Derivation of a model of nonlinear micropolar plate

DERIVATION OF A MODEL OF NONLINEAR MICROPOLARPLATE

JOSIP TAMBACA AND IGOR VELCIC

Abstract. In this paper we derive a two-dimensional model of nonlinear mi-cropolar plate. We start from the equilibrium problem for the three-dimensi-onal nonlinear micropolar elastic plate-like body with nonlinear strains andlinear constitutive law. Using the asymptotic expansion we formally derivethe two-dimensional problem, posed on the middle plane of the plate-like body,for the leading deformation and microrotation. Nonlinear micropolar elasticityand Plate model and Asymptotic expansion 74K20 and 74A35 and 74B20 and35B25

1. Introduction

In this paper we consider the equilibrium problem of nonlinear three-dimensionalmicropolar elasticity for the plate-like bodies. In contrast to the classical elasticitythe unknowns in the problem are the deformation field and the microrotation field (afunction with values in rotations); namely the points are allowed to rotate withoutstretch. Such generalized continuum is introduced by the Cosserat brothers in[7]. For an overview of the micropolar elasticity, which is a special case of themicrostretch continua see [8]. As the geometry of the domain is ”almost” two-dimensional the leading order behavior is expected to be two-dimensional. We applythe asymptotic expansion to derive this two-dimensional model. To simplify thederivation and in order to explicitly derive the model we use the three-dimensionalequations with nonlinear strains and linear constitutive law. This material is ananalogue of the Biot-type material in the classical elasticity (see [15]). Actually, thisconstitution law is derived from the assumption that internal energy is quadraticin strains. Existence theorem for such nonlinear micropolar body can be found in[13].

The asymptotic expansion method is an usual method in derivation of lower-dimensional models of thin structures. A small parameter is recognized in thegeometry of the problem and the expansion of the unknown function, usually beingdisplacement or deformation, and loads is assumed. It is then inserted in thethree-dimensional equations. The coefficients of the same powers of ε then formthe equations for the unknowns in the problem. In that way the leading orderunknown function is identified.

The asymptotic expansion method has been previously applied to derive modelsin linearized micropolar elasticity (see [1] for equilibrium of rods and plates, [2] forequilibrium of shells, [3] for curved rods and [16] for evolution of plates). In classicalnonlinear elasticity the method has been used for derivation of models of rods (see

Josip Tambaca, Department of Mathematics, University of Zagreb, [email protected] Velcic, Department of Mathematics, University of Zagreb, [email protected].

1

2 JOSIP TAMBACA AND IGOR VELCIC

[6]), plates (see [11, 10, 9] or [4]) and shells (for references see [5]). In contrast tothe derivation of the plate model within the classical nonlinear elasticity in [11] inthis paper the loads are scaled with order 0, as in the linearized case. Moreover,linearization of the obtained model leads to the model of linearly micropolar platederived in [1] and is not an assumption of the derivation. This is in contrast to thecase of the classical elasticity where in order to derive the nonlinear plate modelone assumes that the linearization of the obtained model should be the modelof linearly elastic plate, see [11]. The obtained model is written in terms of thedeformation field and the microrotation field which are the leading terms of theasymptotic expansion. These functions are defined on the middle plane of theplate making the obtained model two-dimensional. In the linearized case, see [1],using the first correctors, then one finds the relation between the microrotationand the macrorotation of the cross-sections and writes the model in macroscopicquantities. Unfortunately, we have not been able to do it here. Instead we notethat in the constitutive law of the three-dimensional equations we have neglectedquadratic terms in (nonlinear) strains and do the same in the two-dimensionalmodel. In such a way we derive a nonlinear micropolar plate model in terms ofthe deformation field, macrorotation vector and a function related to the rotationof the transversal segment of the plate around its axis (in plane rotation). Theobtained model clearly measures the change of metrics (capturing the membraneeffects of the plate), fourth order terms capturing the flexure of the plate, the termmeasuring the rotation of the transversal segment around its axis and the termmeasuring shearability of the transversal segment of the plate.

Note that besides the asymptotic expansion method there are other methods ofderivation of lower-dimensional models in elasticity, like Γ-convergence (as in [9] fornonlinear elastic plates) or reducing a given three-dimensional model via physicallyreasonable constitutive assumptions on the kinematics to a lower-dimensional model(for nonlinear micropolar elastic plates see [12]). In contrast to the asymptoticexpansion method Γ-convergence provides a convergence result. It is importantto notice that in the case of classical nonlinear elastic plates the Γ-convergencemethod and the asymptotic expansion method give the same result only if additionalassumptions are made on deformations.1

By Av we denote skew-symmetric matrix associated to its axial vector v, i.e.Avx = v × x. By aT we denote the axial vector of T − TT i.e. T − TT = AaT

.Note that T ·Av = aT ·v. Summation convention for the repeated indices is used. Idenotes a unit matrix of the appropriate dimension, while symA and skewA standfor the symmetric and skew-symmetric part of the matrix A. Finally, εijk standsfor the Levi-Civita symbol.

2. Three-dimensional micropolar elasticity

Let Ω ⊂ R3 be an open bounded domain. We assume Ω to be a micropolarbody. That means, in contrast to the classical elasticity where the motion of amaterial particle is fully described by a vector function called deformation function

1During the review of this paper we have learned that a related derivation of the micropolarplate model based on Γ-convergence has been already given in [14]. The obtained model can beshown to be the same as the one obtained here. We still believe that it is interesting to showthat the two approaches give the same result (in contrast to the classical elasticity) and to see theapproximate plate model in terms of macroscopic unknowns.

DERIVATION OF A MODEL OF NONLINEAR MICROPOLAR PLATE 3

ϕ, that material particles undergo an additional micromotion, corresponding to therotation R of the material particle at the microscale. Such continuum is introducedby the Cosserat brothers in [7]. For an overview of the micropolar elasticity see [8].

As R is a rotation there are vectors ω1, ω2, ω3 such that

∂iR = ωi ×R, i = 1, 2, 3,

where the vector product is taken with respect to the columns of R =(

R1 R2 R3

).

Vectors ωi can be expressed in terms of R by

ωi =12Rj × ∂iRj , i = 1, 2, 3.

The strain measures (deformation tensors) can be given by2

(2.1) U = RT∇ϕ, Γ = R

T (ω1 ω2 ω3

).

Note here that in the rigid motion case U = I and Γ = 0. In the sequel we assumethe strains are small and for the energy of the material consider the only quadraticfunction which is isotropic3 and has a minimum equal to zero for the rigid motion

Σ(U,Γ) =12(λ(tr (U− I))2 + (µ + µc) tr ((U− I)T (U− I))

+(µ− µc) tr (U− I)2 + α(trΓ)2 + β tr (Γ2) + γ tr (ΓT Γ)),

=λ

2(tr (U− I))2 + µ‖ sym (U− I)‖2 + µc‖ skew (U− I)‖2

+α

2(trΓ)2 +

γ + β

2‖ symΓ‖2 +

γ − β

2‖ skewΓ‖2;(2.2)

here the norm used is the Frobenius norm. We will use the first form of the energyin the computations, while the second one is more useful for the analysis of themodel with respect to the change in parameters [13]. This energy combined withthe nonlinear strains correspond to the Biot-type material in the classical elasticity(see [15]). Let us denote by Tϕ the stress tensor and Mϕ the couple stress tensor indeformed configuration. Using the fact that the energy increment on an arbitraryvolume element is in fact work done by the internal forces and couples on theboundary of the element one can obtain (see [8])

(2.3) Tϕ =1

det∇ϕR

∂Σ∂U

∇ϕT , Mϕ =1

det∇ϕR

∂Σ∂Γ

∇ϕT .

Derivatives of the energy can be expressed by

∂Σ∂U

= T (U− I),∂Σ∂Γ

= M(Γ),

using fourth order tensors

T E = λ (trE) I + 2µ symE + 2µc skewE(2.4)

= λ (trE) I + (µ− µc)ET + (µ + µc)E, E ∈ M3(R),

MG = α (trG)I + β GT + γ G, G ∈ M3(R).(2.5)

2It can be shown that Γ =(

12εkmn∂lRm ·Rn

)k,l=1,2,3

, as defined in [8].3Isotropic material should satisfy Σ(QUQT ,QΓQT ) = Σ(U,Γ), for all Q ∈ SO(3) and

Σ(U,−Γ) = Σ(U,Γ) (central symmetry).


The elasticity constants are taken to satisfy

(2.6) 3λ + 2µ > 0, µ > 0, µc ≥ 0,3α + β + γ > 0, γ + β > 0, γ − β > 0,

which ensures positive semidefiniteness of the tensor T and positive definiteness ofthe tensor M. Actually, for the formal asymptotic approach in deriving the platemodel it is enough to assume λ + 2µ > 0, µ + µc > 0, α + β + γ > 0 and γ > 0.

Let us denote F ϕ the volume force density and by Lϕ the couple density in thedeformed configuration. Then the balance of forces and momentum are expressedby

(2.7) div Tϕ + F ϕ = 0, div Mϕ + aTϕ + Lϕ = 0.

In the sequel we assume that the loads are dead loads. Existence of minimizersof the energy functional associated to the problem (2.1)–(2.7) with appropriateboundary conditions is guaranteed by [13]. The special analysis was needed for thecase µc = 0 due to the loss of the positive definitness of the form T .

3. Plate-like body

Let h, ε be positive numbers, S open bounded subset of R2 with the Lipschitzboundary. We define the cylinder Ωε in R3 by

Ωε = S × ε (−h/2, h/2)

and denote parts of its boundary by

Γε0 = ∂S × ε(−h/2, h/2), Γε

l = S × (−εh/2 ∪ εh/2) .

We assume Ωε to be the reference configuration of a three-dimensional, homoge-neous, isotropic micropolar body of the relaxed Biot-type with elasticity constantsλ, µ, µc, α, β, γ independent of ε. We assume that the body is clamped at thepart of the boundary Γε

0, force-free at the rest of the boundary Γεl , and subjected

to volume force with density F ε ∈ L2(Ωε)3, and couple with density Lε ∈ L2(Ωε)3.Then according to the Section 2 the equilibrium deformation ϕε ∈ H1(Ωε)3 andmicrorotation R

ε ∈ H1(Ωε; SO(3)) formally satisfy the following equations (writtenin the reference configuration):

− div Tε = F ε, in Ωε,

− div Mε − aTε(∇ϕε)T = Lε in Ωε,

Tε = RεTU

ε, in Ωε,

Mε = RεMΓε in Ωε,

ϕε = i,Rε

= I on Γε0,

Tεν = Mεν = 0 on Γεl ;

here i denotes the identity function from R3 to R3. The strains Uε

and Γε aredefined in (2.1), the elasticity tensors T and M are given in (2.4) and (2.5), whileν denotes unit normal at the lateral boundary.


Next we rewrite the problem in the weak formulation, as it will be the formsuitable for the asymptotic expansion. The equilibrium deformation ϕε and micro-rotation R

εsatisfy the following variational equations∫

Ωε

Tε · ∇V dy =∫

Ωε

F ε · V dy, V ∈ V0(Ωε),∫

Ωε

(Mε · ∇W − aTε(∇ϕε)T ·W )

dy =∫

Ωε

Lε ·W dy, W ∈ V0(Ωε),

or after some calculations∫

Ωε

T (Uε − I) · (Rε

)T∇V dy =∫

Ωε

F ε · V dy, V ∈ V0(Ωε),(3.1)∫

Ωε

(MΓε · (Rε

)T∇W − T (Uε − I)(∇ϕε)T · (Rε

)T AW

)dy(3.2)

=∫

Ωε

Lε ·W dy, W ∈ V0(Ωε),

where V0(Ωε) = V ∈ H1(Ωε)3 : V |Γε0

= 0.In the sequel we rewrite (3.1), (3.2) on the canonical domain, independent of ε.

The resulting problem will depend on ε explicitly. Let

Ω = S × (−h/2, h/2), Γ0 = ∂S × (−h/2, h/2), Γl = S × (−h/2 ∪ h/2) ,

P ε : Ω → Ωε, y = P ε(z) = (z1, z2, εz3).Let the functions on Ω be related to functions on Ωε by

f(ε) = F ε P ε, l(ε) = Lε P ε, ϕ(ε) = ϕε P ε, R(ε) = Rε P ε.

Then f(ε) and l(ε) belong to L2(Ω)3 and ϕ(ε) ∈ V0(Ω) = V ∈ H1(Ω)3 : V |Γ0 =0. Moreover, ϕ(ε) and R(ε) satisfy equations∫

Ω

T (U(ε)− I) ·R(ε)T∇εv dy =∫

Ω

f(ε) · v dy, v ∈ V0(Ω),(3.3)∫

Ω

(MΓ(ε) ·R(ε)T∇εw − T (U(ε)− I)∇ϕ(ε)T ·R(ε)T Aw

)dy(3.4)

=∫

Ω

l(ε) ·w dy, w ∈ V0(Ω);

here U(ε) = R(ε)T∇εϕ(ε),

Γ(ε) = R(ε)T(

12Rj(ε)× ∂1Rj(ε) 1

2Rj(ε)× ∂2Rj(ε) 12εRj(ε)× ∂3Rj(ε)

),

where we use the notation R(ε) =(

R1(ε) R2(ε) R3(ε)), and

∇ε =1ε∇z +∇y, ∇yv =

∂1v1 ∂2v1 0∂1v2 ∂2v2 0∂1v3 ∂2v3 0

, ∇zv =

0 0 ∂3v1

0 0 ∂3v2

0 0 ∂3v3

.

4. Asymptotic expansion

In this section we perform the asymptotic expansion. We assume that the loadsf(ε) and l(ε) and unknown functions ϕ(ε) and R(ε) allow the expansion of theform

f(ε) = f0 + εf1 + · · · , l(ε) = l0 + εl1 + · · · ,

ϕ(ε) = ϕ0 + εϕ1 + · · · , R(ε) = R0

+ εR1

+ · · · .


We plug these expansions in (3.3) and (3.4) and identify the equations for thecoefficients of the same powers of ε.

In the first step we consider the lowest nontrivial coefficient which is ε−2. Theequations are ∫

Ω

TU−1 · (R0

)T∇zv dy = 0, v ∈ V0(Ω),(4.1)∫

Ωε

MΓ−1 · (R0)T∇zw dy = 0, w ∈ V0(Ω),(4.2)

where U−1

= (R0)T∇zϕ

0 and

Γ−1 =(

0 0 12ej × (R

0)T ∂3R

0

j

)=

0 0 R0

3 · ∂3R0

2

0 0 R0

1 · ∂3R0

3

0 0 R0

2 · ∂3R0

1

.

From the expansion of the Dirichlet boundary condition it follows v(y) = ϕ0(y)−(y1, y2, 0) ∈ V0(Ω). Therefore we are allowed to insert such v(y) as the test functionin (4.1). It follows

∫

Ω

T ((R0)T∇zϕ

0) · (R0)T∇zϕ

0 dy =

(µ + µc)(∂3ϕ0 ·R0

1)2 + (µ + µc)(∂3ϕ

0 ·R0

2)2 + (λ + 2µ)(∂3ϕ

0 ·R0

3)2 = 0.

As λ + 2µ > 0 and µ + µc > 0 it follows U−1

= (R0)T∇zϕ

0 = 0. ExpandingR

T(ε)R(ε) = I we conclude that R

0is a rotation at each argument. Therefore

∇zϕ0 = 0, so ϕ0 is independent of the transversal variable y3.

By the Lions lemma (e.g. see [11] or [4]: if u ∈ L2(Ω)3 and∫Ω

u∂3v = 0, for allv ∈ V0(Ω) then u = 0) from (4.2) it follows that

R0MΓ−1e3 = 0,

where e3 = (0, 0, 1). As R0

is a rotation and using the definition of M one gets

0 = MΓ−1e3 = α

00

R0

2 · ∂3R0

1

+ β

00

R0

2 · ∂3R0

1

+ γ

R0

3 · ∂3R0

2

R0

1 · ∂3R0

3

R0

2 · ∂3R0

1

.

As γ > 0 and α + β + γ > 0 it follows

R0

3 · ∂3R0

2 = R0

1 · ∂3R0

3 = R0

2 · ∂3R0

1 = 0.

This impliesR

0

i · ∂3R0

j = 0, i, j = 1, 2, 3,

and therefore ∂3R0

= 0, so R0

is independent of transversal variable y3 as well.The consequence is also that Γ−1 = 0.

In the second step we identify the equations which are coefficients of ε−1:∫

Ω

T (U0 − I) · (R0

)T∇zv dy = 0, v ∈ V0(Ω),∫

Ωε

MΓ0 · (R0)T∇zw dy = 0, w ∈ V0(Ω);


here

U0

= (R0)T

(∇yϕ0 +∇zϕ1),

Γ0 =12(R

0)T

(R

0

j × ∂1R0

j R0

j × ∂2R0

j R0

j × ∂3R1

j

)

=

R0

3 · ∂1R0

2 R0

3 · ∂2R0

212

(R

0

3 · ∂3R1

2 −R0

2 · ∂3R1

3

)

R0

1 · ∂1R0

3 R0

1 · ∂2R0

312

(R

0

1 · ∂3R1

3 −R0

3 · ∂3R1

1

)

R0

2 · ∂1R0

1 R0

2 · ∂2R0

112

(R

0

2 · ∂3R1

1 −R0

1 · ∂3R1

2

)

.

As before, using the Lions lemma (e.g. see [11]) and R0

being a rotation it followsthat

(4.3) T (U0 − I)e3 = MΓ0e3 = 0.

Therefore only in-plane components of the stress and couple stress tensor arenonzero. These relations are fundamental for the rest of the derivation of themodel. From the first equation one gets

0 = T (U0 − I)e3 = λ (∂1ϕ

0 ·R0

1 + ∂2ϕ0 ·R0

2 + ∂3ϕ1 ·R0

3 − 3)e3

+(µ− µc)

∂1ϕ0 ·R0

3

∂2ϕ0 ·R0

3

∂3ϕ1 ·R0

3 − 1

+ (µ + µc)

∂3ϕ1 ·R0

1

∂3ϕ1 ·R0

2

∂3ϕ1 ·R0

3 − 1

.

We obtain three equations

(µ− µc)∂1ϕ0 ·R0

3 + (µ + µc)∂3ϕ1 ·R0

1 = 0,

(µ− µc)∂2ϕ0 ·R0

3 + (µ + µc)∂3ϕ1 ·R0

2 = 0,

λ (∂1ϕ0 ·R0

1 + ∂2ϕ0 ·R0

2 − 3)− (µ− µc)− (µ + µc) + (λ + 2µ)∂3ϕ1 ·R0

3 = 0.

Using the notation

r = −µ− µc

µ + µc∂1ϕ

0 ·R0

3R0

1 −µ− µc

µ + µc∂2ϕ

0 ·R0

3R0

2

−(

λ

λ + 2µ

(∂1ϕ

0 ·R0

1 + ∂2ϕ0 ·R0

2

)− 3λ + 2µ

λ + 2µ

)R

0

3.(4.4)

the previous system can be written as one equation ∂3ϕ1 = r, which is solved to

obtain the first corrector of the deformation

ϕ1(y1, y2, y3) = r(y1, y2)y3 + π1(y1, y2),

with π1 an arbitrary function of in-plane variables. Now it follows

U0

= (R0)T∇yϕ0 + (R

0)T∇zϕ

1 =

∂1ϕ0 ·R0

1 ∂2ϕ0 ·R0

1 r ·R0

1

∂1ϕ0 ·R0

2 ∂2ϕ0 ·R0

2 r ·R0

2

∂1ϕ0 ·R0

3 ∂2ϕ0 ·R0

3 r ·R0

3

.


From the second equation in (4.3) we obtain

0 = MΓ0e3 =

βR0

2 · ∂1R0

1

βR0

2 · ∂2R0

1

α(R

0

3 · ∂1R0

2 + R0

1 · ∂2R0

3

)

+

γ 0 00 γ 00 0 α + β + γ

12

(R

0

3 · ∂3R1

2 −R0

2 · ∂3R1

3

)

12

(R

0

1 · ∂3R1

3 −R0

3 · ∂3R1

1

)

12

(R

0

2 · ∂3R1

1 −R0

1 · ∂3R1

2

)

.

This actually determines the third column of Γ0, so we have

Γ0 =

R0

3 · ∂1R0

2 R0

3 · ∂2R0

2 −βγ R

0

2 · ∂1R0

1

R0

1 · ∂1R0

3 R0

1 · ∂2R0

3 −βγ R

0

2 · ∂2R0

1

R0

2 · ∂1R0

1 R0

2 · ∂2R0

1 − αα+β+γ

(R

0

3 · ∂1R0

2 + R0

1 · ∂2R0

3

)

.

In the third step we take the test functions from the space H10 (S)3 and identify

the equations which are coefficients of ε0:∫

Ω

T (U0 − I) · (R0

)T∇yv dy =∫

Ω

f0 ·w dy,(4.5)∫

Ω

(MΓ0 · (R0

)T∇yw − T (U0 − I)(∇ϕ0)T · (R0

)T Aw

)dy =

∫

Ω

l0 ·wdy,(4.6)

for all v,w ∈ H10 (S)3. We compute

T (U0 − I) = λ

2µ

λ + 2µ

(∂1ϕ

0 ·R0

1 + ∂2ϕ0 ·R0

2 − 2)

1 0 00 1 00 0 0

+(µ− µc)

∂1ϕ0 ·R0

1 − 1 ∂1ϕ0 ·R0

2 0∂2ϕ

0 ·R0

1 ∂2ϕ0 ·R0

2 − 1 0−µ−µc

µ+µc∂1ϕ

0 ·R0

3 −µ−µc

µ+µc∂2ϕ

0 ·R0

3 0

+(µ + µc)

∂1ϕ0 ·R0

1 − 1 ∂2ϕ0 ·R0

1 0∂1ϕ

0 ·R0

2 ∂2ϕ0 ·R0

2 − 1 0∂1ϕ

0 ·R0

3 ∂2ϕ0 ·R0

3 0

,

MΓ0 = αβ + γ

α + β + γ

(R

0

3 · ∂1R0

2 + R0

1 · ∂2R0

3

)

1 0 00 1 00 0 0

+β

R0

3 · ∂1R0

2 R0

1 · ∂1R0

3 0R

0

3 · ∂2R0

2 R0

1 · ∂2R0

3 0−β

γ R0

2 · ∂1R0

1 −βγ R

0

2 · ∂2R0

1 0

+ γ

R0

3 · ∂1R0

2 R0

3 · ∂2R0

2 0R

0

1 · ∂1R0

3 R0

1 · ∂2R0

3 0R

0

2 · ∂1R0

1 R0

2 · ∂2R0

1 0

.

Let us introduce the notation

f0

=∫ h/2

−h/2

f0dy3, l0

=∫ h/2

−h/2

l0dy3, ∇ =(

∂1 ∂2

), I =

1 00 10 0

,


Γ0

=(

12εkmn∂lR

0

m ·R0

n

)

k=1,2,3,l=1,2

= (R0)T

(ω0

1 ω02

),

and two tensors X ,Y which are positive definite fourth order tensors which act onthe matrices in M3,2(R) and are defined as follows

X(

AaT

)·(

BbT

)=

4µµc

µ + µca · b +

2µλ

λ + 2µtrA trB + (µ− µc)AT ·B

+(µ + µc)A ·B,

=4µµc

µ + µca · b +

2µλ

λ + 2µtrA trB + 2µ symA · symB

+2µc skewA · skewB,

Y(

AaT

)·(

BbT

)=

γ2 − β2

γa · b +

(β + γ)αα + β + γ

trA trB + βAT ·B + γA ·B.

Then the equations (4.5) and (4.6) can be written by∫

S

X ((R0)T∇ϕ0 − I) · (R0

)T∇v dy1dy2 =∫

S

f0 ·w dy1dy2,(4.7)

∫

S

(YΓ

0 · (R0)T∇w −X ((R

0)T∇ϕ0 − I)(∇ϕ0)T · (R0

)T Aw

)dy1dy2

=∫

S

l0 ·w dy1dy2,(4.8)

for all v, w ∈ H10 (S)3. This system is a two-dimensional model of nonlinear mi-

cropolar plate. It is a second order quasilinear system of the same type as thethree-dimensional system. Linearization of this model is expressed in terms of thedisplacement and microrotation vector and is the same as the model obtained bythe asymptotic analysis starting from linearized micropolar elasticity, see [1]. Notethat in performing this analysis we did not need to assume that linearization of thenonlinear model should coincide with the model obtained starting from linearizedthree-dimensional equations of micropolar elasticity, as it is the case in the classicalelasticity (see [11]). Also note that the model is frame indifferent.

In case of zero couple density l the system is formally equivalent to the mini-mization of the energy functional

J(ϕ0,R0) =

12

∫

S

X((R

0)T∇ϕ0 − I

)·((R

0)T∇ϕ0 − I

)(4.9)

+12

∫

S

YΓ0 · Γ0 −

∫

S

f0 ·ϕ0;

here the first two terms are internal energy of the plate. The obtained model, aftera tedious, but straight forward calculation, can be shown to be the same as in [14].The existence of the minimum of the obtained energy functional when µc > 0 canbe shown in the same way as in [12] (coercivity+ convexity in ∇ϕ0 and ∇R

0) or

can be concluded directly from the Γ-convergence.

5. The model in macroscopic variables

In this section from the model (4.7), (4.8) we derive the model in macroscopicquantities. In the linearized case the macroscopic quantities were displacement ofthe middle plane of the plate and the infinitesimal rotation of the cross-section. This


macrorotation, denoted by r in (4.4) is determined by the first corrector ϕ1 andrelates microrotation and the macrorotation. In the linearized case in such a waya form of the Reissner-Mindlin plate model is obtained, the model which includesboth the effects of the change of metric and curvature tensors, i.e. membrane andflexural effects, see [1]. In order to do it exactly in the nonlinear case the relationof r and R

0should be inverted which we have not succeed. But, we neglect higher

order terms in strains (which are nonlinear and which are once already neglectedin the constitutive law for the material) (R

0)T∇ϕ0 − I,Γ

0and obtain a plausible

model.Let us introduce the normal on the deformed middle plane of the plate

n =∂1ϕ

0 × ∂2ϕ0

‖∂1ϕ0 × ∂2ϕ0‖ .

After some computation we obtain (in the sequel h.o.t stands for ”higher orderterms” and relates to the terms which are quadratic in strains or derivatives ofstrains)

∂1ϕ0 × ∂2ϕ

0 = −(∂1ϕ0 ·R0

3)R0

1 − (∂2ϕ0 ·R0

3)R0

2

+(∂1ϕ0 ·R0

1 − 1 + ∂2ϕ0 ·R0

2 − 1 + 1)R0

3 + h.o.t.

Therefore

r · n = 1− λ

λ + 2µ

(∂1ϕ

0 ·R0

1 − 1 + ∂2ϕ0 ·R0

2 − 1)

+ h.o.t.

Since

∂1ϕ0 · ∂1ϕ

0 = 2∂1ϕ0 ·R0

1 − 1 + h.o.t.,

∂1ϕ0 · ∂2ϕ

0 = ∂2ϕ0 ·R0

1 + ∂1ϕ0 ·R0

2 + h.o.t.,

∂2ϕ0 · ∂2ϕ

0 = 2∂2ϕ0 ·R0

2 − 1 + h.o.t.,

after neglecting h.o.t. in strains the product r · n can be expressed in terms of∇ϕ0 only. Therefore r introduces only two new independent unknown functions(r · ∂1ϕ

0 and r · ∂2ϕ0), so in order to replace the microrotation (with three degrees

of freedom) we need one additional unknown variable

ω = ∂1ϕ0 ·R0

2 − ∂2ϕ0 ·R0

1

in order to express the model in macroscopic unknowns. We have

(R0)T∇ϕ0 =

(∂1ϕ0·∂1ϕ0+1

2ω+∂1ϕ0·∂2ϕ0

2µ+µc

2µcr · ∂1ϕ

0

∂1ϕ0·∂2ϕ0−ω2

∂2ϕ0·∂2ϕ0+12

µ+µc

2µcr · ∂2ϕ

0

)T

+ h.o.t.

From the previous computations we have

∂1ϕ0 = ∂1ϕ

0 ·R0

1R0

1 + ∂1ϕ0 ·R0

2R0

2 + ∂1ϕ0 ·R0

3R0

3,

∂2ϕ0 = ∂2ϕ

0 ·R0

1R0

1 + ∂2ϕ0 ·R0

2R0

2 + ∂2ϕ0 ·R0

3R0

3,

n = −(∂1ϕ0 ·R0

3)R0

1 − (∂2ϕ0 ·R0

3)R0

2 + R0

3 + h.o.t.


In order to express R0

1, R0

2, R0

3 in the base ∂1ϕ0, ∂2ϕ

0, n we have to invert

S =

∂1ϕ0 ·R0

1 ∂1ϕ0 ·R0

2 ∂1ϕ0 ·R0

3

∂2ϕ0 ·R0

1 ∂2ϕ0 ·R0

2 ∂2ϕ0 ·R0

3

−∂1ϕ0 ·R0

3 −∂2ϕ0 ·R0

3 1

It can be easily verified that

S−1 =

3−∂1ϕ0·∂1ϕ0

2 −ω+∂1ϕ0·∂2ϕ0

2 −µ+µc

2µcr · ∂1ϕ

0

ω−∂1ϕ0·∂2ϕ0

23−∂2ϕ0·∂2ϕ0

2 −µ+µc

2µcr · ∂2ϕ

0

µ+µc

2µcr · ∂1ϕ

0 µ+µc

2µcr · ∂2ϕ

0 1

+ h.o.t.

Thus we have(R

0

1 R0

2 R0

3

)=

(∂1ϕ

0 ∂2ϕ0 n

)S−T + h.o.t.

In the sequel we will additionally suppose that the derivatives of the h.o.t are alsoh.o.t. Let us denote

M =(

∂1ϕ0 · ∂1ϕ

0 ∂1ϕ0 · ∂2ϕ

0

∂1ϕ0 · ∂2ϕ

0 ∂2ϕ0 · ∂2ϕ

0

)− I,

K =(

∂21ϕ0 · n ∂1∂2ϕ

0 · n∂1∂2ϕ

0 · n ∂22ϕ0 · n

)− µ + µc

2µc

(∂1(r · ∂1ϕ

0) ∂2(r · ∂1ϕ0)

∂1(r · ∂2ϕ0) ∂2(r · ∂2ϕ

0)

).

After some calculations we obtain

Γ0

=

K21 K22

−K11 −K1212 (∂1M12 − ∂2M11 − ∂1ω) 1

2 (∂1M22 − ∂2M12 − ∂2ω)

+ h.o.t.

From (4.9) the approximate internal energy of the plate follows

JP =12

[ ∫

S

Q1(M) + Q2(K) + µcω2 +

µ(µ + µc)µc

((∇ϕ0)T r

)2

+γ2 − β2

4γ

∫

S

((∂1M12 − ∂2M11 − ∂1ω)2 + (∂1M22 − ∂2M12 − ∂2ω)2

) ],

where Q1, Q2 are positive definite quadratic forms given by (here A = (aij))

Q1(A) =2µλ

λ + 2µ(trA)2 + (µ− µc)AT ·A + (µ + µc)A ·A,

Q2(A) =(β + γ)αα + β + γ

(a12 − a21)2 + β(a212 + a2

21 − 2a11a22) + γA ·A.

The matrix M clearly measures the change of metric in the plate (membrane part ofthe energy) while the matrix K contains the curvature and an additional part whichmeasures the change of the projection of the vector r on the plane of the plate.Since the field K contains the material constants it can not be seen as the measureof the deformation. However, note that the material constants in the definition ofK can be removed by introducing the new unknown function r = µ+µc

2µcr instead of

r.

Acknowledgment. The authors are grateful to Patrizio Neff for his valuable sug-gestions and comments.


References

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