andare

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Historical review of Zig-Zag theoriesfor multilayered plates and shells

Erasmo CarreraDepartment of Aeronautics and Aerospace Engineering, Politecnico di Torino,Corso Duca degli Abruzzi 24, 10129 Torino, Italy; carrera@polito.it

This paper gives a historical review of the theories that have been developed for the analysisof multilayered structures. Attention has been restricted to the so-called Zig-Zag theories,which describe a piecewise continuous displacement field in the plate thickness direction andfulfill interlaminar continuityof transverse stresses at each layer interface. Basically, plate andshell geometries are addressed, even though beams are also considered in some cases. Modelsin which the number of displacement variables is kept independent of the number of constitu-tive layers are discussed to the greatest extent. Attention has been restricted to those plate andshell theories which are based on the so-called method of hypotheses or axiomatic approach inwhich assumptions are introduced for displacements and/or transverse stresses. Mostly, thework published in the English language is reviewed. However, an account of a few articlesoriginally written in Russian is also given. The historical review conducted has led to the fol-lowing main conclusions. 1! Lekhnitskii ~1935! was the first to propose a Zig-Zag theory,which was obtained by solving an elasticity problem involving a layered beam. 2! Two otherdifferent and independent Zig-Zag theories have been singled out. One was developed by Am-bartsumian~1958!, who extended the well-known Reissner-Mindlin theory to layered, aniso-tropic plates and shells; the other approach was introduced by Reissner~1984!, who proposeda variational theorem that permits both displacements and transverse stress assumptions. 3! Onthe basis of historical considerations, which are detailed in the paper, it is proposed to refer tothese three theories by using the following three names: Lekhnitskii Multilayered Theory,~LMT !, Ambartsumian Multilayered Theory~AMT !, and Reissner Multilayered Theory~RMT!. As far as subsequent contributions to these three theories are concerned, it can be re-marked that: 4! LMT although very promising, has almost been ignored in the open literature.5! Dozens of papers have instead been presented which consist of direct applications or par-ticular cases of the original AMT. The contents of the original works have very often beenignored, not recognized, or not mentioned in the large number of articles that were publishedin journals written in the English language. Such historical unfairness is detailed in Section3.2. 6! RMT seems to be the most natural and powerful method to analyze multilayered struc-tures. Compared to other theories, the RMT approach has allowed from the beginning devel-opment of models which retain the fundamental effect related to transverse normal stressesand strains. This review article cites 138 references.@DOI: 10.1115/1.1557614#

1 INTRODUCTION

Two-dimensional~2D! modeling of multilayered plates andshells requires appropriate theories. The discontinuity ofphysical/mechanical properties in the thickness directionmakes inadequate those theories which were originally de-veloped for one-layered structures, eg, the Cauchy-Poisson-Kirchhoff-Love thin plate/shell theory @1–4#, or theReissner-Mindlin theory~Reissner@5# and Mindlin @6#!, aswell as higher order models such as the one by Hildebrand,Reissner, and Thomas@7#. These theories are, in fact, not

able to reproduce piecewise continuous displacementtransverse stress fields in the thickness direction, whichexperienced by multilayered structures@8#. In @9#, these two

effects have been summarized by the acronymCz0-

requirements; that is, displacements and transverse stre

must beC0-continuous functions in thez-thickness direction.A qualitative comparison of displacement and stress fielda single-layered and a multi-layered structure is shownFig. 1. This picture clearly shows that theories designedsingle-layered structures are not suitable to analyze multi

Appl Mech Rev vol 56, no 3, May 2003 28

Transmitted by Associate Editor S Adali

© 2003 American Society of Mechanical Engineers7

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288 Carrera: Historical review of Zig-Zag theories Appl Mech Rev vol 56, no 3, May 2003

ered ones. The piecewise form of transverse stress andplacement fields are often described in the open literaturZig-Zag~ZZ! andInterlaminar Continuity~IC!, respectively.The theories which describe these two effects are referreasZig-Zag theoriesin the present study.

A number of refinements of classical models, as welltheories developed for multilayered structures, have bproposed in the literature over the last four decades. Axioatic and asymptotic approaches, along with other continubased ones, have been used to build 2D theories incases of equivalent single layer~ESL! and layer-wise~LW!variable description. Following Reddy@10#, it is intendedthat the number of displacement variables is kept indepdent of the number of constitutive layers in the ESL modewhile the same variables are independent in each layerLW models. Koiter in@11# states, ‘‘A refinement of Love’sfirst approximation theory is indeed meaningless, in geneunless the effects of transverse shear and normal stressetaken into account at the same time.’’ Koiter’s Recommedation~KR! should be taken into account in the developmof Zig-Zag theories.

For a complete review of several approaches, comptional techniques and numerical assessment, readers aferred to the many survey articles available on multilayebeams, plates, and shells. Among these, recommendeviews are the articles by Ambartsumian@12,13#, Grigolyukand Kogan @14#, Librescu and Reddy@15#, Leissa @16#,Grigolyuk and Kulikov@17#, Kapania and Raciti@18#, Kapa-nia @19#, Noor and Burton@20,21#, Jemlelita@22#, Vasilievand Lur’e@23#, Reddy and Robbins@24#, Noor, Burton, andBert @25#, Lur’e and Shumova@26#, Grigorenko@27#, Grig-orenko and Vasilenko@28#, Altenbach@29#, Carrera@30#, aswell as the books by Lekhnitskii@31#, Ambartsumian@32–34#, Librescu@35#, and Reddy@10#.

Although these review works are excellent, in the authoopinion there still exists a need for a historical review wthe aim of giving clear answers to the following question

1! Who first presented a zig-zag theory for a multilayerstructure?

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2! How many different and independent ESL zig-zag theries have been proposed in the open literature?

3! Who first proposed the theories in question 2?4! Are the original works recognized and mentioned c

rectly in the subsequent articles?5! What are the main differences among the available z

zag approaches to multilayered structures?

A response to these five points would be extremely usefuthe analysts of layered structures. It will give an insight inthe early and, at the same time, very interesting ideasmethods, such as those by Lekhnitskii@36#, that could beextended and applied to further problems. Furthermore, ais in any historical note, the author aims to establish a sorhistorical justice, permitting to ‘‘give to Caesar what belonto Caesar and to God what belongs to God.’’

The already mentioned works@12–35# show that a largenumber of different techniques, methods, and ideas hbeen applied to analyze multilayered structures. The attion of the present article has been focused on those zigtheories which fall into the two following categories:

I ! The method of hypotheses is referred to and distritions of displacement and stress fields in the thickndirection of the plate/shell are introduced in an axioatic sense.

II ! With the exception of the approaches proposed byissner~Section 4! only those ZZ theories that have beedeveloped in the framework of ESLM are discussedthis review.

Those analyses, such as the very interesting papers@37–46#which are based on an asymptotic approach have not bdiscussed in the present paper. A few examples of otherproaches not discussed herein follows. Savin and Kho@47# obtained multilayered shell equations by averagingequations over the thickness coordinate. The conceptuniform stress-strain state has been used by Khoroshun@48#.Full mixed methods were applied in@49–53#. Of certain rel-evance are the many developments made in the framewof LWM. Among these, the articles@54–61# are mentionedherein. For a complete and detailed review of the sevetopics and contributions, readers are referred to the oviews given in@10,12–35#.

This paper, therefore, gives a historical review of avaableZig-Zagtheories in the context of questions 1–5, takiinto account the limitations mentioned in pointsI and II .

The answer to question 1 is provided in Section 2, whshows that Lekhnitskii@36# was the first to propose a ZigZag theory that described theCz

0-requirements.As far as questions 2 and 3 are concerned, the author

recognized that apart from the method by Lekhnitskii@36#,two other independent and different theories have beenposed in the literature in the second half of last century. Tfirst of these was given by Ambartsumian in the artic@62,63# while the technique of building a second typetheory was traced later to Reissner@64#. The three ap-proaches are discussed in Sections 2–4, along with adetails and literature. Based on historical reasons, which

be-

Fig. 1 Cz0-requirements. Comparison of transverse stress field

tween a single-layered structure and a three-layered structure

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Appl Mech Rev vol 56, no 3, May 2003 Carrera: Historical review of Zig-Zag theories 289

detailed in the respective sections, it has been proposerefer to these three different and independent contributiby the following names:

• Lekhnitskii Multilayered Theory~LMT !• Ambartsumian Multilayered Theory~AMT !• Reissner Multilayered Theory~RMT!

Concerning question 4, it will be pointed out that thoriginal contribution by Lekhnitskii has been almost totaignored in the subsequent literature~exception has to bemade for the work by Ren@65,66#; see next section!. Theplate/shell theory developed by Ambartsumian has recediffering attentions in the West and in the East. The Russliterature has recognized the Ambartsumian theory andproved it for its application to other problems, including ttreatment of KR. On the other hand, the Western literatuwith the exception made in a few works that appeared inearly 1970s, has not properly recognized Ambartsumiaworks. This story has been detailed in Section 3.2. It appethat contributions made by Lekhnitskii and Ambartsumihave not received proper recognition. Probably becausWorld War II and the subsequent Cold War, these works hhad little impact outside the Soviet Union.

As far as question 5 is concerned, discussion of the thapproaches is given in Section 5. Among the threeproaches discussed, the RMT theory has proved to bethe most versatile to describe completely theCz

0-requirements and the most suitable for computatiostudies.

Concerning notation, the author has tried as much assible to use those symbols that were quoted in the origarticles. Such a choice would permit the readers to compthe formulas reported herein directly with those originagiven. In any case, due to the large amounts of algebra awith the significant numbers of described theories, symbused in a certain section refer only to that section. In thcases in which a given symbol is not defined in a givsection, its definition has been provided on a earlier sectThe present paper refers to beam, plate, and shell structFormulas related to the discussed theories do not consboth flat and curved geometries. For the sake of brevity,geometry is referred to in some cases, while doubly curgeometry has been considered in others.

2 LEKHNITSKII MULTILAYERED THEORY

To the best of the author’s knowledge, Lekhnitskii shouldconsidered the first contributor to the theory of multilayerstructures. In@36#, in fact, Lekhnitskii proposed a splendimethod able to describe the Zig-Zag effect~for both in-planeand through the thickness displacements! and interlaminarcontinuous transverse stresses. To prove this point, Figwhich is taken from the pioneering work of Lekhnitskii@36#,shows an interlaminar continuous transverse shear sfield (t1 andt2 are shear stresses in layers 1 and 2, resptively! with discontinuous derivatives at the layer interfacIn other words,Cz

0-requirements of Fig. 1 were entirely accounted for by Lekhnitskii@36#.

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The author believes it is relevant and of interest to quthe original derivations made by Lekhnitskii. It can, in facprove to be difficult to obtain the original article by Lekhnitskii which has not been translated into English. Furthmore, the theory proposed by Lekhnitskii is very interestinand the method used could represent a starting point forture developments. The following detailed derivationtherefore taken directly from the original paper by Leknitskii, written in Russian. Few changes of notations wemade. A more brief treatment can be found in the Engltranslation of the book by Lekhnitskii@31# ~Section 18 ofChapter III, p 74!.

Let us consider a cantilever beam bent by a forceP and amomentM and made up ofNl isotropic layers~see Fig. 3!; xand z are the longitudinal and thickness directions, resptively. l and H are the length and thickness of the bearespectively.c is the beam width andy is the related coor-dinate. Subscriptk is used to denote variables and paraeters related to thek-th layer. The layers are counted startinfrom the bottom.sk is the layer thickness andhk is the dis-tance of a layer interface from thex-axis.sxx

k , szzk , sxz

k , uk,and vk are the stress and displacement components ofk-th layer. Ek, nk, and Gk are the Young’s modulus, Poisson’s ratio, and shear modulus of thek-th layer, respectively.

The problem considered is anx2z plane stress problemIt can therefore be formulated in terms of a stress functionwk

defined in thek-th layer on the domainx,z. Stresses can becalculated usingwk according to the following well-knownrelations:

sxx5]2w

]z2 , szz5]2w

]x2 , sxz52]2w

]x]z(1)

Compatibility of strains can be written in terms the strefunction according to the following compatibility equation

Fig. 2 Cz0-form of a transverse shear stress in a two-layered st

ture. The interface and neutral axis are shown. The upper and lolayers are made of low and high stiffness materials, respectivThis graph was taken from the original work@36#.

r-

r

ed

290 Carrera: Historical review of Zig-Zag theories Appl Mech Rev vol 56, no 3, May 2003

]4wk

]x4 1]4wk

]x2]z2 1]4wk

]z4 50 (2)

At this stage, the fundamentalaxiomaticassumption of theLekhnitskii’s theory is made, ie, a form for the stress funtion is assumed:

wk~x,z!5ak

6z31

bk

2z21xS Ak

3z31

Bk

2z21CkzD (3)

A different choice for the stress function would lead to dferent results. According to Eq.~1! the chosen form is able todescribe:• Longitudinal stressessxx , which are linear in the thick-

nessz direction as well as in the longitudinal direction x

sxxk 5akz1bk1x~2Akz1Bk! (4)

• Thickness stressesszz, which are zero in eachx,z posi-tions

szzk 50 (5)

• Transverse shear stresssxz , which is parabolic inz andindependent of the longitudinal coordinatex

sxzk 52Akz

22Bkz2Ck (6)

The 53Nl unknown-constantsak ,bk ,Ak ,Bk , andCk mustbe determined according to the relations given in itemsI –Vwhich follow:

I ! Displacements are related to stresses by means ostrain-stress relations written for the constitutive eqtion of the k-th layer ~eg, Hooke’s Law written interms of compliances!:

]uk

]x5

1

E k sxxk 2

n k

E 2k szz

k

]wk

52nk

sk 11

sk (7)

c-

if-

,

f theua-

]z Ek xx Ek zz

]uk

]z1

]wk

]x5

1

Gk sxyk

II ! Compatibility conditions for displacements at the intefaces areZig-Zag effects,

uk215uk, wk215wk, k52,Nl (8)

III ! Homogeneous conditionsat the bottom/top surface fothe transverse stresses,

syy1 5syy

Nl50, sxy1 5sxy

Nl50 at z50,H (9)

IV! Interlaminar equilibriumfor the transverse stresses

syyk215syy

k , sxyk215sxy

k , k52,Nl (10)

V! Equivalence or equilibrium conditions between appliloads (M ,P) and stresses

c(1

Nl Ehk21

hkszzdz50,

c(1

Nl Ehk21

hksxxzdz5

M2Px

H,

c(1

Nl Ehk21

hksxzdz52

P

H(11)

First Eqs.~7! are integrated in thex and z directions. Thefollowing expressions for the displacementsuk and wk areobtained:

uk5Ak

Ek x2z1Bk

2Ek x21~aky1bk!x

Ek

1S nk

Ek 21

GkD S Ak

Ek z31Bk

2z2D2aky1bk

(12)

Fig. 3 Geometry and notations of Lekhnitskii’s cantilever, multilayered beam

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the

h; in

s

te

Appl Mech Rev vol 56, no 3, May 2003 Carrera: Historical review of Zig-Zag theories 291

wk52Ak

3Ek x32ak

2Ek x21F2nk

Ek ~Akz21Bkz!

2Ck

Gk 1akGx2nk

Ek S ak

2z21bkzD1gk

ak , bk , andgk are the three integration constants. By usiII – IV, and after some lengthy algebraic manipulations,following relations are obtained,

ak

Ek 5ak21

Ek21 ,Ak

Ek 5Ak21

Ek21

Ak

Ek hk211Bk

2Ek hk215Ak21

Ek21 hk211Bk21

2Ek21 hk21 (13)

akhk211bk

Ek 5ak21hk211bk21

Ek21

C150, ANlhNl

2 1BNlhNl

1CNl50

Akhk212 1Bkhk211Ck5Ak21hk21

2 1Bk21hk211Ck21(14)

S nk

Ek21 21

Gk21D S Akhk213

31

Bkhk212

2 D 2akhk211bk

5S nk21

Ek 21

GkD S Ak21hk213

31

Bk21hk212

2 D 2ak21hk21

1bk21 (15)

nk

Ek ~Akhk212 1Bkhk21!2

ck

Gk 1ak

5nk21

Ek21 ~Ak21hk212 1Bk21hk21!2

ck21

Gk21

1ak212nk

Ek S ak

2hk21

2 1bkhk21D1gk

52nk21

Ek21 S ak21

2hk21

2 1bk21hk21D1gk21 (16)

The following recursive relations are obtained from Eqs.~13!

Ak5A1Ek

E1 , Bk5B1Ek

E1 , ak5a1Ek

E1 , bk5b1Ek

E1 (17)

while from Eqs.~15! it is found,

Ck51

E1 H A1 F (s51

k21

~hs22hs21

2 !Es2hk21EkG1B1F (

s51

k21

~hs2hs21!Es2hk21EkG Jk52,3,Nl21 (18)

Top and bottom layer values are:

C150, CNl5

Ek

E1 ~A1hNl

2 1B1hNl! (19)

nghe

Eqs. ~11! are used to determine the remaining bottom-laconstants,

A156PE1

Sc, B152

6PE1

ScS2 ,

a15212ME1

ScS1 , b152

6ME1

ScS2 (20)

where

S54(i 51

Nl

~hi32hi 21

3 !Ei(i 51

Nl

~hi2hi 21!Ei23

3F(i 51

Nl~hi22hi 21

2 !Ei G2

(21)

S15(i 51

Nl

~hi32hi 21

3 !Ei , S15(i 51

Nl

~hi22hi 21

2 !Ei

It is noted thath050 andhNl5H.

The final expressions for the stresses are:

sxxk 5

6Ek

Sc~Px1M !~2S1z2S2!, k51,2,..Nl

szzk 50, k51,2,..Nl

sxzk 5

6P

Sc H S1 F (s51

k21

~hs22hs21

2 !Es1~z22hk212 !EkG J

2H S2F (s51

k21

~hs2hs21!Es1~z22hk21!EkG J ,

k52,3,Nl21 (22)

sxz1 52

6PE1

Scz~S1z2S2!

sxzNl52

6PENl

Sc~hNl

2z!@S22~hNl1z!S1#

The corresponding expression for the displacements coulobtained directly from Eqs.~12!. It is noted that the amounof algebraic manipulations is quite impressive. At the prestime, the use of software for symbolic calculation could besome help to derive the above formulas or to apply thekhnitskii method to a different stress function assumption

This section closes by making a few remarks ontheory proposed by Lekhnitskii.

1! Lekhnitskii’s theory described Zig-Zag form of botlongitudinal and through the thickness displacementsparticular:

a! The longitudinal displacementsuk show a cubic or-der in thez-thickness direction

b! The through thickness displacementwk varies ac-cording to a parabolic equation inz

2! Lekhnitskii’s theory furnishes interlaminar continuoutransverse stressesszz andsxz Eqs.~22!

3! Stresses obtained by Lekhnitskii fulfill the 3D indefiniequilibrium equations~this fundamental property is in

vi-ine

adeeandn-er-n’sente ofis

ated

d bye

ts are

g

292 Carrera: Historical review of Zig-Zag theories Appl Mech Rev vol 56, no 3, May 2003

trinsic in the used stress function formulation!4! Stresses and displacements have been obtained by

ploying:

a! Compatibility conditions for stress functionsb! Strain-displacement relationsc! compatibility conditions for displacements at th

interfaces

uk215uk, wk215wk, k52,Nl (23)

d! Homogeneous conditions at the bottom and top sface for the transverse stresses

szz1 5szz

Nl50, sxz1 5sxz

Nl50 inz50,h (24)

e! Interlaminar equilibrium for the transverse stress

szzk215szz

k , sxzk215sxz

k , k52,Nl (25)

5! No post-processing was used to recover transvestresses

6! Thickness normal stressszz has been neglected. Nevetheless, the Poisson effects on thickness displacemwk have been fully retained.

7! Full retainment of Koiter’s recommendation would rquire a different assumption for the stress functions~theauthor is not aware of any work that was made in tdirection!

2.1 Developments of LMT

Although Lekhnitskii’s theory was published in the middthirties of the last century and reported in a short paragrof the English edition of his book@31#, it has been systematically forgotten in the recent literature. An exceptioshould be made for the work by Ren which is documentedthis paragraph.

To the best of the author’s knowledge, Ren is the oscientist who has used Lekhnitskii ’s work. In the two pap@65,66# Ren has, in fact, extended Lekhnitskii ’s theory

em-

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his

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orthotropic and anisotropic plates. Further applications tobration and buckling were made in a third paper writtencollaboration with Owen@67#. These three papers are thonly contributions known to the author that have been min the framework of Lekhnitskii ’s theory. As these threpapers have been published in journals that are easilyworldwide available, the full description of the Ren extesion to plates of LMT has, therefore, been omitted. Nevtheless, it is of interest to make a few remarks on Reworks in order to quote explicitly the stress and displacemfields that were introduced by Ren to analyze the responsanisotropic plates. For the sake of simplicity, referencemade to the derivation made by Ren in@65#, where cross-plyplates were considered. The extension to generally laminplates can be found in@65,67#.

On the basis of the form oftxzk obtained by Lekhnitskii,

see Eq.~22!, it appeared reasonable to Ren, see@65#, to as-sume in an axiomatic sensethe following distribution oftransverse shear stresses in a laminated plate, composeNl orthotropic layers, (x, y, andz are the coordinates of threference system shown in Fig. 4!:

sxzk ~x,y,z!5jx~x,y!ak~z!1hx~x,y!ck~z!

(26)

syzk ~x,y,z!5jy~x,y!bk~z!1hy~x,y!gk~z!

Four independent functions ofx,y have been introduced todescribe transverse shear stresses. The layer constanparabolic functions of the thickness coordinatez accordingto Eqs.~22!, and in view of this Ren adopted the followinexpressions

ak~z!5S1xF (i 51

k21

~hi22hi 21

2 !Q11i ~z22hk21

2 !Q11k G

2S2xF (i 51

k21

~hi2hi 21!Q11i ~z2hk21!Q11

k G

Fig. 4 Multilayered plate

c

d

ir

us

u-othrmned

e-an

or.

redex-

dede-

an-

asear

da-or-y

ate

ma-tsr

ys-

Appl Mech Rev vol 56, no 3, May 2003 Carrera: Historical review of Zig-Zag theories 293

ck~z!5S1nF (i 51

k21

~hi22hi 21

2 !Q11

i

n12i ~z22hk21

2 !Q11

k

n12k G

2S2nF (i 51

k21

~hi2hi 21!Q11

i

n12i ~z2hk21!

Q11k

n12k G

(27)

bk~z!5S1yF (i 51

k21

~hi22hi 21

2 !Q22i ~z22hk21

2 !Q22k G

2S2yF (i 51

k21

~hi2hi 21!Q22i ~z2hk21!Q22

k Ggk~z!5S1nF (

i 51

k21

~hi22hi 21

2 !Q22

i

n21i ~z22hk21

2 !Q22

k

n21k G

2S2nF (i 51

k21

~hi2hi 21!Q22

i

n21i ~z2hk21!

Q22k

n21k G

in which

S1x5 (K51

Nl

~hk2hk21!Q11k , S2x5 (

K51

Nl

~hk2hk21!Q11k

S1y5 (K51

Nl

~hk2hk21!Q22k , S2y5 (

K51

Nl

~hk2hk21!Q22k

(28)

S1n5 (K51

Nl

~hk2hk21!Q22

k

n12k , S2n5 (

K51

Nl

~hk2hk21!Q22

k

n12k

Q11k

n12k 5

Q22k

n21k

E1k ,E2

k ,nk are Young’s modulus of thek-th layer in thex andy directions and Poisson’s ratio, respectively.hk denotes thez-value of thek-th interface measured from the referensurface V. Qi j

k layer-stiffnesses are those that appearHooke’s law, hereby written in the reduced form@10# fororthotropic layers,

5sxx

syy

sxy

sxz

syz

6 5F Q11 Q12 0 0 0

Q21 Q22 0 0 0

0 0 Q66 0 0

0 0 0 Q55 0

0 0 0 0 Q44

G 5 exx

eyy

gxy

gxz

gyz

6 (29)

Eqs. ~26! represent an interlaminar continuous transveshear stress field that is parabolic in each layer. As in Lenitskii @36#, displacement fields are obtained by integratistrain-displacement relations after substituting into Hooklaw, Eq. ~29!. In contrast to the work by Lekhnitskii, it isemphasized that transverse strainsezz have been discardeby Ren. It is noted that such an assumption contrasts withalready mentioned Koiter’s recommendation. Layer costants arising from the integration are determined by imping compatibility conditions for the displacements at theterface. The displacement field assumes the following fo

ein

rsekh-nge’s

then-os-n-m,

uk~x,y,z!5u0~x,y!2w,x~x,y!1jx~x,y!Ak~z!

1hx~x,y!Ck~z!

vk~x,y,z!5v0~x,y!2w,y~x,y!1jy~x,y!Bk~z!

1hy~x,y!Gk~z! (30)

w~x,y,z!5w0~x,y!

whereAzk(z), Bz

k(z), Bzk(z) andGz

k(z) are obtained by inte-grating the correspondingaz

k(z), bzk(z), cz

k(z), and gzk(z).

Thus Eqs.~30! represent a piecewise Zig-Zag continuodisplacement field in the thickness directionz which is cubicin each layer.

The displacement model given by Eqs.~30! can be usedin the framework of known variational statements to formlate the governing equations of anisotropic plates in bstrong end weak form. Strong forms and related closed fosolutions have been discussed in the already mentioworks @65–67#. No weak form solutions, such as finite elment applications, nor attempts to include KR as well asextension to shell geometry of LMT are known to the auth

3 AMBARTSUMIAN MULTILAYERED THEORY

This section has been devoted to those Zig-Zag multilayetheories that have been mostly originated by attempts totend the classical Reissner-Mindlin theory~Reissner@5#,Mindlin @6#! for homogeneous, isotropic plates, to inclutheCz

0-requirements. For convenience, these attempts arescribed here in the following points.

1! To describe plates/shells made of a single-layer of anisotropic materials

2! To extend the single-layer case to the multilayered cby including Zig-Zag effects and satisfying interlamincontinuity for the transverse shear stresses

3! To include what was referred to as Koiter’s recommention in the introduction; that is, to include transverse nmal stress/strainszz,ezz effects, which are discarded bReissner-Mindlin type theories

The classical Reissner-Mindlin theory in the case of a plor shell assumesin an axiomatic sensethe following dis-placement field,

u~a,b,z!5u01zfa S 5u01zS gaz2w,a

A2

u0

Ra D Du~a,b,z!5v01zfb S 5v01zS gbz2

w,b

B2

u0

Rb D D(31)

w~a,b,z!5w0

Reissner-Mindlin theory includes transverse shear defortion, it has five degrees of freedom~three displacemenu0(a,b), v0(a,b), w0(a,b) on the reference surface plutwo rotations fa(a,b),fb(a,b) or two transverse sheastrains gaz(a,b),gaz(a,b). Figure 5 shows the notationused for a doubly curved shell. A curvilinear reference stema, b, z has been considered;a andb are the curvilinearcoordinates defined on a shell reference surfaceV. Ra ,Rb

he-

e

of

mp-ow-

294 Carrera: Historical review of Zig-Zag theories Appl Mech Rev vol 56, no 3, May 2003

are the shell radii of principal curvatures.A and B are theLame shell parameters~see @68# for details!. For conve-nience, the Reissner-Mindlin model has been written in tways; the second one~the one in parentheses! expresses therotation in terms of transverse displacement variationsshear strains. Usual approximations have been introducefar as curvature terms of typez/Ra ,z/Rb , see@68,69#. Theshear stress field related to an isotropic monocoque pshell with shear modulusG has a constant distribution in ththickness directions:

saz~a,b,z!5G~fa1w,a!(32)

saz~a,b,z!5G~fb1w,b!

Ambartsumian has been the first one to work on pointand 2. For that reason we refer to this type of Zig-Zag thries as Ambartsumian Multilayered Theory~AMT !. Ambart-sumian has, in fact, considered in a series of pap@62,63,70,71#, and in the two books@32,33#, refinements ofReissner-Mindlin theory directly to make it suitable for thapplication to anisotropic layered plates and shells. Forpurpose, the classical hypotheses of Reissner-Mindlin thewere reformulated by Ambartsumian as follows,

wo

andd as

late/e

s 1eo-

ers

ethisory

a) The line elements of the plate/shell, normal to tmiddle surfaceV, do not change their length after deformation

b! The normal stressesszz are small as compared to thin-plane onessaa , sbb , sab

c! The transverse shear stressessaz , sbz vary in the di-rection of the thickness~of the entire shell or layer!according to the law of the quadratic parabola

Single-layer case.Orthotropic plates and shells composeda single layer were addressed in@62# and @63#, respectively.The shell case is hereby considered. According to assution c!, the transverse shear field was assumed of the folling type @63#,

saz~a,b,z!51

2~saz

T 2sazB !1

z

h~saz

T 1sazB !

11

2 S z22h2

4 D fa~a,b!

(33)

Fig. 5 Geometry and notations used for multilayered shells

r

ns

e

nd

by

r

late

s-for

ritten

eare.

e

ZZof

lace-

Appl Mech Rev vol 56, no 3, May 2003 Carrera: Historical review of Zig-Zag theories 295

4sbz~a,b,z!51

2~sbz

T 2sbzB !1

z

h~sbz

T 1sbzB !

11

2 S z22h2

4 D fb~a,b!

The applied values of transverse shear at the topT and bot-tom B shell surfaces are considered in the previous traverse shear model as:

sazT 5sazS a,b,

h

2D , sazB 5sazS a,b,2

h

2D ,

sbzT 5sbzS a,b,

h

2D , sbzB 5sbzS a,b,2

h

2Dfa(a,b) and fb(a,b) are the two functions of ReissneMindlin theory. It is noted that Eqs.~32! and~33! introduce atransverse shear stress field that is parabolic in the thickdirection; in addition Eqs.~33! include non-homogeneouconditions on top/bottom surfaces. The related transveshear strains are,

eaz~a,b,z!5eaz! 1

z

heaz* 1

1

2 S z22h2

4 D Fa~a,b! (34)

ebz~a,b,z!5ebz! 1

z

hebz* 1

1

2 S z22h2

4 D Fb~a,b!

where

eaz! 5

1

2@S55~saz

T 2sazB !1S45~sbz

T 2sbzB !#,

ebz! 5

1

2@S44~saz

T 2sazB !1S45~sbz

T 2sbzB !#

eaz* 5S55~sazT 1saz

B !1S45~sbzT 1sbz

B !,

ebz* 5S44~sazT 1saz

B !1S45~sbzT 1sbz

B ! (35)

Fa5S55fa1S45fb , Fb5S55fa1S45fb

in which the following compliances have been introduced

S44k 5

Q55k

Dk , S45k 52

Q45k

Dk , S44k 5

Q44k

Dk ,

Dk5Q44k Q55

k 2~Q45k !2

Upon integration with respect toz of the strain/displacemenrelations by taking into account the assumptions and bytroducing some of the usual approximations concerningterms of the typez/Ra ,z/Rb , Ambartsumian obtained thfollowing displacement field,

u~a,b,z!5S 11z

RaDu,a

0 2z

A1w,a2

zh2

8 S 11z

RaDFa

1z3

6 S 11z

4RaDFa1zS 11

z

2RaD eaz

!

1z2

2h S 11z

3RaD eaz

!

ns-

-

ess

rse

,

tin-the

v~a,b,z!5S 11z

RbDu,b

0 2z

A1w,b2

zh2

8 S 11z

RbDFb

1z3

6 S 11z

4RbDFb1zS 11

z

2RbD ebz

!

1z2

2h S 11z

3RbD ebz

!

w~a,b,z!5w0~a,b! (36)

The important fact is noted that in view of Eqs.~31! theAMT gives in-plane displacement components which depeon bothfa andfb .

Multilayer case. The case of two layers was addressedAmbartsumian in the two papers@70,71# and further docu-mented in@32,33#. It is noted that the extension to multilayeplates was given by Osternik and Barg@72# who first wrotetransverse shear stresses in the form of Eqs.~37!. Herein, it ispreferred to present the Ambartsumian theory for the pgeometries as it has been given in@33#, where due credit isgiven to the work done by Osternik and Barg@72#. Assump-tions a! and b! are kept for the multilayered case while asumption c! for transverse shear stresses is now madeeach layer. The transverse shear stresses are, in fact, win the following layer-form,

sxzk ~x,y,z!5@G13

k f ~z!1Ak#fx~x,y!(37)

syzk ~x,y,z!5@G23

k f ~z!1Bk#fy~x,y!

which consists of an interlaminar continuous transverse shstress field.fx andfy are unknown functions that have thsame meaning asfa and fa introduced in the shell caseThese unknown functions are not affected byk-superscriptsas is the case for any Equivalent Single Layer Theory. In@33#attention was restricted to the case in whichf (z) is a sym-metric function ofz ( f (z) is zero at top/bottom external platsurfaces! and to a symmetrically bent plate (u05v050).G13

k andG23k are the shear modulusk-th layer.Ak andBk are

layer constants to be determined by imposing IC andwith respect to each layer interface. A complete listboundary conditions on transverse stresses and dispments is rewritten,

sxzNl~5sxz

T !50,sxz1 ~5sxz

B !50,

syzNl~5syz

T !50,syz1 ~5syz

B !50

sxzk215sxz

k ,syzk215syz

k , k52,Nl (38)

uk215uk, vk215vk, k52,Nl

which lead to the following layer constants:

A15ANl50, B15BNl

50

Ak5(i 5k

Nl

f ~hi !~G13i 112G13

i 11!,

Bk5(i 5k

Nl

f ~hi !~G23i 112G23

i 11!,

k5~Nl11!/2, . . . ,Nl (39)

ti

tht

n

d

t

qs.ndcon--

296 Carrera: Historical review of Zig-Zag theories Appl Mech Rev vol 56, no 3, May 2003

Ak5ANl112k50, Bk5BNl112k50,

k51, . . .~Nl11!/2

Integration of strains gives the following Zig-Zag form of thdisplacement field,

uk~x,y,z!52zw,x1J0~z!fx1zAi

G13k fx

1R1kfx sign~z!

vk~x,y,z!52zw,x1J0~z!fy1zAi

G23k fy

1R2kfy sign~z! (40)

w~x,y,z!5w0

where

R15A(Nl11)/250, R25A(Nl11)/250

R1k5 (

i 5(Nl11)/2

k21

hi S Ai

G13i 11 2

Ai 11

G13i 11D ,

R2k5 (

i 5(Nl11)/2

k21

hi S Bi

G23i 11 2

Bi 11

G23i 11D , k5~Nl13!/2, . . . ,Nl

(41)

R1k5R1

Nl112k50, R2

k5R2Nl112k

50,

k51, . . .~Nl11!/2

Further details, along with governing equations and discsion, can be found in@33#.

The displacements given in Eqs.~40! are affected by thek-superscripts. This can be formally avoided by usingHeaviside step function. These two ways of writing a ZZag displacement field have been detailed in the Appenfor a simple 1D case.

Among the Zig-Zag theories discussed in the presentticle, the AMT is without doubt the theory that has mosinfluenced the development of multilayered plate and stheories. This is probably due to the fact that AMT usesfindings of a well-known theory such as the ReissnMindlin one. The manners in which Ambartsumian ’s wohas been used, referenced, and extended in the WesterEastern scientific communities are discussed in the twolowing subsections.

3.1 Developments of Ambartsumian theory in the East

Many extensions of Ambartsumian’s works have appearejournals published in the Russian language. These wohave been directed to extend AMT to generally laminaconfigurations, to geometrically nonlinear problems, as was for the inclusion of KR. Refinements and applicationsAmbartsumian Multilayered Theory to single-layer plateshells were given in@73,74#, while applications to multilay-ered plates and shells can be found in@75–78#. Applicationsto geometrically nonlinear problems have been given by Psakov @79,80#. As an example, the developments presen

e

us-

heg-dix

ar-lyellheer-rk

andfol-

inrksedellofs/

ru-ted

by Andreev and Nemirovskii in@77# are briefly recountedhere. These authors gave an extension of the AMT in E~37–40! to include nonsymmetrical multilayered plates anonzero top and bottom surface transverse shear stressditions. The displacement fields were written in the following form,

uk~x,y,z!5lxk1u02zw,x1mkfx

vk~x,y,z!5lyk1v02zw,y1mkfy (42)

w~x,y,z!5w0

where

lxk~x,y!5F (

j 51

k21

~hj2hj 21!S44j 1~z2hk21!S44

k GsxzB

1F (j 51

k21

~hj2hj 21!S45j 1~z2hk21!S45

k GsyzB

11

2h F (j 51

k21

~hj22hj 21

2 !S44j

1~z22hk212 !S44

k G ~sxzT 2sxz

B !

11

2h F (j 51

k21

~hj22hj 21

2 !S45j

1~z22hk212 !S45

k G ~syzT 2syz

B !

lyk~x,y!5F (

j 51

k21

~hj2hj 21!S44j 1 ~z2hk21!S44

k GsxzB

1F (j 51

k21

~hj2hj 21!S45j 1~z2hk21!S45

k GsyzB

11

2h F (j 51

k21

~hj22hj 21

2 !S44j

1~z22hk212 !S44

k G ~sxzT 2sxz

B !

11

2h F (j 51

k21

~hj22hj 21

2 !S45j

1~z22hk212 !S45

k G ~syzT 2syz

B ! (43)

mxk~x,y!5 (

j 51

k21

@~ f ~hj !2 f ~hj 21!#S44j

1@ f ~z!2 f ~hk21!#S44k #sxz

B

1 (j 51

k21

@~ f ~hj !2 f ~hj 21!#S45j

1@ f ~z!2 f ~hk21!#S45k #syz

B

eia

on

s

k

o

hA

l-

d asn

m-

ineninal

ndhat

ed

umed

Appl Mech Rev vol 56, no 3, May 2003 Carrera: Historical review of Zig-Zag theories 297

myk~x,y!5 (

j 51

k21

@~ f ~hj !2 f ~hj 21!#S54j

1@ f ~z!2 f ~hk21!#S54k #sxz

B

1 (j 51

k21

@~ f ~hj !2 f ~hj 21!#S55j

1@ f ~z!2 f ~hk21!#S55k #syz

B

The functionf (z) has the following properties

f ,z~0!5 f ,z~h!50 (44)

The form of this function was taken as general in the thretical developments while it was taken as a cubic functin the examples given in@77#. Related transverse shestresses were give as,

sxzk ~x,y,z!5sxz

B 1z

h~sxz

T 2sxzB !1 f ,z~z! fx

(45)

syzk ~x,y,z!5syz

B 1z

h~syz

T 2syzB !1 f ,z~z! fy

Inclusion of szz and ezz. Many efforts have been made drectly for the inclusion of Koiter’s recommendation in AMTRybov @81,82# first made progress in this direction. He prposed a refined theory, taking the variation of transversemal displacements according to the following form:

wk~x,y,z!5w0~x,y!1 f k!~z!f~x,y! (46)

Only one unknown functionf(x,y) was used to expresboth transverse shear stresses,

sxzk ~x,y,z!5G13

k @ f k!~z!1 f * k,z~z!#Ax

kf ,x~x,y!(47)

syzk ~x,y,z!5G23

k @ f k!~z!1 f * k,z~z!#Ay

kf ,y~x,y!

wheref k!(z) and f * k are two assigned functions of the thic

ness coordinatez in the k-th layer.The most relevant contribution for the inclusion ofszz

andezz in the AMT has been made by Rasskazov and cothors. In fact, Rasskazov and coauthors proposed the insion of bothszz and ezz in a series of papers@83–88# forboth plate and shell geometries, linear and nonlinear prlems, and for analytical solution methods as well as comtational techniques such as the finite element method. Sdetails of the Rasskazov refinement of AMT theory followthe complete derivations can be found in the above mtioned papers by Rasskazov which are published in journEnglish translations of which are available worldwide. Ttransverse shear stress field was taken similar to that bybartsumian,

sxzk ~x,y,z!5G13

k ~z! f 1,z~z!fx~x,y!(48)

syzk ~x,y,z!5G23

k ~z! f 2,z~z!fy~x,y!

An interlaminar piecewise continuous transverse normstress was introduced in the following form,

szz5szzT h11z

h1szz

B h22z

h1 f !~z!w!~x,y! (49)

o-onr

i-.-or-

-

au-clu-

ob-pu-me

s;en-als,em-

al

Transverse normal strains were taken as

ezz5 f 3,zz~z!c!~x,y! (50)

The method used to compute the functionw!(x,y) can befound in @83#. The displacements field was given in the folowing form

uk~x,y,z!5u02zw,x1 f x~z!fx1 f 3~z! c ,x!

vk~x,y,z!5w02zw,y1 f y~z!fy1 f 3~z! c ,y! (51)

w~x,y,z!5w01 f 3~z! ,zc!

The mechanical properties of the layers were considerepiecewise functions ofz. The introduced functions are giveas

f x~z!5E0

zFG1321E

2h1

z

A11~z!~z2dx!dzGdz

f x~z!5E0

zFG2321E

2h1

z

A22~z!~z2dy!dzGdz

(52)

f 3~z!5E0

zS E0

z

zh~z!dz D dz

f !~z!5~z1h1!~z1h2!

where

A1151

2Q11@12n12~z!#1G12~z!,

A2251

2Q22@12n21~z!#1G12~z!

h~z!51

2

n13~z!@11n21~z!#1n23~z!@11n12~z!#

12n12~z!n21~z!(53)

dx5*A11~z!zdz

*A11~z!dz, dy5

*A22~z!zdz

*A22~z!dz

Further developments on the inclusion of transverse copression have been made by Grikoliuk and Vasilenko@89,90#and Vasilenko@91#.

3.2 Developments of Ambartsumian theory in the West

A few articles have appeared in journals written in Englishthe early 1970’s, in which developments of AMT have beconsidered, and which made direct reference to the origworks by Ambartsumian. These are discussed below.

Whitney’s contribution. Whitney @92# first applied and ex-tended AMT to generally anisotropic and symmetrical anonsymmetrical plates. It was clearly stated by Whitney this own work was based on that by Ambartsumian@33#. Forthe sake of simplicity, the case of symmetrically laminatplates is hereby outlined. Details can be found in@92#. Inter-laminar continuous transverse shear stresses were assas follows,

sxzk ~x,y,z!5@Q55

k f ~z!1a55k #fx~x,y!

1@Q45k f ~z!1a45

k #fy~x,y! (54)

l

n

y

ed

dear

yno/orngsor

eenectsand

asor’sim-as

Inn-

thele,hatDcon-in

hearenthis

d asartm-hearin

298 Carrera: Historical review of Zig-Zag theories Appl Mech Rev vol 56, no 3, May 2003

syzk ~x,y,z!5@Q45

k f ~z!1a55k #fx~x,y!

1@Q44k f ~z!1a44

k #fy~x,y!

Equations~37! can be obtained by settingQ45k 5a45

k 50. f (z)is a function of the thickness coordinate, the form of whishould be assumed differently as far as symmetrical andsymmetrical laminated cases. A parabolic form forf (z) hasmostly been considered~explicit formulas for unsymmetricacases were also given by Whitney!. The layer constantsa44

k ,a45k ,a55

k are determined by imposing the continuity coditions of transverse shear stresses at the interfaces, wtop-bottom stress-free conditions are used to determineform of f (z). Explicit forms of layer constants were omitteby Whitney. Transverse shear strains related to the assutransverse shear stress fields are

gxzk ~x,y,z!5@ f ~z!1S55

k a55k 1S45

k a45k #fx~x,y!

1@S55k a45

k 1S45k a44

k #fy~x,y!(55)

gyzk ~x,y,z!5@S44

k a44k 1S45

k a45k #fx~x,y!1@ f ~z!

1S44k a44

k 1S45k a55

k #fy~x,y!

By assuming the transverse displacement constant inthickness direction, ie,ezz50, and integrating the sheastrains, the following Zig-Zag displacement fields were otained:

uk~x,y,z!52zw,x1@J~z!1g1k~z!#fx~x,y!

1g2k~z!fy~x,y!

vk~x,y,z!52zw,y1@J~z!1g3k~z!#fy~x,y!

1g4k~z!fx~x,y! (56)

w~x,y,x!5w0~x,y,z!

where

J~z!5E f ~z!dz

g1k~z!5@S55

k a55k 1S45

k a45k #z1d1

k

g2k~z!5@S55

k a55k 1S45

k a45k #z1d2

k (57)

g3k~z!5@S55

k a55k 1S45

k a45k #z1d3

k

g4k~z!5@S55

k a55k 1S45

k a45k #z1d4

k

whered1k ,d2

k ,d3k ,d4

k are calculated by imposing compatibilitof in-plane displacement at each interface.

Rath and Das’s contribution. A second relevant work onthe application of AMT was given by Rath and Das@93# whoextended the work done by Whitney to doubly curved shand dynamic problems. The transverse shear stress fieleach layer was taken the same as those by Whitney. Intetion in z of related shear strains led to the following displacment fields@93#,

chun-

-hilethe

dmed

therb-

llss ingra-e-

uk~a,b,z!5S 11z

RaDu,a

0 2z

A1w,a1Fz1

z2

Ra24

z3

3h2

2z4

3h2RaD1G1

kS z1z2

RaD1d1

kS 11z

RaDfa

vk~a,b,z!5S 11z

RbD v ,a

0 2z

A2w,b1Fz1

z2

Rb24

z3

3h2

2z4

3h2RbD1G3

kS z1z2

RbD1d3

kS 11z

RbDfb

(58)

w~a,b,z!5w0~a,b!

whereG1k , d1

k , G3k , d3

k are layer constants directly derivefrom those which were assumed for the transverse shstress Eqs.~54!.

Other contributions which make direct reference to Am-bartsumian ’s works. Further to the two papers by Whitne@92# and Rath and Das@93#, a third article coauthored by Suand Whitney@94# compared Whitney’s Zig-Zag theory tsimplified ones which discard interlaminar continuity andZig-Zag effects. Further contributions by Hsu and Wa@95,96# employed the original AMT at layer level as it wagiven in @63#. A layer-wise theory was thereby developed fthe cylindrical shell geometry.

Other contributions which do not make direct referenceto Ambartsumian ’s works. With the exception of the fivepapers@92–96#, the author is not aware of any furtherdirectapplications of the AMT. Dozens of papers have instead bpresented over the last decades that deal with Zig-Zag effand interlaminar continuous transverse shear stresses,which mostly ignore the original work by Ambartsumian,well as those by Whitney, and Rath and Das. In the authopinion, most of these articles should be considered as splified cases of the Ambartsumian Multilayered Theorywell as of the developments in@92,93#. Unfortunately, theoriginal works and authors are not mentioned~or barelycited! in the reference lists of this large amount of articles.order to try to explain such historical unfairness, a recostruction of what happened has been attempted here.

A pioneering article by Yu@97# should first be mentionedin which a Zig-Zag theory was presented. Because ofshort time between Ambartsumian’s works and Yu’s artictogether with the Cold War, it could be surely assumed tthe works of Ambartsumian were unknown to Yu. A 1sandwich plate made of an isotropic core and faces wassidered in@97#. The three slopes of the displacement fieldthe three layers were derived by imposing transverse scontinuity at the two interfaces. The in-plane displacemfields were assumed linear in each layer. Yu did not startderivation by a direct assumption of transverse stress fielwas done by Ambartsumian; Yu, in fact, preferred to stfrom a Reissner-Mindlin type displacement model and copute the faces and core slopes by imposing transverse sstress continuity at the interfaces. It was also mentioned

al

n

a

o

b

a

s

cl

sedan

ers

nsr-

a

Its

Appl Mech Rev vol 56, no 3, May 2003 Carrera: Historical review of Zig-Zag theories 299

@97# that the method could easily be extended to displament fields which are taken as cubic functions ofz. Detailsof this last development were not given by Yu.

Almost 15 years later, Chou and Carleone@98# presenteda Zig-Zag theory of anisotropic plates. As in Yu~even thoughthe original work by Yu was not mentioned! a piecewiselinear displacement field in each layer was considered anZig-Zagtheory was proposed. It seems that Chou and Caone were not aware of the works by Yu and Ambartsumias well as those by Whitney. Chou and Carleone as weYu’s analyses, in fact, consist of a particular case of the AMtheory in which only those terms ina44

k ,a45k ,a55

k of Eqs.~54!which are independent ofz are retained.

Disciuva and coauthors@99–101# employed Yu’s andChou and Carleone’s Zig-Zag displacement field, which wlinear in each layer, by employing the Heaviside step fution, and gave finite element applications. It shouldpointed out, once again, that Yu, Chou and Carleone,Disciuva and co-authors, namely the YCCD analyses, dwith a particular case of what is herein called the AMtheory. As previously mentioned, the linear piecewise ctinuous displacement field used in the YCCD cases canfact be obtained from the AMT by simply neglecting thhigher order terms~which multiply z2 andz3) in Eqs.~58!.By doing this, the resulting YCCD models:

i ! are not able to fulfill homogeneous conditions for tranverse shear stresses at the top/bottom plate/shellfaces;

ii ! are not suitable for unsymmetrically laminated strutures.

Most the the subsequent works on Zig-Zag theoriesnot refer to the work by Ambartsumian, nor to thoseWhitney and Rath and Das. In the same manner, the deopments which appeared in journals in the Russian languas well as those reviewed in the previous subsection wmostly ignored. Subsequent works were instead very minfluenced by Yu, Chou and Carleone, and Disciuva’sticles; that is,reference has been made to simplified analywhile the most complete and exhaustive AMT has not brecognized, nor mentioned!As a matter of fact, most of thesubsequent 15 years of literature were devoted to introduthe improvements i! and ii! in the YCCD studies. The resuof this unuseful workwas that, almost ten years later, thoriginal AMT was re-obtained by Cho and Parmerter@102#who gavethe bestrefinements of YCCD studies. A few details of the story behind this are given below.

First Bhaskar and Varadan@103# and then Savithri andVaradan@104# and Leeet al @105# introduced the top-bottomzero stress conditions mentioned in pointi . This was doneby extending, in a YCCD type theory, the Vlasov@106# thirdorder in-plane displacement fields~which were frequentlyapplied to laminated structures by Reddy@107#!. The result-ing model, which did not have the quadratic terms in texpressions for displacement fields, was still not suitableunsymmetrical laminated plates. A finalbest version, whichincludes what is mentioned in pointsi and i i , was proposedby Cho and Parmerter@102#, and then~among others! by

ce-

d arle-n,

l asT

asc-

bend

ealTn-in

e

s-sur-

c-

didy

vel-ge,ereuchar-eseen

ingte

-

hefor

Soldatos and Timarci@108#, Timarci and Soldatos@109# Leeand Waas@110#, Lee, Waas and Karnopp@111#, Iblidi,Karama, and Touratier@112#, Aitharaju and Averill @113#,Cho and Averill@114#, and Polit and Touratier@115#. Some~but not extremely relevant! differences can be found inthese various articles. The Heaviside step function was uby Cho and Parmerter@102#, whose displacement fields forgenerally laminated plate were written in the form showbelow,

u~x,y,z!5u01 (k50

Nu21

Sx~z2zk!H~z2zk!

1 (k50

Ns21

Tx~z2zk!H~2z1zk!

2z2

2h S (k50

Nu21

Sxk1 (

k50

Ns21

TxkD

2z3

3h2 H w,x11

2 S (k50

Nu21

Sxk1 (

k50

Ns21

TxkD J

v~x,y,z!5v01 (k50

Nu21

Sy~z2zk!H~z2zk!

1 (k50

Ns21

Ty~z2zk!H~2z1zk!

2z2

2h S (k50

Nu21

Sxk1 (

k50

Ns21

TykD

2z3

3h2 H w,x11

2 S (k50

Nu21

Syk1 (

k50

Ns21

TykD J (59)

w~x,y,x!5w0~x,y!

Nu andNs are the number of layers in the upper and lowhalf, respectively.z andz are the two thickness coordinatefor the upper and lower half, respectively;zk andzk are thecorresponding interface values. The mid-plane rotatiofx ,fy at z1 (z1 andz2 are the top and bottom layer intefaces, respectively! are

fx5]u

]zuz501

5Sx0 , fy5

]v]zuz501

5Sy0

and

Sxk5axt

k ~w,t0 1f ,t!1bxt

k w,t0 ,

Syk5ayt

k ~w,t0 1f ,t!1byt

k w,t0 ,

k50,1,..,Nu21, t5x,y

Txk5cxt

k ~w,t0 1f ,t!1dxt

k w,t0 ,

Tyk5cyt

k ~w,t0 1f ,t!1dyt

k w,t0 , k50,1,..,Ns21, t5x,y

whereaxtk ,bxt

k ,cxtk ,dxt

k aytk ,byt

k ,cytk ,dyt

k are layer stiffnesses~see@102# for further details!.The previous displacement field, even though written inmore tedious form, coincides exactly with that of Eqs.~31!.The use of the Heaviside step function is not essential.

o

o

u

r

iyl

cee of

eas-es.

ig-of

chity

hec-

ic bys

300 Carrera: Historical review of Zig-Zag theories Appl Mech Rev vol 56, no 3, May 2003

use is, in fact, preferred by some authors and omittedothers. The author’s opinion is that the use of the Heavisfunction, although it has some formal advantages, is not uful as far as calculations and/or computer implementatiare concerned.

More exhaustive discussions on the developments of ZZag theory in the West can be found in most of the preously mentioned review articles@12–34# and in the papers@108–115#. Those developments that were directed toclude transverse normal strain effects in the AMT are mof an interest. Among these, the contributions by Savithri aVaradan@104# and Cho and Averill@114# should be men-tioned.

4 MULTILAYERED THEORY BASED ONREISSNER MIXED VARIATIONAL THEOREM

A third approach to laminated structures was originatedtwo papers by Reissner@64,116# in which a mixed varia-tional equation, namely Reissner’s Mixed Variational Therem ~RMVT! was proposed. For this reason we denote san approach as Reissner Multilayered Theory. This thirdproach is the only one developed in the West.

Reissner Mixed Variational Theorem.RMVT permits oneto satisfy, completely anda priori, the Cz

0-requirements byassuming two independent fields for displacementsu5$u,v,w%, and transverse stressessn5$sxz ,syz ,szz%,~bold letters denote arrays!. Briefly, RMVT expresses 3Dindefinite equilibrium equations~and related equilibriumconditions at the boundary surfaces which are, for the sakbrevity, not written here! and compatibility equations fotransverse strains in a variational form. The 3D equilibriuequations in the dynamic case are,

s i j ,i2r ui5pi i , j 51,2,3 (60)

r is the mass density and double dots denote acceleratwhile (p1 ,p2 ,p3)5p are body forces. The compatibilitconditions for transverse stresses can be written by evaing transverse strains in two ways: by Hooke’s lawenH

5$exzH,eyzH

,ezzH% and by geometrical relationsenG

5$exzG,eyzG

,ezzG%; the subscript n denotes transverse

normal components. They satisfy the equation

enH2enG50 (61)

RMVT therefore states.

EV~depG

T spH1denGT snM1dsnM

T ~enG2enH!!dV

5EVrd uudV1dLe (62)

The superscriptT signifies an array transpose andV denotesthe 3D multilayered body volume, while the subscriptp de-notes in-plane components, respectively. Therefore,sp

5$sxx ,syy ,sxy% and ep5$exx ,eyy ,exy%. The subscriptHindicates that stresses are computed via Hooke’s law.variation of the internal work has been split into in-plane aout-of-plane parts and involves stress from Hooke’s law a

byidese-ns

ig-vi-

in-rend

by

o-ch

ap-

e of

m

ons,

uat-

/

Thendnd

strain from geometrical relations~subscript G!. dLe is thevirtual variation of the work done by the external layer-forp. SubscriptM indicates that transverse stresses are thosthe assumed model.

RMVT leads to a set of both equilibrium and constitutivequations which are variationally consistent with thesumptions made on displacements and transverse stress

Murakami’s contribution to RMT. The first application ofRMVT was made by Murakami@117,118#, who developed arefinement of Reissner-Mindlin type theories. First, a ZZag form of displacement field was introduced by meanstwo Zig-Zag functions ~the Murakami’s Zig-Zag functionsjk(21)kDx ,jk(21)kDy),

uk~x,y,z!5u0~x,y!1zfx~x,y!1jk~21!kDx~x,y!

vk~x,y,z!5v0~x,y!1zfy~x,y!1jk~21!kDy~x,y! (63)

w~x,y,z!5w0~x,y!

jk52zk /hk is a nondimensioned layer coordinate (zk is thephysical coordinate of thek-layer whose thickness ishk).The exponentk changes the sign of the Zig-Zag term in ealayer. Such a trick permits one to reproduce the discontinuof the first derivative of the displacement variables in tz-direction. The geometrical meaning of the Zig-Zag funtion is explained in Figs. 6 and 7.

Transverse shear stress fields were assumed parabolMurakami @117# in each layer, and interlaminar continuouaccording to the following formula,

sxzk ~x,y,z!5sxz

kt~x,y!F0~zk!1F1~zk!Rxk~x,y!

1sxzkb~x,y!F2~zk! (64)

syzk ~x,y,z!5syz

kt ~x,y!F0~zk!1F1~zk!Ryk~x,y!

1syzkb~x,y!F2~zk!

Fig. 6 Geometrical meaning of Murakami’s Zig-Zag function—linear case

n

n

d

n

o

vari-e in-can

val-

late

is-com-ges

Appl Mech Rev vol 56, no 3, May 2003 Carrera: Historical review of Zig-Zag theories 301

wheresxzkt(x,y), syz

kt (x,y), sxzkb(x,y), syz

kb(x,y) are the topand bottom values of transverse shear stresses, wRx

k(x,y), Ryk(x,y) are layer stress resultants. The introduc

layer thickness coordinate polynomials are given by

F0521/41jk13jk2 ,

F153212jk

2

2hk,

F2521/42jk13jk2

Homogeneous and nonhomogeneous boundary conditiothe top/bottom plate surfaces can be linked to the introdustress field.

Toledano and Murakami@119# introduced transverse normal strain and stress effects by using a third-order displament field for both in-plane and out-of-plane componeand a fourth-order transverse stress field for both shearnormals components. Koiter’s recommendation is retaine@119#.

Carrera’s contribution to RMT. A generalization of RMVTto develop ESL and LW plate/shell theories, as well as finelement applications, has been give by Carrera@9,120–130#.Displacement and transverse stress components weresumed as follows

uk5Ft utk1Fbub

k1Frurk5Ftut

k, t5t,b,r

r 52,3,..,N (65)

snMk 5Ftsnt

k 1Fbsnbk 1Frsnr

k 5Ftsntk , k51,2,..,Nl

The subscriptst and b denote values related to the top abottom layer surface, respectively. These two terms contute the linear part of the expansion. The thickness functiFt(jk) have now been defined at thek-th layer level,

toterms

orex-

ec-

ro-

rserain

enty

hileed

s atced

-ce-tsandin

ite

as-

dsti-ns

Ft5P01P1

2, Fb5

P02P1

2,

Fr5Pr2Pr 22 , r 52,3,..,N (66)

in which Pj5Pj (jk) is the Legendre polynomial of thej -order defined in thejk-domain:21<jk<1. For instance,the first five Legendre Polynomials are

P051, P15jk , P25~3jk221!/2 ,

P355jk

3

22

3jk

2, P45

35jk4

82

15jk2

41

3

8

The chosen functions have the following values:

jk5H 1: Ft51; Fb50; Fr50

21: Ft50; Fb51; Fr50,(67)

The top and bottom values have been used as unknownables. Such a choice makes the model particularly suitablview of the fulfillment of theCz

0-requirements. The interlaminar transverse shear and normal stress continuitytherefore be linked by simply writing:

sntk 5snb

(k11) , k51,Nl21 (68)

In those cases in which the top/bottom plate/shell stressues are prescribed~zero or imposed values!, the followingadditional equilibrium conditions must be satisfied for:

snb1 5snb , snt

Nl5snt (69)

where the over-bar denotes the imposed values on the pboundary surfaces.

The Weak Form of Hooke’s Law. Full use of Reissner’stheorem requires solving a problem in terms of both dplacement and transverse stress variables. This can beputationally expensive. In order to preserve the advantaof a classical displacement formulation, aWeak Form ofHooke’s Law ~WFHL! was proposed in@9#. The WFHL,which was completely inspired by RMVT, permits oneexpress, in a weak sense, transverse stress variables inof the displacement ones.

As shown in @9#, the truncated Legendre expansion fdisplacement and transverse stress variables can bepressed in a weighted residual form in the thickness dirtion,

EAk

Fs~enGk 2enH

k !dz50, s5t,b,2,3,..,N (70)

As in the RMVT, Eqs.~70! impose compatibility of trans-verse strains. The difference is that the integral is now intduced only in thez-direction.

On substitution of given displacement and the transvestress models, as well as a given Hooke’s law and stdisplacement relation, and by integrating alongz, the set ofEqs.~70! leads to a relation between stress and displacemvariables that can be formally written in the following arraform:

Huk uk2Hs

k sk50 (71)

Fig. 7 Geometrical meaning of Murakami’s Zig-Zag function—higher degree case

i

,

a

-

a

tn

a

e

e

r

otd

o

an

tou-

dess

hin-re-

to

eneree

post-or

ted,rec-eneof

etryde:e in-ess-r-

cewhich

eenurara-theas

nressT

in-

edent

forng tond-

ts

302 Carrera: Historical review of Zig-Zag theories Appl Mech Rev vol 56, no 3, May 2003

whereHsk is a square symmetric nonsingular matrix, wh

Huk is rectangular, singular and nonsymmetric. Examples

these matrices are given in@9,120# Under certain conditionssee@9#, Eq. ~71! can be explicitly written as,

sk52Hsk 21 Hu

k uk (72)

which gives the sought relation between stresses andplacement variables.

Other available works based on the RMVT.Examples ofapplication of RMVT to laminated plates in the frame ofEquivalent Single-Layers Model were presented in the pviously mentioned articles of Murakami@117,118# andToledano and Murakami@119#. The results obtained for cylindrical bending of cross-ply, symmetrically laminateplates showed improvement in the description of the in-plresponse with respect to first order shear deformation the@118# and applications to unsymmetrically laminated plawere presented in@119#. Shell applications of the results i@118# were developed by Bhaskar and Varadan@131# andJing and Tzeng@132#. Bhaskar and Varadan@131# underlinedthe severe limitation of transverse shear stresses whichevaluateda priori by the assumed model. Finite element aplications of this model have also been developed. Linanalysis of thick plates were discussed by Rao and Me@133#. Linear and geometrically nonlinear static and dynamanalyses were considered by Carrera and coauthors@120–122#. Systematic application of RMVT to develop plate elments has been provided in a recent work by CarreraDemasi@134#. Partial implementation to shell elements wproposed by Bhaskar and Varadan@135#. Full shell imple-mentation has been given by Brank and Carrera@136#.

The limitations of Equivalent Single-Layer analyses weknown to Toledano and Murakami@137#, who applied theRMVT to Layer-Wise models. A linear in-plane displacment expansion was expressed in terms of the interfaceues in each layer, while transverse shear stresses wersumed parabolic. It was shown that the accuracy ofresulting theories was layout independent. Transverse nostress and related effects were discarded and the anashowed severe limitations to analyze thick plates. A mcomprehensive evaluation of Layer-Wise theories forcase of linear and parabolic expansions was considerethe author in@123# where applications to the static analysof plates were presented. Subsequent works extendedanalysis to dynamics and shell geometry@124–130#. A morecomprehensive review on works based on Reissner’s Threm has been recently provided in@30#.

5 REMARKS ON LEKHNITSKII,AMBARTSUMIAN, AND REISSNERMULTILAYERED THEORIES

From a historical point of view, Lekhnitskii@36# has prob-ably made the first remarkable contribution to multilayerstructure modeling: his work was the first to show the imptance of satisfying theCz

0-requirements.A list of the main features of LMT follows.

leof

dis-

nre-

dneoryes

arep-earyeric

e-ands

re

-val-

as-themallysisrehe

byisthe

eo-

edr-

• LMT-0. Lekhnitskii gave thefirst theory which accountsfor the Cz

0-requirements.• LMT-1. Such a Zig-Zag theory was obtained by solving

elasticity problem related to a layered beam.• LMT-2. Even though Lekhnitskii restricted his analysis

a cantilever, multilayer beam, he presented explicit formlas for transverse stresses and displacement fields~Eqs.~12–22!!, which are valid at any point of the considerebeam. This could result in being extremely useful to assnewly developed analytical/numerical models.

• LMT-3. The work by Lekhnitskii shows a manner in whicmultilayered structure problems can be handled. Forstance, the inclusion of transverse normal stress wouldquire a different choice of stress function in Eq.~3!.

• LMT-4. The use of a stress function formulation leadsin-plane and transverse stress fields which satisfyby defi-nition the 3D equilibrium equations. Stresses have becalculated by Lekhnitskii by solving a boundary valuproblem related to the compatibility equations which wewritten in terms of a stress function. In particular, thevaluation of transverse stresses does not require anyprocessing procedure, such as the use of Hooke’s lawintegration of 3D equilibrium equations.

• LMT-5. Although transverse normal stresses are neglecthe transverse displacement varies in the thickness dition according to a piecewise parabolic distribution givin Eq. ~12!. An attempt for the inclusion of the transversnormal stress effect would require an appropriate choicethe stress function in Eq.~3!.

As far as the extensions made by Ren to the plate geomof LMT are concerned, the following remarks can be ma• LMT-6. Transverse shear stresses are continuous at th

terfaces, and parabolic in each layer. Furthermore, strfree conditions are fulfilled at top and bottom plate sufaces.

• LMT-7. Four independent functions defined on a referensurface are used to express transverse shear stressesare assumed parabolic in each layer.

• LMT-8. The form of transverse shear stresses has bgivena priori by Ren in terms of the above mentioned foindependent functions and layer constants which are pbolic in z. The relation between the layer constants andmechanical and geometrical properties of the layers wexplicitly written by Ren. In other words, their calculatiodoes not require any imposition of transverse shear stcontinuity at each interface, as will be the case for AMand RMT.

• LMT-9. In-plane displacements are continuous at eachterface and are cubic in each layer.

• LMT-10. Seven independent variables, which are definin the regionV, have been used to describe displacemand stress fields in the laminated plates. Four were usedthe transverse shear stresses, plus three correspondithe three values of displacements were given correspoing to the reference surface.

• LMT-11. In agreement with Lekhnitskii, Ren neglectransverse normal stressesszz. In contrast to Lekhnitskii,transverse normal strainsezz are discarded by Ren.

c

r

t

.

a

e

t

m

n

s

er

ory.p-

ares,

ny

isof

aneryes.wasau-intide

er,

ects

e

hatepen-e ac-

reemadeeteradit

atedalribeerse

toisdi-tionthesti-

um-theed

y ofin

li-

Appl Mech Rev vol 56, no 3, May 2003 Carrera: Historical review of Zig-Zag theories 303

• LMT-12. Transverse shear stresses are calculated bydirectly by Eqs.~26!. That is, Hooke’s Law is not used, nois integration of the 3D equilibrium equations required.

As far as the AMT theory is concerned, it could be remarkas follows:• AMT-0. It is a natural extension of Reissner-Mindli

theory which was originally developed for isotropimonocoque structures to multilayered, anisotropic plaand shells.

• AMT-1. As LMT-6, nonzero conditions on top/bottom sufaces could also be fulfilled.

• AMT-2. Two independent functions defined onV are usedto express transverse shear stresses, as in Eqs.~54!, twoless than those used in LMT.

• AMT-3. Layer constants, parabolic in each layer, mustcomputed by imposing transverse shear stress continuieach interface, while the form of thef (z) function couldbe found by imposing top-bottom stress-free conditions

• AMT-4. As LMT-9.• AMT-5. Five independent variables, which are defined

V, are used to describe displacement and stress fieldthe laminated plate/shells, two less than the LMT case

• AMT-6. As LMT-11.• AMT-7. Literature has shown that much better evaluatio

for transverse shear stresses can be obtained via integrof the 3D equilibrium equations, as compared to the useEqs.~54!.

• AMT-8. Extension to a shell requires a reformulationdisplacement models and related layer-constants.

• AMT-9. Additional unknown functions are required to include transverse normal stress/strain effects related to

As far as the RMT is concerned, the following remarks cbe made:• RMT-0. It is the only Zig-Zag approach entirely develop

in the West. It is based on a variational theorem that pmits both displacements and transverse stress assump

• RMT-1. As LMT-6. In this case, zero as well as nonzeconditions for transverse shear and normal stresses caincluded at the top and/or bottom of the plate.

• RMT-2. At least 2Nl11 independent variables must bused for each transverse shear and normal stress conent. Additional constitutive equations are therefore otained by applying RMVT. However, these variables cbe expressed in terms of the displacement ones by usiweak form of Hooke’s Law Eqs.~70!.

• RMT-3. In-plane displacements are continuous at eachterface and can be chosen linear or of higher order in elayer. ESL applications require the introduction ofMu-rakami Zig-Zag functions.

• RMT-4 Layer constants do not appear in the expressiondisplacements and transverse stresses. In practice, ICvariationally imposed by writing the constitutive equationbetween transverse stresses and displacement variabl

• RMT-5. The number of independent variables can be atrarily chosen as explained in RMT-3.

• RMT-6. Interlaminar continuous transverse norm

Renr

ed

n,tes

-

bey at

.

ons in

nstionof

of

-KR.

an

der-ions.ron be

epo-

b-ang a

in-ach

ofare

ss.bi-

al

stresses/strains can be easily described by the RMT theKR was, in fact, already included in the early develoments of RMT.

• RMT-7. Much better evaluations of transverse stressesobtained via integration of the 3D equilibrium equationas compared to the use of assumed forms, eg, Eqs.~64!.

• RMT-8. The extension to shell does not require achanges in displacement and stress fields.

• RMT-9. Among the Zig-Zag theories examined, RMTprobably the most suitable from a computational pointview.

Which of the three theories is the best one? Certainlyanswer to such a question would be of great use to evscientist who works on 2D modelings of layered structurThe present historical note, as stated in the Introduction,not aimed at giving an answer to such a question. Thethor’s opinion is that an unequivocal answer to such a podoes not exist. None of the introduced developments provan exact solution. These are allaxiomatic theoriesand, as aconsequence, they will all violate, in same way or anoththe fundamental equations of 3D elasticity~for instance, ifone takes the 3D equilibrium equations as valid and negltransverse normal stress/strain, it follows that az-parabolictransverse shear stress field requiresz-linearity for the in-plane stresses, eg,z-linear displacement field; this is not thcase for LMT, AMT, or RMT which usez-cubic displace-ment fields!. In this respect, the RMT has the advantage ttransverse stresses and displacements are assumed inddent of each other and that these assumptions are madcording to a desired accuracy.

Exhaustive benchmark problems that compare the thapproaches are not available. Some attempts have beenin @127–129,134#. Available results have shown that, in thframework of ESL theory, the LMT analysis leads to a betdescription than the AMT one, and that the RMT could lea to better description than LMT and AMT. In any case,must be taken into account that a general conclusion relto any ESL theory is the following: available numericevaluations show that although ESL theories can desctransverse shear and normal strains, including transvwarping of cross section, the approach iskinematically ho-mogeneousin the sense that the kinematics are insensitiveindividual layers. If detailed response of individual layersrequired and if significant variations in displacement graents between layers exist, as it is the case for the descripof the local phenomena, this approach will necessitateuse of especially higher order theories in each of the contutive layers along with a correspondent increase in the nber of unknowns in the solution process, as well as incomplexity of the analyses. That is, LW analysis is requirin such cases.

Finally, we mention that Gurtovoi and Piskunov@138#have recently proposed a method to compare the accuractwo different plate theories. The hypotheses employed@138# make the method of Gurtovoi and Piskunov not app

hh

r

f

c

ei

e

h

rt

isheersthehed

nototar-

ghtlti-

ichro-redtivelar,fer-on-sionnotorsor

ar-

hebeRT.x-

ous.im-re-eld.

304 Carrera: Historical review of Zig-Zag theories Appl Mech Rev vol 56, no 3, May 2003

cable for comparing the three approaches considered inwork. However, attempts in this direction should be triedthe future.

6 CONCLUSIONS

The historical review documented in this paper has shothat, within axiomatic framework, three independent waysintroducing Zig-Zag theories have been proposed for tanalysis of multilayered plates and shells. In particular, itbeen established that:• Lekhnitskii @36# was the first to propose a theory for mu

tilayered structures which describe the Zig-Zag behavioa displacement field in the thickness direction and intlaminar equilibrium of transverse stresses. Lekhnitskwork was originally presented as an elasticity solutionlayered, cantilever beams.

• Apart from the method by Lekhnitskii, which was extended to plate structures by Ren, the other two approawere proposed by Ambartsumian~who extended the well-known Reissner-Mindlin theory to anisotropic layerplates and shells!; and by Reissner who proposed a vartional theorem that permits both displacement and traverse stress assumptions.

• Based on the author’s historical considerations, whichdocumented in this paper, it has been suggested to refthese three approaches as:

1! Lekhnitskii Multilayered Theory2! Ambartsumian Multilayered Theory3! Reissner Multilayered Theory

• A point-by-point comparison of the three approachesbeen discussed extensively in Section 5.

Future developments could be directed to extend LMtheory to make it suitable for the inclusion of KR. A moextensive comparison of the three approaches and ofnumerical performances vs elasticity solutions would be wcomed. Benchmark problems could be set out for this ppose. Other theories which are not based on axiomatic mods ~such as asymptotic theories, full mixed formulationetc! should be conveniently included in such a compariso

thisin

wnofeas

l-of

er-ii’sor

-hes

da-ns-

arer to

as

Teheirel-ur-eth-s,n.

As a final remark, it is clearly stated that the authoraware that this historical review could not be complete. Tauthor is aware that other, significant articles and papcould exist on this subject that might have escapedpresent work. In particular, there are many articles publisin the Russian language whose English translations areavailable, and they might report relevant information nconsidered here. However, what has been quoted in thisticle could at least be of some help for assigning the ricredentials as far as contributions and contributors to mulayered theory are concerned.

ACKNOWLEDGMENTS

The author has appreciated very much the manner in whAssociate Editor Sarp Adali has conducted the review pcess for this article. Professor Adali has, in fact, considehimself, referees, and author as one team with the objecof reaching the most complete historical note. In particuthe author acknowledges the suggestions given by the reees as far as the developments of AMT in the East is ccerned. Such developments were missing in the early verof the manuscript. However, what is written above doesremove from the author the responsibility of any omissionforgetfulness. The author finally acknowledges ProfesShifrin, who provided the original article in Russian@36# andD Carrera who made the Italian translation of the sameticle.

APPENDIX: TWO DIFFERENT WAYSOF WRITING THE AMT

This appendix shows two different manners in which tdisplacement fields related to the AMT type theories canderived. The same analysis could also be extended to LFor the sake of simplicity, a 1D flat case is considered. Etension to the 2D case and shell geometry should be obviThe origin of the thickness coordinates, for the sake of splicity, is placed in the bottom layer; attention has beenstricted to a piecewise continuous, linear displacement fi

The displacement fieldu in each layer can be first writtenby using displacement values at the interfaces~see Fig. 8!:

u1~z!5u01zc1 , <z<h1

u2~z!5u1~h1!1~z2h1!c2 , h1<z<h2

• • • • • • •

(73)uk~z!5uk21~hk21!1~z2hk21!ck, hk21<z<hk

• • • • • • •

uNl~z!5uNl21~hNl21!1~z2hNl21!cNl,

hNl21<z<hNl

whereu0 is the value of the displacementu with corresponding

to the bottom surface.uk(hk), k51,Nl are the interface values ofu.

Fig. 8 Geometry and notations employed in the Appendix

i

n

n

et

d

vi-eene

d-ted

ondi-to

ssess.

ted

e

er

i-

he

n

lls,

Appl Mech Rev vol 56, no 3, May 2003 Carrera: Historical review of Zig-Zag theories 305

ck , k51,Nl are the values which identify the rotationsthe layers.Moreover

u1~h1!5u01h1c1

u2~h2!5u1~h1!1~h22h1!c2

••5 • • •

• • • •5 • • • (74)

uk~hk!5uk21~hk21!1~hk2hk21!ck

• • • ••5••

uNl21~z!5uNl22~hNl22!1~hNl212hNl22!cNl21

The generic interface values is then re-written,

uk~hk!5u01 (k51

Nl21

~hk2hk21!ck , k51,Nl (75)

It is noted thath050. It follows that the displacement field ieach layer can be written in the following unified form,

uk~z!5u01 (k51

Nl22

~hk212hk22!ck211~z2hk21!ck ,

k51,Nl (76)

TheNl rotationsck can be expressed in terms of one of the~for instance the rotation in the bottom layer! by imposingthe Nl21 interlaminar continuity conditions for transversshear stresses,

ck5akc1 , k52,Nl21

whereak are layer constants defined by interlaminar traverse stresses. To be more precise,c1 appears as a combination of an in-plane derivative of transverse displacementwx ,and a transverse shear straingxz . The previous relation couldbe written as

ck52wx1akgxz

as it is in Eqs.~33,36!. For the sake of simplicity we take thfirst of these as valid: as a consequence the displacemenu is

uk~z!5u01 (k51

Nl22

~hk212hk22!ak21c1

1~z2hk21!akc1 , k51,Nl (77)

Equations~76! give the displacement field in each layeSuch a displacement field can be written in a form whichapplicable to the whole multilayer by using the Heavisistep function. Such a function is defined as follows

H~z2zk!5H 0 z<zk

1 z>zk(78)

By means ofH, the displacementu can be written in a formwhich is formally not affected byk,

n

m

e

s--

r.ise

u~z!5u01 (k51

Nl21

~z2zk21!ckH~z2zk! (79)

or

u~z!5u01 (k51

Nl21

~z2zk21!akc1H~z2zk! (80)

Judging from the available literature, the use of the Heaside function could be considered questionable. It has bused in early works@99,102#, and abandoned in some of thmore recent papers@109,112#. The Heaviside function hasthe formal advantage of permitting one to writeu in a formwhich is not affected byk index. Nevertheless, such an avantage could be ineffective in the calculation and relacomputer implementations.

It appears clear that a linear, piecewise form foru leads tolayer continuity stiffnessesak which are independent ofz. Infact, top-bottom homogeneous transverse shear stress ctions cannot be imposed in this case. This was knownAmbartsumian@33# and Whitney@92# who assumedak as acubic function of z. In fact, the two additional functionsrelated to quadratic and cubic terms ofz are determined bythe two homogeneous conditions of transverse shear strewith corresponding to top and bottom plate/shell surface

LIST OF ACRONYMS

Acronyms that are used frequently in this article are lisbelow.

AMT - Ambartsumian Multilayered TheoryESLM - Equivalent Single Layer ModelsIC - Interlaminar ContinuityKR - Koiter’s RecommendationLMT - Lekhnitskii Multilayered TheoryLWM - Layer-Wise ModelsRM - Reissner-MindlinRMT - Reissner Multilayered TheoryRMVT - Reissner’s Mixed Variational TheoremWFHL - Weak Form of Hooke’s LawZZ - Zig-Zag

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308 Carrera: Historical review of Zig-Zag theories Appl Mech Rev vol 56, no 3, May 2003

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ecnicothe

erissiles0 he iste fu

and96, he

are:smartf non-whichReviews,al for.

thearbitrary laminated configurations,ASME J. Appl. Mech.54, 181–189.

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After earning two degrees (Aeronautics, 1986 and Aerospace Engineering, 1988) at the Politdi Torino, Erasmo Carrera received his PhD degree in Aerospace Engineering in 1991, atPolitecnico di Milano-Politecnico di Torino-Universita´ di Pisa. He began working as a Researchat the Department of Aerospace of Politecnico di Torino in 1992, also teaching courses on Mand Aerospace Structure Design, Plates and Shells, and Finite Element Method. Since 200Associate Professor of Aerospace Structures and Aeroelasticity. He visited twice the Institu¨rStatik und Dynamik, Universita¨t Stuttgart, the first time as a PhD student (six months in 1991)then as visiting Scientist under a GKKS Grant (18 months in 1995-96). In the summer of 19was Visiting Professor at the ESM Department of Virginia Tech. His main research topicscomposite materials, finite elements, plates and shells, postbuckling and stability by FEM,structures, thermal stress, aeroelasticity, multibody dynamics and design, and analysis oclassical lifting systems. He is an author of more than eighty articles on these topics, many of

have been published in international journals. He serves as a referee for many journals, such as Applied MechanicsAIAA Journal, Journal of Sound and Vibration, International Journal of Solids, and Structures International JournNumerical Methods in Engineering, and as contributing editor for Mechanics of Advanced Materials and Structures