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Asymptotic Behavior for Textiles in von-Karman regime

Georges Griso

, Julia Orliky, Stephan Wackerlez

December 20, 2019

Abstract

The paper is dedicated to the investigation of simultaneous homogenization and dimension reduction of textile structures as elasticity problem with an energy in the von-Karman-regime. An extension for deformations is presented allowing to use the decomposition of plate-displacements. The limit problem in terms of displacements is derived with the help of the unfolding operator and yields in the limit the von-Karman plate with linear elastic cell-problems. It is shown, that for homogeneous isotropic beams in the structure, the resulting plate is orthotropic. As application of the obtained limit plate we study the buckling behavior of orthotropic textiles. Keyword: Homogenization, periodic unfolding method, dimension reduction, von-Karman orthotropic plate, Energy minimization under pre-strain, buckling under homogenized pre-strain Mathematics Subject Classication (2010): 35B27, 35J86, 47H05, 74Q05, 74B05, 74K10, 74K20.

1 Introduction

In contrast to our rst paper about homogenization of textiles [21], where a geometrical linear elasticity is considered, we investigate here textiles with von-Karman energy. The von-Karman model is a nonlinear plate model, which is stated with respect to displacements and widely used by mathe-

maticians and engineers, see [4, 2, 7, 8, 14, 15, 26]. To achieve the von-Karman model in the limit we

consider an elastic energy of orderjje(u")jjL2( ")C"5=2, which is in consensus with [5, 7, 14, 15]. A simultaneous homogenization and dimension reduction of a von-Karman plate was already studied in

[23], however in our paper we give the corrector results related on the periodic topology of the textile.

Since the von-Karman plate arises as -Limit from geometrical nonlinear problems, it is necessary to

consider initially deformations. Hence, the homogenization of the textile for von-Karman energy begins

with the extension of a deformation into the holes of the structure. This extension is applied onto the

deformations of the textile beam structure for glued beams. Due to the fact that the limit-plate is stated

with respect to displacements we directly introduce the decomposition of the displacement associated to

the extended deformation, see [4, 18, 17]. For the elementary displacements we establish the Korn-type

estimates giving rise to the asymptotic behavior of the elds. To derive the homogenized model the

unfolding and rescaling operator (see for instance [10, 11, 20, 21]) accounting for both homogenization

and dimension reduction is used. For the von-Karman-plate studied here it is necessary to investigate

also the nonlinear term in the Green-St.Venant strain tensor. The additional term yield the von- Karman nonlinearities in the limit. The derived asymptotic limits allow to prove with arguments of -convergence to show that the limit energy in in fact of von-Karman-type. It is proven, that the homogenized limit energy admits minima. The uniqueness, though, is not provable.

Sorbonne Universite, CNRS, Universite de Paris, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France,

griso@ljll.math.upmc.fr yFraunhofer ITWM, 67663 Kaiserslautern, Germany, orlik@itwm.fhg.de zFraunhofer ITWM, 67663 Kaiserslautern, Germany, wackerle@itwm.fhg.de 1 Although the initial and the homogenized problem are nonlinear, the cell-problems for the von- Karman plate are linear and in fact the same as achieved for a linear elastic plate. Furthermore, the cell-problems yield for isotropic homogeneous beams an orthotropic plate, which is valid for the von-Karman energy and linear elasticity. The nal part of the paper is devoted to homogenization of the pre-stress in yarns and modeling

of the buckling of the von-Karman plate under pre-strain, like in [22, 6, 26, 1], but for an orthotropic

plate.

2 Preliminary extension results for deformations and displace-

ments

In this section,

and

0are two bounded domains inRncontaining the origin with Lipschitz

boundaries and such that

0. For every" >0, we denote

and 0"="

0. In Lemma 2.1

we prove an extension result for deformations inH1( ")n. The lemma below is based on the rigidity theorem obtained by G. Friesecke, R. James and S. Muller in

[13]. Here, for the starshaped open sets with respect to a ball, we use a variant of this theorem which

explicitly gives the dependence of the constants in the estimates in terms of only two parameters which

depend on the geometry of the domain: its diameter and the radius of the ball (see [3, 19]).

Lemma 2.1.For every deformationvinH1(

")nthere exists a deformationevinH1(

0")nsatisfying

evj "=v; distev;SO(n) L2( 0")C dist(v;SO(n)) L2( "):(2.1)

The constant does not depend on".

Proof.First, some classical recalls and then the proof. (i) Since is a bounded domain with Lipschitz boundary, there existN2N,R1andR2two strictly positive constants and a nite setfO1;:::;ONgof open subsets of , each of diameter less thanR1and starshaped with respect to a ball of radiusR2(B(Ai;R2),Ai2 Oi) such that N[ k=1O k: As a consequence, there existsrsuch that for everyOi,i2 f1;:::;Ngthere exists a chain from O 1toOi O l1=O1;Ol2; :::;Olp=Oi; p2 f1;:::;Ng such that, ifp >1 one hasOlj\ Olj+1,j2 f1;:::;p1g, contains a ball of radiusr. (ii) LetObe an open set inRnincluded in the ballB(A;R1) and starshaped with respect to the ballB(A;R2),R1>0; R2>0. Theorem II.1.1 in [3] claims that for every deformation v2H1(O)n, there exist a matrixR2SO(n) anda2Rnsuch that

The constantCdepends only onR1R

2andn.

TransformOby a dilation of ratio" >0 and centerA, the above result gives: for every deformation v2H1(O")nwhereO":="O, there exist a matrixR2SO(n) anda2Rnsuch that

The constantCdoes not depend on".

2 (iii) and

0being two bounded domains inRnwith Lipschitz boundaries and such that

0. There exists a continuous linear extension operatorP0fromH1( )nintoH1(

0)nsatisfying

8v2H1(

)n;P0(v)j =v;kP0(v)kL2(

0)CkvkL2(

);kP(v)kH1(

0)CkvkH1(

If we transform

and

0by the same dilation of ratio"(and centerO2

), this extension operator induces an extension operatorP0"fromH1( ")3intoH1(

0")3satisfying

8v2H1(

")3;(P0"(v)j "=v;kP0"(v)kL2(

0")CkvkL2(

kP

0"(v)kL2(

0")+"krP0"(v)kL2(

0")CkvkL2(

")+"krvkL2(

The constants do not depend on".

Now, consider a deformationv2H1(

")n. We apply (ii) with the open setsOi;"=Ai+"(OiAi), there exist matricesRi2SO(n) and vectorsai2Rnsuch that

The constantCdoes not depend on".

Then, using the second part of (i), we compareRitoR1as well asaitoa1,i2 f1;:::;Ng. As a consequence, one obtains that kva1R1xkL2( ")C"kdist(rv;SO(3))kL2( ");krvR1kL2( e)Ckdist(rv;SO(n))kL2(

The constants do not depend on".

Now, we dene the extension ofv. We set

ev=P0"(va1R1x) +a1+R1xa.e. in 0":

We easily check (2.1).3 The structure

3.1 Parameterization of the yarns

To see the parametrization of yarns and the structure we refer to [21]. Nevertheless, we shortly

repeat the most important denitions and results. The middle line of a beam is paramtrized by rescaled

function "="(z" ) of (z) =8 >>>:;ifz2[0;];

6(z)2(12)24(z)3(12)31

ifz2[;1]; ifz2[1;1]; (2z) ifz2[1;2]:(3.1)

Then the beams in the structure are dened by

P (1)r:=z2R3jz12(0;L);(z2;z3)2(";")2; P(2)r:=z2R3jz22(0;L);(z1;z3)2(";")2: for the reference beams in the two directions. Then the curved beams are dened by P (1;q)":=n x2R3jx= (1;q)"(z); z2P(1)ro ;P(2;p)":=n x2R3jx= (2;p)"(z); z2P(2)ro 3 1221

Figure 1:

The domain YY= (0;1)2(2;2), a quarter of the periodicity cell of the full structure. with the dieomorphisms (1;q)"(z):=M(1;q)"(z1) +z2e2+z3n(1;q)"(z1); (2;p)"(z):=M(2;p)"(z2) +z1e1+z3n(2;p)"(z2); and the corresponding middle lines M (1;q)"(z1):=z1e1+q"e2+ (1)q+1"(z1)e3; M(2;p)"(z2):=p"e1+z2e2+ (1)p"(z2)e3:

3.2 Parameterization of the whole structure

Denote

"the whole structure (see [21] for details) 2N"[ p=0P (1;q)"[2N"[ q=0P (2;p)" ":=!(2";2"); != (0;L)2:(3.2)

3.3 An extension result

The presented extension heavily depends on the fact that the beams are glued. For a more general

contact condition as in [21] it is necessary to treat the two directions separately and obtain two defor-

mations, which give the same limit forg""4. Nevertheless, the more general case would exceed the bounds of this paper.

Proposition 3.1.For every deformationvinH1(

")3there exists a deformationevinH1( ")3sat- isfying evj "=v; distev;SO(3) L2( ")C dist(v;SO(3)) L2( "):(3.3)

The constant does not depend on".

Proof.Now that the general extension for Lipschitz domains in nonlinear elasticity is recalled in the

above lemma, we specify the extension procedure for the the domain 4

Figure 2: First extension domain

Figure 3: Periodicity cell of the periodic plate with holes.

First, we divide the domain

"into portions included in domains isometric to the parallelotope (0;"+ 2")(0;2")(0;4")1as depicted in Figure 2. These portions include a curved beam and parts of the beams in the perpendicular direction with which the beam is in contact. Besides, after a rotation and/or a re ection, all the portions are of the same form and itself Lipschitz-domains. Furthermore, note that these portions intersect each other and every contact cylinderCpq(2";2") withCpq= (p"";p"+")(q"";q"+") is used in four of such domains. Since every portion is a Lipschitz domain the extension procedure given in Lemma 2.1 is applicable for everyv2H1( ") and yields an extension to the parallelotope (e.g. (p"";(p+1)"+")(q" ";q"+")(2";2")). As second step, we dene new domains included in (p"";(p+1)"+") (q"";(q+ 1)"+")(2";2") by collecting four of the above portions as depicted in Figure 3. Note that the contact cylinders in every corner of the new domain is used by two portions. Obviously this domain is again a Lipschitz domain and hence we extend all the elds into the holes using again

Lemma 2.1.

To obtain the full extension we reassemble the structure. To do this, note that the domains (p" ";(p+1)"+")(q"";(q+1)"+")(2";2") have an overlap. This overlap includes every

beam twice and the contact cylinders again fourfold, i.e. the overlap consists exactly of the domains

before the last extension. Together with the interportions from the step before we obtain that the contact cylinders are the most used domains for the extension, namely eight times. This in uences the estimate and nally give the nal extensionevwhich satises distev;SO(3) L2( ")C dist(v;SO(3)) L2( where the constant does not depend on". By construction, we haveevj "=v.Henceforth, we use the extended deformationv2H1( "), which is a deformation of a periodic

plate without holes. This allows to use the results in the papers [4] and general results of [12], [21], [2].

The structure is clamped on its lateral boundary. Moreover, in contrast to [21] here we assume a glued

contact, which corresponds to the caseg"0 in [21]. This allows to obtain one deformation eld for1 We reduce the parallelotopes that are in contact with the boundary of!. 5 the whole structure "instead of one for each beam as in [21].

Denote

=@!\ fx2= 0g= (0;L) f0g;"= (2";2"):

The set of the admissible deformations are

V ":=n v2H1( ")3jsuch thatv=Ida.e. on@ "\"o D ":=n v2H1( ")3jsuch thatv=Ida.e. on "o :(3.4) Remark 3.2.Every deformation belonging toV"is extended in(0;L)(";0)(2";2")by settingv=Idin this open set. Then, Proposition 3.1 gives an extension ofvwhose restriction to belongs toD"and satises(3.3).

4 The non-linear elasticity problem

Set Y

0:= (0;2)2;Y:= (0;2)2(2;2):

LetY Ybe the reference cell of the beam structure. The cellYis deduced fromY(see Figure 1) after two symmetries with respect to the planesy1= 1 andy2= 1.

Denote

cWthe local elastic energy density, then the total elastic energy is J "(v) =Z "c

W";rvdxZ

"f "(vId)dx;8v2V";(4.1) whereIdis the identity map. The local density energycW:YS3!R+[ f+1gis given by c

W"(;F) =8

:Q" ;12 (FTFI3) if det(F)>0; +1if det(F)0; whereS3is the space of 33 symmetric matrices. The quadratic formQis dened by Q(y;S) =aijkl(y)SijSklfor a.e.y2 Yand for allS2S3; where theaijkl's belong toL1(Y) and are periodic with respect toe1ande2. Moreover, the tensorais symmetric, i.e.,aijkl=ajikl=aklji. Also it is positive denite and satises

9c0>0;such thatc0SijSijaijkl(y)SijSklfor a.e.y2 Yand for allS2S3:(4.2)

Note, that the energy density

c

W"(x;rv(x)) =8

:Qx" ;E(v)(x) if det(rv(x))>0; +1if det(rv(x))0;for a.e.x2 depends on the strain tensorE(v) =12 (rv)TrvI3withI3the unit 33 matrix. Remark 4.1.As a classical example of a local elastic energy satisfying the above assumptions, we mention the following St Venant-Kirchho's law for which c

W(F) =8

:8 tr(FTFI3)2+4 tr(FTFI3)2ifdet(F)>0 +1ifdet(F)0: 6

Now we are in the position to state the problem.

Therefore, set

m "= infv2V"J"(v)2:

5 Preliminary estimates

5.1 Recalls about the plate deformations

Denotex0= (x1;x2)2R2and

U ":=n u2H1( ")3ju= 0 a.e. on "o The deformations and the terms of their decompositions are estimated in terms ofkdist(rv;SO(3))kL2( this is why the lemma below plays a crucial role Lemma 5.1.Letv2V"be a deformation andev2D"the extended deformation given by Proposition

3.1 and Remark 3.2. The associated displacementu=evIdbelongs toU"and satises

ke(u)kL2( ")C0kdist(rv;SO(3))kL2( ")+C1"

5=2kdist(rv;SO(3))k2L2(

")(5.1) The constants do not depend on"andv(they depend on!,Yand). Proof.In [4, Lemma 4.3] it is proved that there exists a constant which does not depend on"andev such that ke(u)kL2( ")Ckdist(rev;SO(3))kL2( 1 +1"

5=2kdist(rev;SO(3))kL2(

Then, Proposition 3.1 gives a constant which does not depend on"andvsuch that distev;SO(3) L2( ")C dist(v;SO(3)) L2( This ends the proof of the lemma.5.2 Recalls about the plate displacements Set H1 (!):=2H1(!)j= 0 a.e. on H 2 (!):=2H1(!)j= 0;r= 0 a.e. on Below we recall a denition from [11, Chapter 11] (see also [20, 16]) Denition 5.2.Elementary displacement are elementsueofH1( ")3satisfying for a.e.x= (x0;x3)2 "(wherex02!) u e;1(x) =U1(x0) +x3R1(x0); u e;2(x) =U2(x0) +x3R2(x0); u e;3(x) =U3(x0): Here

U= (U1;U2;U3)2H1(!)3andR=R1e1+R2e22H1(!)2:2

It is well known that the existence of a minimizer forJ"is still an open problem. 7 The following lemma is proved in [11, Theorem 11.4 and Proposition 11.6] Lemma 5.3.Letube inU". The displacementucan be decomposed as the sum u=ue+u(5.2) of an elementary displacementueand a residual displacementu, both belonging toU"and satisfying U 2H1 (!)3;R 2H1 (!)2;kukL2( ")+"krukL2( ")C"ke(u)kL2( "):(5.3)

Moreover, one has

kU kH1(!)+"kU3kH1(!)+kRkH1(!)C"

1=2ke(u)kL2(

@U3+R

L2(!)C"

1=2ke(u)kL2(

kukL2( ")+"ku3kL2( ")Cke(u)kL2( 2 X ;=1 @u@x L2( @u3@x 3 L2( ")Cke(u)kL2( 2 X =1 @u@x 3 L2( @u3@x L2( C" ke(u)kL2( "):(5.4)

The constants do not depend on".

5.3 Assumptions on the forces

The forces have to admit a certain scaling with respect to the"-scaling of the domain. For the textile we require forces of the type f ";1="2f1; f ";2="2f2; f ";3="3f3;a.e. in ";(5.5) withf2L2(!)3. In order to obtain at the limit a von-Karman model, the applied forces must satisfy a condition kfkL2(!)C:(5.6) This constant depends on the reference cellY, the mid-surface!of the structure and the local elastic energyW(see Lemma 5.4). The scaling of the force gives rise to the order of the energy in the elasticity problem. We prove this, in lemma below. Lemma 5.4.Letv2V"be a deformation such thatJ"(v)0. Assume(5.5)on the forces. There exists a constantCindependent of"and the applied forces such that, ifkfkL2(!)< Cone has kdist(rv;SO(3))kL2( ")C"5=2kfkL2(

The constantCdoes not depend on".

Proof.Using (4.2) gives rise to the estimation

c

0kdist(rv;SO(3))k2L2(

")Z "f "(vId)dx:(5.7) 8 Introduceu=evId2U"the associated displacement to the extended deformation (see Lemma 5.1).

Then, with (5.5) and the estimates (5.4)

3we obtain

Z "f "(vId)dx"5=2kfkL2(!)kukL2( ")+"7=2kf3kL2(!)ku3kL2( "5=2kfkL2(!)kukL2( ")+"7=2kf3kL2(!)ku3kL2(

C2"5=2kfkL2(!)ke(u)kL2(

"):(5.8) Eventually, the above inequality with (5.7) and Lemma 5.1 give c

0kdist(rv;SO(3))k2L2(

")C2C0"5=2kfkL2(quotesdbs_dbs24.pdfusesText_30

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