THE REISSNER MINDLIN PLATE IS THE ¡-LIMIT OF COSSERAT...

THE REISSNER{MINDLIN PLATE

IS THE ¡-LIMIT OF COSSERAT ELASTICITY

PATRIZIO NEFF*

Lehrstuhl f€ur Nichtlineare Analysis und Modellierung,

Fakult€at f€ur Mathematik, Universit€at Duisburg-Essen,Campus Essen, Universit€atsstr. 2, 45141 Essen, Germany

patrizio.ne®@uni-due.de

KWON-IL HONG

University of Science, Pyongyang, Democratic Peoples Republic of Korea

JENA JEONG

�Ecole Sp�eciale des Travaux Publics du Batiment et de l'Industrie (ESTP),

28 Avenue du Pr�esident Wilson, 94234 Cachan Cedex, France

[email protected]

Received 4 July 2008

Revised 6 June 2009

Communicated by S. Müller

The linear Reissner�Mindlin membrane-bending plate model is the rigourous �-limit for zero

thickness of a linear isotropic Cosserat bulk model with symmetric curvature. For this result we

use the natural nonlinear scaling for the displacements u and the linear scaling for the in¯ni-

tesimal microrotations �A 2 soð3Þ. We also provide formal calculations for other combinations ofscalings by retrieving other plate models previously proposed in the literature by formal

asymptotic methods as corresponding �-limits. No boundary conditions on the microrotations

are prescribed.

Keywords: Micropolar; thin-plate; Reissner�Mindlin; Gamma convergence.

AMS Subject Classi¯cation: 74A35, 74A30, 74C05, 74C10

1. Introduction

The relation between three-dimensional elasticity and theories for lower-dimensional

objects such as rods, beams, membranes, plates and shells has been an outstanding

question since the very beginning of the research in elasticity. Recently there has been

substantial progress in the rigourous understanding of this relation through the use of

*Corresponding author

Mathematical Models and Methods in Applied SciencesVol. 20, No. 9 (2010) 1553�1590

#.c World Scienti¯c Publishing Company

DOI: 10.1142/S0218202510004763

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http://dx.doi.org/10.1142/S0218202510004763

variational methods, in particular �-convergence. This notion of convergence assures,

roughly speaking, that absolute minimisers of the three-dimensional theory (subject

to suitable boundary conditions and applied loads) converge to absolute minimisers

of the limiting two-dimensional theory.

Variational convergence is not the only way to proceed to obtain lower-dimensional

models. Since the dimensional reduction of a given continuum-mechanical model is

already an old subject, it has seen many \solutions". For example another way to

proceed is the so-called derivation approach, i.e. reducing a given three-dimensional

model via physically reasonable constitutive assumptions on the kinematics to a two-

dimensional model. This is opposed to either the intrinsic approach which views

the plate/shell from the onset as a two-dimensional surface and invokes concepts

from di®erential geometry or the asymptotic methods which try to establish two-

dimensional equations by formal expansion of the three-dimensional solution in

power series in terms of a small non-dimensional thickness parameter, here the aspect

ratio h > 0. The intrinsic approach is closely related to the direct approach which

takes the shell to be a two-dimensional medium with additional extrinsic directors in

the sense of a restricted Cosserat surface.10 For further information together with

more references let us refer to the introduction in Refs. 24, 26, 25, 27 and 29.

It is well known that �-convergence also needs assumptions which concern the

scaling of ¯elds and energies. A ¯rst major breakthrough in ¯nite elasticity was the

justi¯cation of a nonlinear membrane model given in Ref. 12. Later, a hierarchy of

limiting theories based on �-convergence, distinguished by di®erent scaling-expo-

nents of the energy as a function of the aspect ratio h is developed in Refs. 18, 17, 16

and 19. There the di®erent scaling exponents can be put into e®ect by the corre-

sponding scaling assumptions on the applied forces. A typical feature of �-limit

models based on classical elasticity is their de-coupling into either membrane or

bending problems, depending on the chosen regime for the energy. For example, the

Kirchho®�Love plate bending problem appears as �-limit but is restricted to inex-

tensible deformations. Similarly, one may obtain a membrane energy with no bending

term, having no resistance in compression.12 But in a given three-dimensional pro-

blem the di®erent regimes are hardly separated and one wishes to have a model

comprising both membrane and bending contributions simultaneously.

Let us restrict ourselves to linear elasticity in the following. In that case, using

�-convergence in the weak topology in H 1ð�Þ, together with a certain \linear"

scaling, Ciarlet5 arrives at justifying the membrane plate. This result can be, without

problems, extended to the strong L2 � �-limit, see the Appendix. Remarkable is that

the limit problem is not completely two-dimensional since the admissible set is the

space VKL, see De¯nition A.2.

In Ref. 4 basically the nonlinear scaling of the displacement is considered. Com-

pactness can only be assured by assuming that the scaled energy 1h2 I

]hðu]Þ (see the

Appendix (A.8)) is bounded independent of the thickness h. In that case, it is easy to

see that the limit is purely two-dimensional and the energy coincides with the one

1554 P. Ne®, K.-I. Hong & J. Jeong

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previously given. Using the linear scaling in a ¯nite strain setting is known to lead to

inconsistencies.15 A formal deduction of plate models by scaling can be found in

Ref. 23.

A very prominent model for combined membrane and bending behaviour of plates

is the Reissner�Mindlin model, see (A.1). But in Ref. 3 we read: \For plate bending,

the asymptotic approach leads to the Kirchho®�Love or biharmonic plate equation,

rather than to the Reissner�Mindlin model… To the best of our knowledge there is

no way to obtain Reissner�Mindlin type models of plate bending from the asymp-

totic approach." Similarly, Ciarlet writes8: \Open problems: ¯nding a rigourous

justi¯cation of the Reissner�Mindlin equations." With this contribution we want to

¯ll this gap.a Our main idea in this respect is to use extended continuum mechanical

models, more speci¯cally the linear isotropic Cosserat model as a starting point

for the application of �-convergence methods. The use of Cosserat elasticity as a

\parent" model for �-convergence is quite recent, it initiated presumably with

Ref. 28 immediately for the ¯nite strain case using the nonlinear scaling for defor-

mations and exact rotations ð’; �RÞ 2 R3 � SOð3Þ. The �-limit result is a kind of

Reissner�Mindlin model, but not exactly. In Refs. 2 and 1 a linear Cosserat model is

taken as a starting point and the asymptotic development (not the �-limit) is given

based on the nonlinear scaling for displacement and in¯nitesimal microrotation

ðu; �AÞ 2 R3 � soð3Þ. The result is comparable to the previous one in Ref. 28. A

precursor to that is Ref. 13 where the author also used the asymptotic expansion

method but with linear scaling for both ðu; �AÞ 2 R3 � soð3Þ. His result is comparable

to a formal deduction given much earlier in Ref. 14. Neither of these methods,

however, reproduced the classical Reissner�Mindlin model exactly.

While our method is methodologically rather standard, we want to exhibit the

di®erent limit functionals depending on the assumed choice of scaling for the dis-

placement and the in¯nitesimal microrotation. The major di®erence is in the coupling

term between displacement and microrotations after dimensional reduction. One

speci¯c choice of scaling recovers exactly the Reissner�Mindlin membrane bending

model, another choice recovers the Aganovic/Ne® model and still another choice de-

couples the problems. It is interesting to note that for the scaling we have in mind,

only the symmetric curvature case leads to a local formula for the �-limit: the

Reissner�Mindlin model. Central to our development is therefore the notion of

�-convergence, a powerful theory originally initiated by De Giorgi20 and especially

suited for a variational framework on which in turn the numerical treatment with

¯nite elements is based.

The outline of this paper is as follows: We introduce ¯rst the underlying \parent"

three-dimensional linear isotropic Cosserat model with rotational substructure

embodied by the in¯nitesimal Cosserat rotations �A 2 soð3Þ. Next we specialise the

aWhen ¯nishing this paper we have learned that a related justi¯cation of the Reissner�Mindlin modelbased on �-convergence has already been given in Refs. 30 and 31. Since the authors considered a second-

gradient \parent" linear elasticity model instead of our ¯rst-order Cosserat \parent" model we believe in

the interest of our approach.

The Reissner�Mindlin Plate is the �-Limit of Cosserat Elasticity 1555

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model to a thin domain in Sec. 3. The two basically di®erent scalings: linear and

nonlinear, are introduced in Sec. 4. Then we perform the transformation of the bulk

model in physical space to a non-dimensional thin domain and introduce the further

scaling to a ¯xed reference domain �1 with constant thickness on which the �-con-

vergence procedure is ¯nally based. In Sec. 5, the �-limit model is presented and

Sec. 6 furnishes the proofs. The notation is found at the end of the paper. In the

Appendix we recall the Reissner�Mindlin model, the Koiter model and two other

proposals based on di®erent scalings. Korn's inequality for di®erent scalings together

with a recall on the �-limit for classical linear elasticity ¯nishes this work.

2. The Linear Elastic Cosserat Model in Variational Form

This section does not contain any new results, rather it serves to accommodate the

widespread notations used in Cosserat elasticity and to introduce the problem. It is

assumed that the microrotation ¯eld is kinematically independent from the material

rotation (continuum rotation). In the micropolar continuum theory not only forces

but also moments can be transmitted across the surface of a material element. The

very concept of a micropolar theory involves, in a certain way, the substructure

response into the continuum media but it remains a phenomenological model.

For the displacement u : � � R3 7! R3 and the skew-symmetric in¯nitesimal

microrotation �A : � � R3 7! soð3Þ, we consider the two-¯eld minimisation problem

Iðu; �AÞ ¼Z�

Wmpð�"Þ þWcurvðr axl �AÞ � hf;uidx 7! min:w:r:t: ðu; �AÞ; ð2:1Þ

under the following constitutive requirements and boundary conditions

�" ¼ ru� �A; first Cosserat stretch tensor

uj� ¼ ud; essential displacement boundary conditions

Wmpð�"Þ ¼ �jjsym�"jj2 þ �cjjskew�"jj2 þ �

2tr½sym�"�2 strain energy

¼ �jjdev symrujj2 þ �cjjskewðru� �AÞjj2 þ K

2tr½symru�2;

� :¼ axl �A 2 R3; K ¼ r�;

jjcurl �jj2R 3 ¼ 4jjaxl skewr�jj 2R3 ¼ 2 jjskewr�jj2M 3�3 ;

Wcurvðr�Þ ¼ � þ �

2jjsymr�jj2 þ � � �

2jjskewr�jj2 þ �

2tr½r��2 curvature

¼ � þ �

2jjdev symr�jj2 þ � � �

2jjskewr�jj2 þ kc

2tr½r��2:

Here, f are given volume forces while ud are Dirichlet boundary conditions for the

displacement at � � @�. Surface tractions, volume couples and surface couples

can be included in the standard way. The strain energy Wmp and the curvature


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energy Wcurv are the most general isotropic quadratic forms in the in¯nitesimal non-

symmetric ¯rst Cosserat strain tensor �" ¼ ru� �A and the micropolar curvature

tensor K ¼ r axl �A ¼ r� (curvature-twist tensor). The parameters �; �½MPa� are theclassical Lam�e moduli and �; �; � are additional micropolar moduli with dimension

½Pa �m2� ¼ ½N� of a force. Here, the bulk modulus and curvature bulk modulus are

de¯ned by

K ¼ 2�þ 3�

3; kc :¼

ð� þ �Þ þ 3�

3: ð2:2Þ

The additional parameter �c � 0½MPa� in the strain energy is the Cosserat couple

modulus. For �c ¼ 0 the two ¯elds of displacement and microrotations de-couple and

one is left formally with classical linear elasticity for the displacement u. The reader

should note that even for very weak curvature requirements (� þ � > 0; � � � � 0;

kc � 0) the model is well-posed. This is a new result, proved in Ref. 21 making use of a

new coercive inequality for formally positive quadratic forms. For our dimension

reduction procedure we focus on the symmetric-curvature case with � ¼ � and

kc � 0.

3. The Cosserat Bulk Problem on a Thin Flat Domain

The basic task of any shell theory is a consistent reduction of some presumably

\exact" 3D-theory to 2D. The three-dimensional problem (2.1) de¯ned on the

physical space E3 including units of measurement will now be adapted to a plate-like

theory. Let us therefore assume that the problem is already transformed in non-

dimensional form. This means that we are given a three-dimensional (non-dimen-

sional) thin domain �h � R3

�h :¼ !� � h

2;h

2

� �; ! � R

2; ð3:1Þ

with transverse boundary @�transh ¼ !� f� h

2 ;h2g and lateral boundary @�lat

h ¼ @!�½� h

2 ;h2�, where ! is a bounded open domainb in R2 with smooth boundary @! and

h > 0 is the non-dimensional relative characteristic thickness (aspect ratio) with

h � 1. Moreover, assume we are given a deformation u and microrotation �A,

u : �h � R3 7! R

3; �A : �h � R3 7! soð3Þ; ð3:2Þ

solving the minimisation problem on the thin domain �h:

Iðu; �AÞ ¼Z�h

Wmpð�"Þ þWcurvðr axl �AÞ � hf;uidV

�Z@�trans

h [f�s�½�h2;

h2�ghN;uidS 7! min:w:r:t: ðu; �AÞ;

bFor de¯niteness, one can think of ! ¼ ½0; 1� � ½0; 1� � R2 without units of length.


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�" ¼ ru� �A; uj� h0

¼ udðx; y; zÞ;

�h0 ¼ �0 � � h

2;h

2

� �; �0 � @!; �s \ �0 ¼ ;;

�A : free on @!� � h

2;h

2

� �; Neumann-type boundary condition;

Wmpð�"Þ ¼ �jjsym�"jj2 þ �cjjskew�"jj2 þ �

2tr½�"�2;

WcurvðKÞ ¼ �bL2

cðhÞ2

�1jjsymr axl �Ajj2 þ �2jjskewr axl �Ajj2 þ �3

2tr½r axl �A�2

� �:

ð3:3ÞHere, �1; �2; �3 � 0 are non-dimensional weighting parameters and tractions N on

the transverse boundary of the plate are included. Moreover, the parameter bLc has

the form bLc ¼ LRVEc

L , where LRVEc is related to the characteristic size of the micro-

structure and the interaction strength of the microstructure. L is a characteristic

value of the in-plane elongation of the original, relatively thin domain �rel:thin: ¼½0;L½m�� ½0;L½m�� ½� h

2 L½m�; h2 L½m�� E3.c The \real" thickness of the plate is

accordingly d ¼ hL½m�.For our �-limit development we have in mind a sequence of plates with constant

physical thickness d, increased in-plane elongation L ! 1 and simultaneously

increased microstructure interaction strength LRVEc ! 1 to the e®ect that the aspect

ratio h ! 0 and as additional qualitative requirement

bLcðhÞ �1

C1

ðe�h þ hÞ � 1

C1

; ð3:4Þ

with a constant C1 > 0 independent of h. This means that we assume a non-vanishing

interaction-strength of the microstructure as h ! 0. This is essential to our devel-

opment. In order to arrive formally at the classical Reissner�Mindlin model (which

has a h3-factor in front of the bending energy) the de-scaling of the �-limit is per-

formed formally only for aspect ratios h in the regime where

bLcðhÞ �1

C1

h , C1 � bLcðhÞ � C1 �LRVE

c

L� h; ð3:5Þ

i.e. already for moderately large h. This is consistent with the observation that the

Reissner�Mindlin model is especially appropriate for moderately thick plates. In

Fig. 1 we depict the non-vanishing interaction-strength of the microstructure as

h ! 0 according to (3.4).

4. Scaling of Fields

Scaling of independent and/or dependent variables is the usual ¯rst step when per-

forming a dimensional reduction asymptotic analysis for a relatively thin domain.

cThis is an immediate consequence of the non-dimensionalisation procedure, see Ref. 27.


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The employed scaling is decisive for the application of the �-convergence framework.

The major justi¯cation of the employed scalings comes with the ¯nal convergence

result.

There are basically two scalings at hand, one which we call the nonlinear or

natural scaling and one which we refer to as the linear elasticity scaling. See Ref. 15

for an in-depth discussion of the di®erences generated by these scalings in classical

linear/nonlinear elasticity. The nonlinear or natural scaling for a vector ¯eld z :

�h � R3 7! R3 is just that one, which de¯nes z] : � 2 �1 7! R3 as the \same" ¯eld

on the domain �1 ¼ !� ½�1=2; 1=2� (see (4.4)), only the independent variables are

scaled as

�1 ¼ �1; �2 ¼ �2; �3 ¼ h�3;

z] �1; �2;1

h�3

� �:¼ zð�1; �2; �3Þ; nonlinear scaling

r�zð�1; �2; �3Þ ¼ @�1z]ð�1; �2; �3Þj@�2z

]ð�1; �2; �3Þj1

h@�3z

]ð�1; �2; �3Þ� �

¼

@�1z]1ð�Þ @�2z

]1ð�Þ

1

h@�3z

]1ð�Þ

@�1z]2ð�Þ @�2z

]2ð�Þ

1

h@�3z

]2ð�Þ

@�1z]3ð�Þ @�2z

]3ð�Þ

1

h@�3z

]3ð�Þ

0BBBBBB@

1CCCCCCA ¼: rh�z

]ð�Þ: ð4:1Þ

In linear elasticity, in contrast, it is customary8,13 to use a simultaneous scaling of

independent and dependent variables for the vector ¯eld z : �h � R3 7! R3 by

Fig. 1. Assumed interaction strength LcðhÞ of the microstructure as a function of the aspect ratio

according to (3.4). Descaling of the �-limit is performed in a range h � a according to (3.5) such thatformally the Reissner�Mindlin bending factor h3 is retrieved.


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de¯ning z[ : � 2 �1 7! R3 in the form

�1 ¼ �1; �2 ¼ �2; �3 ¼ h�3;

z [1 �1; �2;1

h�3

� �z [2 �1; �2;

1

h�3

� �z [3 �1; �2;

1

h�3

� �

0BBBBBBB@

1CCCCCCCA :¼z1ð�1; �2; �3Þz2ð�1; �2; �3Þhz3ð�1; �2; �3Þ

0B@1CA; linear scaling: ð4:2Þ

Here, the in-plane components z1; z2 of the vector ¯eld are treated di®erently from

the out-of-plane (transverse) component z3.d The corresponding relation between the

gradient is expressed as

r�zð�1; �2; �3Þ ¼

@�1z[1ð�Þ @�2z

[1ð�Þ

1

h@�3z

[1ð�Þ

@�1z[2ð�Þ @�2z

[2ð�Þ

1

h@�3z

[2ð�Þ

1

h@�1z

[3ð�Þ

1

h@�2z

[3ð�Þ

1

h2@�3z

[3ð�Þ

0BBBBBB@

1CCCCCCA ¼: brh�z

[ð�Þ: ð4:3Þ

The scaling of the dependent variable corresponds to an additional ad hoc assump-

tion on the assumed response. In our case, we deal with the displacement ¯eld u :

�h 7! R3 and the microrotation ¯eld �A : �h 7! soð3Þ. For the displacement ¯eld we

propose not to take any scaling of the dependent variables into account. Thus we do

not restrict the modelling to vertical de°ections in the order of the plate thickness.e

Rather we expect large bending terms. In the axial representation � ¼ axl �A 2 R3 of

the in¯nitesimal microrotation, the component �i; i ¼ 1; 2; 3 corresponds to the

in¯nitesimal rotation with axis ei. Thus the in-plane rotation contribution is mapped

by �3. Since the plate is getting very thin, we expect �3 to be much smaller than �1; �2,

which themselves correspond to the bending rotations (out-of-plane rotations) with

axis e1; e2. In order to re°ect this behaviour, the linear scaling suggests itself for the

microrotations, i.e. h�3ð�1; �2; �3Þ ¼ �[3ð�1; �2; 1h �3Þ.

4.1. Transformation on a ¯xed domain with unit thickness

In order to apply standard techniques of �-convergence, we transform the problem

onto a ¯xed domain with unit thickness �1, independent of the aspect ratio h > 0.

dSince we assume that the unscaled vertical de°ection z3 is bounded (otherwise the physics underlying

linear elasticity is void anyway), the linear scaling implies that the scaled vertical de°ection z [3 should be ofthe order of h, i.e. the vertical de°ection should be in the order of the thickness of the plate (instead of large

vertical de°ections…).e In Ciarlet8 it is clearly said: \Thus, counter to appearance, the linear Kirchho®�Love theory is strictly a`small displacement' theory: In order that it be valid, the transverse displacement should remain of the

order of the thickness of the plate." And in Fonseca et al.15:\ … the limit kinematics that are imposed by

the scaling are too stringent: they force the transverse limit displacement to be 0."


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We therefore de¯ne

�1 ¼ !� � 1

2;1

2

� �� R

3; ! � R2: ð4:4Þ

The scaling transformation

: � 2 �1 � R3 7! R3; ð�1; �2; �3Þ :¼ ð�1; �2;h � �3Þ;�1 : � 2 �h � R3 7! R3; �1ð�1; �2; �3Þ :¼ ð�1; �2; �3=hÞ;

ð4:5Þ

maps �1 into �h and ð�1Þ ¼ �h. We consider the correspondingly scaled function

(subsequently, nonlinearly scaled functions de¯ned on �1 will be indicated with a

superscript ], while linearly scaled ¯elds will get a superscript [) u] : �1 ! R3,

de¯ned by

uð�1; �2; �3Þ ¼ u]ð�1ð�1; �2; �3ÞÞ 8 � 2 �h; u]ð�Þ ¼ uðð�ÞÞ 8 � 2 �1;

r�uð�1; �2; �3Þ ¼ @�1u]ð�1; �2; �3Þj@�2u

]ð�1; �2; �3Þj1

h@�3u

]ð�1; �2; �3Þ� �

¼

@�1u]1ð�Þ @�2u

]1ð�Þ

1

h@�3u

]1ð�Þ

@�1u]2ð�Þ @�2u

]2ð�Þ

1

h@�3u

]2ð�Þ

@�1u]3ð�Þ @�2u

]3ð�Þ

1

h@�3u

]3ð�Þ

0BBBBBB@

1CCCCCCA ¼: rh�u

]ð�Þ: ð4:6Þ

We de¯ne a (linearly) scaled in¯nitesimal microrotation �A[ : �1 � R3 7! soð3Þ by

considering the corresponding axial vector �ð�Þ :¼ axl �Að�Þ 2 R3 and its linearly

scaled correspondence �[ð�Þ through

�ð�1; �2; �3Þ ¼ �[ð�1ð�1; �2; �3ÞÞ 8 � 2 �h; �[ð�Þ ¼ �ðð�ÞÞ 8 � 2 �1;

r��ð�1; �2; �3Þ ¼

@�1�[1ð�Þ @�2�

[1ð�Þ

1

h@�3�

[1ð�Þ

@�1�[2ð�Þ @�2�

[2ð�Þ

1

h@�3�

[2ð�Þ

1

h@�1�

[3ð�Þ

1

h@�2�

[3ð�Þ

1

h2@�3�

[3ð�Þ

0BBBBBB@

1CCCCCCA ¼: brh��

[ð�Þ: ð4:7Þ

This allows us to express the non-symmetric stretches on the unit domain �1 in

terms of the transformed non-symmetric stretches �" ]h : �1 7! glð3Þ and the scaled

second-order curvature tensor K [h : �1 7! glð3Þ

�" ]h :¼ rh

�u] � �A

]h; brh

��[ð�Þ ¼: K [

hð�Þ; ð4:8Þ


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where

rh�u

] � �A]h ¼

@�1u]1ð�Þ @�2u

]1ð�Þ

1

h@�3u

]1ð�Þ

@�1u]2ð�Þ @�2u

]2ð�Þ

1

h@�3u

]2ð�Þ

@�1u]3ð�Þ @�2u

]3ð�Þ

1

h@�3u

]3ð�Þ

0BBBBBB@

1CCCCCCA�0 � 1

h�[3 �[

2

1

h�[3 0 ��[

1

��[2 �[

1 0

0BBB@1CCCA

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}¼: �A

]h

ð4:9Þis expressed in terms of the linearly scaled �[ :¼ axl �A[. Moreover, we de¯ne non-

linearly scaled functions by setting

f ]ð�Þ :¼ fðð�ÞÞ; u ]dð�Þ ¼ udðð�ÞÞ; N ]ð�Þ :¼ Nðð�ÞÞ: ð4:10Þ

In terms of the introduced nonlinearly scaled displacement and the linearly scaled

in¯nitesimal microrotations u] : �1 � R3 7! R3; �A[ : �1 � R3 7! soð3Þ, the scaled

problem solves the following two-¯eld minimisation problem on the ¯xed domain �1:

I ];[ðu ];rh�u

]; �A[; brh� axl �A

[Þ

¼Z�2�1

½Wmpð�" ]hÞ þWcurvðK [

hÞ � hf ];u]i�det½rð�Þ�dV�

�Z@�trans

1 [f�s�½�12;

12�ghN ];u]ijjCofrð�Þ � e3jjdS�

¼ h

Z�2�1

Wmpð�" ]hÞ þWcurvðK [

hÞ � hf ];u]idV�

�Z@�trans

1

hN ];u]i1 dS�

�Z�s�½�1

2;12�hN ];u]ih dS� 7! min:w:r:t: ðu]; �A[Þ: ð4:11Þ

4.2. The rescaled variational Cosserat bulk problem

Since the energy 1h I

];[ would not be ¯nite for h ! 0 if tractions N ] on the transverse

boundary were present, our investigations are in principle restricted to the case of

N ] ¼ 0 on @�trans1 .f For conciseness we investigate the following simpli¯ed and

rescaled (N ]; f ] ¼ 0;udð�1; �2; �3Þ :¼ udð�1; �2Þ) two-¯eld minimisation problem on

�1 with respect to �-convergence (without the factor h > 0 now), i.e. we are inter-

ested in the limiting behaviour of the scaled energy per unit aspect ratio h:

I ];[h ðu];rh

�u]; �A[; brh

� axl �A[Þ ¼

Z�2�1

Wmpð�" ]hÞ þWcurvðK [

hÞdV� 7! min:w:r:t: ðu]; �A[Þ;

fThe thin plate limit h ! 0 obviously cannot support non-vanishing transverse surface loads N.


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�" ]h ¼ rh

�u] � �A

]h; u ]

j� 10

ð�Þ ¼ u ]dð�Þ ¼ udðð�ÞÞ ¼ udð�1; �2;h � �3Þ ¼ udð�1; �2; 0Þ;

�10 ¼ �0 � � 1

2;1

2

� �; �0 � @!; K [

h ¼ brh� axl �A

[ð�Þ; ð4:12Þ

�A[ : free on @!� � 1

2;1

2

� �; Neumann-type boundary condition:

Here we assume for simplicity that the bulk boundary condition ud is already inde-

pendent of the transverse variable and we restrict attention to the weakest response,

the Neumann boundary conditions on the Cosserat rotations �A[.

4.3. Recall on ¡-convergence

Let us brie°y recapitulate the notions involved by using �-convergence. For a

detailed treatment we refer to Refs. 22 and 6. The notion of �-convergence depends

strongly on the topology of the space X, which in our discussion is assumed to be

metrisable. In the following, therefore, X will always denote a metric space such that

sequential compactness and compactness coincide. Moreover, we de¯ne the extended

real numbers �R :¼ R [ f1g. We consider a sequence of energy functionals Ihj:

X 7! �R;hj ! 0.

De¯nition 4.1. (�-convergence) Let X be a metric space. We say that a sequence of

functionals Ihj: X 7! �R �-converges in X to the limit functional I0 : X 7! �R, if for all

x 2 X we have

8x 2 X : 8xhj! x : I0ðxÞ lim inf

hj!0Ihj

ðxhjÞ; ðlim inf inequalityÞ

8 x 2 X : 9 xhi! x : I0ðxÞ � lim sup

hi!0Ihiðxhi

Þ; ðrecovery sequenceÞ:

�-convergence corresponds to convergence of the energy along minimising

sequences for a family of functionals and all continuous perturbations.

5. The \Two-Field" Cosserat ¡-Limit

5.1. The spaces and admissible sets

We proceed to the investigation of the �-limit for the rescaled problem (4.12). We do

not use I ];[h directly in our investigation of �-convergence, since this would imply

working with the weak topology of H 1;2ð�1;R3Þ �H 1;2ð�1; soð3ÞÞ, which does not

give rise to a metric space. Instead, we de¯ne suitable \bulk" spaces X;X 0 andsuitable \two-dimensional" spaces X!;X

0!. To this end de¯ne the spaces

X :¼ fðu; �AÞ 2 L2ð�1;R3Þ � L2ð�1; soð3ÞÞg;

X 0 :¼ fðu; �AÞ 2 H 1;2ð�1;R3Þ �H 1;2ð�1; soð3ÞÞg;

X! :¼ fðu; �AÞ 2 L2ð!;R3Þ � L2ð!; soð3ÞÞg;X 0

! :¼ fðu; �AÞ 2 H 1;2ð!;R3Þ �H 1;2ð!; soð3ÞÞg;

ð5:1Þ


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and the admissible sets

A0 :¼ fðu; �AÞ 2 H 1;2ð�1;R3Þ �H 1;2ð�1; soð3ÞÞ; uj

� 10

ð�Þ ¼ u ]dð�Þg;

A0! :¼ fðu; �AÞ 2 H 1;2ð!;R3Þ �H 1;2ð!; soð3ÞÞ; uj�0ð�1; �2Þ ¼ u ]

dð�1; �2; 0Þg:ð5:2Þ

We note the compact embedding X 0 � X and the natural inclusions X! � X and

X 0! � X 0. Now we extend the rescaled energies to the space X through rede¯ning

I ];[h ðu];rh

�u]; �A[; brh

� axl �A[Þ

¼ I ];[h ðu ];rh

�u]; �A[; brh

� axl �A[Þ if ðu ]; �A[Þ 2 A0

þ1 else in X;

(ð5:3Þ

by abuse of notation. This is a classical trick used in applications of �-convergence. It

has the virtue of incorporating the boundary conditions already in the energy func-

tional. In the following, �-convergence results will be shown with respect to the

encompassing metric space X.

5.2. The ¡-limit variational problem

Our main result is the �-limit for symmetric curvature �2 ¼ 0 and strictly positive

curvature bulk modulus kc > 0.

Theorem 5.1. (�-limit for kc > 0 and �2 ¼ 0) For strictly positive curvature bulk

modulus kc > 0 and symmetric curvature �2 ¼ 0 the �-limit for problem (4.12) in the

setting of (5.3) together with the scaling assumption on the interaction strength (3.4)

is given by the limit energy functional I ];[0 : X 7! �R,

I ];[0 ðv; �AÞ :¼

Z!

W hommp ðrv; axl �AÞ þW hom

curv ðr axl �AÞ � hf; vid! ðv; �AÞ 2 A0!

þ1 else in X;

8<:with W hom

mp and W homcurv de¯ned below.

The proof of this statement will be given in Sec. 6. The limit functions are inde-

pendent of the transverse variable �3. This �-limit determines in fact a purely two-

dimensional minimisation problem for the de°ection of the midsurface v : ! �R2 7! R3 and the in¯nitesimal microrotation of the plate (shell) �A : ! � R2 7! soð3Þon ! under the boundary conditions of place for the midsurface de°ection v on the

Dirichlet part of the lateral boundary �0 � @!,

vj�0 ¼ udðx; y; 0Þ; simply supported ðfixed; weldedÞ: ð5:4Þ


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The boundary conditions for the microrotations �A are automatically determined in

the variational process. The dimensionally homogenised local density is

W hommp ðrv; �Þ :¼ � jjsymr�1;�2ðv1; v2Þjj2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

homogenised shear-stretch energy

þ 2��c

�þ �c

r�1;�2v3 ��2�1

� �� 2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}homogenised transverse shear energy

þ ��

2�þ �tr½r�1;�2ðv1; v2Þ�2|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

homogenised elongational stretch energy

:

The homogenised curvature density is given by

W homcurv ðr�Þ :¼ �

bL2cðhÞ2

�1jjsymr�1;�2ð�1; �2Þjj2 þ�1�3

2�1 þ �3

tr½r�1;�2ð�1; �2Þ�2� �

:

It is clear that the limit functional I ];[0 is weakly lower semicontinuous in the topology

of X 0 ¼ H 1;2ð�;R3Þ �H 1;2ð�; soð3ÞÞ by simple convexity arguments. Note the ap-

pearance of the harmonic mean H,

1

2H �;

�

2

� �¼ ��

2�þ �; Hð�; �cÞ ¼ 2�

�c

�þ �c

;1

2H �1;

�3

2

� �¼ �1�3

2�1 þ �3

:

5.3. Descaled ¡-limit : Reissner�Mindlin membrane-bending model

After descaling, the �-limit minimisation problem turns intoZ!

h �jjsymrðv1; v2Þjj2 þ2��c

�þ �c

rv3 ��2�1

� �� 2 þ ��

2�þ �tr½rðv1; v2Þ�2

� �

þ �bL2

cðhÞ2

h �1jjsymrð�1; �2Þjj2 þ�1�3

2�1 þ �3

tr½rð�1; �2Þ�2� �

� hf; vid! 7! min:w:r:t: ðv; �Þ; vj�0 ¼ udðx; y; 0Þ: ð5:5Þ

Choosing as range for descaling only aspect rations h where C1bLcðhÞ � h (3.5) and

abbreviating ¼ 4�c

�þ�cyields the classical Reissner�Mindlin model (A.1) with

appropriate re-de¯nitions of constants.g

6. Proof for Positive Curvature Bulk Modulus kc > 0

We continue by proving Theorem 5.1, i.e. the claim on the form of the �-limit

for strictly positive curvature bulk modulus by considering micropolar curvature

gThe coupling term 2��c

�þ�crv3 � ��2

�1

� �� 2 is not in error as compared to (A.1). In fact, we obtain the

classical Reissner�Mindlin model provided the curvature of the bulk problem is re-de¯ned through apermutation by

caxlð �AÞ ¼ ð�axlð �AÞ2; axlð �AÞ1; axlð �AÞ3ÞT;which is physically equivalent.


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energies having the form

Wcurvðr axl �AÞ ¼ �bL2

c

2jjdev symr axl �Ajj2 þ kc

2tr½r axl �A�2

� �ð6:1Þ

for kc > 0. Note, however, that the Cosserat bulk problem is well-posed for kc ¼ 0, see

Ref. 21. The proof of �-convergence is subsequently split into several steps.

6.1. Compactness

Theorem 6.1. (Compactness of I ];[hj) Consider a sequence ðu ]

hj; �A

[hjÞ 2 A0 � X

such that jj �A [hj jjL2ð�1;soð3ÞÞ K1 and I ];[

hjðu ]

hj; �A

[hjÞ K2, with constants K1;K2

independent of hj > 0. Then, for positive curvature bulk modulus kc > 0 it holds

jju ]hjjjH 1;2ð�1;R 3Þ K3; jj �A [

hjjjH 1;2ð�1;soð3ÞÞ K4; ð6:2Þ

with constants K3;K4 independent of hj > 0. The sequence ðu ]hj; �A

[hjÞ 2 A0 admits

weakly convergent subsequences (not relabelled) ðu ]hj; �A

[hjÞ * ðu ]

0;�A[0Þ 2 X. In

addition, the weak limit

ðu ]0;

�A[0Þ 2 A0

! ð6:3Þis independent of the transverse variable �3 and ðaxl �A [

0Þ3 ¼ 0 (no in-plane drillrotation).

Proof. Along the sequence ðu ]hj; �A

[hjÞ 2 A0 � X we have

1 > K2 > I ];[hjðu ]

hj; �A

[hjÞ ¼

Z�1

Wmpð�" ]hjÞ þWcurvðK [

hjÞdV� �

Z�1

Wmpð�" ]hjÞdV�

�Z�1

min �c; �;K

2

� �rhj

� u]hj� �A

]hj

�� 2dV�: ð6:4Þ

But with (4.9) we obtain

rhj� u

]hj� �A

]hj

¼

@�1u]1ð�Þ @�2u

]1ð�Þ

1

h@�3u

]1ð�Þ

@�1u]2ð�Þ @�2u

]2ð�Þ

1

h@�3u

]2ð�Þ

@�1u]3ð�Þ @�2u

]3ð�Þ

1

h@�3u

]3ð�Þ

0BBBBBB@

1CCCCCCA

�

0 � 1

hj

�[hj;3 �[

hj;2

1

hj

�[hj;3 0 ��[

hj;1

��[hj;2

�[hj;1

0

0BBBBB@

1CCCCCA;


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rhj� u

]hj� �A

]hj

�� 2 ¼ sym@�1u

]1ð�Þ @�2u

]1ð�Þ

@�1u]2ð�Þ @�2u

]2ð�Þ

!��2

þ 1

h 2j

@�3u]3ð�Þ

�� 2

þ skew@�1u

]1ð�Þ @�2u

]1ð�Þ

@�1u]2ð�Þ @�2u

]2ð�Þ

!�

0 � 1

hj

�[hj;3

1

hj

�[hj;3 0

0BB@1CCA

��2

þ @�1u]3ð�Þ

@�2u]3ð�Þ

!� ��[

hj;2

�[hj;1

!��2

þ1

h@�3u

]1ð�Þ

1

h@�3u

]2ð�Þ

0BB@1CCA� �[

hj;2

��[hj;1

!��2

: ð6:5Þ

Combining (6.4) with (6.5), neglecting the skew-symmetric contribution and using

the assumption that �[hj¼ axl �A

[hj is bounded in L2ð�1;R

3Þ independent of hj we

obtain easily an hj-independent bound for

1 > K5 >

Z�1

sym@�1u

]1ð�Þ @�2u

]1ð�Þ

@�1u]2ð�Þ @�2u

]2ð�Þ

!��2

þ 1

h2j

@�3u]3ð�Þ

�� 2

þ @�1u]3ð�Þ

@�2u]3ð�Þ

!��2

þ1

h@�3u

]1ð�Þ

1

h@�3u

]2ð�Þ

0BB@1CCA

��2

dV�

¼Z�1

sym@�1u

]1ð�Þ @�2u

]1ð�Þ

@�1u]2ð�Þ @�2u

]2ð�Þ

!��2

þ 1

h2j

@�3u]3ð�Þ

�� 2

þ @�1u]3ð�Þ

@�2u]3ð�Þ

!��2

þ 1

h 2j

@�3u]1ð�Þ

@�3u]2ð�Þ

!��2

dV�

�Z�1

sym@�1u

]1ð�Þ @�2u

]1ð�Þ

@�1u]2ð�Þ @�2u

]2ð�Þ

!��2

þ @�3u]3ð�Þ

�� 2

þ @�1u]3ð�Þ

@�2u]3ð�Þ

!��2

þ @�3u]1ð�Þ

@�3u]2ð�Þ

!��2

dV�

� 1

Z�1

sym

@�1u]1ð�Þ @�2u

]1ð�Þ @�3u

]1ð�Þ

@�1u]2ð�Þ @�2u

]2ð�Þ @�3u

]2ð�Þ

@�1u]3ð�Þ @�2u

]3ð�Þ @�3u

]3ð�Þ

0BB@1CCA

��2

dV�; ð6:6Þ


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(for hj > 0 small enough). Korn's ¯rst inequality and the Dirichlet-boundary

condition on u ]hjshow the hj-independent H

1-bound on u ]hj. Thus we may extract a

weakly convergent subsequence (not relabelled) u ]hj* u ]

0 and the weak limit must be

independent of �3 on account of (6.6)2.

Next, combine (6.4) and (6.5) and the boundedness of the in-plane skew-sym-

metric de°ection to see that the boundedness of

Z�1

skew@�1u

]1ð�Þ @�2u

]1ð�Þ

@�1u]2ð�Þ @�2u

]2ð�Þ

!�

0 � 1

hj

�[hj;3

1

hj

�[hj;3 0

0BB@1CCA

��2

dV� ð6:7Þ

implies the boundedness of

Z�1

0 � 1

hj

�[hj;3

1

hj

�[hj;3 0

0BB@1CCA

��2

dV� ð6:8Þ

showing that jj�[hj;3

jjL 2ð�1;RÞ ! 0 for hj ! 0. For the (similar) treatment of the cur-

vature energy we note that

1 > K2 > I ];[hjðu ]

hj; �A

[hjÞ ¼

Z�1

Wmpð�" ]hjÞ þWcurvðK [

hjÞdV� �

Z�1

WcurvðK [hjÞdV�

�Z�1

�L2

c

2min 1;

3kc2

� �symbrh

��[hjð�Þ

�� 2dV�: ð6:9Þ

Now use Theorem A.1 to get the hj-independent H1-bound on �[

hjtogether with the

existence of a weakly convergent subsequence �[hj* �[

0 and the claim that the weak

limit is independent of the transverse variable �3 and �[0;3 ¼ 0.

Remark 6.2. In linear Cosserat models Korn's ¯rst inequality is usually not needed

in showing coercivity.

6.2. Lower bound �� the lim inf condition

If I ];[0 is the �-limit of the sequence of energy functionals I ];[

hjthen we must have

(lim inf-inequality) that

I ];[0 ðu ]

0;�A[0Þ lim inf

hjI ];[hjðu ]

hj; �A

[hjÞ; ð6:10Þ

whenever

u ]hj! u ]

0 in L2ð�1;R3Þ; �A

[hj! �A

[0 in L2ð�1; soð3ÞÞ; ð6:11Þ


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for arbitrary ðu ]0;

�A[0Þ 2 X. Observe that we can restrict attention to sequences ðu ]

hj;

�A[hjÞ 2 X such that I ];[

hjðu ]

hj; �A

[hjÞ < 1 since otherwise the statement is true anyway.

Sequences with I ];[hjðu ]

hj; �A

[hjÞ < 1 are uniformly bounded in the space X 0, as seen

previously. This implies weak convergence of a subsequence in X 0. But we already

know that the original sequences converge strongly in X to the limit ðu ]0;

�A[0Þ 2 X.

Hence we must have as well weak convergence to u ]0 2 H 1;2ð!;R3Þ and �A

[0 2

H 1;2ð!; soð3ÞÞ, independent of the transverse variable �3.

In a ¯rst step we consider now the local energy contribution: along sequences

ðu ]hj; �A

[hjÞ 2 X with ¯nite energy I ];[

hj, the third column of rhj

� u]hj

remains bounded

but otherwise indetermined. Therefore, a really trivial lower bound is obtained by

minimising the e®ect of the derivative in this direction in the local energy Wmp. To

continue our development, we need some calculations: for smooth v : ! � R2 7!R3;�A : ! � R2 7! soð3Þ de¯ne the vector ðb�; ~p �

1Þ 2 R4 formally through

W hommp ðrv; �Þ ¼ Wmp ðrvjb�Þ �

0 �~p �1 �2

~p �1 0 ��1

��2 �1 0

0@ 1A0@ 1A

:¼ infb2R 3;~p12R

Wmp ðrvjbÞ �0 �~p1 �2~p1 0 ��1��2 �1 0

0@ 1A0@ 1A: ð6:12Þ

The vector ðb�; ~p �1Þ, which realises this in¯mum, can be explicitly determined. The

calculation is lengthy but otherwise straightforward. We obtain

b�1b�2b�3~p �1

0BBB@1CCCA ¼

�c � �

�þ �c

@�1v3 þ2�c

�þ �c

�2

�c � �

�þ �c

@�2v3 �2�c

�þ �c

�1

� �

2�þ �ð@�1v1 þ @�2v2Þ

@�1v2 � @�2v12

0BBBBBBBBBBB@

1CCCCCCCCCCCA: ð6:13Þ

Reinserting the result in the energy yields for W hommp ðrv; axl �AÞ

W hommp ðrv; �Þ :¼ � sym r�1;�2ðv1; v2Þ �

0 ��3�3 0

� �� 2þ 2�

�c

�þ �c

r�1;�2v3 ��2�1

� �� 2þ ��

2�þ �tr sym r�1;�2ðv1; v2Þ �

0 ��3�3 0

� �� 2

:

Note that �3 cancels after all (left for clarity to show the coupling).


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Consider next

rhj� u

]hj� �A

]hj

¼

@�1u]1ð�Þ @�2u

]1ð�Þ

1

h@�3u

]1ð�Þ

@�1u]2ð�Þ @�2u

]2ð�Þ

1

h@�3u

]2ð�Þ

@�1u]3ð�Þ @�2u

]3ð�Þ

1

h@�3u

]3ð�Þ

0BBBBBB@

1CCCCCCA

�

0 � 1

hj

�[hj;3 �[

hj;2

1

hj

�[hj;3 0 ��[

hj;1

��[hj;2

�[hj;1

0

0BBBBB@

1CCCCCA;

where �[hj:¼ axl �A

[hj. Along the sequence ðu ]

hj; �A

[hjÞ we have by construction through

the local minimisation step

Wmpðrhj� u

]hj� �A

]hjÞ � W hom

mp ðru ]hj; axl �A

]hjÞ ¼ W hom

mp ðru ]hj; axl �A

[hjÞ; ð6:14Þ

and the last equality holds since the third component of axl �A]hj

is not seen by the

homogenised energy. Hence, integrating and taking the lim inf also

lim infhj

Z�1

Wmpðrhj� u

]hj� �A

]hjÞdV� � lim inf

hj

Z�1

W hommp ðru ]

hj; axl �A

[hjÞdV�: ð6:15Þ

Now we use weak convergence of ðu ]hj; �A

[hjÞ * ðu ]

0;�A[0Þ, together with the convexity

w.r.t. ðrv; axl �AÞ of R�1W hom

mp ðrv; axl �AÞdV� to get lower semicontinuity of the right-

hand side in (6.15) and to obtain altogether

lim infhj

Z�1

Wmpðrhj� u

]hj� �A

]hjÞdV� �

Z�1

W hommp ðru ]

0; axl�A[0ÞdV�: ð6:16Þ

Consider next the curvature energy along the sequence �[hjð�Þ ¼ axl �A

[hjð�Þ with

Wcurvðbrh��

[ð�ÞÞ ¼ Wcurv

@�1�[1ð�Þ @�2�

[1ð�Þ

1

h@�3�

[1ð�Þ

@�1�[2ð�Þ @�2�

[2ð�Þ

1

h@�3�

[2ð�Þ

1

h@�1�

[3ð�Þ

1

h@�2�

[3ð�Þ

1

h2@�3�

[3ð�Þ

0BBBBBB@

1CCCCCCA

0BBBBBB@

1CCCCCCA: ð6:17Þ

This motivates one to get a trivial lower bound by de¯ning

W homcurv ðr�Þ :¼ inf

~p2;~p3;~p4;~p5;~p62RWcurv

@�1�1 @�2�1 ~p2@�1�2 @�2�2 ~p3~p6 ~p5 ~p4

0@ 1A0@ 1A: ð6:18Þ


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The in¯mizing values are obtained as

~p �2 ¼ 0; ~p �

3 ¼ 0; ~p �4 ¼ � �3

2�1 þ �3

ð@�1�1 þ @�2�2Þ; ~p �5 ¼ 0; ~p �

6 ¼ 0; ð6:19Þ

such that the homogenised reduced curvature density is given by

W homcurv ðr�Þ :¼ �

bL2c

2�1jjsymr�1;�2ð�1; �2Þjj2 þ

�1�3

2�1 þ �3

tr½r�1;�2ð�1; �2Þ�2� �

:

By construction we have along the sequence �[hj

Wcurvðbrh��

[hjÞ � W hom

curv ðr�[hjÞ: ð6:20Þ

Integrating the last inequality, taking the lim inf on both sides and using that W homcurv

is convex (quadratic) in its argument, together with weak convergence of the two in-

plane components of the curvature tensor, i.e.

r�1;�2ð�[hj;1; �

[hj;2Þ * r�1;�2ð�[

0;1; �[0;2Þ ¼ r�1;�2 axl

�A[0 2 L2ð�1; glð3ÞÞ; ð6:21Þ

(see Theorem A.1) we obtain

lim infhj

Z�1

Wcurvðbrh� axl �A

[hjÞdV� ¼ lim inf

hj

Z�1

Wcurvðbrh��

[hjÞdV�

� lim infhj

Z�1

W homcurv ðr�[

hjÞdV�

�Z�1

W homcurv ðr axl �A

[0ÞdV�: ð6:22Þ

Then, because Wcurv;Wmp � 0,

lim infhj

Z�1

Wmpðrh�u

]hj� �A

]hjÞ þWcurvðbrh

� axl �A[hjÞdV�

� lim infhj

Z�1

Wmpðrhj� u

]hj� �A

]hjÞdV� þ lim inf

hj

Z�1

Wcurvðbrh� axl �A

[hjÞdV�

�Z�1

W hommp ðru ]

0; axl�A[0ÞdV� þ

Z�1

W homcurv ðr axl �A

[0ÞdV�; ð6:23Þ

where we used (6.16) and (6.22). Now we use that u ]0;

�A[0 are both independent of the

transverse variable �3 to obtain altogether the desired lim inf inequality

I ];[0 ðu ]

0;�A[0Þ lim inf

hj

I ];[hjðu ]

hj; �A

[hjÞ ð6:24Þ

for

I ];[0 ðu0; �A

[0Þ :¼

Z 12

� 12

Z!

W hommp ðru0; axl �A

[0Þ þW hom

curv ðr axl �A[0ÞdV�

¼Z!

W hommp ðru0; �A

[0Þ þW hom

curv ðr axl �A[0Þ d!:


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6.3. Global/local minimisation

Because of the coupling of the ¯elds together with the scaling of the third component

of the microrotation, we have to compute, however, a more complicated minimisation

problem. Looking simultaneously at scaled stretch and scaled curvature we are led to

infb2R 3;p12R4

Z�1

Wmp ðrvjbÞ �0 �p1 �2p1 0 ��1��2 �1 0

0@ 1A0@ 1A264

þWcurv

@�1�1 @�2�1 p2@�1�2 @�2�2 p3@�1p1 @�2p1 p4

0B@1CA

0B@1CA375dV�

¼Z�1

W hommp ðrv; �Þ þW hom

curv ðr�ÞdV�: ð6:25Þ

The minimisation problem is in principle a global PDE-problem, since r�1;�2p1appears in the curvature energy. However, the split minimisation (6.25) is the correct

result in precisely our case where the curvature energy depends only on the sym-

metric part.

Let us use the precise form of the energy to see what is going on. We write

Wmp ðrvjbÞ �0 �p1 �2p1 0 ��1��2 �1 0

0@ 1A0@ 1AþWcurv

@�1�1 @�2�1 p2@�1�2 @�2�2 p3@�1p1 @�2p1 p4

0B@1CA

0B@1CA

264375

¼ �jjsymðrvjbÞjj2 þ �c skewðrvjbÞ �0 �p1 �2p1 0 ��1��2 �1 0

0@ 1A��2

þ �

2tr½ðrvjbÞ�2

þ �1 sym

@�1�1 @�2�1 p2@�1�2 @�2�2 p3@�1p1 @�2p1 p4

0B@1CA

��2

þ �2 skew

@�1�1 @�2�1 p2@�1�2 @�2�2 p3@�1p1 @�2p1 p4

0B@1CA

��2

þ �3

2ð@�1�1 þ @�2�2 þ p4Þ2

¼ � sym

@�1v1 @�2v1 b1@�1v2 @�2v2 b2@�1v3 @�2v3 b3

0B@1CA

��2

þ �c skewðrvjbÞ �0 �p1 �2p1 0 ��1��2 �1 0

0@ 1A��2

þ �

2ð@�1v1 þ @�2v2 þ b3Þ2

þ �1 sym

@�1�1 @�2�1 0

@�1�2 @�2�2 0

0 0 p4

0@ 1A��2

þ �1

2ðp2 þ @�1p1Þ2 þ ðp3 þ @�2p1Þ2


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þ �2 skew

@�1�1 @�2�1 p2@�1�2 @�2�2 p3@�1p1 @�2p1 p4

0B@1CA

��2

þ �3

2ð@�1�1 þ @�2�2 þ p4Þ2

¼ � sym

@�1v1 @�2v1 0

@�1v2 @�2v2 0

0 0 b3

0@ 1A��2

þ �

2ðb1 þ @�1v3Þ2 þ ðb2 þ @�2v3Þ2

þ �c skew@�1v1 @�2v1@�1v2 @�2v2

� �� 0 �p1

p1 0

� �� 2

þ �c skew

0 0 b10 0 b2

@�1v3 @�2v3 0

0B@1CA�

0 0 �20 0 ��1

��2 �1 0

0@ 1A��

��2

þ �

2ð@�1v1 þ @�2v2 þ b3Þ2

þ �1 sym

@�1�1 @�2�1 0

@�1�2 @�2�2 0

0 0 p4

0@ 1A��2

þ �1

2ðp2 þ @�1p1Þ2 þ ðp3 þ @�2p1Þ2

þ �2 skew@�1�1 @�2�1@�1�2 @�2�2

� �� 2 þ �2

2ðp2 � @�1p1Þ2 þ ðp3 � @�2p1Þ2

þ �3

2ð@�1�1 þ @�2�2 þ p4Þ2: ð6:26Þ

Grouping the di®erent expressions together we see that for the symmetric case �2 ¼ 0

the vector ðb�; p�Þ 2 R7, which realises the in¯mum, can be explicitly determined.

The calculation is lengthy but otherwise straightforward. We obtain

b�1b�2b�3p�1

p�2

p�3

p�4

0BBBBBBBBBB@

1CCCCCCCCCCA¼

�c � �

�þ �c

@�1v3 þ2�c

�þ �c

�2

�c � �

�þ �c

@�2 v3 �2�c

�þ �c

�1

� �

2�þ �ð@�1v1 þ @�2v2Þ

@�1v2 � @�2v12

�@�1p�1

�@�2p�1

� �3

2�1 þ �3

ð@�1�1 þ @�2�2Þ

0BBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCA

: ð6:27Þ


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Reinserting the result in the energy yields the claim in (6.25). The importance of this

calculation (while not changing the lower bound trivial limit energy), rests with the

determination of the minimising values (6.27) which are needed in the following

reconstruction procedure.h

6.4. Upper bound �� the recovery sequence

Now we show that the lower bound will actually be reached. A su±cient requirement

for the recovery sequence is that

8 ðu0; �A[0Þ 2 X ¼ L2ð�1;R

3Þ � L2ð�1; soð3ÞÞ9u ]

hj! u0 in L2ð�1;R

3Þ; �A[hj! �A

[0 in L2ð�1; soð3ÞÞ : ð6:28Þ

lim suphj

I ];[hjðu ]

hj; �A

[hjÞ I ];[

0 ðu0; �A[0Þ:

Observe that this is now only a condition if I ];[0 ðu0; �A

[0Þ < 1. In this case the uniform

coercivity of I ];[hjðu ]

hj; �A

[hjÞ over X 0 ¼ H 1;2ð�1;R

3Þ �H 1;2ð�1; soð3ÞÞ implies that we

can restrict attention to sequences ðu ]hj; �A

[hjÞ converging weakly to some ðu0; �A

[0Þ 2

H 1;2ð!;R3Þ �H 1;2ð!; soð3ÞÞ ¼ X 0!, de¯ned over the two-dimensional domain ! only.

Note, however, that ¯nally it is strong convergence in X which matters.

Since

u ]hjð�1; �2; �3Þ ¼ u ]

hjð�1; �2; 0Þ þ @�3u

]hjð�1; �2; 0Þ�3 þ � � � ð6:29Þ

and b�ð�1; �2Þ replaces the term 1hj@�3u

]hjð�1; �2; �3Þ, the natural candidate for the

recovery sequence for the bulk displacement is given by the \reconstruction"

u ]hjð�1; �2; �3Þ :¼ u0ð�1; �2Þ þ hj�3b

�ð�1; �2Þ ¼ u0ð�1; �2Þ þ hj�3b�ð�1; �2Þ

¼ u0ð�1; �2Þ þ hj�3

ð�c � �Þ@�1u0;3 þ 2�c�[0;2

�þ �c

ð�c � �Þ@�2u0;3 � 2�c�[0;1

�þ �c

��ð@�1u0;1 þ @�2u0;2Þ2�þ �

0BBBBBBBB@

1CCCCCCCCA; ð6:30Þ

where we have used the de¯nition of b� given in (6.27). Observe that b� 2 L2ð!;R3Þ.Convergence of u ]

hjin L2ð�1;R

3Þ to the limit u0 as hj ! 0 is obvious.

hIf the curvature energy depends also on the non-symmetric part of the curvature tensor, i.e. if �2 > 0,

then the minimisation step is truly global and no simple solution can be provided. Moreover, the resulting

limit energy would depend on imposed boundary conditions for �. But any useful e®ective two-dimensionalmodel should be boundary condition independent! Thus we get a strong motivation to use only Cosserat

curvature energies depending only on the symmetric part of the curvature tensor in the Cosserat bulk

model.


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The reconstruction for the in¯nitesimal rotation �A[0 is only slightly more com-

plicated. In terms of the axial representation we write

�[hjð�1; �2; �3Þ ¼

�[0;1ð�1; �2Þ

�[0;2ð�1; �2Þ

hjp�1ð�1; �2Þ

0B@1CAþ

hj�3p�2ð�1; �2Þ

hj�3p�3ð�1; �2Þ

h2j�3p

�4ð�1; �2Þ

0B@1CA

¼�[0;1ð�1; �2Þ

�[0;2ð�1; �2Þ

hjp�1ð�1; �2Þ

0B@1CAþ

�hj�3@�1p�1ð�1; �2Þ

�hj�3@�2p�1ð�1; �2Þ

h2j�3p

�4ð�1; �2Þ

0B@1CA

¼�[0;1ð�1; �2Þ

�[0;2ð�1; �2Þ

hj

@�1u0;2 � @�2u0;1

2

0BBB@1CCCAþ

�hj�3@�1

@�1u0;2 � @�2u0;1

2

�hj�3@�2

@�1u0;2 � @�2u0;1

2

�h 2j�3

�3ð@�1�[0;1 þ @�2�

[0;2Þ

2�1 þ �3

0BBBBBBB@

1CCCCCCCA;

ð6:31Þ

where we have used (6.27). Again it is clear that �[hj! �[

0 2 L2ð�1;R3Þ as hj ! 0.

Both reconstructions are completely given in terms of the two-dimensional functions

ðu0; �[0Þ. Since neither b� nor p� need be di®erentiable, we have to consider slightly

modi¯ed recovery sequences, however. With ¯xed " > 0 choose b" 2 H 1;2ð!;R3Þ suchthat jjb" � b�jjL 2ð!;R 3Þ < " and similarly for p� choose p�

" 2 H 2;2ð!;R4Þ such that

jjp�" � p�jjL 2ð!;R 4Þ < ". This allows us to present ¯nally our recovery sequence

u ]hj;"

ð�1; �2; �3Þ :¼ u0ð�1; �2Þ þ hj �3 b"ð�1; �2Þ;

� [hj;"

ð�1; �2; �3Þ :¼�[0;1ð�1; �2Þ

�[0;2ð�1; �2Þ

hjp�1;"ð�1; �2Þ

0B@1CAþ

�hj�3@�1p�1;"ð�1; �2Þ

�hj�3@�2p�1;"ð�1; �2Þ

h2j�3p

�4;"ð�1; �2Þ

0B@1CA; ð6:32Þ

and correspondingly

�A[hj;" :¼

0 �hj p�1;"ð�1; �2Þ �[

0;2ð�1; �2Þhj p

�1;"ð�1; �2Þ 0 ��[

0;1ð�1; �2Þ��[

0;2ð�1; �2Þ �[0;1ð�1; �2Þ 0

0BB@1CCA

þ0 �h 2

j�3p�4;"ð�1; �2Þ �hj�3@�2p

�1;"ð�1; �2Þ

h 2j�3p

�4;"ð�1; �2Þ 0 hj�3@�1p

�1;"ð�1; �2Þ

hj�3@�2p�1;"ð�1; �2Þ �hj�3@�1p

�1;"ð�1; �2Þ 0

0B@1CA;


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�A[;[hj;"

:¼0 �p�

1;"ð�1; �2Þ �[0;2ð�1; �2Þ

p�1;"ð�1; �2Þ 0 ��[

0;1ð�1; �2Þ��[

0;2ð�1; �2Þ �[0;1ð�1; �2Þ 0

0BB@1CCA

þ0 �hj�3p

�4;"ð�1; �2Þ �hj�3@�2p

�1;"ð�1; �2Þ

hj�3p�4;"ð�1; �2Þ 0 hj�3@�1p

�1;"ð�1; �2Þ

hj�3@�2p�1;"ð�1; �2Þ �hj�3@�1p

�1;"ð�1; �2Þ 0

0B@1CA;

�A[;[0;" :¼

0 �p�1;"ð�1; �2Þ �[

0;2ð�1; �2Þp�1;"ð�1; �2Þ 0 ��[

0;1ð�1; �2Þ��[

0;2ð�1; �2Þ �[0;1ð�1; �2Þ 0

0BB@1CCA;

�A[;[0 :¼

0 �p�1ð�1; �2Þ �[

0;2ð�1; �2Þp�1ð�1; �2Þ 0 ��[

0;1ð�1; �2Þ��[

0;2ð�1; �2Þ �[0;1ð�1; �2Þ 0

0BB@1CCA;

�A[0 :¼

0 0 �[0;2ð�1; �2Þ

0 0 ��[0;1ð�1; �2Þ

��[0;2ð�1; �2Þ �[

0;1ð�1; �2Þ 0

0BB@1CCA ¼ antið�[

0Þ: ð6:33Þ

The de¯nition (6.32) implies

ru ]hj;"

ð�1; �2; �3Þ ¼ ðru0ð�1; �2Þjhjb"ð�1; �2ÞÞ þ hj�3 ðrb"ð�1; �2Þj0Þ;

r�[hj;"

ð�1; �2; �3Þ

¼@�1�

[0;1ð�1; �2Þ @�2�

[0;1ð�1; �2Þ 0

@�1�[0;2ð�1; �2Þ @�2�

[0;2ð�1; �2Þ 0

hj@�1p�1;"ð�1; �2Þ hj@�2p

�1;"ð�1; �2Þ 0

0BB@1CCA

þ�hj�3@�1@�1p

�1;"ð�1; �2Þ �hj�3@�2@�1p

�1;"ð�1; �2Þ �hj@�1p

�1;"ð�1; �2Þ

�hj�3@�1@�2p�1;"ð�1; �2Þ �hj�3@�2@�2p

�1;"ð�1; �2Þ �hj@�2p

�1;"ð�1; �2Þ

h 2j�3@�1p

�4;"ð�1; �2Þ h 2

j�3@�2p�4;"ð�1; �2Þ h 2

jp�4;"ð�1; �2Þ

0BB@1CCA:

ð6:34Þ


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Note that by appropriately choosing hj and " > 0 we can arrange that strong con-

vergence of (6.34) to the correct limit still exists. Now abbreviate further

~�E"hj :¼ ½ðru0ð�1; �2Þjb"ð�1; �2ÞÞ þ hj �3 ðrb"ð�1; �2Þj0Þ� � �A

[;[hj;"

2 glð3Þ;~�E"0 :¼ ðru0ð�1; �2Þjb"ð�1; �2ÞÞ � �A

[;[0;" 2 glð3Þ;

~�E :¼ ðru0ð�1; �2Þjb�ð�1; �2ÞÞ � �A[;[0 2 glð3Þ;

~K [hj;"

:¼@�1�

[0;1ð�1; �2Þ @�2�

[0;1ð�1; �2Þ 0

@�1�[0;2ð�1; �2Þ @�2�

[0;2ð�1; �2Þ 0

@�1p�1;"ð�1; �2Þ @�2p

�1;"ð�1; �2Þ 0

0B@1CA

þ�hj�3@�1@�1p

�1;"ð�1; �2Þ �hj�3@�2@�1p

�1;"ð�1; �2Þ �@�1p

�1;"ð�1; �2Þ

�hj�3@�1@�2p�1;"ð�1; �2Þ �hj�3@�2@�2p

�1;"ð�1; �2Þ �@�2p

�1;"ð�1; �2Þ

hj�3@�1p�4;"ð�1; �2Þ hj�3@�2p

�4;"ð�1; �2Þ p�

4;"ð�1; �2Þ

0B@1CA;

~K [0;" :¼

@�1�[0;1ð�1; �2Þ @�2�

[0;1ð�1; �2Þ �@�1p

�1;"ð�1; �2Þ

@�1�[0;2ð�1; �2Þ @�2�

[0;2ð�1; �2Þ �@�2p

�1;"ð�1; �2Þ

@�1p�1;"ð�1; �2Þ @�2p

�1;"ð�1; �2Þ p�

4;"ð�1; �2Þ

0B@1CA;

~K [0 :¼

@�1�[0;1ð�1; �2Þ @�2�

[0;1ð�1; �2Þ �@�1p

�1ð�1; �2Þ

@�1�[0;2ð�1; �2Þ @�2�

[0;2ð�1; �2Þ �@�2p

�1ð�1; �2Þ

@�1p�1ð�1; �2Þ @�2p

�1ð�1; �2Þ p�

4ð�1; �2Þ

0B@1CA;

K [0 :¼

@�1�[0;1ð�1; �2Þ @�2�

[0;1ð�1; �2Þ 0

@�1�[0;2ð�1; �2Þ @�2�

[0;2ð�1; �2Þ 0

0 0 0

0B@1CA ¼ r�[

0 2 glð3Þ: ð6:35Þ

We note that

jj~K [hj;"

� ~K [0;"jjL2ð�1;M 3�3Þ ! 0 if hj ! 0;

jjsym½~K [0;" � ~K [

0�jjL 2ð�1;M3�3Þ ¼ jjp�4;" � p�

4jjL2ð�1;RÞ ! 0 if � ! 0;

jj ~�E "hj� ~�E

"0jjL2ð�1;M 3�3Þ ! 0 if hj ! 0;

jj ~�E "0 � ~�E jjL 2ð�1;M 3�3Þ ! 0 if " ! 0:

ð6:36Þ

The abbreviations in (6.35) imply

I ];[hjðu ]

hj;"; �A

[hj;"Þ ¼

Z�1

Wmpð ~�E "hjÞ þWcurvð~K [

hj;"ÞdV�; ð6:37Þ

where we used that hj � b" in the de¯nition of the recovery deformation gradient

(6.34)1 is cancelled by the factor 1hj

in the de¯nition of I ];[hj, similarly for the other


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components. Whence, adding and subtracting Wmpð ~�E Þ

I ];[hjðu ]

hj;"; �A

[hj;"Þ ¼

Z�1

Wmpð ~�E Þ þWmpð ~�E "hjÞ �Wmpð ~�E Þ þWcurvð~K [

hj;"ÞdV�

¼Z�1

Wmpð ~�E Þ þWmpð ~�E þ ~�E"hj� ~�E Þ �Wmpð ~�E Þ þWcurvðKhjÞdV�

since Wmp and Wcurv are both positive, we get from the triangle inequality

Z�1

Wmpð ~�E Þ þ jWmpð ~�E þ ~�E"hj � ~�E Þ �Wmpð ~�E Þj þWcurvð~K [

hj;"ÞdV�

expanding the quadratic energy Wmp we obtain

¼Z�1

Wmpð ~�E Þ þ jWmpð ~�E Þ þ hDWmpð ~�E Þ; ~�E "hj � ~�E i

þ D2Wmpð ~�E Þ � ð ~�E "hj� ~�E ; ~�E

"hj� ~�E Þ �Wmpð ~�E Þj þWcurvð~K [

hj;"ÞdV�

Z�1

Wmpð ~�E Þ þ jjDWmpð ~�E Þjjjj ~�E "hj� ~�E jj þ C jj ~�E "

hj � ~�E jj2 þWcurvð~K [hj;"

ÞdV�

for jj ~�E "hj� ~�E jj 1 we have

Z�1

Wmpð ~�E Þ þ ðC þ jjDWmpð ~�E ÞjjÞjj ~�E "hj� ~�E jj þWcurvð~K [

hj;"ÞdV�

since jjDWmpð ~�E Þjj C2jj ~�E jj we obtain

Z�1

Wmpð ~�E Þ þ ðC þ C2jj ~�E jjÞjj ~�E "hj� ~�E jj þWcurvð~K [

hj;"ÞdV� ð6:38Þ

and by H€older's inequality we get

Z�1

Wmpð ~�E Þ þWcurvð~K [hj;"

ÞdV� þ ðC þ C2jj ~�E jjL 2ð�1ÞÞjj ~�E"hj � ~�E jjL 2ð�1Þ:

Continuing the estimate with regard to Wcurvð~K [hj;"

Þ, adding and subtracting ~�E"0 we

may obtain

I ];[hjðu ]

hj;"; �A

[hj;"Þ

Z�1

Wmpð ~�E Þ þWcurvð~K [0Þ þWcurvð~K [

hj;"Þ

�Wcurvð~K [0ÞdV�

þðC þ C2jj ~�E jjL 2ð�1ÞÞjj ~�E"hj� ~�E

"0 þ ~�E

"0 � ~�E jjL 2ð�1Þ

Z�1

Wmpð ~�E Þ þWcurvð~K [0ÞdV�

þ jjWcurvð~K [hj;"

Þ �Wcurvð~K [0;"ÞjjL 1ð�1Þ

þ jjWcurvð~K [0;"Þ �Wcurvð~K [

0ÞjjL1ð�1Þ

þ ðC þ C2jj ~�E jjL 2ð�1ÞÞðjj ~�E"hj� ~�E

"0jjL 2ð�1Þ þ jj ~�E "

0 � ~�E jjL 2ð�1ÞÞ: ð6:39Þ


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Now take hj ! 0 to obtain by the continuity of Wcurv (the argument is similar to

(6.41)) and (6.36)3

lim suphj!0

I ];[hjðu ]

hj;"; �A

[hj;"Þ

Z�1


þ jjWcurvð~K [0;"Þ �Wcurvð~K [

0ÞjjL1ð�1Þ

þ ðC þ C2jj ~�E jjL 2ð�1ÞÞjj ~�E"0 � ~�E jjL 2ð�1Þ: ð6:40Þ

Since the curvature energy depends only on the symmetric part, we observe also

Wcurvð~K [0;"Þ �Wcurvð~K [

0Þ ¼ jjp�4;" � p�

4jj2: ð6:41Þ

Since

jj ~�E "0 � ~�E jj2 ¼ jjðru0ð�1; �2Þjb"Þ � ðru0ð�1; �2Þjb�Þ þ ð �A [;[

0;� � �A[;[0 Þjj2

2ðjjb" � b�jj2 þ jj �A [;[0;" � �A

[;[0 jj2Þ ¼ 2ðjjb" � b�jj2 þ 2jjp�

1;" � p�1jj2Þ;

ð6:42Þ

we get, by letting " ! 0 and using (6.41), the bound

lim suphj!0

I ];[hjðu ]

hj;"; �A

[hj;"Þ

Z�1


¼Z�1

W hommp ðru0; �A

[0Þ þW hom

curv ðK [0ÞdV�: ð6:43Þ

Since u0; �A[0 are two-dimensional (independent of the transverse variable), we may

write as well

lim suphj!0

I ];[hjðu ]

hj;"; �A

[hj;"Þ

Z�1

W hommp ðru0; �A

[0Þ þW hom

curv ðK [0ÞdV�

¼Z!

W hommp ðru0; �A

[0Þ þW hom

curv ðK [0Þd! ¼ I ];[

0 ðu0; �A[0Þ;

ð6:44Þ

which shows the desired upper bound. This ¯nishes the proof of Theorem 5.1.

Notation. Let � � R3 be a bounded domain with Lipschitz boundary @� and let �

be a smooth subset of @� with non-vanishing two-dimensional Hausdor® measure.

For a; b 2 R3 we let ha; biR 3 denote the scalar product on R3 with associated vector

norm jjajj 2R3 ¼ ha; aiR 3 . We denote byM3�3 the set of real 3� 3 second-order tensors,

written with capital letters and Sym denotes symmetric second-order tensors.

The standard Euclidean scalar product on M3�3 is given by hX;Y iM 3�3 ¼ tr½XY T�,and thus the Frobenius tensor norm is jjXjj2 ¼ hX;XiM 3�3 . In the following we

omit the index R3;M3�3. The identity tensor on M3�3 will be denoted by 1, so that


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tr½X� ¼ hX;1i. We set symðXÞ ¼ 12 ðXT þXÞ and skewðXÞ ¼ 1

2 ðX �XTÞ such that

X ¼ symðXÞ þ skewðXÞ. For X 2 M3�3 we set for the deviatoric part devX ¼X � 1

3 tr½X�12 slð3Þ where slð3Þ is the Lie-algebra of traceless matrices. The set

Sym ðnÞ denotes all symmetric n� n-matrices. The Lie-algebra of SOð3Þ :¼ fX 2GLð3ÞjXTX ¼ 1; detX ¼ 1g is given by the set soð3Þ :¼ fX 2 M3�3jXT ¼ �Xg of

all skew symmetric tensors. The canonical identi¯cation of soð3Þ and R3 is denoted by

axl �A 2 R3 for �A 2 soð3Þ. Note that ðaxl �AÞ � � ¼ �A � � for all � 2 R3, such that

axl

0 � �

�� 0 �

�� 0

0@ 1A :¼��

��

0@ 1A; �Aij ¼X3k¼1

� �ijk � axl �Ak;

jj �Ajj2M 3�3 ¼ 2jjaxl �Ajj 2R3 ; h �A; �BiM3�3 ¼ 2haxl �A; axl �BiR 3 ;

where "ijk is the totally antisymmetric permutation tensor. Here, �A � � denotes the

application of the matrix �A to the vector � and a� b is the usual cross-product.

Moreover, the inverse of axl is denoted by anti and de¯ned by

0 � �

�� 0 �

�� 0

0@ 1A :¼ anti

��

��

0@ 1A; axlðskewða� bÞÞ ¼ � 1

2a� b;

and 2 skewðb� aÞ ¼ antiða� bÞ ¼ antiðantiðaÞ � bÞ. Moreover, curl u¼ 2 axlðskewruÞ.Notation for plates and shells. Let ! � R2 always be a bounded open domain

with Lipschitz boundary @! and let �0 be a smooth subset of @! with non-vanishing

one-dimensional Hausdor® measure. The aspect ratio of the plate is h > 0. We denote

byMm�n the set of matrices mapping Rn 7! Rm. For H 2 M3�2 and � 2 R3 we write

ðHj�Þ 2 M3�3 for the matrix composed of H and the column �. Likewise ðvj�j�Þ is thematrix composed of the columns v; �; �. This allows us to write for u 2 C 1ðR3;R3Þ :ru ¼ ðuxjuyjuzÞ ¼ ð@xuj@yuj@zuÞ. The identity tensor on M2�2 is 12. The mapping

m : ! � R2 7! R3 is the deformation of the midsurface, rm is the corresponding

deformation gradient and nm is the outer unit normal on m. A matrix X 2 M3�3 can

now be written asX ¼ ðX � e2jX � e2jX � e3Þ ¼ ðX1jX2jX3Þ. We write v : R2 7! R3 for

the de°ection of the midsurface, such that mðx; yÞ ¼ ðx; y; 0ÞT þ vðx; yÞ. The stan-

dard volume element is dx dy dz ¼ dV ¼ d!dz.

Appendix

A.1. The in¯nitesimal Reissner�Mindlin

membrane-bending model

The Reissner�Mindlin membrane-bending plate model is a ¯ve-parameter model:

three midsurface de°ections v : ! � R2 7! R3 and two out-of-plane rotation par-

ameters � : ! 7! R2 describing the in¯nitesimal increment of the director. It should

be noted that in the classical Reissner�Mindlin model drill rotations are absent.


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The model reads

Z!

h � jjsymrðv1; v2Þjj2 þ�

2rv3 � �1

�2

� �� 2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}transverse shear energy

þ ��

2�þ �tr½symrðv1; v2Þ�2

0BB@1CCA

þ h3

12� jjsymrð�1; �2Þjj2 þ

��

2�þ �tr½symrð�1; �2Þ�2

� �� hf; vid! 7! min: w:r:t: ðv; �Þ;

vj�0 ¼ udðx; y; 0Þ; simply supported

��j�0 ¼ ðud1;z;u

d2;z; 0ÞT; rigid director prescription: ðA:1Þ

Here 0 < 1 is the so-called shear correction factor. The model is very popular

among engineers and can be found, e.g. in Ref. 7. i

A.2. The classical in¯nitesimal-displacement Kirchho®�Love plate

(Koiter model)

For the convenience of the reader we also supply the similar system of equations for

the classical in¯nitesimal-displacement Kirchho®�Love plate (also the Koiter

model). In terms of the midsurface de°ection v : ! � R2 7! R3 we have to ¯nd a

solution of the minimisation problemZ!

h � jjsymrðv1; v2Þjj2 þ��

2�þ �tr½symrðv1; v2Þ�2

� �

þ h3

12� jjD2v3jj2 þ

��

2�þ �tr½D2v3�2

� �� hf; vi d! 7! min:w:r:t: v;

vj�0 ¼ udðx; y; 0Þ; simply supported

�rv3j�0 ¼ ðud1;z;u

d2;z; 0ÞT; typical rigid prescription of the infinitesimal normal:

ðA:2ÞThis energy can also be obtained formally from (A.1) by constraining the linearised

director to the linearised normal of the plate, i.e. setting � ¼ rv3.

A.3. Aganovic's and Ne®'s model based on simultaneous

nonlinear scaling u ], �A]

In Ref. 1 a shell model is proposed based on asymptotic analysis of the linear isotropic

micropolar model, the assumption of nonlinear scaling for displacements u ] and

iHence the shear correction factor is directly determined by the Cosserat couple modulus �c comparedwith (5.5). For rather thick plates, it is known that the shear energy in RMlin is overestimated, therefore,

one is led to reduce the shear energy contribution a posteriori by taking < 1.


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in¯nitesimal microrotations �A] and uniform positivity assumption for the curvature

together with homogeneous Dirichlet conditions on the microrotations. We focus on

this model from shell to plate, rewrite its weak form into a minimisation problem and

adapt it to our notation. Then the problem reads: ¯nd the de°ection of the mid-

surface v : ! � R2 7! R3 and the microrotation vector � : ! � R2 7! R3 such that

I asymp0 ðv; �Þ ¼

Z!

W asympmp ðrv; �Þ þW asymp

curv ðr�Þ

� hf; vid! 7! min: w:r:t: ðv; �Þ ðA:3Þand the boundary conditions of place for the midsurface de°ection v on the Dirichlet

part of the lateral boundary �0 � @!,

vj�0 ¼ udðx; y; 0Þ simply supported

and the homogeneous boundary condition for the microrotation

�j@! ¼ 0; completely clamped:

The asymptotically reduced local density is

W asympmp ðrv; �Þ :¼ � sym r�1;�2ðv1; v2Þ �

0 ��3�3 0

� �� 2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}shear-stretch energy

þ�c skew r�1;�2ðv1; v2Þ �0 ��3�3 0

� �� 2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}in-plane drill energy

þ 2��c

�þ �c

r�1;�2v3 ��2�1

� �� 2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}asymptotic transverse shear energy

þ ��

2�þ �tr sym r�1;�2ðv1; v2Þ �

0 ��3�3 0

� �� 2

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}asymptotic elongational stretch energy

: ðA:4Þ

The asymptotically correct curvature density is given by

W asympcurv ðr�Þ :¼ �

bL2c

2�1 jjsymr�1;�2ð�1; �2Þjj2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

I-energy

þ �2 jjskewr�1;�2ð�1; �2Þjj2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}II-energy

0BBB@

þ 2�1

�2

�1 þ �2

jjr�1;�2�3jj2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}III-energy

þ �1�3

2�1 þ �3

tr½r�1;�2ð�1; �2Þ�2|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}IV-energy

1CCCA:

ðA:5Þ


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While �2 ¼ 0 would give formally the Reissner�Mindlin model, the proof of

asymptotic convergence in Ref. 2 needs decisively the uniform positive curvature

assumption kc > 0; �2 > 0. The limit model is well-posed for kc > 0; �2 ¼ 0.

The conformal curvature case is retrieved for �1 ¼ 1; �2 ¼ 0; �3

2 ¼ � 13 in which

case the dimensionally reduced curvature turns into

W asymp; confcurv ðr�Þ :¼ �

bL2c

2jjdev2 symr�1;�2ð�1; �2Þjj2: ðA:6Þ

This case is not 2D�well-posed! It is straightforward to show that this asymptotic

limit model coincides with the �-limit for simultaneous nonlinear scaling u]; �A] in the

strong topology of L2 for both ¯elds under the conditions kc > 0; �2 > 0. This

asymptotic limit model coincides with the linearisation of the �-limit for nonlinear

Cosserat plates in Refs. 28 and 29 were also based on the simultaneous nonlinear

scaling of deformations and rotations (note that in the nonlinear regime, dealing with

exact rotations, it is di±cult to scale the rotations with a linear scaling). Also here,

�2 > 0 is implicitly assumed. Thus, the presence of the in-plane drill component �3cannot be avoided and therefore, this is not the Reissner�Mindlin model, for no

choice of (derivation-) admissible Cosserat parameters.

A.4. A model based on linear scaling of u and nonlinear scaling

of �A, i.e. u [, �A]

The �-limit can be established along the presented lines provided that �2 � 0. Note

that the local minimisation step for linear and nonlinear scaling with respect to the

displacement u yields the same homogenized energyj since

infb2R3

W ððrvjbÞÞ ¼ infp2R 3

W

@�1v1 @�2v1 p1@�1v2 @�2v2 p2p1 p2 p3

0@ 1A0@ 1Afor W ðXÞ ¼ �jjsymXjj2 þ �

2tr½X�2: ðA:7Þ

The model is: ¯nd the de°ection of the midsurface v : ! � R2 7! R3 and the micro-

rotation vector � : ! � R2 7! R3 such that

I [;]0 ðv; �Þ ¼

Z!


curv ðr�Þ � hf; vid! 7! min: w:r:t: ðv; �Þ; ðA:8Þ

and the boundary conditions of place for the midsurface de°ection v on the Dirichlet


vj�0 ¼ udðx; y; 0Þ ¼ udðx; y; 0Þ; simply supported ðfixed;weldedÞ:and the homogeneous boundary condition for the microrotation

�j�0 ¼ 0; partially clamped:

jNot true for the curvature energy depending also on anti-symmetric terms for �2 > 0.


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W asympmp ðrv; �Þ :¼ �jjsymr�1;�2ðv1; v2Þjj2

þ�c skew r�1;�2ðv1; v2Þ �0 ��3�3 0

� �� 2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}in-plane drill energy

þ ��

2�þ �tr½r�1;�2ðv1; v2Þ�2: ðA:9Þ



bL2c

2�1 jjsymr�1;�2ð�1; �2Þjj2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

I-energy


0BBB@

þ 2�1

�2

�1 þ �2


þ �1�3

2�1 þ �3

tr½r�1;�2ð�1; �2Þ�2|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}IE-energy

1CCCA:

ðA:10ÞThe limit model already de-couples the bending rotations �1; �2 from the in-plane

de°ections v1; v2. For the nonlinear scaling of the microrotations it is necessary to

have �2 > 0 for the �-limit result.

A.5. A model based on linear scaling of u and linear

scaling of �A, i.e. u [, �A[

The problem is: ¯nd the de°ection of the midsurface v : ! � R2 7! R3 and the

microrotation vector � : ! � R2 7! R3 such that

I [;[0 ðv; �Þ ¼

Z!


curv ðr�Þ � hf; vid! 7! min:w:r:t: ðv; �Þ; ðA:11Þ

and the boundary conditions of place for the midsurface de°ection v on the Dirichlet


vj�0 ¼ udðx; y; 0Þ ¼ udðx; y; 0Þ; simply supported ðfixed;weldedÞ:

and the homogeneous boundary condition for the microrotation

�j�0 ¼ 0; partially clamped:


W asympmp ðrv; �Þ :¼ �jjsymr�1;�2ðv1; v2Þjj2 þ

��

2�þ �tr½symr�1;�2ðv1; v2Þ�2:


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bL2c

2�1jjsymr�1;�2ð�1; �2Þjj2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

I-energy


0BBB@

þ 2�1

�2

�1 þ �2


þ �1�3

2�1 þ �3

tr½r�1;�2ð�1; �2Þ�2|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}IV-energy

1CCCA:

ðA:12ÞIf we identify ð�1; �2Þ ¼ rv3 we recover the linear Koiter model (A.2). Erbay writes13:

\An examination of these equations shows that, as in the classical plate theory, the

equations governing the °exural (bending) and the extensional (stretching) motions

of the plate are independent of each other." The presented �-limit reproduces this

de-coupling.

A.6. Korn's inequality and the linear scaling for �A[

The major merit of the linear scaling (4.2) is that it respects the in¯nitesimal strain

structure and allows one to derive estimates independent of the scaling parameter

h > 0 in the case where one controls only symmetrised gradients in the curvature

energy. To see this, abbreviate �[ :¼ axl �A[ and consider

sym brh��

[ð�Þ

¼

@�1�[1ð�Þ

1

2½@�2�

[1ð�Þ þ @�1�

[2ð�Þ�

1

2h½@�3�

[1ð�Þ þ @�1�

[3ð�Þ�

1

2½@�1�

[2ð�Þ þ @�2�

[1ð�Þ� @�2�

[2ð�Þ

1

2h½@�3�

[2ð�Þ þ @�2�

[3ð�Þ�

1

2h½@�3�

[1ð�Þ þ @�1�

[3ð�Þ�

1

2h½@�3�

[2ð�Þ þ @�2�

[3ð�Þ�

1

h2@�3�

[1ð�Þ

0BBBBBB@

1CCCCCCA;

jjsym brh��

[ð�Þjj2 ¼ sym

@�1�[1ð�Þ @�2�

[1ð�Þ

1

h@�3�

[1ð�Þ

@�1�[2ð�Þ @�2�

[2ð�Þ

1

h@�3�

[2ð�Þ

1

h@�1�

[3ð�Þ

1

h@�2�

[3ð�Þ

1

h2@�3�

[1ð�Þ

0BBBBBB@

1CCCCCCA

��

��

2

¼ sym@�1�

[1ð�Þ @�2�

[1ð�Þ

@�1�[2ð�Þ @�2�

[2ð�Þ

!��2

þ 1

h2sym

0 0 @�3�[1ð�Þ

0 0 @�3�[2ð�Þ

@�1�[3ð�Þ @�2�

[3ð�Þ 0

0B@1CA

��2

þ 1

h4ð@�3�

[3ð�ÞÞ2 � jjsymr��

[ð�Þjj2: ðA:13Þ


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With this preparation we show

Theorem A.1. (Scaled Korn's inequality for the microrotation vector) For hj ! 0

as j ! 1 consider the linearly scaled sequence �[hj: �1 7! R3 and assume that it

satis¯es either

(1) 8hj > 0 : jj�[hjjjL2ð�1;R 3Þ K1; jj�[

hj;3jjL 2ð�1;RÞ ! 0 as hj ! 0 or

(2) �[hjj� 1

0¼ 0.

Assume in addition the boundedness of scaled strains along hjZ�1

jjsym brh��

[hjð�Þjj2dV� K2;

where the constants K1;K2 are independent of hj. Then

jj�[hj jjH 1;2ð�1;R 3Þ K3;

with a constant K3 independent of hj and there exists a weakly convergent sub-

sequence in H 1ð�;R3Þ (without relabelling), such that

�[hj

* �[0 2 H 1ð�;R3Þ; hj ! 0;

� [hj

! �[0 2 L2ð�1;R

3Þ; hj ! 0:

In the ¯rst case we obtain moreover that �[0ð�1; �2; �3Þ ¼ �[

0ð�1; �2Þ, independent of thetransverse variable and for the limit of the third component �[

0;3 ¼ 0.

In the second case (Dirichlet-boundary case) we obtain for the weak limit (only)

�[0ð�1; �2; �3Þ 2 VKLð�1Þ.

Proof. In the ¯rst case, we may use the estimate (A.13) and Korn's second

inequality without boundary conditions. In the second case for Dirichlet-boundary

conditions, we may use Korn's ¯rst inequality with boundary conditions. Then the

existence of a weakly convergent subsequence is clear from boundedness in

H 1ð�;R3Þ. Rellich's compact embedding provides us with strong convergence in

L2ð�;R3Þ. In the second case, the weak limit satis¯es the boundary condition

�[0j� 1

0¼ 0. Boundedness of scaled strains and weak convergence of r��

[hj

implies as

well (compare with Ciarlet8)Z�1

jj½symr��[hj� � e3jj2 Kh2

j ! 0 ) ½symr��[hj � � e3 * ½symr��

[0� � e3 ¼ 0: ðA:14Þ

Thus the weak limit �[0 of the scaled microrotation vector is found in the space VKL.

In the ¯rst case we know more, namely that h�[0; e3i ¼ 0 in L2ð�1;RÞ which gives the

result.

De¯nition A.2. (Space of scaled Kirchho®�Love displacements VKL) Following

Ciarlet, we de¯ne the space

VKLð�1Þ :¼ f� 2 H 1;2ð�1;R3Þ : �j�0 ¼ 0; ½symr��ð�Þ� � e3 ¼ 0 for � 2 �1g: ðA:15Þ


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This space is equivalently characterised by (Refs. 8 or 15 or 11)

VKLð�1Þ :¼ f� 2 H 1;2ð�1;R3Þ : �j

� 10

¼ 0;

�1ð�1; �2; �3Þ ¼ w1ð�1; �2Þ � �3@�1�3ð�1; �2Þ;�2ð�1; �2; �3Þ ¼ w2ð�1; �2Þ � �3@�2�3ð�1; �2Þ;�3ð�1; �2; �3Þ ¼ w3ð�1; �2Þ;w1;w2 2 H 1ð!;RÞ;w3 2 H 2ð!;RÞ;

w1;w2;w3j�0 ¼ 0; @�w3j�0 ¼ 0g: ðA:16Þ

We ¯rst remark that the in-plane components �1; �2 are not necessarily two-

dimensional, although they are determined by two-dimensional functions.

A.7. The ¡-limit for linear elasticity and linear scaling u [hj

To put our result into further perspective let us relate it to classical linear elasticity.

Using Theorem A.1 for the scaled displacement u [hj

with Dirichlet boundary con-

ditions allows us to establish the suitable bounds. The �-limit of I [hjðu [

hjÞ in the strong

topology of L2ð�1;R3Þ is given by the limit energy functional I [

0 : L2ð�1;R

3Þ 7! �R,

I [0ðvÞ :¼

Z�1

W homðrvÞ � hf; vidV� v 2 VKLð�1Þþ1 else in H 1;2ð�1;R

3Þ;

8<: ðA:17Þ

with

W homðrvÞ :¼ �jjsymr�1;�2ðv1; v2Þjj2 þ��

2�þ �tr½r�1;�2ðv1; v2Þ�2:

This result is a slight variation of the statements in Refs. 8 and 5. One might be

tempted to think that this de¯nes a membrane model. However, the limit is not truly

two-dimensional but in the space VKL. It is therefore possible to insert the limit into

the integral and to perform the integration over the thickness analytically. The result

is, after descaling, surprisingly, the Kirchho®�Love membrane-bending plate (A.2)

written in the de°ection �v : ! � R2 7! R3 and setting �vð�1; �2Þ :¼ Av � v for v 2VKLð�1Þ. Note again that the vertical de°ection should be of the order of the

thickness of the plate for this result to make sense.

A.8. An inequality for linear elasticity with nonlinear scaling u ]hj

Assuming linear elastic behaviour and simply considering the nonlinear scaling, the

following inequality can be established:

Theorem A.3. (hj-dependent Korn's ¯rst inequality and nonlinear scaling) For

hj ! 0 consider a sequence u ]hj2 A0. Then there exists a constant C independent of

hj > 0 such that

C

h2j

Z�1

jjsymrh�u

]hjð�Þjj2dV� � jju ]

hjð�Þjj2H 1;2ð�1;R 3Þ:


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Proof. Can be found in Ref. 4, see also Ref. 9.

Remark A.4. With this (essentially sharp) inequality, it is di±cult to continue the

�-limit development in classical linear elasticity based on the nonlinear scaling

without further assumptions on the scaling of energies. This is one of the reasons, why

Ciarlet uses the linear scaling in the case of plates (the inequality can be improved to

be independent of hj in case of a shell with elliptic surface).

Assume, however, that the scaled energy satis¯es (this is a strong assumption on

the data in disguise)

1

h2j

I ]ðu ]hjÞ C: ðA:18Þ

Then Theorem A.3 allows one to establish weak compactness of u ]hjin H 1;2ð�1;R

3Þ.The �-limit of 1

h 2j

I ] in the strong topology of L2ð�1;R3Þ is given by the limit energy

functional I ]0 : L

2ð�1;R3Þ 7! �R,

I ]0ðvÞ :¼

Z!

W homðrvÞ � hf; vid! v 2 H 1;2ð!;R3Þþ1 else in L2ð�1;R

3Þ:

8<: ðA:19Þ

For this result compare to Ref. 4. For sequences bounded in H 1 it is easy to see that

the weak limit is actually independent of �3 and thus the limit problem is a mem-

brane-plate.

Remark A.5. In the ¯nite strain setting the assumption 1h 2

j

I ]ð’]hjÞ C leads to the

classical plate-bending problem.18

Acknowledgements

K.-I.H. was supported by DAAD, Grant A/07/98690/. K.-I.H. is grateful for a three-

month grant from DAAD making possible his stay at the Fachbereich Mathematik,

Technische Universität Darmstadt in spring 2008. P.N. thanks J. Tambaca and an

unknown reviewer for pointing out inconsistencies in a preliminary version of the

paper. The authors thank the editor S. Müller for help in clarifying the meaning of the

scaling of the interaction strength (3.4).

References

1. I. Aganovic, J. Tambaca and Z. Tutek, Derivation and justi¯cation of the model ofmicropolar elastic shells from three-dimensional linearized micropolar elasticity, Asympt.Anal. 51 (2007) 335�361.

2. I. Aganovic, J. Tambaca and Z. Tutek, Derivation and justi¯cation of the models of rodsand plates from linearized three-dimensional micropolar elasticity, J. Elasticity 84 (2007)131�152.


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3. S. M. Alessandrini, D. N. Arnold, R. S. Falk and A. L. Madureira, Derivation and jus-ti¯cation of plate models by variational models, in Plates and Shells, Vol. 21, ed. M. Fortin(Amer. Math. Soc., 1999), pp. 1�20.

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THE REISSNER MINDLIN PLATE IS THE ¡-LIMIT OF COSSERAT...

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