Mathematical Models and Methods in Applied Sciences
Vol. 20, No. 9 (2010) 15531590
#
.c World Scienti¯c Publishing Company
DOI: 10.1142/S0218202510004763
{
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
THE REISSNER MINDLIN PLATE
IS THE ¡-LIMIT OF COSSERAT ELASTICITY
PATRIZIO NEFF*
€ Nichtlineare Analysis und Modellierung,
Lehrstuhl fur
€ Mathematik, Universit€
Fakult€
at fur
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
Speciale des Travaux Publics du B^
atiment et de l'Industrie (ESTP),
28 Avenue du President Wilson, 94234 Cachan Cedex, France
[email protected]
Received 4 July 2008
Revised 6 June 2009
Communicated by S. Müller
The linear ReissnerMindlin 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 2 soð3Þ. We also provide formal calculations for other combinations of
tesimal microrotations A
scalings 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; ReissnerMindlin; 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
1553
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
1554
P. Ne®, K.-I. Hong & J. Jeong
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 twodimensional 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 twodimensional 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-exponents 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 corresponding 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 inextensible deformations. Similarly, one may obtain a membrane energy with no bending
term, having no resistance in compression.12 But in a given three-dimensional problem 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 1 ðÞ, together with a certain \linear"
scaling, Ciarlet5 arrives at justifying the membrane plate. This result can be, without
problems, extended to the strong L 2 -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. Compactness can only be assured by assuming that the scaled energy h12 I h] ðu ] 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
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1555
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 ReissnerMindlin 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 ReissnerMindlin model… To the best of our knowledge there is
no way to obtain ReissnerMindlin type models of plate bending from the asymptotic approach." Similarly, Ciarlet writes8: \Open problems: ¯nding a rigourous
justi¯cation of the ReissnerMindlin 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 2 R 3 SOð3Þ. The -limit result is a kind of
mations and exact rotations ð’; RÞ
ReissnerMindlin 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
2 R 3 soð3Þ. The result is comparable to the previous one in Ref. 28. A
ðu; AÞ
precursor to that is Ref. 13 where the author also used the asymptotic expansion
2 R 3 soð3Þ. His result is comparable
method but with linear scaling for both ðu; AÞ
to a formal deduction given much earlier in Ref. 14. Neither of these methods,
however, reproduced the classical ReissnerMindlin 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 displacement 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 ReissnerMindlin membrane bending
model, another choice recovers the Aganovic/Ne® model and still another choice decouples 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
ReissnerMindlin 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
a When
¯nishing this paper we have learned that a related justi¯cation of the ReissnerMindlin model
based on -convergence has already been given in Refs. 30 and 31. Since the authors considered a secondgradient \parent" linear elasticity model instead of our ¯rst-order Cosserat \parent" model we believe in
the interest of our approach.
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
1556
P. Ne®, K.-I. Hong & J. Jeong
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 -convergence 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 ReissnerMindlin 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 : R 3 7! R 3 and the skew-symmetric in¯nitesimal
microrotation A : R 3 7! soð3Þ, we consider the two-¯eld minimisation problem
Z
hf; uidx 7! min: w:r:t: ðu; AÞ;
¼
Wmp ð" Þ þ Wcurv ðr axl AÞ
ð2:1Þ
Iðu; AÞ
under the following constitutive requirements and boundary conditions
first Cosserat stretch tensor
" ¼ ru A;
uj ¼ ud ; essential displacement boundary conditions
Wmp ð" Þ ¼ jjsym " jj 2 þ c jjskew " jj 2 þ tr½sym " 2 strain energy
2
2 þ K tr½sym ru 2 ;
¼ jjdev sym rujj 2 þ c jjskewðru AÞjj
2
3
:¼ axl A 2 R ; K ¼ r;
jjcurl jj 2R 3 ¼ 4jjaxl skew rjj 2R 3 ¼ 2 jjskew rjj 2M 33 ;
þ
jjsym rjj 2 þ
jjskew rjj 2 þ tr½r 2 curvature
2
2
2
þ
k
jjdev sym rjj 2 þ
jjskew rjj 2 þ c tr½r 2 :
¼
2
2
2
Wcurv ðrÞ ¼
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
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1557
energy Wcurv are the most general isotropic quadratic forms in the in¯nitesimal non and the micropolar curvature
symmetric ¯rst Cosserat strain tensor " ¼ ru A
tensor K ¼ r axl A ¼ r (curvature-twist tensor). The parameters ; ½MPa are the
classical Lame moduli and ; ; are additional micropolar moduli with dimension
½Pa m 2 ¼ ½N of a force. Here, the bulk modulus and curvature bulk modulus are
de¯ned by
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
K¼
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 E 3 including units of measurement will now be adapted to a plate-like
theory. Let us therefore assume that the problem is already transformed in nondimensional form. This means that we are given a three-dimensional (non-dimensional) thin domain h R 3
h h
h :¼ ! ; ; ! R 2 ;
ð3:1Þ
2 2
¼ ! f h2 ; h2 g and lateral boundary @ lat
with transverse boundary @ trans
h
h ¼ @! ½ h2 ; h2 , where ! is a bounded open domainb in R 2 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 R 3 7! R 3 ;
: h R 3 7! soð3Þ;
A
solving the minimisation problem on the thin domain h :
Z
hf; uidV
Iðu; AÞ ¼
Wmp ð" Þ þ Wcurv ðr axl AÞ
h
Z
@ trans
[f s ½ h2 ; h2 g
h
b For
hN; uidS 7! min: w:r:t: ðu; AÞ;
de¯niteness, one can think of ! ¼ ½0; 1 ½0; 1 R 2 without units of length.
ð3:2Þ
1558
P. Ne®, K.-I. Hong & J. Jeong
uj ¼ ud ðx; y; zÞ;
" ¼ ru A;
h
0
h
h
h
0 ¼ 0 ; ; 0 @!; s \ 0 ¼ ;;
2 2
h h
A : free on @! ; ; Neumann-type boundary condition;
2 2
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
Wmp ð" Þ ¼ jjsym " jj 2 þ c jjskew " jj 2 þ
Wcurv ðKÞ ¼ tr½" 2 ;
2
b 2c ðhÞ L
2 :
2 þ 2 jjskew r axl Ajj
2 þ 3 tr½r axl A
1 jjsym r axl Ajj
2
2
ð3:3Þ
Here, 1 ; 2 ; 3 0 are non-dimensional weighting parameters and tractions N on
b c has
the transverse boundary of the plate are included. Moreover, the parameter L
RVE
c
b c ¼ L RVE
the form L
,
where
L
is
related
to
the
characteristic
size
of
the
microc
L
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: ¼
½0; L½m ½0; L½m ½ h2 L½m; h2 L½m E 3 .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 L RVE
! 1 to the e®ect that the aspect
c
ratio h ! 0 and as additional qualitative requirement
b c ðhÞ 1 ðe h þ hÞ 1 ;
L
ð3:4Þ
C1
C1
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 development. In order to arrive formally at the classical ReissnerMindlin model (which
has a h 3 -factor in front of the bending energy) the de-scaling of the -limit is performed formally only for aspect ratios h in the regime where
RVE
b c ðhÞ 1 h , C1 L
b c ðhÞ C1 L c
h;
L
C1
L
ð3:5Þ
i.e. already for moderately large h. This is consistent with the observation that the
ReissnerMindlin 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 performing a dimensional reduction asymptotic analysis for a relatively thin domain.
c This
is an immediate consequence of the non-dimensionalisation procedure, see Ref. 27.
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1559
^ c ðhÞ of the microstructure as a function of the aspect ratio
Fig. 1. Assumed interaction strength L
according to (3.4). Descaling of the -limit is performed in a range h a according to (3.5) such that
formally the ReissnerMindlin bending factor h 3 is retrieved.
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 R 3 7! R 3 is just that one, which de¯nes z ] : 2 1 7! R 3 as the \same" ¯eld
on the domain 1 ¼ ! ½1=2; 1=2 (see (4.4)), only the independent variables are
scaled as
1 ¼ 1 ; 2 ¼ 2 ; 3 ¼ h3 ;
1
]
z 1 ; 2 ; 3 :¼ zð1 ; 2 ; 3 Þ; nonlinear scaling
h
1
r zð1 ; 2 ; 3 Þ ¼ @ 1 z ] ð1 ; 2 ; 3 Þj@ 2 z ] ð1 ; 2 ; 3 Þj @ 3 z ] ð1 ; 2 ; 3 Þ
h
1
0
1
@ 3 z ]1 ðÞ
@ 1 z ]1 ðÞ @ 2 z ]1 ðÞ
C
B
h
C
B
C
B
1
]
]
]
C ¼: r h z ] ðÞ:
¼B
@
@
z
ðÞ
@
z
ðÞ
z
ðÞ
2 2
3 2
C
B 1 2
h
C
B
A
@
1
]
]
]
@ 3 z 3 ðÞ
@ 1 z 3 ðÞ @ 2 z 3 ðÞ
h
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 R 3 7! R 3 by
1560
P. Ne®, K.-I. Hong & J. Jeong
de¯ning z [ : 2 1 7! R 3 in the form
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
1 ¼ 1 ;
2 ¼ 2 ; 3 ¼ h3 ;
1
0 1
[
z
;
;
B 1 1 2 h 3 C
0
1
B z1 ð1 ; 2 ; 3 Þ
C
C
B
1
C
B
C
B [
B z 2 1 ; 2 ; 3 C :¼ @ z2 ð1 ; 2 ; 3 Þ A;
C
B
h
B C
hz3 ð1 ; 2 ; 3 Þ
A
@
1
[
z 3 1 ; 2 ; 3
h
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
1
0
1
@ 3 z [1 ðÞ
@ 2 z [1 ðÞ
@ 1 z [1 ðÞ
C
B
h
C
B
C
B
1
[
[
[
bh [
B
r zð1 ; 2 ; 3 Þ ¼ B @ 1 z 2 ðÞ
ð4:3Þ
@ 3 z 2 ðÞ C
@ 2 z 2 ðÞ
C ¼: r z ðÞ:
h
C
B
A
@1
1
1
[
@ 1 z [3 ðÞ
@ 2 z [3 ðÞ
@
z
ðÞ
3
h
h
h2 3
The scaling of the dependent variable corresponds to an additional ad hoc assumption on the assumed response. In our case, we deal with the displacement ¯eld u :
: h 7! soð3Þ. For the displacement ¯eld we
h 7! R 3 and the microrotation ¯eld A
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 R 3 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. h3 ð1 ; 2 ; 3 Þ ¼ [3 ð1 ; 2 ; h1 3 Þ.
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.
d Since we assume that the unscaled vertical de°ection z is bounded (otherwise the physics underlying
3
linear elasticity is void anyway), the linear scaling implies that the scaled vertical de°ection z [3 should be of
the 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."
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1561
We therefore de¯ne
1 1
1 ¼ ! ;
R 3;
2 2
! R 2:
ð4:4Þ
ð1 ; 2 ; 3 Þ :¼ ð1 ; 2 ; h 3 Þ;
1 ð1 ; 2 ; 3 Þ :¼ ð1 ; 2 ; 3 =hÞ;
ð4:5Þ
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
The scaling transformation
: 2 1 R 3 7! R 3 ;
1 : 2 h R 3 7! R 3 ;
maps 1 into h and ð1 Þ 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 ! R 3 ,
de¯ned by
uð1 ; 2 ; 3 Þ ¼ u ] ð 1 ð1 ; 2 ; 3 ÞÞ 8 2 h ; u ] ðÞ ¼ uððÞÞ 8 2 1 ;
1
r uð1 ; 2 ; 3 Þ ¼ @ 1 u ] ð1 ; 2 ; 3 Þj@ 2 u ] ð1 ; 2 ; 3 Þj @ 3 u ] ð1 ; 2 ; 3 Þ
h
1
0
1
@ 3 u ]1 ðÞ
@ 1 u ]1 ðÞ @ 2 u ]1 ðÞ
C
B
h
C
B
C
B
1
ð4:6Þ
]
]
]
C ¼: r h u ] ðÞ:
¼B
@
@
u
ðÞ
@
u
ðÞ
u
ðÞ
2 2
3 2
C
B 1 2
h
C
B
A
@
1
]
]
]
@ 3 u 3 ðÞ
@ 1 u 3 ðÞ @ 2 u 3 ðÞ
h
[ : 1 R 3 7! soð3Þ by
We de¯ne a (linearly) scaled in¯nitesimal microrotation A
considering the corresponding axial vector ðÞ :¼ axl AðÞ 2 R 3 and its linearly
scaled correspondence [ ðÞ through
ð1 ; 2 ; 3 Þ ¼ [ ð 1 ð1 ; 2 ; 3 ÞÞ 8 2 h ;
0
@ 2 [1 ðÞ
@ [ ðÞ
B 1 1
B
B
[
r ð1 ; 2 ; 3 Þ ¼ B
@ 2 [2 ðÞ
B @ 1 2 ðÞ
B
@1
1
@ [ ðÞ
@ [ ðÞ
h 1 3
h 2 3
[ ðÞ ¼ ððÞÞ 8 2 1 ;
1
1
@ 3 [1 ðÞ
C
h
C
C
1
[
bh [
@ 3 2 ðÞ C
C ¼: r ðÞ:
h
C
A
1
[
@
ðÞ
3
3
h2
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 r h u ] A
h
b h [ ðÞ ¼: K [h ðÞ;
r
ð4:8Þ
1562
P. Ne®, K.-I. Hong & J. Jeong
where
0
@ 1 u ]1 ðÞ
@ 2 u ]1 ðÞ
B
B
B
]
]
r h u ] A h ¼ B
B @ 1 u 2 ðÞ
B
@
@ 1 u ]3 ðÞ
@ 2 u ]2 ðÞ
@ 2 u ]3 ðÞ
1
1
0
1
@ 3 u ]1 ðÞ
1 [
[
C
h
0
C B
2 C
h 3
C B
1
C
1 [
B
@ 3 u ]2 ðÞ C
[ C
C @ 3
0
A
1
h
C
h
A
[
[
1
2
1
0
@ u ] ðÞ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
ffl}
h 3 3
]
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
¼:A h
ð4:9Þ
[
is expressed in terms of the linearly scaled [ :¼ axl A . Moreover, we de¯ne nonlinearly scaled functions by setting
f ] ðÞ :¼ fððÞÞ;
u ]d ðÞ ¼ ud ððÞÞ;
N ] ðÞ :¼ NððÞÞ:
ð4:10Þ
In terms of the introduced nonlinearly scaled displacement and the linearly scaled
[ : 1 R 3 7! soð3Þ, the scaled
in¯nitesimal microrotations u ] : 1 R 3 7! R 3 ; A
problem solves the following two-¯eld minimisation problem on the ¯xed domain 1 :
[; r
[Þ
b h axl A
I ];[ ðu ] ; r h u ] ; A
Z
¼
½Wmp ð" ]h Þ þ Wcurv ðK [h Þ hf ] ; u ] idet½rðÞdV
21
Z
@ trans
[f s ½ 12 ; 12 g
1
Z
¼h
21
hN ] ; u ] ijjCof rðÞ e3 jjdS
Wmp ð" ]h Þ þ Wcurv ðK [h Þ hf ] ; u ] idV
Z
Z
@ trans
1
hN ] ; u ] i1 dS
s ½ 12 ; 12 [ Þ:
hN ] ; u ] ih dS 7! min: w:r:t: ðu ] ; A
ð4:11Þ
4.2. The rescaled variational Cosserat bulk problem
Since the energy h1 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 @ trans
.f For conciseness we investigate the following simpli¯ed and
1
]
]
rescaled (N ; f ¼ 0; ud ð1 ; 2 ; 3 Þ :¼ ud ð1 ; 2 Þ) two-¯eld minimisation problem on
1 with respect to -convergence (without the factor h > 0 now), i.e. we are interested in the limiting behaviour of the scaled energy per unit aspect ratio h:
Z
];[
[
h ] [ b h
]
I h ðu ; r u ; A ; r axl A Þ ¼
Wmp ð" ]h Þ þ Wcurv ðK [h ÞdV 7! min: w:r:t: ðu ] ; A [ Þ;
21
f The
thin plate limit h ! 0 obviously cannot support non-vanishing transverse surface loads N.
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
] ;
" ]h ¼ r h u ] A
h
u ]j 1ðÞ ¼ u ]d ðÞ ¼ ud ððÞÞ ¼ ud ð1 ; 2 ; h 3 Þ ¼ ud ð1 ; 2 ; 0Þ;
0
1 1
[ ðÞ;
b h axl A
¼ 0 ; ; 0 @!; K [h ¼ r
2 2
[ : free on @! 1 ; 1 ; Neumann-type boundary condition:
A
2 2
10
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
1563
ð4:12Þ
Here we assume for simplicity that the bulk boundary condition ud is already independent 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
:¼ R [ f1g. We consider a sequence of energy functionals Ih :
real numbers R
j
X 7! R; hj ! 0.
De¯nition 4.1. (-convergence) Let X be a metric space. We say that a sequence of
-converges in X to the limit functional I0 : X 7! R,
if for all
functionals Ihj : X 7! R
x 2 X we have
8 x 2 X : 8 xh j ! x :
8 x 2 X : 9 xh i ! x :
I0 ðxÞ lim inf Ihj ðxhj Þ;
ðlim inf inequalityÞ
I0 ðxÞ lim sup Ihi ðxhi Þ;
ðrecovery sequenceÞ:
hj !0
hi !0
-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;2 ð1 ; R 3 Þ H 1;2 ð1 ; soð3ÞÞ, which does not
give rise to a metric space. Instead, we de¯ne suitable \bulk" spaces X; X 0 and
suitable \two-dimensional" spaces X! ; X !0 . To this end de¯ne the spaces
2 L 2 ð1 ; R 3 Þ L 2 ð1 ; soð3ÞÞg;
X :¼ fðu; AÞ
2 H 1;2 ð1 ; R 3 Þ H 1;2 ð1 ; soð3ÞÞg;
X 0 :¼ fðu; AÞ
2 L 2 ð!; R 3 Þ L 2 ð!; soð3ÞÞg;
X! :¼ fðu; AÞ
0
2 H 1;2 ð!; R 3 Þ H 1;2 ð!; soð3ÞÞg;
X ! :¼ fðu; AÞ
ð5:1Þ
1564
P. Ne®, K.-I. Hong & J. Jeong
and the admissible sets
2 H 1;2 ð1 ; R 3 Þ H 1;2 ð1 ; soð3ÞÞ; uj ðÞ ¼ u ] ðÞg;
A 0 :¼ fðu; AÞ
d
1
0
2 H 1;2 ð!; R 3 Þ H 1;2 ð!; soð3ÞÞ; uj ð1 ; 2 Þ ¼ u ] ð1 ; 2 ; 0Þg:
A 0! :¼ fðu; AÞ
d
0
ð5:2Þ
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
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
[; r
b h axl A [ Þ
I h];[ ðu ] ; r h u ] ; A
(
[; r
[ Þ if ðu ] ; A
[Þ 2 A 0
b h axl A
I h];[ ðu ] ; r h u ] ; A
¼
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 functional. 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 ];[ : X 7! R,
0
:¼
I 0];[ ðv; AÞ
8Z
<
hom
hom
þ W curv
hf; vid! ðv; AÞ
2 A 0!
W mp
ðrv; axl AÞ
ðr axl AÞ
:
þ1
!
else in X;
hom
hom
with W mp
and W curv
de¯ned below.
The proof of this statement will be given in Sec. 6. The limit functions are independent of the transverse variable 3 . This -limit determines in fact a purely twodimensional minimisation problem for the de°ection of the midsurface v : ! R 2 7! R 3 and the in¯nitesimal microrotation of the plate (shell) A : ! R 2 7! soð3Þ
on ! under the boundary conditions of place for the midsurface de°ection v on the
Dirichlet part of the lateral boundary 0 @!,
vj0 ¼ ud ðx; y; 0Þ;
simply supported ðfixed; weldedÞ:
ð5:4Þ
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1565
The boundary conditions for the microrotations A are automatically determined in
the variational process. The dimensionally homogenised local density is
2
c 2 hom
2
W mp ðrv; Þ :¼ jjsym r1 ;2 ðv1 ; v2 Þjj
þ 2
r
v 1 þ c 1 ;2 3
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
homogenised shear-stretch energy
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
þ
2 þ homogenised transverse shear energy
tr½r1 ;2 ðv1 ; v2 Þ 2
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
:
homogenised elongational stretch energy
The homogenised curvature density is given by
b 2 ðhÞ 1 3
L
hom
W curv
1 jjsym r1 ;2 ð1 ; 2 Þjj 2 þ
ðrÞ :¼ c
tr½r1 ;2 ð1 ; 2 Þ 2 :
2
21 þ 3
It is clear that the limit functional I 0];[ is weakly lower semicontinuous in the topology
of X 0 ¼ H 1;2 ð; R 3 Þ H 1;2 ð; soð3ÞÞ by simple convexity arguments. Note the appearance of the harmonic mean H,
1
c
1 1 3
H ;
; Hð; c Þ ¼ 2
H 1 ; 3 ¼
;
:
¼
2
2
2 þ 2
þ c
2
21 þ 3
5.3. Descaled ¡-limit : ReissnerMindlin membrane-bending model
After descaling, the -limit minimisation problem turns into
2
Z 2 2c 2
2
tr½rðv1 ; v2 Þ
h jjsym rðv1 ; v2 Þjj þ
rv þ
2 þ þ c 3
1 !
b 2 ðhÞ 1 3
L
h 1 jjsym rð1 ; 2 Þjj 2 þ
þ c
tr½rð1 ; 2 Þ 2
2
21 þ 3
hf; vid! 7! min: w:r:t: ðv; Þ;
vj0 ¼ u d ðx; y; 0Þ:
ð5:5Þ
b c ðhÞ h (3.5) and
Choosing as range for descaling only aspect rations h where C1 L
4c
abbreviating ¼ þc yields the classical ReissnerMindlin 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
2
rv3 2 is not in error as compared to (A.1). In fact, we obtain the
1
classical ReissnerMindlin model provided the curvature of the bulk problem is re-de¯ned through a
permutation by
g The
coupling term
2c
þc
1 ; axlðAÞ
3Þ T;
¼ ðaxlðAÞ
2 ; axlðAÞ
c AÞ
axlð
which is physically equivalent.
1566
P. Ne®, K.-I. Hong & J. Jeong
energies having the form
b 2c kc
L
2
2
jjdev sym r axl Ajj þ tr½r axl A
Wcurv ðr axl AÞ ¼ 2
2
ð6:1Þ
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
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
[h Þ 2 A 0 X
Theorem 6.1. (Compactness of I h];[j ) Consider a sequence ðu ]hj ; A
j
[
[
];[
]
h Þ K2 , with constants K1 ; K2
such that jjA hj jjL 2 ð1 ;soð3ÞÞ K1 and I hj ðu hj ; A
j
independent of hj > 0. Then, for positive curvature bulk modulus kc > 0 it holds
[
jju ]hj jjH 1;2 ð1 ;R 3 Þ K3 ;
h jjH 1;2 ð ;soð3ÞÞ K4 ;
jjA
j
1
ð6:2Þ
[h Þ 2 A 0 admits
with constants K3 ; K4 independent of hj > 0. The sequence ðu ]hj ; A
j
[
]
] [
weakly convergent subsequences (not relabelled) ðu hj ; A hj Þ * ðu 0 ; A 0 Þ 2 X. In
addition, the weak limit
[
0 Þ 2 A 0!
ðu ]0 ; A
ð6:3Þ
[0 Þ3 ¼ 0 (no in-plane drill
is independent of the transverse variable 3 and ðaxl A
rotation).
[h Þ 2 A 0 X we have
Proof. Along the sequence ðu ]hj ; A
j
Z
Z
]
[
[h Þ ¼
1 > K2 > I h];[j ðu ]hj ; A
W
ð
"
Þ
þ
W
ðK
ÞdV
Wmp ð" ]hj ÞdV
mp
curv
hj
hj
j
Z
1
1
1
K hj ]
] 2
min c ; ;
r u hj A hj dV :
2
But with (4.9) we obtain
0
h
r j u ]hj
]
A hj
@ 1 u ]1 ðÞ
B
B
B
]
¼B
B @ 1 u 2 ðÞ
B
@
@ 1 u ]3 ðÞ
0
0
B
B
B
B 1 [
B h hj ;3
@ j
[hj ;2
@ 2 u ]1 ðÞ
@ 2 u ]2 ðÞ
@ 2 u ]3 ðÞ
1 [
hj hj ;3
0
[hj ;1
1
1
@ 3 u ]1 ðÞ
C
h
C
C
1
]
@ u ðÞ C
h 3 2 C
C
A
1
]
@ 3 u 3 ðÞ
h
1
[hj ;2
C
C
C
[
;
hj ;1 C
C
A
0
ð6:4Þ
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1567
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
! 2
2
@ 1 u ]1 ðÞ @ 2 u ]1 ðÞ 1 hj ]
] 2
]
þ
@
u
ðÞ
r u hj A hj ¼ sym
3
3
h 2j
@ 1 u ]2 ðÞ @ 2 u ]2 ðÞ 0
1 2
1
!
0
[hj ;3 ]
]
B
C
@ 1 u 1 ðÞ @ 2 u 1 ðÞ
hj
C
þ
B
skew
@ 1 [
A
]
]
@ 1 u 2 ðÞ @ 2 u 2 ðÞ
0
h hj ;3
j
! 2
!
@ u ] ðÞ
[hj ;2 1 3
þ
[
@ u ] ðÞ
hj ;1
3
2
2
0
1
1
]
!
B h @ 3 u 1 ðÞ C
[hj ;2 :
B
C
þ @
A [
1 @ u ] ðÞ
hj ;1 3 2
h
ð6:5Þ
Combining (6.4) with (6.5), neglecting the skew-symmetric contribution and using
[h is bounded in L 2 ð1 ; R 3 Þ independent of hj we
the assumption that [hj ¼ axlA
j
obtain easily an hj -independent bound for
! 2
Z 2
@ 1 u ]1 ðÞ @ 2 u ]1 ðÞ 1 ]
þ
@
u
ðÞ
1 > K5 >
sym
3 3
2
]
]
h
@ 1 u 2 ðÞ @ 2 u 2 ðÞ
1
j
0
1 2
1
]
! 2 ]
@
u
ðÞ
@ u ðÞ 3 1
B
C
1 3
h
dV
B
C
þ
þ
@
A
@ u ] ðÞ 1
]
3
2
@ 3 u 2 ðÞ h
! 2
Z 2
@ 1 u ]1 ðÞ @ 2 u ]1 ðÞ 1 ]
¼
þ
@
u
ðÞ
sym
3
3
h 2j
@ u ]2 ðÞ @ u ]2 ðÞ 1 1
2
!
! 2
]
@ u ] ðÞ 2
1 1 3
@ 3 u 1 ðÞ þ
þ
dV
@ u ] ðÞ h 2j @ u ]2 ðÞ 3
2
3
! 2
Z 2
@ 1 u ]1 ðÞ @ 2 u ]1 ðÞ ]
þ
@
u
ðÞ
sym
3
3
]
]
@ u ðÞ @ u ðÞ
1
2
1
!
@ u ] ðÞ 2 1 3
þ
þ
@ u ] ðÞ 3
2
0
@ u ]1 ðÞ
Z B 1 ]
symB @ u ðÞ
1
@ 1 2
1 @ u ] ðÞ
1
3
2
2
! 2
@ 3 u ]1 ðÞ dV
]
@ u ðÞ 3
2
@ 2 u ]1 ðÞ
@ 2 u ]2 ðÞ
@ 2 u ]3 ðÞ
1 2
C
]
C
@ 3 u 2 ðÞ A
dV ;
]
@ u ðÞ @ 3 u ]1 ðÞ
3
3
ð6:6Þ
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
1568
P. Ne®, K.-I. Hong & J. Jeong
(for hj > 0 small enough). Korn's ¯rst inequality and the Dirichlet-boundary
condition on u ]hj show 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-symmetric de°ection to see that the boundedness of
0
1 [ 1 2
!
0
Z B
@ 1 u ]1 ðÞ @ 2 u ]1 ðÞ
hj hj ;3 C
dV
skew
B
C
ð6:7Þ
@
A
]
]
1
[
@ 1 u 2 ðÞ @ 2 u 2 ðÞ
1 0
;3
h
hj j
implies the boundedness of
0
1 2
1
0
[hj ;3 Z B
C
h
j
B
C dV
@ 1 [
A
1 0
h hj ;3
ð6:8Þ
j
showing that jj [hj ;3 jjL 2 ð1 ;RÞ ! 0 for hj ! 0. For the (similar) treatment of the curvature energy we note that
Z
Z
]
[
[h Þ ¼
1 > K2 > I h];[j ðu ]hj ; A
W
ð
"
Þ
þ
W
ðK
ÞdV
Wcurv ðK [hj ÞdV
mp hj
curv
hj
j
1
1
2
3kc b h [h ðÞ
min 1;
symr
dV :
j
2
2
1
Z
L 2c
ð6:9Þ
Now use Theorem A.1 to get the hj -independent H 1 -bound on [hj together 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 h];[j then we must have
(lim inf-inequality) that
[0 Þ lim inf I ];[ ðu ] ; A
[h Þ;
I 0];[ ðu ]0 ; A
hj
hj
j
hj
ð6:10Þ
whenever
u ]hj ! u ]0
in L 2 ð1 ; R 3 Þ;
[h ! A [0
A
j
in L 2 ð1 ; soð3ÞÞ;
ð6:11Þ
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1569
[0 Þ 2 X. Observe that we can restrict attention to sequences ðu ] ;
for arbitrary ðu ]0 ; A
hj
[
[h Þ < 1 since otherwise the statement is true anyway.
A hj Þ 2 X such that I h];[j ðu ]hj ; A
j
[h Þ < 1 are uniformly bounded in the space X 0 , as seen
Sequences with I ];[ ðu ] ; A
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
hj
hj
j
previously. This implies weak convergence of a subsequence in X 0 . But we already
[0 Þ 2 X.
know that the original sequences converge strongly in X to the limit ðu ]0 ; A
]
[0 2
Hence we must have as well weak convergence to u 0 2 H 1;2 ð!; R 3 Þ and A
H 1;2 ð!; soð3ÞÞ, independent of the transverse variable 3 .
In a ¯rst step we consider now the local energy contribution: along sequences
h
[
ðu ]hj ; A hj Þ 2 X with ¯nite energy I h];[j , the third column of r j 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 : ! R 2 7! R 3 ;
A : ! R 2 7! soð3Þ de¯ne the vector ðb ; p~ 1 Þ 2 R 4 formally through
0
0
11
0 p~ 1 2
hom
W mp
ðrv; Þ ¼ Wmp @ðrvjb Þ @ p~ 1
0 1 AA
2 1
0
0
0
11
0 p~1 2
ð6:12Þ
:¼ inf
Wmp @ðrvjbÞ @ p~1
0 1 AA:
b2R 3 ;p~1 2R
2 1
0
The vector ðb ; p~ 1 Þ, which realises this in¯mum, can be explicitly determined. The
calculation is lengthy but otherwise straightforward. We obtain
1
0
c 2c
@ 1 v3 þ
B
þ c 2 C
C
0 1 B þ c
C
B b1
2c
C
B c
B b C B þ @ 2 v3 þ 1 C
C
B 2C B
c
c
ð6:13Þ
C:
B C¼B
C
@ b3 A B
C
B
B 2 þ ð@ 1 v1 þ @ 2 v2 Þ C
p~ 1
C
B
A
@
@ 1 v2 @ 2 v1
2
hom
Reinserting the result in the energy yields for W mp
ðrv; axl AÞ
2
0 3
hom
W mp ðrv; Þ :¼ sym r1 ;2 ðv1 ; v2 Þ 3 0
2
c r ; v3 2 þ 2
1 2
þ c
1 2
0 3
tr sym r1 ;2 ðv1 ; v2 Þ þ
:
2 þ 3 0
Note that 3 cancels after all (left for clarity to show the coupling).
1570
P. Ne®, K.-I. Hong & J. Jeong
Consider next
0
]
r j u ]hj A hj
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
h
@ 1 u ]1 ðÞ
1
1
@ 3 u ]1 ðÞ
C
h
C
C
1
@ 3 u ]2 ðÞ C
@ 2 u ]2 ðÞ
C
h
C
A
1
]
]
@ 3 u 3 ðÞ
@ 2 u 3 ðÞ
h
1
1
[hj ;3 [hj ;2
C
hj
C
C
[
;
0
hj ;1 C
C
A
[hj ;1
0
@ 2 u ]1 ðÞ
B
B
B
]
¼B
B @ 1 u 2 ðÞ
B
@
@ 1 u ]3 ðÞ
0
0
B
B
B
B 1 [
B h hj ;3
@ j
[hj ;2
[h . Along the sequence ðu ] ; A
[h Þ we have by construction through
where [hj :¼ axl A
hj
j
j
the local minimisation step
]
]
[
hom
hom
Þ W mp
Þ ¼ W mp
Wmp ðr j u ]hj A
ðru ]hj ; axl A
ðru ]hj ; axl A hj Þ;
hj
hj
h
ð6:14Þ
]
is not seen by the
and the last equality holds since the third component of axl A
hj
homogenised energy. Hence, integrating and taking the lim inf also
Z
Z
h
[
hom
] ÞdV lim inf
lim inf
Wmp ðr j u ]hj A
W mp
ðru ]hj ; axl A hj ÞdV : ð6:15Þ
hj
hj
hj
1
1
[h Þ * ðu ] ; A
[0 Þ, together with the convexity
Now we use weak convergence of ðu ]hj ; A
0
j
R
hom
of
w.r.t. ðrv; axl AÞ
W mp
ðrv; axl AÞdV
to get lower semicontinuity of the right1
hand side in (6.15) and to obtain altogether
Z
Z
h
[
hom
] ÞdV lim inf
Wmp ðr j u ]hj A
W mp
ðru ]0 ; axl A 0 ÞdV :
hj
hj
1
ð6:16Þ
1
[h ðÞ with
Consider next the curvature energy along the sequence [hj ðÞ ¼ axl A
j
00
11
1
@ 3 [1 ðÞ
@ [ ðÞ
@ 2 [1 ðÞ
BB 1 1
CC
h
BB
CC
B
CC
B
1
h
[
[
[
[
b
C
B
B
Wcurv ðr ðÞÞ ¼ Wcurv BB @ 1 2 ðÞ
@ 3 2 ðÞ C
@ 2 2 ðÞ
CC: ð6:17Þ
h
BB
CC
@@ 1
AA
1
1
[
@ 1 [3 ðÞ
@ 2 [3 ðÞ
@
ðÞ
h
h
h 2 3 3
This motivates one to get a trivial lower bound by de¯ning
00
@ 1 1 @ 2 1
hom
@
@
W curv ðrÞ :¼
inf
Wcurv
@ 1 2 @ 2 2
p~2 ;p~3 ;p~4 ;p~5 ;p~6 2R
p~6
p~5
11
p~2
p~3 AA:
p~4
ð6:18Þ
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
The in¯mizing values are obtained as
3
p~ 2 ¼ 0; p~ 3 ¼ 0; p~ 4 ¼ ð@ þ @ 2 2 Þ;
21 þ 3 1 1
p~ 5 ¼ 0;
p~ 6 ¼ 0;
1571
ð6:19Þ
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
such that the homogenised reduced curvature density is given by
b2 1 3
L
hom
W curv
ðrÞ :¼ c 1 jjsym r1 ;2 ð1 ; 2 Þjj 2 þ
tr½r1 ;2 ð1 ; 2 Þ 2 :
2
21 þ 3
By construction we have along the sequence [hj
hom
b h [h Þ W curv
Wcurv ðr
ðr [hj Þ:
j
ð6:20Þ
hom
Integrating the last inequality, taking the lim inf on both sides and using that W curv
is convex (quadratic) in its argument, together with weak convergence of the two inplane components of the curvature tensor, i.e.
[0 2 L 2 ð1 ; glð3ÞÞ;
r1 ;2 ð [hj ;1 ; [hj ;2 Þ * r1 ;2 ð [0;1 ; [0;2 Þ ¼ r1 ;2 axl A
(see Theorem A.1) we obtain
Z
Z
[h ÞdV ¼ lim inf
b h axl A
b h [ ÞdV
lim inf
Wcurv ðr
Wcurv ðr
hj
j
hj
hj
1
Z 1
hom
lim inf
W curv
ðr [hj ÞdV
hj
1
Z
[
hom
W curv
ðr axl A 0 ÞdV :
ð6:21Þ
ð6:22Þ
1
Then, because Wcurv ; Wmp 0,
Z
] Þ þ Wcurv ðr
[h ÞdV
b h axl A
lim inf
Wmp ðr h u ]hj A
hj
j
hj
1
Z
Z
h
]
[h ÞdV
b h axl A
lim inf
Wmp ðr j u ]hj A hj ÞdV þ lim inf
Wcurv ðr
j
Z
hj
1
1
[
hom
W mp
ðru ]0 ; axl A 0 ÞdV þ
hj
Z
1
1
[
hom
W curv
ðr axl A 0 ÞdV ;
ð6:23Þ
[0 are both independent of the
where we used (6.16) and (6.22). Now we use that u ]0 ; A
transverse variable 3 to obtain altogether the desired lim inf inequality
[0 Þ lim inf I ];[ ðu ] ; A
[h Þ
I 0];[ ðu ]0 ; A
hj
hj
j
ð6:24Þ
hj
for
[0 Þ
I 0];[ ðu0 ; A
Z
1
2
Z
:¼
Z
1
2
!
[
hom
hom
[0 Þ þ W curv
W mp
ðru0 ; axl A
ðr axl A 0 ÞdV
[
¼
!
[
hom
hom
0 Þ þ W curv
0 Þ d!:
W mp
ðru0 ; A
ðr axl A
1572
P. Ne®, K.-I. Hong & J. Jeong
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
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
2
0
0
11
0 p1 2
Z
6
0 1 AA
inf
4Wmp @ðrvjbÞ @ p1
b2R 3 ;p1 2R 4 1
2 1
0
00
@ 1 1
BB @ þ Wcurv @@ 1 2
@ 1 p1
@ 2 1
@ 2 2
@ 2 p1
1 13
p2
C7
p3 C
AA5dV
p4
Z
¼
1
hom
hom
W mp
ðrv; Þ þ W curv
ðrÞdV :
ð6:25Þ
The minimisation problem is in principle a global PDE-problem, since r1 ;2 p1
appears 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 symmetric part.
Let us use the precise form of the energy to see what is going on. We write
2
00
113
0
0
11
@ 1 1 @ 2 1 p2
0 p1 2
6
BB
CC7
0 1 AA þ Wcurv @@ @ 1 2 @ 2 2 p3 AA5
4Wmp @ðrvjbÞ @ p1
@ 1 p1 @ 2 p1 p4
2 1
0
0
1
0 p1 2 2
2
@ p1
¼ jjsymðrvjbÞjj 2 þ c 0 1 A
þ 2 tr½ðrvjbÞ
skewðrvjbÞ 2 1
0
0
1 2
0
1 2
@ 1 1 @ 2 1 p2 @ 1 1 @ 2 1 p2 B
C
B
C
þ 1 sym@ @ 1 2 @ 2 2 p3 A þ 2 skew@ @ 1 2 @ 2 2 p3 A
@ 1 p1 @ 2 p1 p4 @ 1 p1 @ 2 p1 p4 3
ð@ 1 1 þ @ 2 2 þ p4 Þ 2
2
0
1 2
0
1
@ 1 v1 @ 2 v1 b1 0 p1 2 2
B
C
¼ sym@ @ 1 v2 @ 2 v2 b2 A þ c skewðrvjbÞ @ p1
0 1 A
@ 1 v3 @ 2 v3 b3 2 1
0
ð@ v þ @ 2 v2 þ b3 Þ 2
þ
2 1 1
0
1
0 2
@ 1 1 @ 2 1
1 2
2
@
þ 1 0 A
þ 2 ðp2 þ @ 1 p1 Þ þ ðp3 þ @ 2 p1 Þ
sym @ 1 2 @ 2 2
0
0
p4 þ
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
0
@ 1 1
B@ þ 2 skew@ 1 2
@ 1 p1
@ 2 1
@ 2 2
@ 2 p1
1573
1 2
p2 3
p3 C
ð@ 1 1 þ @ 2 2 þ p4 Þ 2
A þ
2
p4 0
1
@ 1 v1 @ 2 v1
0 2
2
2
@
¼ 0 A
þ 2 ðb1 þ @ 1 v3 Þ þ ðb2 þ @ 2 v3 Þ
sym @ 1 v2 @ 2 v2
0
0
b3 2
@ 1 v1 @ 2 v1
0 p1 þ c skew
@ 1 v2 @ 2 v2
p1 0 0
0
B 0
þ c skew@
@ 1 v3
0
0
@ 2 v3
1 0
1 2
b1
0
0 2 b2 C
0 1 A
A@ 0
0
2 1 0
ð@ v þ @ 2 v2 þ b3 Þ 2
2 1 1
0
1
0 2
@ 1 1 @ 2 1
1 2
2
@
þ 1 0 A
þ 2 ðp2 þ @ 1 p1 Þ þ ðp3 þ @ 2 p1 Þ
sym @ 1 2 @ 2 2
0
0
p4 2
@ 1 1 @ 2 1 þ 2 ðp2 @ p1 Þ 2 þ ðp3 @ p1 Þ 2
þ 2 skew
1
2
@ 1 2 @ 2 2
2
þ
þ
3
ð@ þ @ 2 2 þ p4 Þ 2 :
2 1 1
ð6:26Þ
Grouping the di®erent expressions together we see that for the symmetric case 2 ¼ 0
the vector ðb ; p Þ 2 R 7 , which realises the in¯mum, can be explicitly determined.
The calculation is lengthy but otherwise straightforward. We obtain
1
c 2c
@ 1 v3 þ
2
C
B þ c
þ c
C
B
C
B
C
0 1 B c 2c
C
B
b1
@
v
B þ c 2 3 þ c 1 C
C
B b C B
C
B 2C B
C
B C B
B b3 C B ð@ 1 v1 þ @ 2 v2 Þ C
C
B C B
2 þ C:
B p1 C ¼ B
C
B C B
C
B p C B
@ 1 v2 @ 2 v1
C
B 2C B
C
B C B
2
C
@ p3 A B
C
B
C
B
@ 1 p 1
p 4
C
B
C
B
@
p
2 1
C
B
A
@
3
ð@ 1 1 þ @ 2 2 Þ
21 þ 3
0
ð6:27Þ
1574
P. Ne®, K.-I. Hong & J. Jeong
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
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
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
[0 Þ 2 X ¼ L 2 ð1 ; R 3 Þ L 2 ð1 ; soð3ÞÞ
8 ðu0 ; A
9 u ]hj ! u0
in L 2 ð1 ; R 3 Þ;
[h ! A [0
A
j
in L 2 ð1 ; soð3ÞÞ :
ð6:28Þ
[h Þ I ];[ ðu0 ; A
[0 Þ:
lim sup I h];[j ðu ]hj ; A
0
j
hj
[0 Þ < 1. In this case the uniform
Observe that this is now only a condition if I 0];[ ðu0 ; A
[
];[
]
h Þ over X 0 ¼ H 1;2 ð1 ; R 3 Þ H 1;2 ð1 ; soð3ÞÞ implies that we
coercivity of I hj ðu hj ; A
j
[h Þ converging weakly to some ðu0 ; A
[0 Þ 2
can restrict attention to sequences ðu ] ; A
hj
j
H 1;2 ð!; R 3 Þ H 1;2 ð!; 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 ]hj ð1 ; 2 ; 3 Þ ¼ u ]hj ð1 ; 2 ; 0Þ þ @ 3 u ]hj ð1 ; 2 ; 0Þ3 þ ð6:29Þ
and b ð1 ; 2 Þ replaces the term h1j @ 3 u ]hj ð1 ; 2 ; 3 Þ, the natural candidate for the
recovery sequence for the bulk displacement is given by the \reconstruction"
u ]hj ð1 ; 2 ; 3 Þ :¼ u0 ð1 ; 2 Þ þ hj 3 b ð1 ; 2 Þ ¼ u0 ð1 ; 2 Þ þ hj 3 b ð1 ; 2 Þ
1
0
ðc Þ@ 1 u0;3 þ 2c [0;2
C
B
þ c
C
B
C
B
B ð Þ@ u 2 [ C
c
2 0;3
c 0;1 C;
¼ u0 ð1 ; 2 Þ þ hj 3 B
C
B
þ c
C
B
C
B
@ ð@ 1 u0;1 þ @ 2 u0;2 Þ A
ð6:30Þ
2 þ where we have used the de¯nition of b given in (6.27). Observe that b 2 L 2 ð!; R 3 Þ.
Convergence of u ]hj in L 2 ð1 ; R 3 Þ to the limit u0 as hj ! 0 is obvious.
h If the curvature energy depends also on the non-symmetric part of the curvature tensor, i.e. if > 0,
2
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-dimensional
model 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.
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1575
[0 is only slightly more comThe reconstruction for the in¯nitesimal rotation A
plicated. In terms of the axial representation we write
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
0
1
1
hj 3 p 2 ð1 ; 2 Þ
B
C B
C
[hj ð1 ; 2 ; 3 Þ ¼ @ [0;2 ð1 ; 2 Þ A þ @ hj 3 p 3 ð1 ; 2 Þ A
h 2j 3 p 4 ð1 ; 2 Þ
hj p 1 ð1 ; 2 Þ
0 [
1 0
1
hj 3 @ 1 p 1 ð1 ; 2 Þ
0;1 ð1 ; 2 Þ
B
C B
C
¼ @ [0;2 ð1 ; 2 Þ A þ @ hj 3 @ 2 p 1 ð1 ; 2 Þ A
[0;1 ð1 ; 2 Þ
0
h 2j 3 p 4 ð1 ; 2 Þ
hj p 1 ð1 ; 2 Þ
0
B
B
¼B
@
1
0
@ 1 u0;2 @ 2 u0;1
2
@ 1 u0;2 @ 2 u0;1
hj 3 @ 2
2
hj 3 @ 1
1
C
B
C
B
C
B
C
C
C B
[0;2 ð1 ; 2 Þ
C;
CþB
C
B
A
@ 1 u0;2 @ 2 u0;1
C
B
[
[
hj
@
3 ð@ 1 0;1 þ @ 2 0;2 Þ A
2
2
h j 3
21 þ 3
[0;1 ð1 ; 2 Þ
ð6:31Þ
where we have used (6.27). Again it is clear that [hj ! [0 2 L 2 ð1 ; R 3 Þ as hj ! 0.
Both reconstructions are completely given in terms of the two-dimensional functions
ðu0 ; [0 Þ. 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;2 ð!; R 3 Þ such
that jjb" b jjL 2 ð!;R 3 Þ < " and similarly for p choose p " 2 H 2;2 ð!; R 4 Þ such that
jjp " p jjL 2 ð!;R 4 Þ < ". This allows us to present ¯nally our recovery sequence
u ]hj ;" ð1 ; 2 ; 3 Þ :¼ u0 ð1 ; 2 Þ þ hj 3 b" ð1 ; 2 Þ;
0
1
1
hj 3 @ 1 p 1;" ð1 ; 2 Þ
B
C B
C
[hj ;" ð1 ; 2 ; 3 Þ :¼ @ [0;2 ð1 ; 2 Þ A þ @ hj 3 @ 2 p 1;" ð1 ; 2 Þ A;
h 2j 3 p 4;" ð1 ; 2 Þ
hj p 1;" ð1 ; 2 Þ
and correspondingly
0
0
B
[h ;" :¼ B h p ð ; Þ
A
j
@ j 1;" 1 2
[0;2 ð1 ; 2 Þ
0
0
[0;1 ð1 ; 2 Þ
hj p 1;" ð1 ; 2 Þ
0
[0;1 ð1 ; 2 Þ
0
[0;2 ð1 ; 2 Þ
ð6:32Þ
1
C
[0;1 ð1 ; 2 Þ C
A
0
h 2j 3 p 4;" ð1 ; 2 Þ
B
þ @ h 2j 3 p 4;" ð1 ; 2 Þ
0
hj 3 @ 2 p 1;" ð1 ; 2 Þ hj 3 @ 1 p 1;" ð1 ; 2 Þ
hj 3 @ 2 p 1;" ð1 ; 2 Þ
1
C
hj 3 @ 1 p 1;" ð1 ; 2 Þ A;
0
1576
P. Ne®, K.-I. Hong & J. Jeong
0
0
B [;[
A hj ;" :¼ B
@ p 1;" ð1 ; 2 Þ
[0;2 ð1 ; 2 Þ
0
p 1;" ð1 ; 2 Þ
0
1
C
[0;1 ð1 ; 2 Þ C
A
[0;1 ð1 ; 2 Þ
0
hj 3 p 4;" ð1 ; 2 Þ
0
B
þ @ hj 3 p 4;" ð1 ; 2 Þ
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
[0;2 ð1 ; 2 Þ
hj 3 @ 2 p 1;" ð1 ; 2 Þ
hj 3 @ 1 p 1;" ð1 ; 2 Þ C
A;
0
hj 3 @ 2 p 1;" ð1 ; 2 Þ hj 3 @ 1 p 1;" ð1 ; 2 Þ
0
0
B
[;[
B A
0;" :¼ @ p 1;" ð 1 ; 2 Þ
[0;2 ð1 ; 2 Þ
0
p 1;" ð1 ; 2 Þ
0
[0;1 ð1 ; 2 Þ
[0;2 ð1 ; 2 Þ
1
0
1
C
[0;1 ð1 ; 2 Þ C
A;
0
0
p 1 ð1 ; 2 Þ
[0;2 ð1 ; 2 Þ
0
0
[0;2 ð1 ; 2 Þ
0
0
[0;2 ð1 ; 2 Þ
[0;1 ð1 ; 2 Þ
1
B C
[;[
A 0 :¼ B
0
[0;1 ð1 ; 2 Þ C
@ p 1 ð1 ; 2 Þ
A;
0
[0;2 ð1 ; 2 Þ [0;1 ð1 ; 2 Þ
0
B
[
A 0 :¼ B
@
1
C
[
[0;1 ð1 ; 2 Þ C
A ¼ antið 0 Þ:
ð6:33Þ
0
The de¯nition (6.32) implies
ru ]hj ;" ð1 ; 2 ; 3 Þ ¼ ðru0 ð1 ; 2 Þjhj b" ð1 ; 2 ÞÞ þ hj 3 ðrb" ð1 ; 2 Þj0Þ;
r [hj ;" ð1 ; 2 ; 3 Þ
0
1
@ 1 [0;1 ð1 ; 2 Þ
@ 2 [0;1 ð1 ; 2 Þ 0
B
C
[
¼B
@ 2 [0;2 ð1 ; 2 Þ 0 C
@ @ 1 0;2 ð1 ; 2 Þ
A
hj @ 1 p 1;" ð1 ; 2 Þ hj @ 2 p 1;" ð1 ; 2 Þ 0
0
hj 3 @ 1 @ 1 p 1;" ð1 ; 2 Þ hj 3 @ 2 @ 1 p 1;" ð1 ; 2 Þ hj @ 1 p 1;" ð1 ; 2 Þ
1
B
C
C
þB
@ hj 3 @ 1 @ 2 p 1;" ð1 ; 2 Þ hj 3 @ 2 @ 2 p 1;" ð1 ; 2 Þ hj @ 2 p 1;" ð1 ; 2 Þ A:
h 2j 3 @ 1 p 4;" ð1 ; 2 Þ
h 2j 3 @ 2 p 4;" ð1 ; 2 Þ
h 2j p 4;" ð1 ; 2 Þ
ð6:34Þ
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1577
Note that by appropriately choosing hj and " > 0 we can arrange that strong convergence of (6.34) to the correct limit still exists. Now abbreviate further
"
[;[ 2 glð3Þ;
E~ hj :¼ ½ðru0 ð1 ; 2 Þjb" ð1 ; 2 ÞÞ þ hj 3 ðrb" ð1 ; 2 Þj0Þ A
hj ;"
"
[;[
E~ 0 :¼ ðru0 ð1 ; 2 Þjb" ð1 ; 2 ÞÞ A
2
glð3Þ;
0;"
[;[
~
E :¼ ðru ð ; Þjb ð ; ÞÞ A 2 glð3Þ;
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
0
0
1
2
1
2
0
@ 1 [0;1 ð1 ; 2 Þ @ 2 [0;1 ð1 ; 2 Þ 0
1
C
~ [ :¼ B
K
@ @ 1 [0;2 ð1 ; 2 Þ @ 2 [0;2 ð1 ; 2 Þ 0 A
hj ;"
@ 1 p 1;" ð1 ; 2 Þ @ 2 p 1;" ð1 ; 2 Þ 0
0
1
hj 3 @ 1 @ 1 p 1;" ð1 ; 2 Þ hj 3 @ 2 @ 1 p 1;" ð1 ; 2 Þ @ 1 p 1;" ð1 ; 2 Þ
B
C
þ @ hj 3 @ 1 @ 2 p 1;" ð1 ; 2 Þ hj 3 @ 2 @ 2 p 1;" ð1 ; 2 Þ @ 2 p 1;" ð1 ; 2 Þ A;
hj 3 @ 1 p 4;" ð1 ; 2 Þ
hj 3 @ 2 p 4;" ð1 ; 2 Þ
p 4;" ð1 ; 2 Þ
0
@ 1 [0;1 ð1 ; 2 Þ @ 2 [0;1 ð1 ; 2 Þ @ 1 p 1;" ð1 ; 2 Þ
1
C
~ [0;" :¼ B
K
@ @ 1 [0;2 ð1 ; 2 Þ @ 2 [0;2 ð1 ; 2 Þ @ 2 p 1;" ð1 ; 2 Þ A;
p 4;" ð1 ; 2 Þ
@ 1 p 1;" ð1 ; 2 Þ @ 2 p 1;" ð1 ; 2 Þ
0
@ 1 [0;1 ð1 ; 2 Þ @ 2 [0;1 ð1 ; 2 Þ @ 1 p 1 ð1 ; 2 Þ
1
C
~ [0 :¼ B
K
@ @ 1 [0;2 ð1 ; 2 Þ @ 2 [0;2 ð1 ; 2 Þ @ 2 p 1 ð1 ; 2 Þ A;
p 4 ð1 ; 2 Þ
@ 1 p 1 ð1 ; 2 Þ @ 2 p 1 ð1 ; 2 Þ
0
@ 1 [0;1 ð1 ; 2 Þ @ 2 [0;1 ð1 ; 2 Þ 0
1
B
C
K [0 :¼ @ @ [0;2 ð1 ; 2 Þ @ [0;2 ð1 ; 2 Þ 0 A ¼ r [0 2 glð3Þ:
1
2
0
0
0
ð6:35Þ
We note that
~[ K
~ [0;" jjL 2 ð ;M 33 Þ ! 0 if hj ! 0;
jjK
hj ;"
1
~ [0;" K
~ [0 jjL 2 ð ;M 33 Þ ¼ jjp 4;" p 4 jjL 2 ð ;RÞ ! 0
jjsym½K
1
1
"
"
if h ! 0;
jjE~ E~ jj 2
33 ! 0
hj
"
~
jjE 0 0 L ð1 ;M
Þ
E~ jjL 2 ð1 ;M 33 Þ ! 0
if ! 0;
ð6:36Þ
j
if " ! 0:
The abbreviations in (6.35) imply
[h ;" Þ
I h];[j ðu ]hj ;" ; A
j
Z
¼
1
"
~ [ ÞdV ;
Wmp ðE~ hj Þ þ Wcurv ðK
hj ;"
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 h1j in the de¯nition of I h];[j , similarly for the other
1578
P. Ne®, K.-I. Hong & J. Jeong
components. Whence, adding and subtracting Wmp ðE~ Þ
Z
"
[
~ [ ÞdV
I h];[j ðu ]hj ;" ; A hj ;" Þ ¼
Wmp ðE~ Þ þ Wmp ðE~ hj Þ Wmp ðE~ Þ þ Wcurv ðK
hj ;"
Z
1
¼
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
1
"
Wmp ðE~ Þ þ Wmp ðE~ þ E~ hj E~ Þ Wmp ðE~ Þ þ Wcurv ðKhj ÞdV
since Wmp and Wcurv are both positive, we get from the triangle inequality
Z
"
~ [ ÞdV
Wmp ðE~ Þ þ jWmp ðE~ þ E~ hj E~ Þ Wmp ðE~ Þj þ Wcurv ðK
hj ;"
1
expanding the quadratic energy Wmp we obtain
Z
"
¼
Wmp ðE~ Þ þ jWmp ðE~ Þ þ hDWmp ðE~ Þ; E~ hj E~ i
1
"
"
~ [ ÞdV
þ D 2 Wmp ðE~ Þ ðE~ hj E~ ; E~ hj E~ Þ Wmp ðE~ Þj þ Wcurv ðK
hj ;"
Z
"
"
~ [ ÞdV
Wmp ðE~ Þ þ jjDWmp ðE~ ÞjjjjE~ hj E~ jj þ C jjE~ hj E~ jj 2 þ Wcurv ðK
hj ;"
1
"
for jjE~ hj E~ jj 1 we have
Z
"
~ [ ÞdV
Wmp ðE~ Þ þ ðC þ jjDWmp ðE~ ÞjjÞjjE~ hj E~ jj þ Wcurv ðK
hj ;"
1
since jjDWmp ðE~ Þjj C2 jjE~ jj we obtain
Z
"
~ [ ÞdV
Wmp ðE~ Þ þ ðC þ C2 jjE~ jjÞjjE~ hj E~ jj þ Wcurv ðK
hj ;"
ð6:38Þ
1
and by H€
older's inequality we get
Z
~ [ ÞdV þ ðC þ C2 jjE~ jjL 2 ð Þ ÞjjE~ "h E~ jjL 2 ð Þ :
Wmp ðE~ Þ þ Wcurv ðK
hj ;"
1
j
1
1
~ [ Þ, adding and subtracting E~ "0 we
Continuing the estimate with regard to Wcurv ðK
hj ;"
may obtain
Z
[
~ [0 Þ þ Wcurv ðK
~[ Þ
I h];[j ðu ]hj ;" ; A hj ;" Þ Wmp ðE~ Þ þ Wcurv ðK
hj ;"
1
~ [0 ÞdV
Wcurv ðK
"
"
"
þ ðC þ C2 jjE~ jjL 2 ð1 Þ ÞjjE~ hj E~ 0 þ E~ 0 E~ jjL 2 ð1 Þ
Z
~ [0 ÞdV
Wmp ðE~ Þ þ Wcurv ðK
1
~ [ Þ Wcurv ðK
~ [0;" ÞjjL 1 ð Þ
þ jjWcurv ðK
hj ;"
1
~ [0;" Þ Wcurv ðK
~ [0 ÞjjL 1 ð Þ
þ jjWcurv ðK
1
"
"
"
þ ðC þ C2 jjE~ jjL 2 ð1 Þ ÞðjjE~ hj E~ 0 jjL 2 ð1 Þ þ jjE~ 0 E~ jjL 2 ð1 Þ Þ:
ð6:39Þ
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1579
Now take hj ! 0 to obtain by the continuity of Wcurv (the argument is similar to
(6.41)) and (6.36)3
Z
[
];[
]
~ [0 ÞdV
lim sup I hj ðu hj ;" ; A hj ;" Þ Wmp ðE~ Þ þ Wcurv ðK
hj !0
1
~ [0;" Þ Wcurv ðK
~ [0 ÞjjL 1 ð Þ
þ jjWcurv ðK
1
"
~
~
~
þ ðC þ C2 jjE jjL 2 ð Þ ÞjjE 0 E jjL 2 ð Þ :
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
1
1
ð6:40Þ
Since the curvature energy depends only on the symmetric part, we observe also
~ [0;" Þ Wcurv ðK
~ [0 Þ ¼ jjp 4;" p 4 jj 2 :
Wcurv ðK
ð6:41Þ
Since
"
[;[
[;[
jjE~ 0 E~ jj 2 ¼ jjðru0 ð1 ; 2 Þjb" Þ ðru0 ð1 ; 2 Þjb Þ þ ðA 0; A 0 Þjj 2
[;[
[;[
2ðjjb" b jj 2 þ jjA 0;" A 0 jj 2 Þ ¼ 2ðjjb" b jj 2 þ 2jjp 1;" p 1 jj 2 Þ;
ð6:42Þ
we get, by letting " ! 0 and using (6.41), the bound
Z
[h ;" Þ ~ [0 ÞdV
lim sup I h];[j ðu ]hj ;" ; A
Wmp ðE~ Þ þ Wcurv ðK
j
hj !0
Z
1
¼
1
hom
hom
[0 Þ þ W curv
W mp
ðru0 ; A
ðK [0 ÞdV :
ð6:43Þ
[0 are two-dimensional (independent of the transverse variable), we may
Since u0 ; A
write as well
Z
hom
hom
[h ;" Þ [0 Þ þ W curv
lim sup I h];[j ðu ]hj ;" ; A
W mp
ðru0 ; A
ðK [0 ÞdV
j
hj !0
Z
1
¼
!
hom
hom
[0 Þ þ W curv
[0 Þ;
W mp
ðru0 ; A
ðK [0 Þd! ¼ I 0];[ ðu0 ; A
ð6:44Þ
which shows the desired upper bound. This ¯nishes the proof of Theorem 5.1.
Notation. Let R 3 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 R 3 we let ha; biR 3 denote the scalar product on R 3 with associated vector
norm jjajj 2R 3 ¼ ha; aiR 3 . We denote by M 33 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 M 33 is given by hX; Y iM 33 ¼ tr½XY T ,
and thus the Frobenius tensor norm is jjXjj 2 ¼ hX; XiM 33 . In the following we
omit the index R 3 ; M 33 . The identity tensor on M 33 will be denoted by 1, so that
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
1580
P. Ne®, K.-I. Hong & J. Jeong
tr½X ¼ hX; 1i. We set symðXÞ ¼ 12 ðX T þ XÞ and skewðXÞ ¼ 12 ðX X T Þ such that
X ¼ symðXÞ þ skewðXÞ. For X 2 M 33 we set for the deviatoric part dev X ¼
X 13 tr½X1 2 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 2
GLð3ÞjX T X ¼ 1; det X ¼ 1g is given by the set soð3Þ :¼ fX 2 M 33 jX T ¼ Xg of
all skew symmetric tensors. The canonical identi¯cation of soð3Þ and R 3 is denoted by
¼A
for all 2 R 3 , such that
axl A 2 R 3 for A 2 soð3Þ. Note that ðaxl AÞ
0
1
0
1
0
3
X
axl@ 0 A :¼ @ A; Aij ¼
ijk axl Ak ;
k¼1
0
2 3;
2 33 ¼ 2jjaxl Ajj
jjAjj
M
R
Bi
axl Bi
M 33 ¼ 2haxl A;
R3 ;
hA;
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
1
0
1
0
1
@ 0 A :¼ anti@ A; axlðskewða bÞÞ ¼ a b;
2
0
and 2 skewðb aÞ ¼ antiða bÞ ¼ antiðantiðaÞ bÞ. Moreover, curl u ¼ 2 axlðskew ruÞ.
Notation for plates and shells. Let ! R 2 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
by M mn the set of matrices mapping R n 7! R m . For H 2 M 32 and 2 R 3 we write
ðHjÞ 2 M 33 for the matrix composed of H and the column . Likewise ðvjjÞ is the
matrix composed of the columns v; ; . This allows us to write for u 2 C 1 ðR 3 ; R 3 Þ :
ru ¼ ðux juy juz Þ ¼ ð@ x uj@ y uj@ z uÞ. The identity tensor on M 22 is 12 . The mapping
m : ! R 2 7! R 3 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 M 33 can
now be written as X ¼ ðX e2 jX e2 jX e3 Þ ¼ ðX1 jX2 jX3 Þ. We write v : R 2 7! R 3 for
the de°ection of the midsurface, such that mðx; yÞ ¼ ðx; y; 0Þ T þ vðx; yÞ. The standard volume element is dx dy dz ¼ dV ¼ d! dz.
Appendix
A.1. The in¯nitesimal ReissnerMindlin
membrane-bending model
The ReissnerMindlin membrane-bending plate model is a ¯ve-parameter model:
three midsurface de°ections v : ! R 2 7! R 3 and two out-of-plane rotation parameters : ! 7! R 2 describing the in¯nitesimal increment of the director. It should
be noted that in the classical ReissnerMindlin model drill rotations are absent.
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1581
The model reads
0
1
Z
2
B
C
1 2C
2
hB
@ jjsym rðv1 ; v2 Þjj þ 2 rv3 2 þ 2 þ tr½sym rðv1 ; v2 Þ A
!
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
transverse shear energy
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
þ
3
h
tr½sym rð1 ; 2 Þ 2
jjsym rð1 ; 2 Þjj 2 þ
2 þ 12
hf; vid! 7! min: w:r:t: ðv; Þ;
vj0 ¼ u d ðx; y; 0Þ;
j0 ¼ ðu d1;z ; u d2;z ; 0Þ T ;
simply supported
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 : ! R 2 7! R 3 we have to ¯nd a
solution of the minimisation problem
Z 2
2
tr½sym rðv1 ; v2 Þ
h jjsym rðv1 ; v2 Þjj þ
2 þ !
h3
2
2
2
2
tr½D v3 hf; vi d! 7! min: w:r:t: v;
jjD v3 jj þ
þ
2 þ 12
vj0 ¼ u d ðx; y; 0Þ;
rv3j0 ¼ ðu d1;z ; u d2;z ; 0Þ T ;
simply supported
typical rigid prescription of the infinitesimal normal:
ðA:2Þ
This 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
the shear correction factor is directly determined by the Cosserat couple modulus c compared
with (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.
i Hence
1582
P. Ne®, K.-I. Hong & J. Jeong
] and uniform positivity assumption for the curvature
in¯nitesimal microrotations A
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 midsurface v : ! R 2 7! R 3 and the microrotation vector : ! R 2 7! R 3 such that
Z
asymp
asymp
ðrv; Þ þ W curv
ðrÞ
I 0asymp ðv; Þ ¼ W mp
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
!
hf; vid! 7! min: w:r:t: ðv; Þ
ðA:3Þ
and the boundary conditions of place for the midsurface de°ection v on the Dirichlet
part of the lateral boundary 0 @!,
vj0 ¼ ud ðx; y; 0Þ simply supported
and the homogeneous boundary condition for the microrotation
j@! ¼ 0;
completely clamped:
The asymptotically reduced local density is
2
0 3
asymp
W mp ðrv; Þ :¼ sym r1 ;2 ðv1 ; v2 Þ 3 0
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
shear-stretch energy
2
0 3
þ c skew r1 ;2 ðv1 ; v2 Þ 3 0
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
in-plane drill energy
2
c 2 þ 2
r
v 1 þ c 1 ;2 3
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
asymptotic transverse shear energy
2
0 3
tr sym r1 ;2 ðv1 ; v2 Þ þ
:
3 0
2 þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ðA:4Þ
asymptotic elongational stretch energy
The asymptotically correct curvature density is given by
0
asymp
ðrÞ :¼ W curv
b 2c B
L
B
B jjsym r1 ;2 ð1 ; 2 Þjj 2 þ 2 jjskew r1 ;2 ð1 ; 2 Þjj 2
2 @ 1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
I-energy
II-energy
1
C
2
1 3
C
þ 21
jjr1 ;2 3 jj 2 þ
tr½r1 ;2 ð1 ; 2 Þ 2 C:
1 þ 2
21 þ 3 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}A
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
IV-energy
III-energy
ðA:5Þ
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1583
While 2 ¼ 0 would give formally the ReissnerMindlin 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; 23 ¼ 13 in which
case the dimensionally reduced curvature turns into
b 2c
L
jjdev2 sym r1 ;2 ð1 ; 2 Þjj 2 :
ðA:6Þ
2
This case is not 2Dwell-posed! It is straightforward to show that this asymptotic
] in the
limit model coincides with the -limit for simultaneous nonlinear scaling u ] ; A
strong topology of L 2 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 3
cannot be avoided and therefore, this is not the ReissnerMindlin model, for no
choice of (derivation-) admissible Cosserat parameters.
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
asymp; conf
W curv
ðrÞ :¼ A.4. A model based on linear scaling of u and nonlinear scaling
i.e. u [ , A
]
of 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
00
11
@ 1 v1 @ 2 v1 p1
inf W ððrvjbÞÞ ¼ inf3 W @@ @ 1 v2 @ 2 v2 p2 AA
b2R 3
p2R
p1
p2
p3
ðA:7Þ
for W ðXÞ ¼ jjsym Xjj 2 þ tr½X 2 :
2
The model is: ¯nd the de°ection of the midsurface v : ! R 2 7! R 3 and the microrotation vector : ! R 2 7! R 3 such that
Z
[;]
asymp
asymp
I 0 ðv; Þ ¼ W mp
ðrv; Þ þ W curv
ðrÞ 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
part of the lateral boundary 0 @!,
vj0 ¼ ud ðx; y; 0Þ ¼ ud ðx; y; 0Þ;
simply supported ðfixed; weldedÞ:
and the homogeneous boundary condition for the microrotation
j ¼ 0;
0
j Not
partially clamped:
true for the curvature energy depending also on anti-symmetric terms for 2 > 0.
1584
P. Ne®, K.-I. Hong & J. Jeong
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
The asymptotically reduced local density is
asymp
W mp
ðrv; Þ :¼ jjsym r1 ;2 ðv1 ; v2 Þjj 2
2
0 3
þ c skew r1 ;2 ðv1 ; v2 Þ 3 0
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
in-plane drill energy
tr½r1 ;2 ðv1 ; v2 Þ 2 :
þ
2 þ ðA:9Þ
The asymptotically correct curvature density is given by
0
asymp
W curv
ðrÞ :¼ b 2c B
L
B
B jjsym r1 ;2 ð1 ; 2 Þjj 2 þ 2 jjskew r1 ;2 ð1 ; 2 Þjj 2
2 @ 1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
I-energy
II-energy
1
C
2
1 3
C
þ 21
jjr1 ;2 3 jj 2 þ
tr½r1 ;2 ð1 ; 2 Þ 2 C:
1 þ 2
21 þ 3 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}A
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
IE-energy
III-energy
ðA:10Þ
The 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
i.e. u [ , A
[
scaling of A,
The problem is: ¯nd the de°ection of the midsurface v : ! R 2 7! R 3 and the
microrotation vector : ! R 2 7! R 3 such that
Z
asymp
asymp
I 0[;[ ðv; Þ ¼ W mp
ðrv; Þ þ W curv
ðrÞ 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
part of the lateral boundary 0 @!,
vj0 ¼ ud ðx; y; 0Þ ¼ ud ðx; y; 0Þ;
simply supported ðfixed; weldedÞ:
and the homogeneous boundary condition for the microrotation
j0 ¼ 0;
partially clamped:
The asymptotically reduced local density is
asymp
ðrv; Þ :¼ jjsym r1 ;2 ðv1 ; v2 Þjj 2 þ
W mp
tr½sym r1 ;2 ðv1 ; v2 Þ 2 :
2 þ The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1585
The asymptotically correct curvature density is given by
0
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
asymp
ðrÞ :¼ W curv
b 2c B
L
B
B jjsym r1 ;2 ð1 ; 2 Þjj 2 þ 2 jjskew r1 ;2 ð1 ; 2 Þjj 2
2 @ 1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
I-energy
II-energy
1
C
2
1 3
C
þ 21
jjr1 ;2 3 jj 2 þ
tr½r1 ;2 ð1 ; 2 Þ 2 C:
1 þ 2
21 þ 3 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}A
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
IV-energy
III-energy
ðA:12Þ
If we identify ð1 ; 2 Þ 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
[ and consider
energy. To see this, abbreviate [ :¼ axl A
b [ ðÞ
sym r
0
h
@ 1 [1 ðÞ
B
B
B 1
[
[
¼B
B 2 ½@ 1 2 ðÞ þ @ 2 1 ðÞ
B
@ 1
½@ [ ðÞ þ @ 1 [3 ðÞ
2h 3 1
1
½@ [ ðÞ þ @ 1 [2 ðÞ
2 2 1
@ 2 [2 ðÞ
1
½@ [ ðÞ þ @ 2 [3 ðÞ
2h 3 2
1
1
½@ 3 [1 ðÞ þ @ 1 [3 ðÞ
C
2h
C
C
1
½@ 3 [2 ðÞ þ @ 2 [3 ðÞ C
C;
2h
C
A
1
[
@
ðÞ
h 2 3 1
1 2
0
1
@ 1 [1 ðÞ
@ 2 [1 ðÞ
@ 3 [1 ðÞ C
B
h
C
B
C
B
1
[
[
b h [ ðÞjj 2 ¼ symB @ [ ðÞ
C
jjsym r
@
@
ðÞ
ðÞ
2 2
3 2
C
B 1 2
h
C
B
A
@1
1
1
[
[
[
@ 1 3 ðÞ
@ 2 3 ðÞ
@ 3 1 ðÞ 2
h
h
h
! 2
@ 1 [1 ðÞ @ 2 [1 ðÞ ¼ sym
@ 1 [2 ðÞ @ 2 [2 ðÞ 0
1 2
0
0
@ 3 [1 ðÞ 1 B
C
þ 2 sym@
0
0
@ 3 [2 ðÞ A
h 0
@ 1 [3 ðÞ @ 2 [3 ðÞ
þ
1
ð@ [ ðÞÞ 2 jjsym r [ ðÞjj 2 :
h 4 3 3
ðA:13Þ
1586
P. Ne®, K.-I. Hong & J. Jeong
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! R 3 and assume that it
satis¯es either
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
(1) 8 hj > 0 : jj [hj jjL 2 ð1 ;R 3 Þ K1 ; jj [hj ;3 jjL 2 ð1 ;RÞ ! 0 as hj ! 0 or
(2) [hj j 10 ¼ 0.
Assume in addition the boundedness of scaled strains along hj
Z
b h [h ðÞjj 2 dV K2 ;
jjsym r
j
1
where the constants K1 ; K2 are independent of hj . Then
jj [hj jjH 1;2 ð1 ;R 3 Þ K3 ;
with a constant K3 independent of hj and there exists a weakly convergent subsequence in H 1 ð; R 3 Þ (without relabelling), such that
[hj * [0 2 H 1 ð; R 3 Þ;
hj ! 0;
[hj
hj ! 0:
!
[0
2
L 2 ð1 ; R 3 Þ;
In the ¯rst case we obtain moreover that [0 ð1 ; 2 ; 3 Þ ¼ [0 ð1 ; 2 Þ, independent of the
transverse 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)
[
0 ð1 ; 2 ; 3 Þ 2 VKL ð1 Þ.
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 1 ð; R 3 Þ. Rellich's compact embedding provides us with strong convergence in
L 2 ð; R 3 Þ. In the second case, the weak limit satis¯es the boundary condition
[0 j 10 ¼ 0. Boundedness of scaled strains and weak convergence of r [hj implies as
well (compare with Ciarlet8)
Z
jj½sym r [hj e3 jj 2 Kh 2j ! 0 ) ½sym r [hj e3 * ½sym r [0 e3 ¼ 0: ðA:14Þ
1
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 ; e3 i ¼ 0 in L 2 ð1 ; RÞ which gives the
result.
De¯nition A.2. (Space of scaled Kirchho®Love displacements VKL ) Following
Ciarlet, we de¯ne the space
VKL ð1 Þ :¼ f 2 H 1;2 ð1 ; R 3 Þ : j0 ¼ 0; ½sym r ðÞ e3 ¼ 0 for 2 1 g: ðA:15Þ
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1587
This space is equivalently characterised by (Refs. 8 or 15 or 11)
VKL ð1 Þ :¼ f 2 H 1;2 ð1 ; R 3 Þ : j 1 ¼ 0;
0
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
1 ð1 ; 2 ; 3 Þ ¼ w1 ð1 ; 2 Þ 3 @ 1 3 ð1 ; 2 Þ;
2 ð1 ; 2 ; 3 Þ ¼ w2 ð1 ; 2 Þ 3 @ 2 3 ð1 ; 2 Þ;
3 ð1 ; 2 ; 3 Þ ¼ w3 ð1 ; 2 Þ; w1 ; w2 2 H 1 ð!; RÞ; w3 2 H 2 ð!; RÞ;
w1 ; w2 ; w3j0 ¼ 0; @ w3j0 ¼ 0g:
ðA:16Þ
We ¯rst remark that the in-plane components 1 ; 2 are not necessarily twodimensional, although they are determined by two-dimensional functions.
A.7. The ¡-limit for linear elasticity and linear scaling u h[ j
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 conditions allows us to establish the suitable bounds. The -limit of I h[ j ðu [hj Þ in the strong
topology of L 2 ð1 ; R 3 Þ is given by the limit energy functional I 0[ : L 2 ð1 ; R 3 Þ 7! R,
8Z
< W hom ðrvÞ hf; vidV v 2 V ð Þ
KL
1
ðA:17Þ
I 0[ ðvÞ :¼
: 1
þ1
else in H 1;2 ð1 ; R 3 Þ;
with
W hom ðrvÞ :¼ jjsym r1 ;2 ðv1 ; v2 Þjj 2 þ
tr½r1 ;2 ðv1 ; v2 Þ 2 :
2 þ 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 : ! R 2 7! R 3 and setting vð1 ; 2 Þ :¼ Av v for v 2
VKL ð1 Þ. 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 h] j
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 ]hj 2 A 0 . Then there exists a constant C independent of
hj > 0 such that
Z
C
jjsym r h u ]hj ðÞjj 2 dV jju ]hj ðÞjj 2H 1;2 ð1 ;R 3 Þ :
h 2j 1
1588
P. Ne®, K.-I. Hong & J. Jeong
Proof. Can be found in Ref. 4, see also Ref. 9.
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
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 ] ]
I ðu hj Þ C:
h 2j
ðA:18Þ
Then Theorem A.3 allows one to establish weak compactness of u ]hj in H 1;2 ð1 ; R 3 Þ.
The -limit of h12 I ] in the strong topology of L 2 ð1 ; R 3 Þ is given by the limit energy
j
functional I 0] : L 2 ð1 ; R 3 Þ 7! R,
8Z
<
W hom ðrvÞ hf; vid! v 2 H 1;2 ð!; R 3 Þ
I 0] ðvÞ :¼
ðA:19Þ
: !
þ1
else in L 2 ð1 ; R 3 Þ:
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 membrane-plate.
Remark A.5. In the ¯nite strain setting the assumption
classical plate-bending problem.18
1
h 2j
I ] ð’ ]hj Þ C leads to the
Acknowledgements
K.-I.H. was supported by DAAD, Grant A/07/98690/. K.-I.H. is grateful for a threemonth 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 of
micropolar elastic shells from three-dimensional linearized micropolar elasticity, Asympt.
Anal. 51 (2007) 335361.
2. I. Aganovic, J. Tambaca and Z. Tutek, Derivation and justi¯cation of the models of rods
and plates from linearized three-dimensional micropolar elasticity, J. Elasticity 84 (2007)
131152.
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
The ReissnerMindlin Plate is the -Limit of Cosserat Elasticity
1589
3. S. M. Alessandrini, D. N. Arnold, R. S. Falk and A. L. Madureira, Derivation and justi¯cation of plate models by variational models, in Plates and Shells, Vol. 21, ed. M. Fortin
(Amer. Math. Soc., 1999), pp. 120.
4. G. Anzellotti, S. Baldo and D. Percivale, Dimension reduction in variational problems,
asymptotic development in -convergence and thin structures in elasticity, Asympt. Anal.
9 (1994) 61100.
5. F. Bourquin, P. G. Ciarlet, G. Geymonat and A. Raoult, -convergence et analyse
asymptotique des plaques minces, C. R. Acad. Sci. Paris, Ser. I 315 (1992) 10171024.
6. A. Braides, -Convergence for Beginners (Oxford Univ. Press, 2002).
7. D. Chapelle and K. J. Bathe, The Finite Element Analysis of Shells Fundamentals
(Springer, 2003).
8. P. G. Ciarlet, Mathematical Elasticity, Vol. II: Theory of Plates, 1st edn. (North-Holland,
1997).
9. P. G. Ciarlet, V. Lods and B. Miara, Asymptotic analysis of linearly elastic shells. II.
Justi¯cation of °exural shell equations, Arch. Rational Mech. Anal. 136 (1996)
1631190.
10. E. Cosserat and F. Cosserat, Theorie des corps deformables, Librairie Scienti¯que A.
Hermann et Fils [Engl. translation by D. Delphenich 2007, pdf available at http://www.
mathematik.tudarmstadt.de/fbereiche/analysis/pde/sta®/ne®/patrizio/Cosserat.html],
Paris, 1909.
11. P. Destuynder and M. Salaun, Mathematical Analysis of Thin Plate Models (Springer,
1996).
12. H. Le Dret and A. Raoult, The membrane shell model in nonlinear elasticity: A variational
asymptotic derivation, J. Nonlinear Sci. 6 (1996) 5984.
13. H. A. Erbay, An asymptotic theory of thin micropolar plates, Int. J. Engrg. Sci. 38 (2000)
14971516.
14. A. C. Eringen, Theory of micropolar plates, Z. Angew. Math. Phys. 18 (1967) 1230.
15. I. Fonseca and G. Francfort, On the inadequacy of the scaling of linear elasticity for
3d2d asymptotics in a nonlinear setting, J. Math. Pures Appl. 80 (2001) 547562.
16. G. Friesecke, R. D. James, M. G. Mora and S. Müller, Derivation of nonlinear bending
theory for shells from three-dimensional nonlinear elasticity by -convergence, C. R.
Math. Acad. Sci. Paris 336 (2003) 697702.
17. G. Friesecke, R. D. James and S. Müller, The F€
opplvon K
arm
an plate theory as a low
energy -limit of nonlinear elasticity, C. R. Math. Acad. Sci. Paris 335 (2002) 201206.
18. G. Friesecke, R. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl.
Math. 55 (2002) 14611506.
19. G. Friesecke, R. D. James and S. Müller, A hierarchy of plate models derived from
nonlinear elasticity by Gamma-convergence, Arch. Rational Mech. Anal. 180 (2006)
183236.
20. E. De Giorgi, -convergenza e G-convergenza, Boll. Un. Mat. Ital. 5 (1977) 213220.
21. J. Jeong and P. Ne®, Existence, uniqueness and stability in linear Cosserat elasticity for
weakest curvature conditions, preprint 2550, http://www3.mathematik.tudarmstadt.de/
fb/mathe/bibliothek/preprints.html, Math. Mech. Solids, doi:10.1177/1081286508093581.
22. G. Dal Maso, Introduction to -Convergence (Birkhäuser, 1992).
23. B. Miara and P. Podio-Guidigli, Deduction by scaling: A uni¯ed approach to classic plate
and rod theories, Asympt. Anal. 51 (2007) 113131.
24. P. Ne®, A geometrically exact Cosserat-shell model including size e®ects, avoiding
degeneracy in the thin shell limit. Part I: Formal dimensional reduction for elastic plates
1590
25.
Math. Models Methods Appl. Sci. 2010.20:1553-1590. Downloaded from www.worldscientific.com
by UNIVERSITY OF DUISBURG-ESSEN on 03/04/13. For personal use only.
26.
27.
28.
29.
30.
31.
P. Ne®, K.-I. Hong & J. Jeong
and existence of minimizers for positive Cosserat couple modulus, Cont. Mech. Thermodyn. 16 (2004) 577628.
P. Ne®, A geometrically exact viscoplastic membrane-shell with viscoelastic transverse
shear resistance avoiding degeneracy in the thin-shell limit. Part I: The viscoelastic
membrane-plate, preprint 2337, http://www3.mathematik.tu-darmstadt.de/fb/mathe/
bibliothek/preprints.html, Z. Angew. Math. Phys. (ZAMP), DOI 10.1007/s00033-0044065-0, 56 (2005) 148182.
P. Ne®, The -limit of a ¯nite strain Cosserat model for asymptotically thin domains
versus a formal dimensional reduction, in Shell-Structures: Theory and Applications,
W. Pietraszkiewiecz and C. Szymczak, eds. (Taylor and Francis, 2006), pp. 149152.
P. Ne®, A geometrically exact planar Cosserat shell-model with microstructure. Existence
of minimizers for zero Cosserat couple modulus, Math. Models Methods Appl. Sci. 17
(2007) 363392.
P. Ne® and K. Chełmiński, A geometrically exact Cosserat shell-model including size
e®ects, avoiding degeneracy in the thin shell limit. Rigourous justi¯cation via -convergence for the elastic plate, preprint 2365, http://www3.mathematik.tudarmstadt.de/fb/
mathe/bibliothek/preprints.html, 10/2004.
P. Ne® and K. Chełmiński, A geometrically exact Cosserat shell-model for defective elastic
crystals. Justi¯cation via -convergence, Interfaces and Free Boundaries 9 (2007)
455492.
R. Paroni, P. Podio-Guidugli and G. Tomassetti, The ReissnerMindlin plate theory via
-convergence, C. R. Acad. Sci. Paris, Ser. I 343 (2006) 437440.
R. Paroni, P. Podio-Guidugli and G. Tomassetti, A justi¯cation of the ReissnerMindlin
plate theory through variational convergence, Anal. Appl. 5 (2007) 165182.