model, which is based on the comparison with the three-dimensional Cosserat elastodynamics exact solutions. Based on the presented results of the computer simulations we were able to detect and classify the additional high resonance frequencies of a plate. We have shown that the frequencies depend on the shape and orientation of microelements (ball-shaped elements, horizontally and vertically stretched ellipsoids) incorporated into the Cosserat plates. We also have been able to identify that micro frequencies associated with the micro rotatory inertia and its transverse variation of the ellipsoid elements have higher micro frequencies than the ball-shaped elements. We also showed the dependence of the eigenfrequencies on the angles of rotation of the horizontal ellipsoid micro-elements. These results can be used to identify the characteristics of the plate micro-elements.

A.1 Conventions

We use the following notation convention:

  1. the values of the Latin subindex itake values in the set 1,2,3

  2. the values of the Greek indices αand βtake values in the set 12

  3. the Einstein summation notation is used throughout the chapter

A.2 Notations

xi

artesian coordinates

P

Cosserat thin plate

h

plate thickness

μ,λ

Lamé parameters

α,β,γ,ε

Cosserat elasticity parameters

ρ

material density

JjiorJ

rotatory inertia

σjiorσ

the stress tensor

μjiorμ

the couple stress tensor

γjiorγ

strain tensor

χjiorχ

bend-twist tensor

uioru

displacement vector

ϕiorϕ

microrotation vector

piorp

linear momentum

qiorq

angular momentum

εijk

Levi-Civita tensor

UC

strain stored energy

UK

stress energy

TC

stored kinetic energy

TW

work of inertia forces

S

Cosserat plate stress set

U

Cosserat plate displacement set

E

Cosserat plate strain set

η

splitting parameter

p

pressure

ω

natural frequency of plate vibration

θ

angle of microelement orientation

Mαβ

bending and twisting moments

Qα

shear forces

Qα,Q̂α

transverse shear forces

Rαβ

micropolar bending moments

Rαβ

micropolar twisting moments

Sα

micropolar couple moments

Ψα

rotations of the middle plane around xαaxis

W,W

vertical deflections of the middle plate

Ωα0

microrotations in the middle plate around xαaxis

Ω3

rate of change of the microrotation

Notes

  • In the following formulas a subindex β = 1 if α = 2 and β = 2 if α = 1.

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Lev Steinberg and Roman Kvasov (July 10th 2019). Distinctive Characteristics of Cosserat Plate Free Vibrations [Online First], IntechOpen, DOI: 10.5772/intechopen.87044. Available from:

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