Numerical modeling of bending of micropolar plates

Numerical modeling of bending of micropolar plates

Roman Kvasov, Lev Steinberg n

University of Puerto Rico at Mayagüez, Department of Mathematical Sciences, Call Box 9000, Mayagüez, Puerto Rico 00681-9018, USA

a r t i c l e i n f o

Article history:

Received 9 January 2013

Received in revised form

7 March 2013

Accepted 2 April 2013Available online 20 May 2013

Keywords:

Cosserat elasticity

Micropolar plates

Finite Element modeling

a b s t r a c t

In this paper we present the Finite Element modeling of the bending of micropolar elastic plates. Based

on our recently published enhanced mathematical model for Cosserat plate bending, we present the

micropolar plate field equations as an elliptic system of nine differential equations in terms of

the kinematic variables. The system includes an optimal value of the splitting parameter, which is the

minimizer of the micropolar plate stress energy. We present the efficient algorithm for the estimation of

the optimal value of this parameter and discuss the approximations of stress and couple stress

components. The numerical algorithm also includes the method for finding the unique solution of the

micropolar plate field equations corresponding to the optimal value of the splitting parameter.

We provide the results of the numerical modeling for the plates made of polyurethane foam used in

structural insulated panels. The comparison of the numerical values of the vertical deflection for the

square plate made of dense polyurethane foam with the analytical solution of the three-dimensional

micropolar elasticity confirms the high order of approximation of the three-dimensional (exact) solution.

The size effect of micropolar plate theory predicts that plates made of smaller thickness will be more

rigid than would be expected on the basis of the Reissner plate theory. We present the numerical results

for plates of different shapes, including shapes with rectangular holes, under different loads.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In the classical theory of elasticity the material particles of a

continuum body are assumed to have three degrees of freedom,

which are responsible for their macrodisplacement within the

body. The size effects of the particles and their mutual rotational

interactions are ignored. The surface loads are assumed to be

completely determined by the force vector, which results in the

symmetry of the stress tensor. Classical theory of elasticity is

widely used in engineering and is usually applied to the linear

elastic materials under small deformations (stainless steel, alumi-

nium, plastic, etc.). However, many modern materials possess

certain microstructure (cellular solids, pores, macromolecules,

fibers, grains, voids, etc.) and exhibit experimental behavior that

cannot be adequately described by the classical elasticity.

For example the influence of size effect on the deformation

reported in [21] can be only predicted by micropolar elasticity.

Micropolar theory of elasticity describes the influence of the

microstructure on the deformation of the material. The mate-

rial particles are allowed to have three additional degrees of

freedom, which represent their microrotations. The surface loads

are described by the force and moment vectors, which leads to

asymmetry of the stress tensor and the introduction of the couple

stress tensor. Such important engineering materials as chopped

fiber composites, platelet composites, particulate composites,

sandwich and grid structures, trusses and honeycombs are just a

few examples of the materials that exhibit nonclassical behavior.

Rocks, soils, concrete with sand, ferroelectric and phononic crys-

tals, polyfoams, human bones and some biological tissues are also

considered to be micropolar materials [24,1,18,5,19,2,27,21,17].

The development of the micropolar elasticity can be traced

back to 1887 when Voigt made an assumption that the surface

loads should be described by both force and moment vectors, thus

introducing the concept of couple stress. In 1909 Cosserat brothers

presented the equations of local balance of momenta for stress

and couple stress, and the expressions for surface tractions and

couples [7]. Eringen developed micromorphic and micropolar

theories of solids, fluids, memory-dependent media, microstretch

solids and fluids and solved several problems in these fields [11].

Eringen also introduced a theory of plates in the framework of

the linear micropolar elasticity [13]. The theory was based on a

technique of integration of the micropolar elasticity and assumes

no transverse variation of micropolar rotations. More remarks on

the history of the modeling of classic linear elastic plates can be

found in [23,30,31].

The micropolar plate theory based on the Reissner plate theory

was first proposed in [32] and in the final form has been

developed in [33]. The preliminary computations showed that

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journal homepage: www.elsevier.com/locate/tws

Thin-Walled Structures

0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.tws.2013.04.001

n Corresponding author. Tel.: +1 7878324040; fax: +1 7872655454.

E-mail addresses: [email protected], [email protected] (L. Steinberg).

Thin-Walled Structures 69 (2013) 67–78

the precision of the micropolar plate theory [33] is compatible

with the precision of the Reissner elastic plate theory [10].

In this paper we present the Finite Element analysis of

micropolar plate bending. In Section 2 we make an overview of

the micropolar plate theory, its assumptions, expression for

micropolar plate stress energy and the bending system of

equations. We also present the field equations in terms of the

kinematic variables and derive the explicit expression for

the optimal value of the splitting parameter, which minimizes

the micropolar plate stress energy. In Section 3 we provide the

numerical analysis of the bending of micropolar plates. We

describe the Finite Element method used to solve the micropolar

field equations and discuss the approximations of stress and

couple stress components. We also provide the efficient algo-

rithm for the optimal value of the splitting parameter. The

comparison of the displacement for the square plate made of

dense polyurethane foam with the analytical solution of the

three-dimensional micropolar elasticity presented in [33] shows

the high order of approximation of the three-dimensional (exact)

solution. We also provide the Finite Element analysis of the plates

of different construction shapes and plates with rectangular

holes, under different loads.

2. Enhanced model for Cosserat plate bending

In this section we will reproduce the main micropolar plate

equations and the general results of the micropolar plate theory

presented in [33]. We will develop the micropolar plate field

equations in terms of the kinematic variables and discuss the

ellipticity of the obtained system. We will obtain the explicit

expression for the optimal value of the splitting parameter for

plates of arbitrary shape.

2.1. Micropolar plate equations

The equations of balance of momentum and moment of mome-

ntum of the linear theory of micropolar elasticity are respectively

given by

sji;j þ ρðf i−∂2ui=∂t

2Þ ¼ 0;

εijksjk þ μji;j þ ρðli−j∂2φi=∂t

2Þ ¼ 0;

where sij is the stress tensor, μij is the couple stress tensor, εijk is

the Levi-Civita symbol, ρ is the mass density, f is the body force, uiis the displacement vector, li is the body couple, j is the micro-

inertia and φi is the microrotation vector. In this theory ρ and j are

assumed to be constant [13,12].

For static case the equilibrium equations of the linear micro-

polar theory without body forces and body moments have the

following form:

sji;j ¼ 0; ð1Þ

εijksjk þ μji;j ¼ 0: ð2Þ

The constitutive equations of the linear theory are given in the

form [26]:

sij ¼ ðμþ αÞγij þ ðμ−αÞγji þ λδijγkk; ð3Þ

μij ¼ ðγ þ ϵÞχ ij þ ðγ−ϵÞχji þ βδijχkk; ð4Þ

where γij and χij are the micropolar strain and torsion tensors

respectively, μ and λ are the Lamé parameters, α, β, γ and ϵ are the

asymmetric elasticity constants.

The strain–displacement and torsion–rotation relations are

given as in [12]:

γij ¼ uj;i−εijkφk; ð5Þ

χij ¼ φj;i: ð6Þ

Let us consider the body B0 to be a plate of thickness

h and x3 ¼ 0 containing its middle plane P0. The sets T and B

are the top and bottom surfaces contained in the planes x3 ¼ h=2

and x3 ¼ −h=2 respectively and the curve Γ is the boundary

of the middle plane of the plate. We consider the vertical

load and the boundary conditions at the top and bottom of the

plate:

s33ðx1; x2;h=2Þ ¼ stðx1; x2Þ; s33ðx1; x2;−h=2Þ ¼ s

bðx1; x2Þ; ð7Þ

s3βðx1; x2;7h=2Þ ¼ 0; ð8Þ

μ33ðx1; x2;h=2Þ ¼ μtðx1; x2Þ; μ33ðx1; x2;−h=2Þ ¼ μbðx1; x2Þ; ð9Þ

μ3βðx1; x2;7h=2Þ ¼ 0; ð10Þ

where ðx1; x2Þ∈P0:

We briefly review our approach presented in [33]. As it is

assumed in the standard theory of plates, we use the following

approximation for the stress components:

sαβ ¼h

2ζmαβðx1; x2Þ; ð11Þ

where ζ¼ ð2=hÞx3; and α; β∈f1;2g. Based on (11) and by means of

the first two equations of stress equilibrium (1) written in the

component form

sjβ;j ¼ 0

we obtain for the shear stress components

s3β ¼ qβðx1; x2Þð1−ζ2Þ; ð12Þ

We use expression for the stress components [33]:

sβ3 ¼ qn

βðx1; x2Þð1−ζ2Þ þ q̂βðx1; x2Þ: ð13Þ

Substituting Eq. (13) in the remaining equilibrium differential

equation for stress

sj3;j ¼ 0

we obtain the expression for the transverse normal stress

s33 ¼ ζ 13ζ

2−1

! "

knðx1; x2Þ þ ζl

nðx1; x2Þ þmnðx1; x2Þ: ð14Þ

The next step is to accommodate approximations (14) to the

boundary (7). By direct substitution to (7) it easy to obtain that

s33 ¼−34 ð

13ζ

3−ζÞp1ðx1; x2Þ þ ζp2ðx1; x2Þ þ s0ðx1; x2Þ; ð15Þ

where

p1ðx1; x2Þ þ 2p2ðx1; x2Þ ¼ pðx1; x2Þ

We consider the parametric solution of the last equation in the

form:

p1ðx1; x2Þ ¼ ηpðx1; x2Þ;

p2ðx1; x2Þ ¼ð1−ηÞ

2pðx1; x2Þ

and η∈½0;1( is a parameter, which we call the splitting parameter.

This allows us to split the bending pressure on the plate pðx1; x2Þ

into two parts corresponding to different orders of stress approx-

imations. The optimal value of the splitting parameter shows the

contribution of different types of approximation in the plate

bending. This approach gives a more accurate description of the

mechanical phenomenon of bending.

R. Kvasov, L. Steinberg / Thin-Walled Structures 69 (2013) 67–7868

Note that for

pðx1; x2Þ ¼ stðx1; x2Þ−s

bðx1; x2Þ

and

s0ðx1; x2Þ ¼12ðs

tðx1; x2Þ þ sbðx1; x2ÞÞ

the expression (15) satisfies the boundary condition requirements.

Note that in the case of η¼ 1 expression (15) is identical to the

expression of s33 given in [32].

We use the following approximation for the couple stress

components [33]:

μαβ ¼ ð1−ζ2Þrαβðx1; x2Þ þ rnαβðx1; x2Þ: ð16Þ

and couple stress

μβ3 ¼ ζsnβðx1; x2Þ: ð17Þ

Note that the first two equations of (2) can be written in the

form

ϵβjksjk þ μjβ;j ¼ 0; ð18Þ

and substituting the couple stress (16) into (18) and taking into

account (12) and (13) we obtain the expression for the transverse

shear couple stress

μ3β ¼ ð13ζ3−ζÞsβðx1; x2Þ: ð19Þ

Substituting (19) into boundary (8) we obtain that

sβðx1; x2Þ ¼ 0;

i.e. the transverse shear couple stress [32]

μ3β ¼ 0: ð20Þ

We use the assumption from [32], i.e. μ33 is a first order

polynomial

μ33 ¼ ζbnðx1; x2Þ þ cnðx1; x2Þ: ð21Þ

which satisfies the remaining differential equation of the equili-

brium of angular momentum (2)

ϵ3jksjk þ μj3;j ¼ 0: ð22Þ

This assumption is also consistent with the equilibrium equa-

tion (22) and allows us to proceed as we did for the determination

of transverse loading stress (15) from the stress boundary condi-

tions. The boundary conditions (9) are sufficient to determine μ33,

which must be of the form [32]

μ33 ¼ ζvþ t; ð23Þ

where the function vðx1; x2Þ and tðx1; x2Þ are given as

vðx1; x2Þ ¼12ðμ

tðx1; x2Þ−μbðx1; x2ÞÞ;

tðx1; x2Þ ¼12ðμ

tðx1; x2Þ þ μbðx1; x2ÞÞ:

The choice of kinematic assumptions is based on simplicity and

their compatibility with the constitutive relationships of stress and

couple stress assumptions are the following [33]:

uα ¼h

2ζΨ αðx1; x2Þ; ð24Þ

u3 ¼wðx1; x2Þ þ ð1−ζ2Þwnðx1; x2Þ: ð25Þ

The terms Ψ αðx1; x2Þ in (24) represent the rotations in the

middle plane.

We also use the microrotation φα in the following form [33]:

φα ¼5

4Ω0

αðx1; x2Þð1−ζ2Þ þ Ω̂αðx1; x2Þ; ð26Þ

φ3 ¼5h

8ζ 1−

1

3ζ2

# $

Ω3ðx1; x2Þ: ð27Þ

The constitutive formulas motivate us to chose the forms (26)

and (27), which produce expressions for φα;β and φ3;3 similar to

what we have for couple stress approximations (16).

Thus, we summarize the resulting assumptions for stress sij

and couple stress μij components across the thickness [33]

sαβ ¼h

2ζmαβðx1; x2Þ; ð28Þ

s3β ¼ qβðx1; x2Þð1−ζ2Þ; ð29Þ

sβ3 ¼ qn

βðx1; x2Þð1−ζ2Þ þ q̂βðx1; x2Þ; ð30Þ

s33 ¼−3

4

1

3ζ3−ζ

# $

p1ðx1; x2Þ þ ζp2ðx1; x2Þ þ s0ðx1; x2Þ; ð31Þ

μαβ ¼ ð1−ζ2Þrαβðx1; x2Þ þ rnαβðx1; x2Þ; ð32Þ

μβ3 ¼h

2ζsnβðx1; x2Þ; ð33Þ

μ3β ¼1

3ζ3−ζ

# $

sβðx1; x2Þ; ð34Þ

μ3β ¼ 0; ð35Þ

μ33 ¼ ζvþ t; ð36Þ

where

pðx1; x2Þ ¼ stðx1; x2Þ−s

bðx1; x2Þ;

s0ðx1; x2Þ ¼1

2ðstðx1; x2Þ þ s

bðx1; x2ÞÞ;

p1ðx1; x2Þ ¼ ηpðx1; x2Þ;

p2ðx1; x2Þ ¼ð1−ηÞ

2pðx1; x2Þ;

vðx1; x2Þ ¼1

2ðμtðx1; x2Þ−μ

bðx1; x2ÞÞ;

tðx1; x2Þ ¼1

2ðμtðx1; x2Þ þ μbðx1; x2ÞÞ:

The kinematic assumptions are given as

uα ¼h

2ζΨ αðx1; x2Þ; ð37Þ

u3 ¼Wðx1; x2Þ þ ð1−ζ2ÞWnðx1; x2Þ; ð38Þ

φα ¼5

4Ω0

αðx1; x2Þð1−ζ2Þ þ Ω̂αðx1; x2Þ; ð39Þ

φ3 ¼5h

8ζ 1−

1

3ζ2

# $

Ω3ðx1; x2Þ: ð40Þ

We also introduce the set of kinematic variables defined as

UðηÞ ¼ ½Ψ α;W ;Ω3;Ω

0α ;W

n; Ω̂α(T ;

and the micropolar plate stress set

SðηÞ ¼ ½Mαβ ;Qα;Q

n

α; Q̂ α;Rαβ;Rn

αβ; Sn

β (;

where

Mαβ ¼h

2

# $2 Z 1

−1ζ3sαβ dζ3 ¼

h3

12mαβ ; ð41Þ

Qα ¼h

2

Z 1

−1s3α dζ3 ¼

2h

3qα; ð42Þ

Qn

α ¼h

2

Z 1

−1qn

αð1−ζ2Þ dζ3 ¼

2h

3qn

α; ð43Þ

R. Kvasov, L. Steinberg / Thin-Walled Structures 69 (2013) 67–78 69

Q̂ α ¼h

2

Z 1

−1q̂αð1−ζ

2Þ dζ3 ¼2h

3q̂α ð44Þ

Rαβ ¼h

2

Z 1

−1rαβð1−ζ

2Þ dζ3 ¼2h

3rαβ ; ð45Þ

Rn

αβ ¼h

2

Z 1

−1rnαβð1−ζ

2Þ dζ3 ¼2h

3rnαβ ; ð46Þ

Snα ¼h

2

# $2 Z 1

−1ζ3μα3 dζ3 ¼

h3

12snα; ð47Þ

here M11 and M22 are the bending moments, M12 and M21 the

twisting moments, Qα the shear forces, Qn

α; and Q̂ α the transverse

shear forces, R11;R22;Rn

11;Rn

22 the micropolar bending moments,

R12;R21;Rn

12;Rn

21 the micropolar twisting moments, Snα the micro-

polar couple moments, all defined per unit length. [33]

The components of the micropolar plate strain set

EðηÞ ¼ ½eαβ;ωβ;ω

n

a; ω̂α; τ3α; ταβ; τn

αβ(

are defined as in [33]

eαβ ¼3

h

Z 1

−1ζ3γαβ dζ3; ð48Þ

ωα ¼3

4

Z 1

−1γ3αð1−ζ

2Þ dζ3; ð49Þ

ωn

α ¼3

4

Z 1

−1γα3ð1−ζ

2Þ dζ3; ð50Þ

ω̂α ¼3

4

Z 1

−1γα3 dζ3; ð51Þ

τ3α ¼3

h

Z 1

−1ζ3χ3α dζ3; ð52Þ

ταβ ¼3

4

Z 1

−1χαβð1−ζ

2Þ dζ3; ð53Þ

τnαβ ¼3

4

Z 1

−1χαβ dζ3: ð54Þ

The density of the work done by the stress and couple stress

over the micropolar strain field defined as

WðηÞ ¼

h

2

Z 1

−1ðr ) γ þ μ ) χ Þ dζ3 ð55Þ

is equal to the micropolar plate stress energy density [33].

The expression for the density of the work is obtained by

substituting the stress and couple stress assumptions (28)–(36)

and performing the integration along the thickness of the

plate [33]

WðηÞ ¼ S

ðηÞ ) EðηÞ ¼Mαβeαβ þ Qαωα

þQn

αωn

α þ Q̂ αω̂α þ Rαβταβ þ Rn

αβτn

αβ þ Snατ3α: ð56Þ

The components of the micropolar plate strain set EðηÞ are

represented in terms of the components of the set of kinematic

variables UðηÞ by the following micropolar plate strain–displace-

ment relations1:

eαβ ¼Ψ β;α−ε3αβΩ3; ð57Þ

ωα ¼ Ψ α−ε3αβðΩ0β þ Ω̂βÞ; ð58Þ

ωn

α ¼W ;α þ4

5Wn

;α þ ε3αβðΩ0β þ Ω̂βÞ; ð59Þ

ω̂α ¼3

2W ;α þWn

;α þ ε3αβð5

4Ω0

β þ3

2Ω̂βÞ; ð60Þ

τ3α ¼Ω3;α; ð61Þ

ταβ ¼Ω0β;α þ Ω̂β;α; ð62Þ

τnαβ ¼5

4Ω0

β;α þ3

2Ω̂β;α; ð63Þ

The equilibrium and constitutive equations can be obtained in

accordance with the Generalized Hellinger–Prange–Reissner (HPR)

principle for the micropolar plate, which requires

δΘðs; ηÞ ¼ 0;

where Θ is a functional on A defined by

Θðs; ηÞ ¼ USK ðηÞ−

Z

P0

ðS ) E−p )W þ vΩ03Þ da

þ

Z

Γs

So ) ðU−UoÞ dsþ

Z

Γu

Sn ) U ds; ð64Þ

A is the set of all admissible states that satisfy the micropolar plate

strain–displacement and torsion–rotation relations (5) and (6),

p¼ ðp̂1; p̂2Þ, W ¼ ðW ;WnÞ and for every s¼ ½U ; E;S(∈A.

The optimization is equivalent to [33]

(A) The equilibrium system of equations for the plate bending

Mαβ;α−Qβ ¼ 0; ð65Þ

Qn

α;α þ p̂1 ¼ 0; ð66Þ

Rαβ;α þ ε3βγðQn

γ−Q γÞ ¼ 0; ð67Þ

ε3βγMβγ þ Snα;α ¼ 0; ð68Þ

Q̂ α;α þ p̂2 ¼ 0; ð69Þ

Rn

αβ;α þ ε3βγQ̂ γ ¼ 0; ð70Þ

where p̂1 ¼ ηp, p̂2 ¼23 ð1−ηÞp; and η is the splitting parameter;

(B) The zero variation of the stress energy with respect to the

splitting parameter

δUSK ðηÞ ¼ 0;

(C) The constitutive formulas are given in the following reverse

form2:

Mαα ¼h3μðλþ μÞ

3ðλþ 2μÞΨ α;α þ

λμh3

6ðλþ 2μÞΨ β;β þ

ð3p1 þ 5p2Þλh2

30ðλþ 2μÞ; ð71Þ

Mβα ¼ðμ−αÞh

3

12Ψ α;β þ

h3ðα þ μÞ

12Ψ β;α þ ð−1Þβ

αh3

6Ω3; ð72Þ

Rβα ¼5ðγ−ϵÞh

6Ω0

β;α þ5hðγ þ ϵÞ

6Ω0

α;β; ð73Þ

Rαα ¼10hγðβ þ γÞ

3ðβ þ 2γÞΩ0

α;α þ5hβγ

3ðβ þ 2γÞΩ0

β;β; ð74Þ

Rn

βα ¼2ðγ−ϵÞh

3Ω̂β;α þ

2ðγ þ ϵÞh

3Ω̂α;β; ð75Þ

Rn

αα ¼8γðγ þ βÞh

3ðβ þ 2γÞΩ̂α;α þ

4γβh

3ðβ þ 2γÞΩ̂β;β; ð76Þ

1 Eqs. (57) and (58) are the correct versions of the equations developed in [33]. 2 In the following formulas a subindex β¼ 1 iff α¼ 2 and β¼ 2 iff α¼ 1.


Qα ¼5hðαþ μÞ

6Ψ α þ

5ðμ−αÞh

6W ;α þ

2ðμ−αÞh

3Wn

;α ð77Þ

þð−1Þβ5hα

3ðΩ0

β þ Ω̂βÞ; ð78Þ

Qn

α ¼5ðμ−αÞh

6Ψ α þ

5ðμ−αÞ2h

6ðμþ αÞW ;α þ

2ðμþ αÞh

3Wn

;α

þð−1Þα5hα

3ðΩ0

β þðμ−αÞ

ðμþ αÞΩ̂βÞ; ð79Þ

Q̂ α ¼8αμh

3ðμþ αÞW ;α þ ð−1Þα

8αμh

3ðμþ αÞΩ̂β; ð80Þ

Snα ¼5γϵh3

3ðγ þ ϵÞΩ3;α: ð81Þ

2.2. Micropolar plate field equations

In order to obtain the micropolar plate bending field equations

in terms of the kinematic variables, we substitute the constitutive

formulas in the reverse form (71)–(80) into the bending system of

Eqs. (65)–(70). The micropolar plate bending field equations can be

written in the following form:

LU ¼F ðηÞ ð82Þ

where

L¼

L11 L12 L13 L14 0 L16 kL13 0 L16

L12 L22 L23 L24 L16 0 kL23 L16 0

−L13 −L23 L33 0 L35 L36 L77 L38 L39

L41 L42 0 L44 0 0 0 0 0

0 −L16 −L38 0 L55 L56 −kL35 L58 0

L16 0 −L39 0 L56 L66 −kL36 0 L58

−L13 −L14 L73 0 L35 L36 L77 L78 L79

0 −L16 −L78 0 L85 L56 −kL35 L88 kL56

L16 0 −L79 0 L56 L55 −kL36 kL56 L99

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

;

U ¼ Ψ1; Ψ 2; W ; Ω3; Ω01; Ω0

2; Wn; Ω01; Ω0

2 (T;

h

F ðηÞ ¼ −3h2

λð3p1;1 þ 5p2;1Þ

3ðλþ 2μÞ; −

3h2λð3p1;2 þ 5p2;2Þ

30ðλþ 2μÞ; −p1; 0; 0; 0;

h2ð3p1 þ 4p2Þ

24; 0; 0

" #T

:

The operators Lij are defined as follows:

L11 ¼ c1∂2

∂x21þ c2

∂2

∂x22−c3; L12 ¼ ðc1−c2Þ

∂2

∂x1x2;

L13 ¼ c11∂

∂x1; L14 ¼ c12

∂

∂x2;

L16 ¼ c13; L17 ¼ kc11∂

∂x1; L22 ¼ c2

∂2

∂x21þ c1

∂2

∂x22−c3;

L23 ¼ c11∂

∂x2; L24 ¼ −c12

∂

∂x1;

L33 ¼ c4∂2

∂x21þ

∂2

∂x22

!

; L35 ¼−c13∂

∂x2; L36 ¼ c13

∂

∂x1;

L38 ¼−c10∂

∂x2; L39 ¼ c10

∂

∂x1;

L41 ¼−c12∂

∂x2; L42 ¼ c12

∂

∂x1; L44 ¼ c6

∂2

∂x21þ

∂2

∂x22

!

−2c12;

L55 ¼ c7∂2

∂x21þ c8

∂2

∂x22−2c13; L56 ¼ ðc7−c8Þ

∂2

∂x1x2; L58 ¼−c9;

L66 ¼ c8∂2

∂x21þ c7

∂2

∂x22−2c13; L73 ¼ c5

∂2

∂x21þ

∂2

∂x22

!

;

L77 ¼ c3∂2

∂x21þ

∂2

∂x22

!

;

L78 ¼ −c14∂

∂x2; L79 ¼ c14

∂

∂x1; L85 ¼ c7

∂2

∂x21þ c8

∂2

∂x22−2c13;

L88 ¼ kc7∂2

∂x21þ kc8

∂2

∂x22−c15; L99 ¼ kc8

∂2

∂x21þ kc7

∂2

∂x22−c15;

where

k¼4

5; c1 ¼

h3μðλþ μÞ

3ðλþ 2μÞ; c2 ¼

h3ðαþ μÞ

12;

c3 ¼5hðαþ μÞ

6; c4 ¼

5hðα−μÞ2

6ðα þ μÞ;

c5 ¼hð5α2 þ 6αμþ 5μ2Þ

6ðα þ μÞ; c6 ¼

h3γϵ

3ðγ þ ϵÞ;

c7 ¼10hγðβ þ γÞ

3ðβ þ 2γÞ; c8 ¼

5h γ þ ϵð Þ

6;

c9 ¼10hα2

3ðαþ μÞ; c10 ¼

5hαðα−μÞ


5hðα−μÞ

6; c12 ¼

h3α

6;

c13 ¼5hα

3; c14 ¼

hαð5αþ 3μÞ


2hαð5αþ 4μÞ

3ðα þ μÞ:

The parametric system (82) is an order two elliptic system of

nine partial differential equations, where L is a linear differential

operator acting on the set of kinematic variables U . The ellipticity

of the system (82) can be easily verified by calculating the

determinant of the principle symbol LðξÞ

det LðξÞ ¼256αμ

375ðαþ μÞc1c2c3c6c

27ðξ

21 þ ξ22Þ

9;

where ξ¼ ðξ1; ξ2Þ is an arbitrary vector from R2. For nonzero elastic

constants the determinant of LðξÞ is nonzero if and only if ξ≠0, and

thus the system of equations (82) is elliptic. The ellipticity

condition for the system suggests the invertibility of the principal

part of its symbol LðξÞ for all non-zero values of ξ [9].

The right-hand side, and therefore the solution U are the

functions of the splitting parameter η. The optimal value of the

parameter η corresponds to the minimum of elastic energy over

the micropolar strain field (55).

The system (82) should be complemented with the boundary

conditions. We describe the case of the hard simply supported

boundary conditions in the numerical analysis section.

2.3. Optimal value of the splitting parameter

As we have mentioned earlier, the equilibrium systems of

partial differential equations correspond to a state of the system

(82) where the minimum of the energy is reached. The optimiza-

tion of the splitting parameter appears as a result of the

Generalized Hellinger–Prange–Reissner (HPR) principle for the

micropolar (64). The bending system of Eqs. (65)–(70) depends

on the splitting parameter and therefore its solution is parametric.

The minimization procedure for the elastic energy allows us to

find the optimal value of this parameter, which corresponds to the

unique solution of the bending problem.

Taking advantage of the linearity of the system (82) we

consider two cases:

LU0 ¼F ð0Þ; ð83Þ

LU1 ¼F ð1Þ; ð84Þ

where U0 is a solution of the bending system of Eqs. (82) when

η¼ 0, and U1 when η¼ 1.


Let us define Uη as a linear combination of U0 and U1

Uη ¼ ð1−ηÞU0 þ ηU1: ð85Þ

Notice that since p1 ¼ ηp, p2 ¼1−η2

1 2

p, F ðηÞ ¼ ð1−ηÞF ð0Þ þ ηF ð1Þ

and therefore due to the linearity of system (82)

LUη ¼ Lðð1−ηÞU0 þ ηU1Þ ¼ ð1−ηÞLU0 þ ηLU1

¼ ð1−ηÞF ð0Þ þ ηF ð1Þ ¼F ðηÞ:

The set of kinematic variables Uη defined as (85) is a solution of

the micropolar plate bending system of Eqs. (82).

The optimal value of the splitting parameter η is a minimizer of

the density of the work WðηÞ done by the stress and couple stress

over the micropolar strain field. Let us obtain an explicit expres-

sion for the optimal value of the parameter η.

Let us represent each component of the stress set SðηÞ and strain

set EðηÞ as a linear combinations of the stress sets Sð0Þ and Sð1Þ, and

the strain sets Eð0Þ and E

ð1Þ respectively

SðηÞ ¼ ð1−ηÞSð0Þ þ ηSð1Þ

and

EðηÞ ¼ ð1−ηÞEð0Þ þ ηEð1Þ:

Therefore

WðηÞ ¼ S

ðηÞ ) EðηÞ ¼ ðð1−ηÞSð0Þ þ ηSð1ÞÞ ) ðð1−ηÞEð0Þ þ ηEð1ÞÞ:

The zero of the derivative ∂WðηÞ=∂η gives the optimal value η0 of

the splitting parameter η. Thus

∂WðηÞ

∂η¼ ð−Sð0Þ þ S

ð1ÞÞ ) ðð1−ηÞEð0Þ þ ηEð1ÞÞ þ ðð1−ηÞSð0Þ þ ηSð1ÞÞ

)ðEð1Þ−E

ð0ÞÞ;

and the optimal value is

η0 ¼2Wð00Þ

−Wð10Þ

−Wð01Þ

2ðWð11Þ þWð00Þ

−Wð10Þ

−Wð01ÞÞ

; ð86Þ

where

Wð00Þ ¼ S

ð0Þ ) Eð0Þ; ð87Þ

Wð01Þ ¼ S


Wð10Þ ¼ S


Wð11Þ ¼ S

ð1Þ ) Eð1Þ: ð90Þ

3. Numerical analysis

In this section we describe the Finite Element algorithm used to

solve the micropolar field equations. We develop the algorithm for

the calculation of the optimal value of the splitting parameter.

We compare the main kinematic variables for the square plate

made of dense polyurethane foam with the analytical solution of

the three-dimensional micropolar elasticity presented in [33] and

provide the analysis of the plates of different shapes.

3.1. Comparison of the analytical solutions

In our computations we consider plates made of polyurethane

foam—a material reported in the literature to be micropolar [21].

Insulation materials made from rigid polyurethane foam are used

in the construction of large industrial and agricultural buildings.

It is known that in the plaster supports in wooden structures and

bridging large spans between the top chords in flat roofing, the

insulation materials are exposed to bending stresses [15].

Therefore the prediction of the behavior of the polyurethane foam

under bending stress is extremely important. Another example of

the polyurethane foam plates can be found in the structural

insulated panels – a high-performance walling materials, where

the sides of the panel are made of cement, plywood, or oriented

strand board, and the middle part is made of highly dense

polyurethane foam. The structural insulated panels are widely

used in walls, floor slabs and roofs of houses and commercial

buildings.

In our calculations we will use the following technical para-

meters: Young's modulus E, Poisson's ratio ν, the characteristic

length for bending lb, the characteristic length for torsion lt, the

coupling number N.

The conversion formulas between the technical constants and

elastic parameters [17]

E¼μð3λþ 2μÞ

λþ μ; ν¼

λ

2ðλþ μÞ;

lb ¼1

2

ffiffiffiffiffiffiffiffiffiffiffi

γ þ ϵ

μ

r

; lt ¼

ffiffiffi

γ

μ

r

; N¼

ffiffiffiffiffiffiffiffiffiffiffiffi

α

μþ α

r

;

imply the following inverse conversion formulas:

λ¼Eν

2ν2 þ ν−1; μ¼

E

2ðνþ 1Þ; α¼

EN2

2ðνþ 1ÞðN2−1Þ

;

β¼Elt

2ðνþ 1Þ; γ ¼

El2t

2ðνþ 1Þ; ϵ¼

Eð4l2b−l2t Þ

2ðνþ 1Þ:

Let us consider a plate of thickness h¼0.1 m made of polyur-

ethane foam, and the ratio a=h varying from 5 to 30. The values of

the technical elastic parameters for the polyurethane foam are

reported in [21]: E¼ 299:5 MPa, ν¼ 0:44, lt ¼ 0:62 mm, lb ¼ 0:327,

N2 ¼ 0:04. These values correspond to the following values

of Lamé and asymmetric parameters: λ¼ 762:616 MPa, μ¼

103:993 MPa, α¼ 4:333 MPa, β¼ 39:975 MPa, γ ¼ 39:975 MPa,

ϵ¼ 4:505 MPa (the ratio β=γ is equal to 1 for the bending).

Let us first analyze the case of the square plate and compare the

analytical solution for the field equations (82) and the analytical

solution of the corresponding three-dimensional micropolar elas-

ticity problem presented in [33]. Let us consider the following

hard simply supported boundary conditions:

W ¼ 0; Wn ¼ 0; Ψ ) s¼ 0; n )Mn¼ 0; Sn) n¼ 0; ð91Þ

Q n) s¼ 0; Q̂ ) s¼ 0; s ) Rn¼ 0; s ) Rnn¼ 0: ð92Þ

where n and s are the normal and the tangent vectors to the

boundary ∂B0 respectively.

Let the body B0 be a square plate a* a of thickness

h: B0 ¼ ½0; a( * ½0; a( * ½−h=2;h=2(. The boundary of the body is

given by

G¼ G1∪G2

where

G1 ¼ fðx1; x2Þ : x1∈I; x2∈½0; a(g;

G2 ¼ fðx1; x2Þ : x1∈½0; a(; x2∈Ig;

and I¼ f0; ag is the set of two numbers: 0 and a.

For the square plate, the hard simply supported boundary

conditions (90) and (91) can be represented in the following

mixed Dirichlet–Neumann form:

G1 : W ¼ 0; Wn ¼ 0; Ψ 2 ¼ 0; Ω01 ¼ 0; Ω̂

0

1 ¼ 0;

G1 : Ω3 ¼ 0;∂Ψ 1

∂n¼ 0;

∂Ω02

∂n¼ 0;

∂Ω̂0

2

∂n¼ 0;

G2 : W ¼ 0; Wn ¼ 0; Ψ 1 ¼ 0; Ω02 ¼ 0; Ω̂

0

2 ¼ 0;


G2 : Ω3 ¼ 0;∂Ψ2

∂n¼ 0;

∂Ω01

∂n¼ 0;

∂Ω̂0

1

∂n¼ 0:

The initial distribution of the pressure is sinusoidal and is given

in MPa units:

pðx1; x2Þ ¼ sinπx1a

1 2

sinπx2a

1 2

: ð93Þ

The analytical solution of the three-dimensional micropolar

bending problem is presented in [33]. The two-dimensional field

equations (82) can be solved by applying the method of separation

of variables. We obtain the kinematic variables in the following

form:

Ψ1 ¼ A1 cosπx1a

1 2

sinπx2a

1 2

;

Ψ2 ¼ A2 sinπx1a

1 2

cosπx2a

1 2

; ð94Þ

W ¼ A3 sinπx1a

1 2

sinπx2a

1 2

;

Ω03 ¼ A4 cos

πx1a

1 2

cosπx2a

1 2

; ð95Þ

Ω01 ¼ A5 sin

πx1a

1 2

cosπx2a

1 2

; ð96Þ

Ω02 ¼ A6 cos

πx1a

1 2

sinπx2a

1 2

; ð97Þ

Wn ¼ A7 sinπx1a

1 2

sinπx2a

1 2

;

Ω̂0

1 ¼ A8 sinπx1a

1 2

cosπx2a

1 2

; ð98Þ

Ω̂0

2 ¼ A9 cosπx1a

1 2

sinπx2a

1 2

:

We find Ai by substituting the expressions (96) into the system

of Eqs. (82) and solving the obtained system of nine linear

equations for Ai.

The comparison of the analytical solutions for the maximum

vertical deflection and the maximum microrotation of the two-

dimensional model versus the analytical solutions for three-

dimensional micropolar elasticity is provided in Tables 1 and 2.

The comparison of the numerical values of the kinematic

variables for the square plate made of dense polyurethane foam

with the analytical solution of the three-dimensional micropolar

elasticity confirms the high order of approximation of three-

dimensional (exact) solution. The relative error of order 1% for

the maximum vertical deflection is compatible with the precision

of the Reissner plate theory [10] (Fig. 1).

The comparison of the transverse variations of the components

of the displacement of the two-dimensional micropolar plate

theory versus three-dimensional micropolar elasticity theory is

given in Fig. 2.

3.2. Finite element analysis

Let us look for the solution of the micropolar plate bending

system of Eqs. (82) in the Hilbert space V defined as

V ¼ V1 * V2 * V3 * V4 * V5 * V6 * V7 * V8 * V9;

where V1 ¼ V6 ¼ V9 ¼ fv∈H1ðB0Þ; v¼ 0 on G1g, V2 ¼ V5 ¼ V8 ¼

fv∈H1ðB0Þ; v¼ 0 on G2g, V3 ¼ V7 ¼ fv∈H1ðB0Þ; v¼ 0 on Gg,

V4 ¼H1ðB0Þ, and H1ðB0Þ is a standard Hilbert space of func-

tions that are square-integrable together with their first partial

derivatives.

The weak formulation of the micropolar plate bending problem

is obtained as follows. Let V∈H1ðB0Þ9. By multiplying both sides of

(82) by V we have

VLU ¼ VF ðηÞ

and then integrating over the plate B0 we obtainZ

B0

VLU ds¼

Z

B0

VF ðηÞ ds:

Let us introduce the following notation:

aðU;VÞ ¼

Z

B0

VLU ds¼

Z

B0

V iLijU j ds;

f ðVÞ ¼

Z

B0

VF ðηÞ ds:

The corresponding weak problem is formulated as follows: find

all U∈V such that for all V∈H1ðB0Þ9

aðU;VÞ ¼ f ðVÞ: ð99Þ

Note that the form aðU;VÞ ¼R

B0V iLijU j ds is a summation of the

weak forms of the operators Lij. In order to obtain the expression

for the weak formulation (99) we observe that there are only three

types of linear operators present in the expression for aðU ;VÞ—

operators of order zero, one and two:

Lð0Þ ¼ 1: ð100Þ

Table 1

Comparison of the maximum vertical deflection u3 of the polyurethane foam square plate.

a=h 10 15 20 25 30

Optimal value of η 0.2383156 0.1280473 0.0795667 0.0548782 0.0407998

Micropolar plate model (m) 0.0236306 0.0622810 0.1205956 0.2011537 0.3076744

Micropolar 3D elasticity (m) 0.0237443 0.0624282 0.1207626 0.2013360 0.3078709

Relative error (%) 0.47 0.23 0.13 0.09 0.06

Table 2

Comparison of the maximum microrotation φ1 of the polyurethane foam square plate.

a=h 10 15 20 25 30

Optimal value of η 0.2383156 0.1280473 0.0795667 0.0548782 0.0407998

Micropolar Plate Model 0.0013365 0.0052564 0.0132366 0.0266257 0.0467707

Micropolar 3D Elasticity 0.0013430 0.0052686 0.0132545 0.0266490 0.0467995

Relative Error (%) 0.48 0.23 0.14 0.09 0.06


Lð1Þ ¼∂

∂xi: ð101Þ

Lð2Þ ¼∇ ) A∇: ð102Þ

where A¼ A11A21

A12A22

1 2

.

The weak forms of these operators are given as follows:

Z

B0

vðLð0ÞuÞ ds¼ c

Z

B0

vu da;

Fig. 1. Comparison of the vertical deflection for the square plate made of dense polyurethane foam: analytical solution of the proposed plate theory WP and the analytical

solution of the three-dimensional micropolar elasticity WE .

Fig. 2. The comparison of the analytical solutions of the transverse variation for the displacement components for the square plate made of polyurethane foam: solid red line

– the two-dimensional micropolar plate theory, dashed black line – three-dimensional micropolar elasticity ða=h¼ 30Þ.


Z

B0

vðLð1ÞuÞ ds¼

Z

B0

v∂u

∂xida;

Z

B0

vðLð2ÞuÞ ds¼

Z

B0

∇ ) A∇uv da¼

Z

∂B0

ðA∇u ) nÞv dτ−

Z

B0

A∇u )∇v da;

where the weak form of the second order operator is obtained by

performing the corresponding integration by parts.

Note that all second order operators in the weak formulation

(99) can be represented in the form (102):

∂2

∂x21þ

∂2

∂x22¼∇ )

1 0

0 1

# $

∇;

∂2

∂x1x2¼∇ )

0 12

12 0

!

∇;

c1∂2

∂x21þ c2

∂2

∂x22¼∇ )

c1 0

0 c1

!

∇:

In the previous section we showed that the system (82)

represents an elliptic system of partial differential equations. It is

possible to prove that the operator L is in fact strongly elliptic. This

will imply the coerciveness of the operator L and its bilinear form

on V, which is important for the verification of the hypothesis of

Babuška–Lax–Milgram theorem (see [25,3] for details). The corre-

sponding convergence analysis of finite element approximations

of the elliptic systems and the analytic estimates on the domains

with corners are given in [8].

Theorem 1 (Strong ellipticity of the bending system of equations).

The micropolar plate bending system of Eqs. (82) is strongly elliptic.

Proof. Let ξ¼ ðξ1; ξ2Þ and υ¼ ðυ1; υ2; υ3; υ4; υ5; υ6; υ7; υ8; υ9Þ be some

arbitrary vectors from R2 and R9 respectively. Let us assume that

ξ¼ ðξ1; ξ2Þ≠0 and denote the principle symbol of the system of

equations (82) by LðξÞ. The bilinear form υLðξÞυT can be expressed as

υLðξÞυT ¼ l1 þ l2 þ l3 þ l4 þ l5;

where

l1 ¼ c1ðυ1ξ1 þ υ2ξ2Þ2 þ c2ðυ2ξ1−υ1ξ2Þ

2;

l2 ¼ c6υ24ðξ

21 þ ξ22Þ;

l3 ¼ c4υ23ðξ

21 þ ξ22Þ

2 þ4

5c3υ

27ðξ

21 þ ξ22Þ

2;

l4 ¼ c7ð45ðυ8ξ1 þ υ9ξ2Þ

2 þ ðυ5ξ1 þ υ6ξ2Þðυ8ξ1 þ υ9ξ2Þ þ ðυ5ξ1 þ υ6ξ2Þ2Þ;

l5 ¼ c8ð45ðυ9ξ1−υ8ξ2Þ

2 þ ðυ6ξ1−υ5ξ2Þðυ9ξ1−υ8ξ2Þ þ ðυ6ξ1−υ5ξ2Þ2Þ:

It is easy to check that l1, l2, and l3 are always nonnegative.

Indeed, since c1; c240 then l1≥0 for all ξ∈R2, and l1 ¼ 0 if and only

if υ1 ¼ υ2 ¼ 0. Since c640 then l2≥0 for all ξ∈R2, and l2 ¼ 0 if and

only if υ4 ¼ 0. Since c3; c440 then l3≥0 for all ξ∈R2, and l3 ¼ 0 if

Fig. 3. 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 without load (left) and under the load

(right). The applied sinusoidal load is represented by the dashed black line.


and only if υ3 ¼ υ7 ¼ 0. The analysis of l4 and l5 is very similar

since they have quadratic form: 45 x

2 þ xyþ y2 with non-positive

discriminant. Since c740 and c840 then l4≥0 and l5≥0; l3 ¼ 0 if

and only if υ8 ¼ υ9 ¼ υ5 ¼ υ6 ¼ 0, and l4 ¼ 0 if and only if

υ8 ¼ υ9 ¼ υ5 ¼ υ6 ¼ 0. Since li≥0, then υLðξÞυT ¼ 0 if and only if all

li ¼ 0. The last is equivalent to υ¼ 0 and therefore we conclude

that the operator L is strongly elliptic.

The strong ellipticity of the operator L implies the coerciveness

of the bilinear form AðU;VÞ and therefore by Babuška–Lax–

Milgram theorem there exists a unique solution U∈V to the weak

(99).

The proposed minimization of the plate stress energy allows us

to introduce the following effective algorithm for the optimal

value of the splitting parameter η; which can be used for the plate

of an arbitrary shape.

Fig. 4. Hard simply supported square plate 2.0 m*2.0 m*0.1 m made of polyurethane foam: the initial mesh and the isometric view of the resulting vertical deflection

of the plate.

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 5. Nonrectangular hard simply supported plate made of polyurethane foam: (a) the area where the sinusoidal load is applied, (b) initial mesh, (c) the density plot

and (d) the isometric view of the resulting vertical deflection of the plate.


Algorithm 2 (Optimal value of the splitting parameter).

1. Solve the systems (82) and (83) for U0 and U1 respectively.

2. Calculate the components of the micropolar plate stress sets

Sð0Þ and S

ð1Þ from the sets of kinematic variables U0 and U1,

using the constitutive formulas in the reverse form (71)–(80).

3. Calculate the components of the micropolar plate strain sets.

Eð0Þ and E

ð1Þ from the sets of kinematic variables U0 and U1,

using the strain–displacement relations (57)–(63).

4. Find the work densities Wð00Þ, W

ð11Þ, Wð10Þ and W

ð01Þ by

substituting the micropolar plate stress and strain sets Sð0Þ,

Sð1Þ, Eð0Þ and E

ð1Þ into the definitions (86)–(89).

5. Substitute the values of the work densities Wð00Þ, Wð11Þ, Wð10Þ

and Wð01Þ into the expression for the optimal value of the

splitting parameter η0 (86).

The above algorithm can be efficiently used to find the solution

of the system of equations (82). Indeed, once the optimal value of

the splitting parameter η0 is found, the solution Uη0of the bending

system of Eqs. (82) is found from (85) as a linear combination of

the sets of kinematic variables U0 and U1

Uη0¼ ð1−η0ÞU0 þ η0U1:

The numerical modeling of the vertical deflection of the square

plate made of polyurethane foam is given in Figs. 3 and 4.

The distribution of the vertical deflection for nonrectangular

plates and plates with rectangular holes is given in Figs. 5–7.

The effect of the asymmetric part on the maximal vertical deflec-

tion of the plate made of polyurethane foam is shown in Table 3 and

Fig. 8 for a¼1.0 m and the ratio a=h varying from 10 to 30.

Fig. 8 provides information for comparison between micropolar

and classical elasticity. The energy of microrotation becomes a part

of the total elastic energy. This causes the redistribution of the

elastic energy depending on the value of the asymmetric part.

We perform computations for different levels of the asymmetric

microstructure by reducing the values of the elastic asymmetric

parameters. In the case when 0.1% of real values of asymmetric

parameters are used, i.e. the microstructure is almost irrelevant,

the solution of the proposed model converges to the correspond-

ing Reissner solution. Fig. 8 illustrates the size effect of micropolar

plate theory which predicts that plates of smaller thickness will be

more rigid than would be expected on the basis of the Reissner

plate theory. Similar experimental behavior was reported in [21]

for torsion and bending of cylindrical rods of a Cosserat solid.

4. Conclusion

In this paper we presented the Finite Element modeling of the

bending of micropolar elastic plates. Based on our enhanced math-

ematical model for Cosserat plate bending [33] we introduced the

micropolar plate field equations as an elliptic system of nine differ-

ential equations in terms of the kinematic variables. We have devel-

oped the efficient algorithm for finding the optimal value of the

splitting parameter and discussed the approximations of stress and

couple stress components. The numerical algorithm includes the

efficient method for finding the solution of the micropolar plate field

10 15 20 25 300.994

0.995

0.996

0.997

0.998

0.999

1.000

1.001

Deflection 3D Elasticity

Deflection Proposed Plate Theory

Fig. 6. Rectangular plate made of polyurethane foam with hard simply supported

rectangular holes: (a) the area where the sinusoidal load is applied, (b) initial mesh,

(c) the density plot and (d) the isometric view of the resulting vertical deflection of

the plate.

0.04 0.02 0.00 0.02 0.04

0.010

0.005

0.000

0.005

0.010

0.015

Thickness x3 m

Dis

pla

cem

entu

1m

0.04 0.02 0.00 0.02 0.04

0.015

0.010

0.005

0.000

0.005

0.010

0.015

Thickness x3 m

Dis

pla

cem

entu

2m

0.04 0.02 0.00 0.02 0.04

0.3072

0.3073

0.3074

0.3075

0.3076

0.3077

0.3078

Thickness x3 m

Dis

pla

cem

entu

3m

0.015

Fig. 7. Rectangular plate made of polyurethane foam with hard simply supported rectangular holes: (a) the area where the sinusoidal load is applied, (b) initial mesh,

(c) the density plot and (d) the isometric view of the resulting vertical deflection of the plate.

Table 3

Effect of the asymmetric part on the vertical deflection of the plate.

a=h 10 15 20 25 30

Micropolar Plate (m) 0.02363 0.04071 0.05738 0.07370 0.08980

Micropolar Plate 10 of asymmetric part (m) 0.06835 0.17631 0.31858 0.47802 0.64428

Micropolar Plate 1% of asymmetric part (m) 0.08509 0.26975 0.60504 1.11329 1.80130

Micropolar Plate 0.1% of asymmetric part (m) 0.08723 0.28491 0.66508 1.28470 2.19753

Reissner Solution (m) 0.08748 0.28670 0.67250 1.30707 2.25260


equations corresponding to the optimal value of the splitting para-

meter. The comparison of the numerical values of the kinematic

variables for the square plate made of the dense polyurethane foam

with the analytical solution of the three-dimensional micropolar

elasticity confirmed the high order of approximation of three-

dimensional solution. The size effect of micropolar plate theory

predicts that plates made of smaller thickness will be more rigid than

would be expected on the basis of the Reissner plate theory. The

numerical modeling of the bending nonrectangular plates and plates

with rectangular holes made of the polyurethane foam under different

loads have been provided.

Acknowledgments

We would like to thank the reviewers for useful suggestions

and Dr. Paul Castillo, Dr. Robert Acar and Dr. Frederick Just for

helpful discussions in the preparation of the paper.

References

[1] Altenbach H, Eremeyev VA. Thin-walled structures made of foams. In: H.Cellular and porous materials: modeling, testing, application, CISM Coursesand Lecture Notes, vol. 521. Springer; 2010. p. 167–242.

[2] Anderson W, Lakes R. Size effectes due to cosserat elasticity and surfasedamage in closed-cell polymenthacrylimide foam. Journal of Materials Science1994;29:6413–9.

[3] Babuška I. Error-bounds for finite element method. Numerische Mathematik1971;16:322–33.

[5] Chang T, Lin H, Wen-Tse C, Hsiao J. Engineering properties of lightweightaggregate concrete assessed by stress wave propagation methods. Cement andConcrete Composites 2006;28:57–68.

[7] Cosserat E, Cosserat F. Theorie des corps deformables; 1909.[8] Costabel M, Dauge M, Nicaise S. Analytic regularity for linear elliptic systems

in polygons and polyhedra; 2011.[9] Costabel M, Dauge M, Nicaise S. Corner singularities and analytic regularity for

linear elliptic systems; 2010.

[10] Donnell L, Beams. Plates and shells engineering societies monographs.

McGraw-Hill Inc.; 1976.[11] Eringen AC. Microcontinuum field theories. Springer; 1999.[12] Eringen AC. Theory of micropolar elasticity. Fracture 1968;2:621–729.[13] Eringen AC. Theory of micropolar plates. Journal of Applied Mathematics and

Physics (ZAMP) 1967;18:12–30.[15] Federation of European Rigid Polyurethane Foam Associations Report no. 1.

Thermal insulation materials made of rigid polyurethane foam (PUR/PIR)

BING; 2006.[17] Gauthier R, Jahsman W. A quest for micropolar elastic constants. Journal of

Applied Mechanics 1975;42:369–74.[18] Gerstle W., Sau N., Aguilera E., Micropolar peridynamic constitutive model for

concrete. In Transactions of the 19th International Conference of Structural

Mechanics in Reactor Technology (SMiRT 19). Toronto Canada 2007; August

12-17, SMiRT19-B02/1: 1–8.[19] Kumar R. Wave propagation in micropolar viscoelastic generalized thermo-

elastic solid. International Journal of Engineering Science 2000;38.[21] Lakes R. Experimental microelasticity of two porous solids. International

Journal of Solid Structures 1986;22(1):55–63.[23] Love AEH. A treatise on the mathematical theory of elasticity. New York:

Dover; 1986.[24] Merkel A, Tournat V, Gusev V. Experimental evidence of rotational elastic

waves in granular phononic crystals. Physical Review Letters 2011;107.[25] McLean W. Strongly elliptic systems and boundary integral equations. Cam-

bridge: Cambridge University Press; 2000.[26] Nowacki W. Theory of asymmetric elasticity. Oxford, New York: Pergamon

Press; 1986 (2007).[27] Purasinghe R, Tan S, Rencis J. Micropolar elasticity model the stress analysis of

human bones. In: Proceedings of the 11th annual international conference;

1989.[30] Reissner E. Reflections on the theory of elastic plates. Applied Mechanics

Reviews 1985;38:1453–64.[31] Rössle A, Bischoff M, Wendland W, Ramm E. On the mathematical foundation

of the (1,1,2)-plate model. International Journal of Solids and Structures

1999;36:2143–68.[32] Steinberg L. Deformation of micropolar plates of moderate thickness. Inter-

national Journal of Applied Mathematics and Mechanics 2010;6(17):1–24.[33] Steinberg L, Kvasov R. Enhanced mathematical model for Cosserat plate

bending. Thin-Walled Structures 63; 2013:51–62.

10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

a h

WMWR

Reissner Plate

Micropolar Plate with 0.1 asymmetric part

Micropolar Plate with 1 asymmetric part

Micropolar Plate with 10 asymmetric part

Micropolar Plate

Fig. 8. Comparison of the vertical deflections of the micropolar plate WM with Reissner plate WR , which illustrates the influence of the thickness of the micropolar plate.


Documents

Numerical modeling of bending of micropolar plates