EVOLUTION MODEL FOR LINEARIZED MICROPOLARPLATES BY THE FOURIER METHOD

JOSIP TAMBACA AND IGOR VELCIC

Department of MathematicsUniversity of Zagreb

10000 ZagrebCroatia

Abstract. In this paper we justify a two-dimensional evolution and eigen-value model for micropolar plates starting from three-dimensional linearly mi-cropolar elasticity. A small parameter representing the thickness of the plate-like body is introduced in the problem. The asymptotics of the evolution andeigenvalue problems is then developed as this small parameter tends to zero.First the appropriate convergences of the eigenpairs of the three-dimensionalproblem to the eigenpairs of the two-dimensional eigenvalue problem for mi-cropolar plates is shown. Then these convergences are used in the Fouriermethod to obtain the convergences of the solution of the three-dimensional evo-lution problem to the solution of the two-dimensional evolution plate model.micropolar elasticity and asymptotic analysis and plate model and the Fouriermethod 74K20 and 74A35

1. Introduction

The asymptotic derivation and justification of the lower dimensional models,equilibrium and dynamic, of rods, curved rods, plates, shells in linearized elasticityis well established (see [3], [4] and references therein). Recently, the technique hasbeen applied to the derivation and justification of the equilibrium models (of rodsand plates in [1], plates in [9], [22], curved rods in [2]) and the evolution models(of plates by a variational technique in [26]) starting from linearized micropolarelasticity. The analysis of geometrically exact static micropolar plates is given in[17, 20]. A rigorous justification via Γ-convergence is given in [21], even for thecase of zero Cosserat couple modulus (see Remark 2.1 below). The same model isobtained in [27] via formal asymptotics.

Micropolar elasticity, in contrast to classical elasticity, uses two kinematical vec-tor fields, the displacement and the microrotation of the material points; the mate-rial points are allowed to rotate without stretch. Such generalized continua were in-troduced by the Cosserat brothers in [7], so the theory is also known as the Cosserattheory. Eringen extended the theory and introduced the notion of microinertia (formore details see [10]).

One of the essential features of the micropolar continua is that the stress tensoris not necessarily symmetric and thus the angular momentum balance equation hasto be added to the linear momentum balance equation. Prototypes of micropolar

tambaca@math.hr.ivelcic@math.hr.

1

2 JOSIP TAMBACA AND IGOR VELCIC

materials are metallic foams and bones (see [15, 16]). For an application of thetheory to continuous solids see [18] and for an application of the theory in con-junction with elasto-plasticity see [19]. One can also use micropolar continua as ahomogenized replacement for materials with periodic microstructure (see [11]). In[23] the extended micromorphic continua is analyzed and applied to metallic foams.

In this paper we apply asymptotic analysis to the eigenvalue and evolution prob-lem for plate-like micropolar elastic bodies. In the analysis we let the thickness ofthe plate tend to zero. When dealing with granular materials there is no sense toinvestigate ever smaller samples since there is a length scale below which nothingexists. In that case we consider plates of constant thickness which are increas-ing in plane-length. We start with a derivation of the two-dimensional eigenvaluemodel for micropolar plates following a similar considerations for classical plates in[6]. The convergence results are in Section 5. We then proceed, using the Fouriermethod, to justify the evolution model, previously derived and justified using avariational method in [26].

In contrast to the variational technique, the Fourier method includes explicitdependence of the solution of the three-dimensional problem on the time vari-able. This enables us to obtain sharper convergence results than was obtainedusing a variational technique in [26]. Namely, we are able to prove the weak con-vergence (Theorem 7.3) and then the strong convergence for the solution of thethree-dimensional problem to the solution of the evolution plate model for everytime t (Theorem 7.5).

Typically, in the lower-dimensional models derived from linearized elasticity thenormal (cross-section) to the middle surface remains approximately perpendicularto the deformed middle surface forming so called Kirchhoff-Love or Bernoulli-Navierdisplacements. In contrast, some lower-dimensional models (Timoshenko beammodel, Reissner-Mindlin plate model, Naghdi shell model) alow shearing of normals(cross-sections) and are usually formulated in terms of the displacement of themiddle surface and the rotation of the normals. Micropolar elasticity allows one toobtain these terms using asymptotic analysis from three-dimensional theory. Themodel that is obtained in the equilibrium case is of the same form as the Reissner–Mindlin plate model, while in the dynamic case some additional terms appear, seeRemark 7.8.

Throughout the paper Av denotes the skew-symmetric matrix with axial vector v(Avx = v×x); AT denotes the axial vector of the matrix T; derivative with respectto t is denoted by vt = dv

dt = v′. · denotes the Euclidean scalar product of vectorsand tensors A ·B = tr (BT A), where tr denotes the trace of a matrix. ‖ · ‖ denotesthe L2 norm on an appropriate domain. symA denotes the symmetric part of A(symA = (A + AT )/2), skew A its skew-symmetric part (skew A = (A−AT )/2).

2. The evolution and eigenvalue problem

Let h, ε be positive, and let S be a simply connected, open, bounded subset ofR2 with the Lipschitz boundary. We define a cylinder Ωε in R3 by

Ωε = S × ε(−h/2, h/2).

Let Sεb = ∂S × ε(−h/2, h/2), Γε = S × (−εh/2, εh/2). We assume that Ωε

is an isotropic, homogeneous (with mass density ρ), linearized micropolar elasticbody fixed at Sε

b , free at the remainder of the boundary and subjected to a volume

EVOLUTION OF MICROPOLAR PLATES 3

force with density F ε : Ωε × [0, T ] → R3 and subjected to a spin with densityLε : Ωε × [0, T ] → R3. Then the deformation at time t is described by two vectorfields, displacement U ε(t) and microrotation $ε(t). They both belong to the space

V0(Ωε) = V ∈ H1(Ωε)3 : V |Sεb =0

and satisfy variational equations

ρ

∫

Ωε

∂2U ε

∂t2· V dy +

∫

Ωε

Tε · ∇V dy =∫

Ωε

F ε · V dy, V ∈ V0(Ωε),(1)

ρ

∫

Ωε

J∂2$ε

∂t2·W dy +

∫

Ωε

(Mε · ∇W −ATε ·W )

dy =∫

Ωε

Lε ·W dy,

W ∈ V0(Ωε),(2)U ε|t=0 = U ε

0, U εt |t=0 = U ε

1, $ε|t=0 = $ε0, $ε

t |t=0 = $ε1.(3)

Here, J is a positive definite matrix J ∈ M3(R). Tensor Tε is the force stress tensorand Mε the couple stress tensor. In the scope of linearized micropolar elasticitythey are related to the linear strain tensors

Eε= ∇U ε −A$ε , Gε = ∇$ε

through the constitutive relations

Tε = T Eε, Mε = MGε,

where the elasticity tensors T and M are defined by

T E = λ(tr E)I + 2µ sym E + 2µc skew E , E ∈ M3(R),

MG = α(trG)I + βGT + γG

= α(trG)I + (γ + β) symG + (γ − β) skew G, G ∈ M3(R).

The elasticity constants are taken to satisfy (see [10]):

(4) 3λ + 2µ > 0, µ > 0, µc > 0, 3α + β + γ > 0, γ + β > 0, γ − β > 0,

which guaranties positivity of the elasticity tensors T and M (see [1]): there aremT > 0 and mM > 0 such that

(5) T E · E ≥ mTE2, E ∈ M3(R), MG ·G ≥ mMG2, G ∈ M3(R).

Remark 2.1. By thermodynamical argument one obtains the nonnegativity ofthe material coefficients in (4). Strict inequalities in (4) allow us to invert thestress/strain and couple stress/strain relations and imply positive definiteness in(5). These are not necessary for existence of solutions neither in the static case([18, 14]), nor in the dynamic case. There are two particularly interesting cases.In the case µc = 0 the equations for the displacement and microrotation decouplewhich results with classical linearized elasticity problem for the displacement. Inthis case one can use the Korn inequality to prove the coercivity of the internalpotential energy form. The other interesting case is µc > 0 and γ = β, α = −2β

3

because it implies conformally invariant curvature energy density MEε · Eεand

avoids unbounded stiffness behavior in the torsion and bending of a slender cylinder(see [24]). In this case one can also use the Korn inequality for the microrotationto prove the coercivity of the internal potential energy form and consequently theexistence of solutions.

In the asymptotic analysis we perform, the strict inequalities in (4) are essentialto obtain that the families ∇εu(ε) and ∇εω(ε) are L2 bounded which implies the

4 JOSIP TAMBACA AND IGOR VELCIC

uniform boundedness of u(ε) and ω(ε) in H1 (see e.g., Lemma 4.2) for loads oforder ε0. In the remaining cases we need the scaling of loads different from ε0 (asknown in the case of classical linearized elasticity, i.e., for µc = 0) in order toobtain uniform bounds on the unknowns. This also requires different analysis thenthe one presented in this paper and is left for the future work.

Taking the sum of equations (1) and (2) and using the definitions of the stresstensors, the problem can be written in the form

ρ

∫

Ωε

∂2U ε

∂t2· V dy + ρ

∫

Ωε

J∂2$ε

∂t2·W dy

+∫

Ωε

T (∇U ε −A$ε) · (∇V −AW ) dy +∫

Ωε

M∇$ε · ∇W dy

=∫

Ωε

F ε · V dy +∫

Ωε

Lε ·W dy, V ∈ V0(Ωε),(6)

U ε|t=0 = U ε0, U ε

t |t=0 = U ε1, $ε|t=0 = $ε

0, $εt |t=0 = $ε

1.(7)

Let us now search for all solutions of the associated homogeneous equation (6)in the form sε(t, y) = T ε(t)Y ε(y). It follows that

(T ε)′′(t)ρ∫

Ωε

Y ε1 · V dy + (T ε)′′(t)ρ

∫

Ωε

JY ε2 ·W dy

+T ε(t)∫

Ωε

T (∇Y ε1 −AY ε

2) · (∇V −AW )dy

+T ε(t)∫

Ωε

M∇Y ε2 · ∇W dy = 0.

Therefore,

(T ε)′′(t)T ε(t)

= −∫Ωε T (∇Y ε

1 −AY ε2) · (∇V −AW ) +

∫Ωε M∇Y ε

2 · ∇W

ρ∫Ωε Y ε

1 · V + ρ∫Ωε JY ε

2 ·W.

On the left hand side is a function of the variable t, while on the right hand side isa constant. That constant we denote by −αε. Therefore we obtain the eigenvalueproblem for a three-dimensional isotropic, homogeneous, linearized micropolar elas-tic body: find αε and 0 6= Y ε = (Y ε

1, Yε2) ∈ V0(Ωε)2 such that

∫

Ωε

T (∇Y ε1 −AY ε

2) · (∇V −AW ) +

∫

Ωε

M∇Y ε2 · ∇W

= αερ

(∫

Ωε

Y ε1 · V +

∫

Ωε

JY ε2 ·W

),(8)

for all V , W ∈ V0(Ωε). This problem is the starting point for the asymptotic anal-ysis when the thickness of the plate-like three-dimensional body tends to zero. Inorder to perform this analysis we use the usual approach by Ciarlet and Destuynder[5] and rescale the domain to a canonical one, independent of ε. Let

Ω = S × (−h/2, h/2), Sb = ∂S × (−h/2, h/2), Γ = S × (−h/2 ∪ h/2),Rε : Ω → Ωε, Rε(z) = (z1, z2, εz3).

Then u(ε) = U ε Rε, ω(ε) = $ε Rε belong to

V0(Ω) = v = (v1, v2, v3) ∈ H1(Ω)3 : v|Sb= 0.

EVOLUTION OF MICROPOLAR PLATES 5

We equip V0(Ω) with the usual H1(Ω)3 norm. Let us define the forma(ε)((u, ω), (v, w)) : V0(Ω)× V0(Ω) → R by

(9) a(ε)((u, ω), (v, w)) =∫

Ω

T (∇εu−Aω) · (∇εv−Aw)dz +∫

Ω

M∇εω ·∇εwdz,

where

∇ε =1ε∇z +∇y, ∇yv =

∂1v1 ∂2v1 0∂1v2 ∂2v2 0∂1v3 ∂1v3 0

, ∇zv =

0 0 ∂3v1

0 0 ∂3v2

0 0 ∂3v3

,

and b : H ×H → R, where H = (L2(Ω)3)2, by

(10) b((u,ω), (v,w)) = ρ

∫

Ω

u · vdz + ρ

∫

Ω

Jω ·wdz.

Then the equation (8), with Y i(ε) = Y εi Rε, i = 1, 2 and α(ε) = αε, becomes

a(ε)((Y 1, Y 2), (v, w)) = α(ε)b((Y 1,Y 2), (v, w)), v, w ∈ V0(Ω).

An essential role in the asymptotic behavior of the corresponding equilibrium prob-lem (see [1] for a proof by simple use of the Poincare and the Young inequality)is played by the uniform (with respect to ε) ellipticity of the form a(ε): there areconstants Cp > 0 and εp > 0 such that for all 0 < ε ≤ εp one has

(11) a(ε)((v, w), (v, w)) ≥ Cp(‖∇εv‖+ ‖∇εw‖)2, v, w ∈ V0(Ω).

We also know that (σ(J) is the spectrum of J)

(12) b((v, w), (v, w)) ≥ ρ min σ(J)(‖v‖2 + ‖w‖2).Now we consider the problem in an abstract way. Let V and H be Hilbert spaces

with the scalar multiplication denoted by (·|·)V and (·|·)H and let the imbedding

V → H

be dense and compact. Let a and b be two bilinear forms

a : V × V → R, b : H ×H → R

which are symmetric, continuous and elliptic on the spaces where they are defined.We look for a solution of the following problem.

Problem 2.2. Find all α ∈ R and 0 6= u ∈ V such that

(13) a(u, v) = αb(u, v), v ∈ V.

If we denote the Rayleighy quotient by

R(v) =a(v, v)b(v, v)

, v ∈ V

we have the following theorem.

Theorem 2.3. There are countable many eigenvalues of the problem (13), whichhave finite dimensional eigenspaces. We can order them in the nondecreasing se-quence

0 < α1 ≤ α2 ≤ · · · , limn→∞

αn = ∞.

6 JOSIP TAMBACA AND IGOR VELCIC

The associated eigenvectors un, n ∈ N are orthonormal with respect to the scalarmultiplication defined by b(u, v) and orthogonal with respect to the scalar multipli-cation defined by a(u, v) and they form complete systems in the spaces V and H.We also have the following characterizations

αn = minR(v) : b(v, uk) = 0, k = 1, . . . , n− 1,(14)αn = min

L∈V nmaxv∈L

R(v),(15)

where V n is the family of all n− dimensional subspaces of V .

Now the eigenvalue problem for linearized micropolar elasticity can be writtenas

Problem 2.4. Find all α(ε) ∈ R and 0 6= s(ε) ∈ V0(Ω)2 such that b(s(ε), s(ε)) = 1and

a(ε)(s(ε), (v, w)) = α(ε)b(s(ε), (v,w)), v,w ∈ V0(Ω).

From Theorem 2.3 it follows that there is a countable number of eigenvalueswhich all have finite dimensional eigenspaces. The set of eigenvalues doesn’t havefinite limit points. We order eigenvalues in the nondecreasing sequence

0 < α1(ε) ≤ α2(ε) ≤ α3(ε) ≤ · · · , limn→∞

αn(ε) = ∞and to each eigenvalue αn(ε) associate an eigenvector (eigenfunction) sn(ε) =(un(ε),ωn(ε)) such that b(sn(ε), sn(ε)) = 1.

3. The limit eigenvalue problem for micropolar plates

In this section we formulate the eigenvalue problem for micropolar plates andapply the Theorem 2.3 to it. Let

b0((u, ω), (v, w)) = ρh

∫

S

u · v + ρh

∫

S

Jω ·w,(16)

a0((u, ω), (v, w)) = h

∫

S

X (∇u + (e1 × ω e2 × ω)) ··(∇v + (e1 ×w e2 ×w))

+h

∫

S

Y∇ω · ∇w u,ω, v, w ∈ H10 (S)3,(17)

where the elasticity tensors X : M3,2(R) × M3,2(R) → R and Y : M3,2(R) ×M3,2(R) → R are defined as

X(

AaT

)·(

BbT

)=

4µµc

µ + µca · b +

2µλ

λ + 2µtrAtrB

+2µ symA · symB + 2µc skew A · skew B,

Y(

AaT

)·(

BbT

)=

γ2 − β2

γa · b +

(β + γ)αα + β + γ

trAtrB

+(γ + β) symA · symB + (γ − β) skew A · skew B.

It is easy to prove (see [1]) that X and Y are positive definite.

Lemma 3.1. There are mX and mY such that

XA ·A ≥ mXA2, YA ·A ≥ mYA2, A ∈ M3,2(R).

EVOLUTION OF MICROPOLAR PLATES 7

The proof of H10 (S)2– ellipticity of the form a0 follows in the same fashion as

the proof of ellipticity of the form a(ε) (see [1]). More precisely, there are constantsC0

p > 0 and ε0p > 0 such that for all 0 < ε ≤ ε0

p one has

(18) a0((v, w), (v, w)) ≥ C0p(‖∇v‖+ ‖∇w‖)2, v, w ∈ V0(Ω).

The two dimensional eigenvalue problem for linearly micropolar plates is now givenas

Problem 3.2. Find α0 ∈ R and 0 6= s ∈ (H10 (S)3)2 such that b(s, s) = 1 and

a0(s, (v, w)) = α0b0(s, (v, w)), v, w ∈ H10 (S)3.

It follows from Theorem 2.3, as in three-dimensional case, that there are count-ably many eigenvalues which have finite dimensional eigenspaces. The set of eigen-values does not have limit points. We order them in the nondecreasing sequence

0 < α10 ≤ α2

0 ≤ α30 ≤ · · · , lim

n→∞αn

0 = ∞and to each eigenvalue αn

0 associate an eigenvector sn0 = (un

0 , ωn0 ) such that b(sn

0 , sn0 ) =

1.

4. A priori estimates

In order to prove the convergence result the first step is to find an a prioriestimate (uniform with respect to ε) for the nth eigenpair (αn(ε), sn(ε)). We startwith an a priori estimate for αn(ε).

Lemma 4.1. For every n ∈ N there exists a constant δn, independent of ε, suchthat

(19) αn(ε) ≤ δn.

Proof. We prove this lemma by using the characterization (15) of the nth eigen-value. Let (u, ω) ∈ H1

0 (S)3 ×H10 (S)3. We have

a(ε)((u,ω), (u,ω)) ≤≤ max‖T ‖, ‖M‖

( ∫

Ω

(∇εu−Aω)(∇εu−Aω)dy +∫

Ω

∇εω · ∇εωdy)

= max‖T ‖, ‖M‖h∫

S

(‖∂1u + e1 × ω‖2 + ‖∂2u + e2 × ω‖2

+‖∇yω‖2 + (ω21 + ω2

2))dy

≤ max‖T ‖, ‖M‖(C2P + 1)h∫

S

(‖∂1u + e1 × ω‖2 + ‖∂2u + e2 × ω‖2) + ‖∇yω‖2

)

≤ max‖T ‖, ‖M‖(C2P + 1)max

1

mX,

1mY

a0((u,ω), (u,ω)),

where CP is the constant from the Poincare inequality for the domain S, and

b((u,ω), (u,ω)) = b0((u, ω), (u, ω)).

If we denote C = max‖T ‖, ‖M‖max1/mX , 1/mY(C2P + 1) it follows that

R(ε)((u,ω), (u, ω)) ≤ Ca0((u,ω), (u, ω))b0((u, ω), (u, ω))

.

8 JOSIP TAMBACA AND IGOR VELCIC

Let us denote by V n00 the family of all n-dimensional subspaces of H1

0 (S)3 ×H1

0 (S)3. As the natural inclusion H10 (S)3 × H1

0 (S)3 ⊂ V0(Ω)2 holds, using thecharacterization (15) from Theorem 2.3 twice, once for the three-dimensional Prob-lem 2.4 and once for the limit Problem 3.2, we obtain

αn(ε) ≤ minL1∈V n

00

max(u,ω)∈L1

Ca0((u, ω), (u, ω))b0((u, ω), (u,ω))

= C minL1∈V n

00

max(u,ω)∈L1

a0((u, ω), (u, ω))b0((u, ω), (u,ω))

= Cαn0 .

Therefore we obtain the desired result with δn = Cαn0 . ¤

Now we derive a priori estimates for eigenfunctions.

Lemma 4.2. There exists εp such that for every n ∈ N there exists C such thatfor every 0 < ε ≤ εp one has

‖∇εsn(ε)‖L2(Ω)9 ≤ C,(20)‖sn(ε)‖V0(Ω)2 ≤ C.(21)

Proof. For n ∈ N the pair (αn(ε), sn(ε)) satisfies equality

a(ε)(sn(ε), sn(ε)) = αn(ε)b(sn(ε), sn(ε)) = αn(ε).

Using the V0(Ω)2–ellipticity of the form a(ε) from (11) we obtain

Cp(‖∇εun(ε)‖2 + ‖∇εωn(ε)‖2) ≤ a(ε)(sn(ε), sn(ε)) = αn(ε).

From Lemma 4.1 we obtain the first estimate (20). The second one is a consequenceof the first estimate and the Poincare inequality. ¤

5. The convergence results

In this section, we first prove, that there is a subsequence such that the limitsof all eigenpairs (αn(ε), sn(ε)) for Problem 2.4 are the eigenpairs from the limitProblem 3.2. Then we prove that limits of eigenfunctions form a complete set,comprising all eigenfunctions of the limit problem. A monotonicity argument thenimplies the convergence of the whole families (αn(ε), ε > 0), for all n ∈ N. The con-vergence of the eigenfunctions is then showed for the eigenvalues from Problem 3.2with multiplicity 1 and a kind of convergence is determined for the eigenspaceprojectors of eigenvalues of multiplicity more than 1.

Theorem 5.1. For every n ∈ N, for every sequence in ε, ε > 0 which tends tozero, there exists a sub-subsequence, which we still denote by ε, a real number αn,functions sn = (un,ωn) ∈ (H1

0 (S)3)2 and (En,Gn) ∈ (L2(S)9)2 such that

αn(ε) → αn,

sn(ε) → sn in V0(Ω)2 strongly,(22)

∇εsn(ε) → (En,Gn) in (L2(Ω)9)2 strongly,(23)

where the pair (αn, sn) satisfies

(24) a0(sn, (v, w)) = αnb0(sn, (v, w)), v, w ∈ H10 (S)3

EVOLUTION OF MICROPOLAR PLATES 9

and

(25) En=

∂1un1 ∂2u

n1 −µ−µc

µ+µc∂1u

n3 + 2µc

µ+µcωn

2

∂1un2 ∂2u

n2 −µ−µc

µ+µc∂2u

n3 − 2µc

µ+µcωn

1

∂1un3 ∂2u

n3 − λ

λ+2µ (∂1un1 + ∂2u

n2 )

,

(26) Gn =

∂1ωn1 ∂2ω

n1 −β

γ ∂1ωn3

∂1ωn2 ∂2ω

n2 −β

γ ∂2ωn3

∂1ωn3 ∂2ω

n3 − α

α+β+γ (∂1ωn1 + ∂2ω

n2 )

.

Proof. From the a priori estimates from Lemma 4.1 and Lemma 4.2 follows thatthere exist αn ∈ R, sn = (un, ωn) ∈ (V0(Ω))2 and γn = (γn

u, γnω) ∈ (L2(Ω)9)2, and

a sequence in ε, ε > 0, which we still denote by ε, such that

αn(ε) → αn,(27)sn(ε) → sn in (V0(Ω))2 weakly,(28)

∇εsn(ε) → (γnu, γn

ω) in (L2(Ω)9)2 weakly.(29)

From (29) and the definition of ∇ε one has

∇zun(ε) → 0, ∇zω

n(ε) → 0, strongly in L2(Ω)9.

Continuity of the distributional derivative and uniqueness of the limit implies∇zs

n = (∇zun,∇zω

n) = 0. Therefore sn ∈ (H10 (S)3)2 and

(γnu )i,η = ∂ηun

i (γnω)i,η = ∂ηωn

i , η = 1, 2, i = 1, 2, 3.

The nth eigenpair (αn(ε), sn(ε)) satisfies

∫

Ω

T (∇εun(ε)−Aωn(ε)) · (∇εv −Aw)dz +∫

Ω

M∇εωn(ε) · ∇εwdz

= αn(ε)ρ( ∫

Ω

un(ε) · vdz +∫

Ω

Jωn(ε) ·wdz) ∀v, w ∈ V0(Ω).(30)

Multiplying this equation with ε and letting ε → 0, using the convergence (27)-(29),we obtain that

∫

Ω

T (γnu −Aωn) · ∇zvdz +

∫

Ω

Mγnω∇zwdz = 0, v, w ∈ V0(Ω).

For test functions of the form v(z) = p(z1, z2)z3, w(z) = π(z1, z2)z3, p,π ∈ H10 (S)3

for i = 1, 2, 3 we obtain

(31)∫ h/2

−h/2

(T (γnu −Aωn))i,3dz3 =

∫ h/2

−h/2

(T (γnu −Aωn))i,3dz3 = 0.

10 JOSIP TAMBACA AND IGOR VELCIC

Now, using definitions of T ,M and the above identities, we conclude

∫ h/2

−h/2

((γnu)1,3 − ωn

2 ) = −µ− µc

µ + µch(∂1u

n2 + ωn

2 ),(32)

∫ h/2

−h/2

((γnu)2,3 + ωn

1 ) = −µ− µc

µ + µch(∂2u

n3 − ωn

1 ),(33)

∫ h/2

−h/2

(γnu)3,3 = − λ

λ + 2µh(∂1u

n1 + ∂2u

n2 ),(34)

∫ h/2

−h/2

(γnω)1,3 = −β

γh∂1ω

n3 ,(35)

∫ h/2

−h/2

(γnω)2,3 = −β

γh∂1ω

n3 ,(36)

∫ h/2

−h/2

(γnω)3,3 = − α

α + β + γh(∂1ω

n1 + ∂2ω

n2 ).(37)

Now, for v,w ∈ H10 (S)3, take the limit ε → 0 in the equation (30) to obtain

∫

Ω

T (γnu −Aωn) · (∇yv −Aw) +

∫

Ω

Mγnω · ∇yw = αnρ

( ∫

Ω

un · v +∫

Ω

Jωn ·w).

Using (34)-(36) and the fact that v, w depend only on the first two variables onehas that the pair (αn, (un, ωn)) satisfies the equation (24).

Now we obtain the strong convergence result. Let us define the family

Λε :=∫

Ω

T (∇εun(ε)−Aωn(ε) − (γnu −Aωn)

) ·

·(∇εun(ε)−Aωn(ε) − (γnu −Aωn)

)

+∫

Ω

M(∇εωn(ε)− γnω

) · (∇εωn(ε)− γnω

)

and let En

H = γnu − E

n,Gn

H = γnω −Gn. Using (30) we get

Λε = αn(ε)ρ( ∫

Ω

un(ε) · un(ε) +∫

Ω

Jωn(ε) · ωn(ε))

+∫

Ω

T ((γn

u −Aωn)− 2(∇εun(ε)−Aωn(ε))) · (γn

u −Aωn

)

+∫

Ω

M(γnω − 2∇εωn(ε)) · γn

ω.

Compactness of the imbedding V0(Ω) → L2(Ω)3 and the a priori estimate (20)imply the strong convergence un(ε) → un, ωn(ε) → ωn in L2(Ω)3. Thus, using theconvergence (27)-(29), we have the convergence of Λε to the constant Λ = limε→0 Λε

EVOLUTION OF MICROPOLAR PLATES 11

given by

Λ = αnρ( ∫

Ω

un · un +∫

Ω

Jωn · ωn)−

∫

Ω

T (γn

u −Aωn

) · (γnu −Aωn

)

−∫

Ω

Mγnω · γn

ω

= −3∑

i=1

∫

Ω

(T (γnu −Aωn)

)i,3· (γn

u −Aωn

)i,3−

3∑

i=1

∫

Ω

(Mγnω)i,3 · (γn

ω)i,3

(using the fact that (T (En −Aωn))i,3 = 0, (MGn)i,3 = 0, i = 1, 2, 3)

= −3∑

i=1

(∫

Ω

(T En

H)i,3(En

H)i,3 + (MGnH)i,3 · (Gn

H)i,3

)

(using the fact that (En

H)i,j = 0, (GnH)i,j = 0, j = 1, 2, i = 1, 2, 3)

= −(∫

Ω

T En

H · En

H +MGnH ·Gn

H

).

Using positivity of T and M (5) we obtain

Λε ≥ mT

∫

Ω

(∇εun(ε)−Aωn(ε) − (γnu −Aωn)

)2 + mM

∫

Ω

(∇εωn − γnω

)2 ≥ 0.

Positivity of T and M in the limit Λ implies that Λ = 0 and thus En

H = 0,GnH = 0.

Therefore∇ε(un(ε), ωn(ε)) → γn in (L2(Ω)9)2 strongly.

and consequently

(un(ε), ωn(ε)) → (un, ωn) in V0(Ω)2 strongly.

Finally the equalities (25) and (26) follow directly from the fact that En

H = GnH =

0. ¤

Using a diagonal procedure we can extract a subsequence for which all eigenvaluesand eigenfunctions converge simultaneously.

Corollary 5.2. For every sequence in ε, ε > 0 converging to zero we can find asubsequence, still denoted by ε, such that for all n ∈ N one has

αn(ε) → αn,

sn(ε) = (un(ε), ωn(ε)) → sn = (un, ωn) in V0(Ω)2 strongly,

where (αn, sn) is an eigenpair from Problem 3.2.

Lemma 5.3. The family αn : n ∈ N contains all eigenvalues from Problem 3.2while the associated eigenfunctions (un, ωn) ∈ N form a complete set in H1

0 (S)3 ×H1

0 (S)3 and L2(S)3 × L2(S)3 and satisfy the equalities

(38) b0(sn, sk) = δnk, a0(sn, sk) = αnδnk, n, k ∈ N,

where δnk is the Kroneker delta symbol.

Proof. This follows in the same fashion as in the case of the classical plate [6] orthe classical curved rod [25]. ¤

12 JOSIP TAMBACA AND IGOR VELCIC

Lemma 5.4. For every n ∈ N the whole family αn(ε) converges to αn0 .

If for some n ∈ N, αn0 is a simple eigenvalue, then there is ε0(n) such that for

every 0 < ε ≤ ε0(n), αn(ε) is a simple eigenvalue and there are eigenfunctionssn(ε) such that the convergence ((22) in Theorem 5.1) is satisfied for the wholefamily.

When αn0 is of multiplicity m then there exists δ0 > 0 and ε0(n) such that for all

0 < δ < δ0 and every 0 < ε ≤ ε0(n) there are, in a δ neighborhood of αn0 , exactly

m eigenvalues from Problem 2.4 (each counted with its multiplicity). Moreover, forevery s(ε) ∈ V0(Ω)2 such that

s(ε) → s, weakly in (L2(Ω)3)2,

where s ∈ (H10 (S)3)2 one has

(39) Pn(ε)s(ε) → Pn0 s strongly in V0(Ω)2;

here Pn0 is an orthogonal projector with respect to the scalar product b0 onto the

eigenspace corresponding to αn0 and Pn(ε) is an orthogonal projector with respect

to the scalar product b onto the space spanned by sn(ε), . . . , sn+m−1(ε).Proof. Let n ∈ N. From Corollary 5.2 and Lemma 5.3 we know that every sub-sequence of αn(ε) has a convergent subsequence. As the limit is precisely the ntheigenvalue from Problem 3.2, it is unique. Thus, limε→∞ αn(ε) = αn

0 .Let now αn

0 be a simple eigenvalue. Using the fact that for all n, αn(ε) → αn0 we

have that for ε small enough the multiplicity of αn(ε) is 1. The eigenvalue αn(ε)has two normal eigenvectors which are ±sn(ε). We choose such normal eigenvectorsn(ε) that b(sn, sn(ε)) > 0. Then every convergent subsequence of sn(ε) necessarilyconverges to sn; thus we have uniqueness of the limit.

Let us now turn to the case of an eigenvalue αn0 of multiplicity m > 1. Suppose

the opposite: for every δ0 > 0 and ε0(n), there are 0 < δ(δ0, ε0(n)) < δ0 and0 < ε(δ0, ε0(n)) < ε0(n) such that in a δ(δ0, ε0(n)) neighborhood of αn

0 there arel eigenvalues of Problem 2.4, l 6= m. It follows that there are l eigenvalues fromProblem 2.4 with parameter εk, that converge to αn

0 . This is in contradiction withthe first statement of this lemma.

Let s(ε) → s strongly in (L2(Ω)3)2 and let αn0 be with multiplicity m. Then as

the eigenvectors are orthonormal with respect to the scalar product defined by bone has

Pn(ε)s(ε) =m∑

i=1

b(s(ε), sn+i−1(ε))sn+i−1(ε).

According to Corollary 5.2 for every sequence in ε : ε > 0 converging to zerowe can find a subsequence such that all eigenvectors simultaneously converge.Therefore sn+i−1(ε) → sn+i−1 strongly in V0(Ω)2 for all i ∈ 1, . . . , m, for somesn+i−1 ∈ V0(Ω), i = 1, . . . ,m. According to Lemma 5.3 sn+i−1 ∈ (H1

0 (S)3)2,i = 1, . . . , m span the m-dimensional eigenspace of αn

0 . This implies

b(s(ε), sn+i−1(ε)) → b0(s, sn+i−1), i = 1, . . . , m.

Therefore

Pn(ε)s(ε) →m∑

i=1

b0(s, sn+i−1)sn+i−1 = Pn0 s

strongly in V0(Ω)2; here we have used that sn+i−1, i = 1, . . . ,m are orthonormalwith respect to b0. ¤

EVOLUTION OF MICROPOLAR PLATES 13

6. The evolution problem

The three-dimensional problem given by (6) with the initial conditions (7) canbe written on the canonical domain Ω in the form

d

dtb(s(ε)t, r) + a(ε)(s(ε), r) = (p(ε), r)H , r = (v, w) ∈ V0(Ω)2,(40)

s(ε)|t=0 = s0(ε), st(ε)|t=0 = s1(ε),(41)

where a(ε) and b(ε) are given in (9) and (10), s(ε) = (u(ε), ω(ε)) and

p(ε) = (f(ε), l(ε)), f(ε) = F ε Rε, l(ε) = Lε Rε.

Now we use abstract results from [8]. Let V, H, V ′ be real separable Hilbert spaces,where ( | )H is the scalar product on H, V ′〈 | 〉V is the dual product and V →H → V ′ (all the embeddings are continuous and dense). Let a and b be symmetriccontinuous bilinear forms such that a is V –elliptic and b is H–elliptic (with constantαb). By the theorem for the representation of bilinear functionals there exist regularoperators A ∈ L(V, V ′) and B ∈ L(H) such that

a(s, r) = V ′〈As, r〉V , s, r ∈ V, b(s, r) = (Bs|r)H , u, v ∈ H.

The operator B is hermitian and positive definite. We consider the following ab-stract problem.

Problem 6.1. Let T > 0. For given p ∈ L2(0, T ; H), s0 ∈ V and s1 ∈ H find ssuch that (here st = ds

dt )

s ∈ C([0, T ];V ),ds

dt∈ C([0, T ]; H),

d

dtB

ds

dt∈ L2(0, T ; V ′),

d

dtb(st, r) + a(s, r) = (p|r)H , r ∈ V,(42)

s(0) = s0,ds

dt(0) = s1.(43)

Note that for a function that satisfies (42) one has b(st, r), a(s, r) ∈ Lq(0, T )for 1 ≤ q ≤ ∞. Thus d

dtb(st, r) ∈ D′(0, T ) so (42) is satisfied in the sense ofdistributions.

Using the definition of operators A and B, (42) can be written in differentialform d

dtBdsdt + As = p. Therefore for s ∈ L2(0, T ; V ) and p ∈ L2(0, T ; H) we have

that p − As ∈ L2(0, T ; V ′), so we conclude that ddtBst ∈ L2(0, T ; V ′). Therefore,

the equation (42) is meaningful in the space

WB =

s; s ∈ L∞(0, T ; V ) :ds

dt∈ L∞(0, T ; H),

d

dtB

ds

dt∈ L2(0, T ;V ′)

.

We still have to interpret the initial conditions in WB . Since we know that s ∈L2(0, T ;V ), st ∈ L2(0, T ;H) implies s ∈ C([0, T ];H) the initial condition s(0) = s0

is well posed. In a similar way we can interpret the initial condition (Bst)(0) = Bs1

using the fact that st ∈ L2(0, T ; H) and ddtBst ∈ L2(0, T ;V ′).

Theorem 6.2. Let a, b, V,H, WB , s0 ∈ H, s1 ∈ V be as above.(i) For a solution s of (42), (43) in WB one has

αb‖st‖2H + a(s, s) ≤ (‖p‖2L2(0,T ;H) + b(s1, s1) + a(s0, s0))e

Tαb .

(ii) The solution of (42), (43) is unique in WB.(iii) There exists a solution of Problem 6.1.

14 JOSIP TAMBACA AND IGOR VELCIC

Proof. The statement (iii) is a consequence of Theorem 4 in [8, XVIII 5.5]. Thestatement (ii) is a consequence of the estimate (i). The estimate (i) is proved in[25]. ¤

Now we apply this abstract theory to the problem (40), (41). We take V =V0(Ω)2 (using the induced topology of (H1(Ω)3)2) and H = (L2(Ω)3)2. Note aswell that in our case B is a constant invertible operator (from H to H and V ′ toV ′). Therefore WB = WI =: W . The positivity of the forms a(ε) and b(ε) followsdirectly from the inequalities (11) and (12). Now we can apply Theorem 6.2 toobtain the existence and uniqueness of solutions of (40), (41).

Theorem 6.3. Let p(ε) ∈ L2(0, T ; H), s0(ε) ∈ V and s1(ε) ∈ H. Then there is asolution s(ε) of (40), (41) such that

(44) s(ε) ∈ C([0, T ];V ),ds(ε)

dt∈ C([0, T ];H),

d2s(ε)dt2

∈ L2(0, T ; V ′).

The solution s(ε) is unique in the larger space W .

Now we can establish the existence and uniqueness result for the limit problem.

Theorem 6.4. Let V0 = (H10 (S)3)2, H0 = (L2(S)3)2, p ∈ L2(0, T ; H0), s0 ∈ V0,

s1 ∈ H0 and let b0 and a0 be given by (16) and (17). Then there is a solution ofthe problem

d

dtb0(st, r) + a0(s, r) = (p|r)H0 , r = (v, w) ∈ V0,(45)

s|t=0 = s0, st|t=0 = s1,(46)

such that

(47) s ∈ C([0, T ];V0),ds

dt∈ C([0, T ]; H0),

d2s

dt2∈ L2(0, T ; (V0)′).

This solution is unique in the larger space W0 given by

W0 =

s; s ∈ L∞(0, T ;V0) :ds

dt∈ L∞(0, T ; H0),

d2s

dt2∈ L2(0, T ; V ′

0)

.

7. The Fourier method

In this section we consider the asymptotics of the evolution problem for microp-olar plates, when instead of the variational technique applied in [26], we use theFourier method. As a benefit we obtain the convergence of solutions pointwise intime. We first show that the solution of the evolution problem can be written as theseries (48) and then prove first weak and then strong convergence results pointwisein time t.

The Fourier method, or separation of variables, consists of finding the solutionof the evolution problem in the form of a series with respect to an eigenbasis of theproblem, in our case sn(ε) : n ∈ N

(48) s(ε)(t, z) =∞∑

n=1

Tn(ε)(t)sn(ε)(z).

In the following analysis we will extensively use the following algebraic charac-terization of spaces V0(Ω)2, (L2(Ω)3)2 and the topological dual of V0(Ω)2, namely

EVOLUTION OF MICROPOLAR PLATES 15

(V0(Ω)2)′, see [8]:

(49) r =∞∑

i=1

Rnsn(ε) ∈ V0(Ω)2 ⇐⇒ a(ε)(r, r) =∞∑

n=1

(Rn)2αn(ε) < ∞,

(50) r =∞∑

i=1

Rnsn(ε) ∈ (L2(Ω)3)2 ⇐⇒ b(r, r) =∞∑

n=1

(Rn)2 < ∞,

(51) r =∞∑

i=1

Rnsn(ε) ∈ (V0(Ω)2)′ ⇐⇒ ‖r‖(V0(Ω)2)′ =∞∑

n=1

(Rn)2

αn(ε)< ∞.

Moreover, the convergence of the above series on the left hand side is strong in theassociated function space.

In order to apply the Fourier method we need to expand the initial conditionsin the eigenbasis sn(ε) : n ∈ N:

s0(ε) =∞∑

n=1

sn0 (ε)√αn(ε)

sn(ε) converges strongly in V0(Ω)2

s1(ε) =∞∑

n=1

sn1 (ε)sn(ε) converges strongly in (L2(Ω)3)2.

From the previous characterizations it follows that

a(ε)(s0(ε), s0(ε)) =∞∑

n=1

sn0 (ε)2 < ∞, b(s1(ε), s1(ε)) =

∞∑n=1

sn1 (ε)2 < ∞.

We also define βn(ε) = (p(ε)|sn(ε))(L2(Ω)3)2 which is an element of L2(0, T ).

Lemma 7.1. The solution of the problem (40), (41) is given by the series (48)where Tn(ε) is given by

Tn(ε)(t) =sn0 (ε)√αn(ε)

cos(√

αn(ε)t) +sn1 (ε)√αn(ε)

sin(√

αn(ε)t)

+1√

αn(ε)

∫ t

0

sin(√

αn(ε)(t− τ))βn(ε)(τ)dτ,

i.e., it is the unique solution of the initial value problem

Tn(ε)′′ + αn(ε)Tn(ε) = βn(ε), Tn(ε)|t=0 =sn0 (ε)√αn(ε)

, Tn(ε)′|t=0 = sn1 (ε).

The series (48) converges uniformly in V0(Ω)2, while the series for st(ε) convergesuniformly in the strong topology of (L2(Ω)3)2.

Proof. Let us first show that the series (48) with Tn(ε) defined as above converges.As βn(ε) belongs to L2(0, T ) it follows that Tn(ε) ∈ C1([0, T ]) and Tn(ε)′′ ∈L2(0, T ). Moreover, for βn(ε) the following estimate holds

∞∑n=1

∫ T

0

βn(ε)(τ)2dτ =∫ T

0

b(B−1p(ε)(τ), B−1p(ε)(τ))dτ

≤ ‖B−1‖‖p(ε)‖2L2(0,T ;H).(52)

16 JOSIP TAMBACA AND IGOR VELCIC

Therefore, one has

a(ε)(s(ε)(t), s(ε)(t)) =∞∑

n=1

αn(ε)(Tn(ε)(t))2

=∞∑

n=1

αn(ε)(

sn0 (ε)√αn(ε)

cos(√

αn(ε)t) +sn1 (ε)√αn(ε)

sin(√

αn(ε)t)

+1√

αn(ε)

∫ t

0

sin(√

αn(ε)(t− τ))βn(ε)(τ)dτ

)2

≤ 3∞∑

n=1

(sn0 (ε)2 + sn

1 (ε)2 + T

∫ T

0

(βn(ε)(τ))2dτ)

≤ 3( ∞∑

n=1

sn0 (ε)2 +

∞∑n=1

sn1 (ε)2 + T‖B−1‖‖p(ε)‖2L2(0,T ;H)

)=: M1(ε, T ).

Thus s(ε)(t, ·) ∈ V0(Ω)2, s(ε) ∈ L∞(0, T ; V0(Ω)2) and the series (48) convergesuniformly in the strong topology of V0(Ω)2.

In order to prove that the derivative of s(ε) is given by

(53) st(ε)(t, z) =∞∑

n=1

Tn(ε)′(t)sn(ε)(z)

it is, because of the continuity of the derivative in the space of distributions, enoughto prove that the sum in (53) is well defined in L∞(0, T ; (L2(Ω)3)2). As before onehas

b(st(ε)(t), st(ε)(t)) =∞∑

n=1

(Tn(ε)′(t))2

=∞∑

n=1

(− sn

0 (ε) sin(√

αn(ε)t) + sn1 (ε) cos(

√αn(ε)t)

+∫ t

0

cos(√

αn(ε)(t− τ))βn(ε)(τ)dτ)2

≤ M1(ε, T ).

Therefore st(ε)(t, ·) ∈ (L2(Ω)3)2, s(ε) ∈ L∞(0, T ; (L2(Ω)3)2) and the series (53)converges uniformly in the strong topology of (L2(Ω)3)2.

In the same way we show that the series differentiated twice with respect to t isconvergent and that the limit is equal to stt(ε). Using (51) we have

∫ T

0

‖stt(ε)(t)‖2(V0(Ω)2)′dt =∫ T

0

∞∑n=1

1αn(ε)

(βn(ε)− αn(ε)Tn(ε)(t)

)2

dt

≤ 2∫ T

0

∞∑n=1

(βn(ε)(t)2

αn(ε)+ αn(ε)(Tn(ε)(t))2

)dt

using the fact that α1(ε) → α1 > 0 when ε → 0

≤ M2

∫ T

0

( ∞∑n=1

βn(ε)(t)2 + M1(ε, T )

)dt

≤ M2

(‖B−1‖‖p(ε)‖2L2(0,T ;H) + TM1(ε, T )

).

EVOLUTION OF MICROPOLAR PLATES 17

Therefore stt(ε) ∈ L2(0, T ; (V0(Ω)2)′).In order to prove that the function defined by (48) satisfies the equation (40) we

need to fulfill the equation only for test functions that are eigenvectors sn(ε). Asthe series (48) converges strongly in V0(Ω)2 for each t one has

a(ε)(s(ε), sn(ε)) = αn(ε)Tn(ε).

As the series (53) converges strongly in (L2(Ω)3)2 for each t one has

b(st(ε), sn(ε)) = Tn(ε)′.

Therefore (40) is fulfilled.By (48) and (53) the initial conditions are fulfilled as the series converge for each

t ∈ [0, T ]. ¤The continuity of the limit functions, i.e.,

s(ε) ∈ C([0, T ];V0(Ω)2), st(ε) ∈ C([0, T ]; (L2(Ω)3)2).

is a consequence of the uniform convergence in (48) and (53) and continuity ofTn ∈ C([0, T ]), T ′n ∈ C([0, T ]).

Remark 7.2. Let us expand the initial conditions of the limit problem in the basissn : n ∈ N:

s0 =∞∑

n=1

sn0√αn

sn converges in (H10 (S)3)2,

s1 =∞∑

n=1

sn1sn converges in (L2(S)3)2.

By the same reasoning as in the proof of Lemma 7.1 one can prove that the solutionof (45), (46) can be written as

(54) s(t, z1, z2) =∞∑

n=1

Tn(t)sn(z1, z2),

where Tn are given by

Tn(t) =sn0√αn

cos(√

αnt) +sn1√αn

sin(√

αnt) +1√αn

∫ t

0

sin(√

αn(t− τ))βn(τ)dτ,

i.e., it is the unique solution of the initial value problem

(Tn)′′ + αnTn = βn, Tn|t=0 =sn0√αn

, (Tn)′|t=0 = sn1 ,

where βn = (p|sn)(L2(S)3)2 . The series (54) converges uniformly in the space(H1

0 (S)3)2, while the series for st(ε) (differentiated term by term) converges uni-formly in the strong topology of (L2(S)3)2.

Theorem 7.3. Let us suppose that s0(ε) → s0 weakly in V0(Ω)2, p(ε) → p weaklyin L2([0, T ]; L2(Ω)3), ‖∇εs0(ε)‖(L2(Ω)9)2 ≤ M and s1(ε) → s1 weakly in (L2(Ω)3)2,where s1 depends only on the first two variables. Let s(ε) be the solution of (40),(41). Then for every t ∈ [0, T ] one has

s(ε)(t) → s(t) weakly in V0(Ω)2,(55)st(ε)(t) → st(t) weakly in (L2(Ω)3)2,(56)

where s is the unique solution of (45), (46).

18 JOSIP TAMBACA AND IGOR VELCIC

Proof. For an arbitrary sequence in ε : ε > 0 we chose a subsequence as inCorollary 5.2.

Boundedness of ∇εs0(ε) and the Banach-Alaouglu-Bourbaki theorem imply thaton a subsequence ∇εs0(ε) → γs0

= (γu0,γω0

). Using the continuity of the distri-butional derivative we have that

(γu0)i,η = ∂η(u0)i (γω0)i,η = ∂η(ω0)i, η = 1, 2, i = 1, 2, 3

and ∇zu0 = ∇zω0 = 0. Thus we conclude that s0 ∈ (H10 (S)3)2, which together

with T (En −Aωn)i,3 = 0, (MGn)i,3 = 0, i = 1, 2, 3 implies that

sn0 (ε) =

1√αn(ε)

a(ε)(s0(ε), sn(ε)) → 1√αn

a0(s0, sn) = sn

0 ,(57)

sn1 (ε) = b(s1(ε), sn(ε)) → b0(s1, s

n) = sn1 .(58)

Let us fix t ∈ [0, T ]. Using the energy inequality (i) from Theorem 6.2 we obtain

αb‖st(ε)(t)‖2(L2(Ω)3)2 + a(ε)(s(ε)(t), s(ε)(t))

≤ (‖p(ε)‖2L2(0,T ;(L2(Ω)3)2) + b(s1(ε), s1(ε)) + a(ε)(s0(ε), s0(ε)))e

Tαb ,

where αb = ρ minσ(J). The assumptions of the theorem imply that the righthand side is uniformly bounded with respect to ε. Therefore there is M > 0 suchthat

(59) ‖s(ε)(t)‖V0(Ω)2 ≤ M, ‖∇εs(ε)(t)‖(L2(Ω)9)2 ≤ M, ‖st(ε)(t)‖2(L2(Ω)3)2 ≤ M.

Thus for fixed t there are s(t) and γ s(t) such that (on a sequence in ε, ε > 0 thattends to 0)

s(ε)(t) → s(t) weakly in V0(Ω)2, ∇εs(ε)(t) → γ s(t) weakly in (L2(Ω)3)2.

Again, concluding in the same way as for s0(ε) we have that s(t) = (u(t), ω(t))belongs (H1

0 (S)3)2 and

(60) (γu)i,η(t) = ∂ηui((t)), (γω)i,η(t) = ∂ηωi(t), η = 1, 2, i = 1, 2, 3.

The above with (57)-(58), (22)-(23) and T (En −Aωn)i,3 = 0, (MGn)i,3 = 0, i =1, 2, 3 implies

a(ε)(sn(ε), s(ε)(t)) → a0(sn, s(t)).On the other hand

a(ε)(sn(ε), s(ε)(t)) = sn0 (ε)

√αn(ε) cos(

√αn(ε)t)

+sn1 (ε)

√αn(ε) sin(

√αn(ε)t)

+√

αn(ε)∫ t

0

sin(√

αn(ε)(t− τ))βn(ε)(τ)dτ,

so using (57), (58) and the convergence p(ε) → p we obtain

a(ε)(sn(ε), s(ε)(t)) → sn0

√αn cos(

√αnt) + sn

1

√αn sin(

√αnt)

+√

αn

∫ t

0

sin(√

αn(t− τ))βn(τ)dτ

= a0(sn, s(t)).

Uniqueness of the limit then implies

a0(sn, s(t)) = a0(sn, s(t)), n ∈ N.

EVOLUTION OF MICROPOLAR PLATES 19

Completeness of the basis sn, n ∈ N in V0(S)2 then implies that s(t) = s(t) whichproves (55).

For a fixed t from the estimate (59) it follows that (on a sequence in ε, ε > 0that tends to 0) there is r(t) such that

st(ε)(t) → r(t) weakly in (L2(Ω)3)2.

It follows thatb(sn(ε), st(ε)(t)) → b0(sn, r(t)).

On the other hand

b(sn(ε), st(ε)(t)) = −sn0 (ε) sin(

√αn(ε)t) + sn

1 (ε) cos(√

αn(ε)t)

+∫ t

0

cos(√

αn(ε)(t− τ))βn(ε)(τ)dτ,

so using (57), (58) and the convergence p(ε) → p we obtain

b(sn(ε), st(ε)(t)) → b0(sn, st(t)).

Uniqueness of the limit then implies

b0(sn, r(t)) = b0(sn, st(t)), n ∈ N.

Completeness of the basis sn, n ∈ N in (L2(S)3)2 then implies that st(t) = r(t)which proves (56).

The limits in (55) and (56) are unique for all subsequences of ε so one has theconvergence of the whole family in (55) and (56). ¤

For the strong convergence result we need the following lemma which is a slightmodification of the Lebesgue dominated convergence theorem.

Lemma 7.4. Let fn(ε) → fn in R as ε → 0 and let for all ε,∑∞

n=1 |fn(ε)| < ∞,∑∞n=1 |fn| < ∞. Let gn(ε) be a sequence of numbers such that for all n |fn(ε)| ≤

gn(ε), gn(ε) → gn in R, and for all ε,∑∞

n=1 |gn(ε)| < ∞,∑∞

n=1 |gn| < ∞ and∑∞n=1 gn(ε) → ∑∞

n=1 gn in R. Then∑∞

n=1 fn(ε) → ∑∞n=1 fn in R.

Let us state the strong convergence result.

Theorem 7.5. Let us suppose that s0(ε) → s0 strongly in V0(Ω)2, ∇εs0(ε) →(E0,G0) strongly in (L2(Ω)9)2, s1(ε) → s1 strongly in (L2(Ω)3)2 where s1 de-pends only on the first two variables, and let us suppose that p(ε) → p strongly inL2([0, T ]; L2(Ω)3). Let s(ε) be the solution of (40), (41). Then for every t ∈ [0, T ]one has

s(ε)(t) → s(t) strongly in (V0(Ω))2,(61)st(ε)(t) → st(t) strongly in (L2(Ω)3)2,(62)

∇εs(ε)(t) → (E(t),G(t)) strongly in (L2(Ω)9)2,(63)

where s = (u, ω) is the unique solution of (45), (46), where E and G are given by

E(t) =

∂1u1(t) ∂2u1(t) −µ−µc

µ+µc∂1u3(t) + 2µc

µ+µcω2(t)

∂1u2(t) ∂2u2(t) −µ−µc

µ+µc∂2u3(t)− 2µc

µ+µcω1(t)

∂1u3(t) ∂2u3(t) − λλ+2µ (∂1u1(t) + ∂2u2(t))

,

20 JOSIP TAMBACA AND IGOR VELCIC

G(t) =

∂1ω1(t) ∂2ω1(t) −βγ ∂1ω3(t)

∂1ω2(t) ∂2ω2(t) −βγ ∂2ω3(t)

∂1ω3(t) ∂2ω3(t) − αα+β+γ (∂1ω1(t) + ∂2ω2(t))

,

and E0 and G0 are given by the corresponding expressions with s(t) = (u(t), ω(t))replaced by s0 = (u0, ω0).

Proof. For E0 and G0 defined as above one has T (E0 −Aω0)i,3 = 0, (MG0)i,3 =0, i = 1, 2, 3. Moreover, from the strong convergence of s0(ε) and ∇εs0(ε) one has

∞∑n=1

sn0 (ε)2 = a(ε)(s0(ε), s0(ε)) → a0(s0, s0) =

∞∑n=1

(sn0 )2,(64)

∞∑n=1

sn1 (ε)2 = b(s1(ε), s1(ε)) → b(s1, s1) =

∞∑n=1

(sn1 )2.(65)

From Lemma 7.4, convergence in (64)-(65) and the estimate (52) we have

a(ε)(s(ε)(t), s(ε)(t)) =∞∑

n=1

(sn0 (ε) cos(

√αn(ε)t) + sn

1 (ε) sin(√

αn(ε)t)

+∫ t

0

sin(√

αn(ε)(t− τ))βn(ε)(τ)dτ)2

→∞∑

n=1

(sn0 cos(

√αnt) + sn

1 sin(√

αnt)(66)

+∫ t

0

sin(√

αn(t− τ))βn(τ)dτ)2

= a0(s(t), s(t)).

Now using (T (E(t)−Aω(t)))i,3 = 0, (MG(t))i,3 = 0, i = 1, 2, 3 we have that∫

Ω

T (E(t)−Aω(t)) · (E(t)−Aω(t)) +∫

Ω

MG(t) ·G(t) = a0(s(t), s(t)).

Thus (66) is in fact∫

Ω

T (∇εu(ε)(t)−Aω(ε)(t)) · (∇εu(ε)(t)−Aω(ε)(t))

+∫

Ω

M∇εω(ε)(t) · ∇εω(ε)(t)

→∫

Ω

T (E(t)−Aω(t)) · (E(t)−Aω(t)) +∫

Ω

MG(t) ·G(t).(67)

Let us define the family

Λε(t) =∫

Ω

T ((∇εu(ε)(t)−Aω(ε)(t))− (E(t)−Aω(t))) ·

·((∇εu(ε)(t)−Aω(ε)(t))− (E(t)−Aω(t)))

+∫

Ω

M(∇εω(ε)(t)−G(t)) · (∇εω(ε)(t)−G(t)).

Now, using (67) we take the limit in quadratic terms (in ε); using (55), (T (E(t)−Aω(t)))i,3 = 0, (MG(t))i,3 = 0, i = 1, 2, 3 and (60) we take the limit in linear terms

EVOLUTION OF MICROPOLAR PLATES 21

(in ε) to conclude that for all t, Λε(t) → 0. The estimate

Λε(t) ≥ mT ‖(∇εu(ε)(t)−Aω(ε)(t))− (E(t)−Aω(t))‖(L2(Ω)9)2

+mM‖∇εω(ε)(t)−G(t)‖(L2(Ω)9)2

then implies (61) and (63). Similarly as in (66) we have that

(68) b(st(ε)(t), st(ε)(t)) → b(st(t), st(t)).

Using the weak convergence (56) and (68) it follows that

b(st(ε)(t)− st(t), st(ε)(t)− st(t)) → 0.

Using positivity of the form b the convergence in (62) follows. ¤

Remark 7.6. By the energy equality for the abstract problem (42) (given in [8,XVIII, p. 578]) applied to the three-dimensional problem (40), (41) on the canonicaldomain we obtain

12b(st(ε)(t), st(ε)(t)) +

12a(ε)(s(ε)(t), s(ε)(t))

=12b(s1(ε), s1(ε)) +

12a(ε)(s0(ε), s0(ε)) +

∫ t

0

(p, st(ε))L2dτ.

Terms on the left hand side represent the kinetic and potential energy of the three-dimensional body Ω, respectively. Terms on the right hand side represent the initialkinetic and potential energy and the work of external loads. Under the assumptionsof Theorem 7.5 we showed that

b(st(ε)(t), st(ε)(t)) → b(st(t), st(t)), a(ε)(s(ε)(t), s(ε)(t)) → a0(s(t), s(t))

for every t.

Remark 7.7. As mentioned in Remark 2.1 it is interesting to consider the behaviorof micropolar elasticity problem for µc → 0, since for µc = 0 the problem reduces tothe classical elasticity problem. Let us first consider the case of three-dimensionalbodies. Using the same analysis as before one can obtain sµc(ε)(t) → s0(ε)(t)and sµc

t (ε)(t) → s0t (ε)(t) strongly in V0(Ω)2 and (L2(Ω)3)2 respectively, where the

displacement part of s0(ε) is the same as in classical linearized elasticity (i.e.,corresponds to the case µc = 0). Here, we have denoted by s(ε)µc the solution of(40) and (41), for µc ≥ 0. Since a(ε) = ad(ε) + µcap(ε), where ap(ε) is dominatedby ad(ε), we can, using perturbation techniques (see [13]), obtain estimates for theconvergence of eigenvalues and eigenprojectors. In fact, by using (14), we candirectly conclude that αn,0(ε) ≤ αn,µc(ε) ≤ αn,0(ε) + µcC(Ωε)αn,0(ε). Still, theestimates for eigenprojectors do not imply uniform convergence of the evolutionproblem on the time interval [0, T ].

In the limit two-dimensional model (45), (46) the form a0 looses its coercivityfor µc = 0 and we loose control of the displacement in transversal direction unlesswe assume loads are small compared to µc. Even in this case one would obtain asecond order equation for the transversal displacements which is different from thecase of classical linearized elasticity.

Remark 7.8. Our model is the same as the model obtained in [12]. It decouplesflexural and extensional behavior of the plate (see [26] and also [1] for the equilibriummodel). The flexural part of the model is of the Reissner Mindlin type, while theextensional part of the model also contains in-plane rotation ω3 which cannot be

22 JOSIP TAMBACA AND IGOR VELCIC

removed from the model for any choice of micropolar material parameters, see also[22]. If we assume J = diag(σ1, σ2, σ3) and introduce ~φ = (−ω2, ω1)T and h =(−l2, l1)T we obtain the following equations for the flexure of plates

ρh(u3)tt = A div(∇u3 − ~φ) + f3,

ρhJp~φtt = B∆~φ + C∇ div ~φ + A(∇u3 − ~φ) + h,

where Jp = diag(σ1, σ2) and

A =4µµc

µ + µch, B =

(β + γ)(2α + β + γ)(α + β + γ)

h, C = −(

α(β + γ)α + β + γ

+ β

)h.

These equations have the form of the Reissner–Mindlin plate model for transversaloscillations. The obtained shear-correction factor is k = 4µc

µ+µc.

Still, note that the unknown ~φ is the limit microrotation and has no clear macro-scopic interpretation, while in the Reissner–Mindlin plate model it is the infinitesi-mal (macroscopic) rotation of cross-sections. In order to bridge this gap we can usethe first correctors to build the first order approximation to u(ε) (for details see [26]or [1]). Namely, we search for the function s(ε) = s+εs1 such that s(ε)− s(ε) → 0and st(ε)− st(ε) → 0. Then s1 is determined (up to a function of t, x1, x2) usingE(t) and G(t) from Theorem 7.5. Then u(ε) is rewritten on the thin domain Ωε inthe form

u(t, y1, y2) + ω(ε)× y3e3 + s(t, y1, y2)y3e3 + εξ1(t, y1, y2);

here ξ1 is a smaller order correction of the displacement of the middle plane of theplate u (it is the limit of u(ε)), ω is the infinitesimal rotation of the cross-section

ω(t, y1, y2) =

2µc

µ+µcω1(t, y1, y2) + µ−µc

µ+µc∂2u3(t, y1, y2)

2µc

µ+µcω2(t, y1, y2)− µ−µc

µ+µc∂1u3(t, y1, y2)

0

and s is the radial stretch of the cross-section s = − λλ+2µ (∂1u1 + ∂2u2). Using

the definition of ω (for µc > 0) we can express ω1 and ω2 in the model by themacroscopic quantities ω1 and ω2 to obtain the model for the flexure of plates writtenin macroscopic quantities. With the notation φ = (−ω2, ω1)T we obtain

ρh(u3)tt = A div(∇u3 − φ) + f3,

ρhµ + µc

2µcJpφtt = B∆φ + C∇ div φ + A(∇u3 − φ) + D(∇u3)tt + E∇f3 + h,

where

A = 2µh, B =(β + γ)(2α + β + γ)

(α + β + γ)µ + µc

2µch, D = ρh

µ− µc

2µc(Jp − γ

2µI),

C = −((

α(β + γ)α + β + γ

+ β

)µ + µc

2µc+ γ

µ− µc

2µc

)h, E =

γ

2µ

µ− µc

2µc.

We see some additional terms when comparing this model with the Reissner-Mindlinplate model, namely (∇u3)tt and ∇f3. Note that their coefficients vanish if and onlyif µ = µc. Moreover, the predicted shear correction factor is 2 (which is in contrastto the classical Reissner-Mindlin plate model assumptions (≤ 1)).

EVOLUTION OF MICROPOLAR PLATES 23

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