partially zig-zag advanced higher order shear deformation ...demasi/articles/... · of zig-zag form...

Composite Structures xxx (2011) xxx–xxx

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Composite Structures

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Partially Zig-Zag Advanced Higher Order Shear Deformation Theories Based on theGeneralized Unified Formulation

Luciano DemasiDepartment of Aerospace Engineering and Engineering Mechanics, San Diego State University, College of Engineering, 5500 Campanile Drive, San Diego, USA

a r t i c l e i n f o

Article history:Available online xxxx

Keywords:Composite structuresCarrera’s Unified FormulationCompact NotationsGeneralized Unified FormulationPartially Zig-Zag Advanced Higher OrderShear Deformation TheoriesMurakami’s Zig-Zag Function

0263-8223/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruct.2011.07.022

E-mail address: [email protected]: http://www.lucianodemasi.com

Please cite this article in press as: Demasi L. Parmulation. Compos Struct (2011), doi:10.1016/j.

a b s t r a c t

The Generalized Unified Formulation (GUF) is a multi-theory and a multi-fidelity architecture for the gen-eration of a virtually infinite class of Advanced Higher Order Shear Deformation Theories (AHSDT) or Zig-Zag theories or Layer-Wise (LW) theories with any order of expansion for each of the primary variables.This work will present, for the first time in the literature, an extension of GUF to address problems inwhich every single variable can have either an Equivalent Single Layer (ESL) or a Zig-Zag-enhanced ESLdescription [Partially Zig-Zag Advanced Higher Order Shear Deformation Theories (PZZAHSDT)]. Applica-tions to the case of thick sandwich structures are presented: starting from a baseline fourth-order AHSDTwhich also includes the transverse strain effects, all the possible types of PZZAHSDT are generated andcompared with the baseline and with a fourth-order fully Zig-Zag theories.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The analysis of composite and sandwich structures requires amodel that can take into account the strong anisotropy along thethickness and the shear deformation effects. Classical Plate Theory[33] is in general not adequate especially for moderately thick orthick plates. The relaxation of the Kirchhoff assumptions led tothe formulation of the First Order Shear Deformation Theory (FSDT)[44,35,32] in which a generic planar cross-section initially orthog-onal to the mid-plane of the plate after the deformation takes placeis still planar but no longer perpendicular to the mid-surface of theplate. An initially planar cross-section can be allowed to deform in ageneric shape with the inclusion (usually for the in-plane displace-ments only) of higher order terms in the axiomatic expansion alongthe thickness. These approaches are the so called Higher OrderShear Deformation Theories (HSDT) [54,7,30,56]. Transverse straineffects may be added by including higher order terms in the thick-ness expansion for the out-of-plane displacement uz. The resultingtheories are here indicated as Advanced Higher Order Shear Derfor-mation Theories (AHSDT). However, as will be demonstrated indetail in this paper, due to both the interlaminar equilibrium ofthe transverse stresses and anisotropy of the mechanical propertiesalong the thickness, all the displacements and stresses (with theonly exception of the transverse normal stress) present a disconti-nuity of the first spatial derivative with respect to the thicknesscoordinate z. The discontinuity of the displacement variables is

ll rights reserved.

tially Zig-Zag Advanced Highercompstruct.2011.07.022

what people refer to ‘‘Zig-Zag form of the displacements’’[6,1,31,53]. Based on this physical evidence it has been proposedby many researchers to include the Zig-Zag form of the displace-ments a priori with a model that is still a computationally inexpen-sive formulation but has a significant improvement of the resultsdue to a better physical representation of the real deformation ofthe anisotropic composite structure. Following the historical recon-struction attempted in Ref. [13] on this subject, the Zig-Zag theoriescan be subdivided into three major groups:

� Lekhnitskii Multilayered Theory (LMT)� Ambartsumian Multilayered Theory (AMT)� Reissner Multilayered Theory (RMT)

In the Lekhnitskii Multilayered Theory [34] (originally formu-lated for multilayered beams) the Zig-Zag form of the displace-ments and continuity of the transverse stresses were enforced.LMT was extended to the case of plates by Ren [48,47].

In the Ambartsumian Multilayered Theory [5,4,2,3] (formulatedfor both plates and shells) an interlaminar continuous transverseshear stress field is a priori enforced. The displacement fields pres-ent a discontinuity of the first derivatives in the thickness direction(Zig-Zag form). Later the effects of transverse normal strain/stresswere also included [51,52,38–40]. Whitney [55] applied AMT toanisotropic and non-symmetrical plates. Later [41] Rath and Dasextended Whitney’s work to shells and dynamic problems. Otherauthors such as Yu [57], Chou and Carleone [18] and Di Sciuva[29] worked on similar (but less general) approaches. Cho and Par-merter refined [17] these alternative approaches and obtained aZig-Zag formulation which was equivalent to AMT.

Order Shear Deformation Theories Based on the Generalized Unified For-

http://dx.doi.org/10.1016/j.compstruct.2011.07.022

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2 L. Demasi / Composite Structures xxx (2011) xxx–xxx

In the Reissner Multilayered Theory the transverse stresses areprimary unknowns as well as the displacement variables [45,46].The variational statement is Reissner’s Mixed Variational Theorem.Murakami [36] proposed to take into account the Zig-Zag effects byenhancing the corresponding displacement variable with a Zig-Zagfunction denoted here as Murakami’s Zig-Zag Function (MZZF).Applications of the concept of enhancing the displacement fieldwith MZZF were presented in several works [15,19,21,26,10,9,8,50] in the last few years. The main advantage of Zig-Zagapproaches is in the significant improvement of the accuracy withlittle increment of the computational cost with respect to the inex-pensive (but often inaccurate) classical formulations.

For more challenging problems (for example a sandwich struc-ture with a very high Face-to-Core Stiffness Ratio) a LW[16,37,42,49,43,12,11] approach is necessary and represents avaluable alternative to the computationally very expensive FiniteElement discretizations based on solid elements. All of these LWand ESL approaches can be unified with the adoption of the socalled Compact Notations (CN). Example of Compact Notations areCarrera’s Unified Formulation (CUF) [14] and its generalization rep-resented by the Generalized Unified Formulation (GUF) [22–28].The main idea behind GUF is the writing of each displacement var-iable (or/and stress variable in the case of mixed formulations)independently from the other unknowns. With this approach,any combination of orders can be achieved. For example, an AHSDTwith cubic thickness expansion for the in-plane displacements anda parabolic expansion for the out-of-plane displacement can berepresented as well as a LW theory with parabolic expansion ofthe in-plane displacement variables and linear expansion for thetransverse displacement uz.

1.1. What are the new contributions of this work

Up to now the Generalized Unified Formulation could handleany combination of orders for the displacement unknowns. How-ever, the type of description was the same for all the displacements(e.g., LW or ESL description for all the variables). With this workand for the first time, GUF is further generalized. In particular, itis presented the case in which some variables can be described inan ESL form and others will be enhanced with MZZF. This work firstdiscusses the physical and mathematical justifications of the needof Zig-Zag form of the displacements and stresses (with the onlyexception represented by the transverse normal stress rzz as willbe discussed later) and then presents the main theoretical aspectsand results.

2. Zig-Zag form of the displacements

A mathematical explanation of the Zig-Zag form of the displace-ments is discussed. Classical Form of Hooke’s Law (CFHL) in platecoordinates [43,23] is:

rxx

ryy

rxy

rzx

rzy

rzz

2666666664

3777777775¼

eC11eC12

eC16 0 0 eC13eC12eC22

eC26 0 0 eC23eC16eC26

eC66 0 0 eC36

0 0 0 eC55eC45 0

0 0 0 eC45eC44 0eC13

eC23eC36 0 0 eC33

26666666664

37777777775

exx

eyy

cxy

czx

czy

ezz

2666666664

3777777775ð1Þ

where the explicit expressions for the coefficients can be found inRef. [43]. With the exception of the Zig-Zag term (in the Zig-Zagterm k represents an actual exponent as will be specifically seenlater), whenever superscript/subscript k is present it means thatthe corresponding quantity is referred to layer k. From CFHL andthe geometric relations, which relate the strains to the displace-

Please cite this article in press as: Demasi L. Partially Zig-Zag Advanced Highermulation. Compos Struct (2011), doi:10.1016/j.compstruct.2011.07.022

ments, the transverse shear stress rzx evaluated at the top surfaceðz ¼ ztopk

Þ of layer k can be written as

½rzx�z¼ztopk� rk t

zx ¼ eCk55c

k tzx þ eC k

45ck tzy

¼ eCk55

@uk tz

@xþ @uk t

x

@z

� �þ eC k

45@uk t

z

@yþ@uk t

y

@z

!ð2Þ

the superscript t is used to highlight the fact that the top surface oflayer k is considered. The same relationship written at the bottomsurface ðz ¼ zbotðkþ1Þ Þ of layer k + 1 can be written as

½rzx�z¼zbotðkþ1Þ� rðkþ1Þ b

zx

¼ eC ðkþ1Þ55

@uðkþ1Þ bz

@xþ @uðkþ1Þ b

x

@z

!

þ eC ðkþ1Þ45

@uðkþ1Þ bz

@yþ @uðkþ1Þ b

y

@z

!ð3Þ

the superscript b is used to highlight the fact that the bottom surfaceof layer k + 1 is considered. For the transverse shear stress rzy sim-ilar finding can be obtained from CFHL (see Eq. (1)):

rðkþ1Þ bzy ¼ eC ðkþ1Þ

45@uðkþ1Þ b

z

@xþ @uðkþ1Þ b

x

@z

!þ eC ðkþ1Þ

44@uðkþ1Þ b

z

@yþ @uðkþ1Þ b

y

@z

!

rk tzy ¼ eCk

45@uk t

z

@xþ @uk t

x

@z

� �þ eCk

44@uk t

z

@yþ@uk t

y

@z

!ð4Þ

In the most general case the fiber orientations and/or material prop-erties of layer k are different than the ones relative to layer k + 1.Thus, in the general case it can be inferred thateC ðkþ1Þ

ij – eCkij ð5Þ

where eC ðkþ1Þij is the generic Hooke’s coefficient relative to layer k + 1

whereas eCkij is the generic Hooke’s coefficient relative to layer k. At

any arbitrarily given point P1 � (x1,y1) located at the interfacebetween two consecutive layers the displacement uz must be a con-tinuous function in the thickness direction: uðkþ1Þ b

z ðx1; y1Þ ¼uk t

z ðx1; y1Þ. At any other arbitrary point P2 � (x2,y2) at the interfacebetween the same two layers the displacement uz must still be acontinuous function: uðkþ1Þ b

z ðx2; y2Þ ¼ uk tz ðx2; y2Þ. Since the points P1

and P2 can be selected anywhere in the x, y plate domain (note thatthe z coordinates of the two points correspond to zbotðkþ1Þ which iscoincident with ztopk

by definition of interface) it is deduced thatall the in-plane derivatives of the displacement uz must also be con-tinuous functions in the thickness direction (otherwise a satisfiedcontinuity of the displacement uz in P1 would not imply the conti-nuity of the same displacement in another point P2 selected onthe interface). Thus, the compatibility of the displacement uz

implies that any derivative of uz of any order with respect to anydirection contained in the plane at the interface between any twoconsecutive layers must be a continuous function along the thick-ness. A particular case of this statement is the continuity of the firstderivatives (but as discussed above any order of in-plane deriva-tives must be continuous functions):

½uz�z¼zbotðkþ1Þ¼ ½uz�z¼ztopk

() uðkþ1Þ bz ¼ uk t

z )@uðkþ1Þ b

z@x ¼ @uk t

z@x

@uðkþ1Þ bz@y ¼ @uk t

z@y

8<: ð6Þ

where for example the following definition has been used:

@uz

@x

� �z¼zbotðkþ1Þ

� @uðkþ1Þ bz

@xð7Þ



L. Demasi / Composite Structures xxx (2011) xxx–xxx 3

For the displacements ux and uy a formally identical relation (see Eq.(6)) is valid. The equilibrium condition rðkþ1Þ b

zx ¼ rk tzx implies (see

Eqs. (2) and (3)):

rðkþ1Þ bzx ¼ rk t

zx

) eC ðkþ1Þ55

@uðkþ1Þ bz

@xþ @uðkþ1Þ b

x

@z

!

þ eC ðkþ1Þ45

@uðkþ1Þ bz

@yþ @uðkþ1Þ b

y

@z

!

¼ eCk55

@uk tz

@xþ @uk t

x

@z

� �þ eC k

45@uk t

z

@yþ@uk t

y

@z

!ð8Þ

The equilibrium condition rðkþ1Þ bzy ¼ rk t

zy implies (see Eq. (4)):

rðkþ1Þ bzy ¼ rk t

zy

) eC ðkþ1Þ45

@uðkþ1Þ bz

@xþ @uðkþ1Þ b

x

@z

!

þ eC ðkþ1Þ44

@uðkþ1Þ bz

@yþ @uðkþ1Þ b

y

@z

!

¼ eCk45

@uk tz

@xþ @uk t

x

@z

� �þ eC k

44@uk t

z

@yþ@uk t

y

@z

!ð9Þ

Substituting Eq. (6) into the condition for rzx (Eq. (8)):

eC ðkþ1Þ55

@uk tz

@xþ @uðkþ1Þ b

x

@z

!þ eC ðkþ1Þ

45@uk t

z

@yþ @uðkþ1Þ b

y

@z

!

¼ eCk55

@uk tz

@xþ @uk t

x

@z

� �þ eCk

45@uk t

z

@yþ@uk t

y

@z

!ð10Þ

Similarly, starting from the condition relative to rzy (Eq. (9)) andusing Eq. (6):

eC ðkþ1Þ45

@uk tz

@xþ @uðkþ1Þ b

x

@z

!þ eC ðkþ1Þ

44@uk t

z

@yþ @uðkþ1Þ b

y

@z

!

¼ eCk45

@uk tz

@xþ @uk t

x

@z

� �þ eCk

44@uk t

z

@yþ@uk t

y

@z

!ð11Þ

If Eq. (11) is multiplied by eC ðkþ1Þ55 and the resulting expression is sub-

tracted from Eq. (10) after multiplying it by eC ðkþ1Þ45 , it is possible to

obtain the following relation:

eC ðkþ1Þ45

eC ðkþ1Þ45 � eC ðkþ1Þ

55eC ðkþ1Þ

44 þ eC ðkþ1Þ55

eCk44 � eC ðkþ1Þ

45eCk

45

� � @uk tz

@y

þ eC ðkþ1Þ45


55eC ðkþ1Þ

44

� � @uðkþ1Þ by

@z

þ eC ðkþ1Þ55


45eCk

45

� � @uk ty

@z

þ eC ðkþ1Þ55


45eCk

55

� � @uk tz

@xþ @uk t

x

@z

� �¼ 0 ð12Þ

Suppose that the corresponding coefficients of layer k and k + 1 arethe same. In that case Eq. (12) becomes:

eCk45eCk

45 � eCk55eCk

44

� � @uðkþ1Þ by

@z� eCk

45eCk

45 � eC k55eC k

44

� � @uk ty

@z¼ 0

) @uðkþ1Þ by

@z¼@uk t

y

@zð13Þ

which means that no discontinuity of the derivative of uy in thethickness direction is present. If, however, Hooke’s coefficients aredifferent (this is the usual general case for composite structures)

then from Eq. (12) it is clear that@uðkþ1Þ b

y

@z –@uk t

y

@z which means that


uy presents a Zig-Zag form (discontinuity of the first derivative inthe thickness direction). To demonstrate the discontinuity of the firstderivative of ux with respect to z it is sufficient to multiply Eq. (10)

by eC ðkþ1Þ44 and Eq. (11) by eC ðkþ1Þ

45 and subtract the two resultingequations:

eC ðkþ1Þ44


45eC ðkþ1Þ

45 þ eC ðkþ1Þ45


44eC k

55

� � @uk tz

@x

þ eC ðkþ1Þ44


45eC ðkþ1Þ

45

� � @uðkþ1Þ bx

@z

þ eC ðkþ1Þ45

eC k45 � eC ðkþ1Þ

44eCk

55

� � @uk tx

@z

þ eC ðkþ1Þ45

eC k44 � eC ðkþ1Þ

44eCk

45

� � @uk tz

@yþ@uk t

y

@z

!¼ 0

Suppose that Hooke’s coefficients of layer k are equal to the onesrelative to layer k + 1. In that case Eq. (14) leads to

eC k44eC k

55 � eCk45eCk

45

� � @uðkþ1Þ bx

@z� eCk

44eCk

55 � eCk45eCk

45

� � @uk tx

@z¼ 0

) @uðkþ1Þ bx

@z¼ @uk t

x

@zð15Þ

If, however, Hooke’s coefficients are different (usual general case for

composite structures) then from Eq. (14) it is clear that @uðkþ1Þ bx@z – @uk t

x@z

which means that ux presents a Zig-Zag form (discontinuity of thefirst derivative in the thickness direction).

To demonstrate the Zig-Zag form of the displacement uz theequilibrium of the transverse normal stress has to be considered.From CFHL (see Eq. (1)) and the geometric relations it is possibleto calculate rk t

zz which represents the transverse normal stressevaluated at the top surface of layer k:

rk tzz ¼ eCk

13@uk t

x

@xþ eCk

23

@uk ty

@yþ eCk

36@uk t

x

@yþ@uk t

y

@x

!þ eC k

33@uk t

z

@zð16Þ

Similar equation can be written for the transverse normal stressevaluated at the bottom surface of layer k + 1:

rðkþ1Þ bzz ¼ eC ðkþ1Þ

13@uðkþ1Þ b

x

@xþ eC ðkþ1Þ

23@uðkþ1Þ b

y

@y

þ eC ðkþ1Þ36

@uðkþ1Þ bx

@yþ @uðkþ1Þ b

y

@x

!þ eC ðkþ1Þ

33@uðkþ1Þ b

z

@zð17Þ

The compatibility of the displacements ux and uy lead to the equal-ity of their in-plane derivatives at the interface (this is the sameconcept that allowed the writing of Eq. (6) from the compatibilityof the displacement uz). This means that the transverse normalstress evaluated at the bottom surface of layer k + 1 (see Eq. (17))can be rewritten as

rðkþ1Þ bzz ¼ eC ðkþ1Þ

13@uk t

x

@xþ eC ðkþ1Þ

23

@uk ty

@yþ eC ðkþ1Þ

36@uk t

x

@yþ@uk t

y

@x

!

þ eC ðkþ1Þ33

@uðkþ1Þ bz

@zð18Þ

Because of equilibrium it has to be rðkþ1Þ bzz ¼ rk t

zz . Thus, from Eqs.(16) and (18) it is possible to deduce:

eC k13� eC ðkþ1Þ

13

� �@uktx

@xþ eCk

23� eC ðkþ1Þ23

� �@ukty

@y

þ eCk36� eC ðkþ1Þ

36

� � @uktx

@yþ@ukt

y

@x

!þ eCk

33@ukt

z

@z� eC ðkþ1Þ

33@uðkþ1Þb

z

@z¼0 ð19Þ




It is immediate to recognize from Eq. (19) that if the two consecu-tive layers have the same Hooke’s coefficients, then it has to be@uk t

z@z ¼

@uðkþ1Þ bz@z . It is also clear from the same relation that in the most

general case it is @uk tz@z – @uðkþ1Þ b

z@z . In other words, uz presents a discon-

tinuity of the first derivative in the thickness direction (Zig-Zagform of the displacement uz).

3. Discontinuity of the in-plane stresses rxx, rxy, and ryy

The in-plane stress rxx can be evaluated at the top surface oflayer k (see CFHL, Eq. (1)):

rk txx ¼ eCk

11@uk t

x

@xþ eC k

12

@uk ty

@yþ eCk

16@uk t

x

@yþ@uk t

y

@x

!þ eC k

13@uk t

z

@zð20Þ

The stress rxx can also be evaluated at the bottom of layer k + 1:

rðkþ1Þ bxx ¼ eC ðkþ1Þ

11@uðkþ1Þ b

x

@xþ eC ðkþ1Þ

12@uðkþ1Þ b

y

@y

þ eC ðkþ1Þ16

@uðkþ1Þ bx

@yþ @uðkþ1Þ b

y

@x

!þ eC ðkþ1Þ

13@uðkþ1Þ b

z

@zð21Þ

However, the compatibility of displacements implies that theirin-plane derivatives of an arbitrary order must be continuous func-tions at the interface. This means that Eq. (21) can be written as

rðkþ1Þ bxx ¼ eC ðkþ1Þ

11@uk t

x

@xþ eC ðkþ1Þ

12

@uk ty

@yþ eC ðkþ1Þ

16@uk t

x

@yþ@uk t

y

@x

!

þ eC ðkþ1Þ13

@uðkþ1Þ bz

@zð22Þ

Direct comparison between Eqs. (22) and (20) shows that in generalthe in-plane stress rxx is a discontinuous function at the interface.Note that when all Hooke’s coefficients are the same, which also

implies that @uk tz@z ¼

@uðkþ1Þ bz@z , the stress rxx is a continuous function.

Similar demonstration can be reproduced for the other in-planestresses. No other mathematical insight can be derived by showingthe explicit derivations and, therefore, the details are here omitted.

4. Non-Zig-Zag form of the transverse normal stress rzz

The transverse normal stress rzz does not present a Zig-Zagform. In other words, @rzz

@z is a continuous function along the thickness.To show that, consider the third equilibrium equation written atthe top surface of layer k:

@rk tzz

@z¼ � @rk t

zx

@xþ@rk t

zy

@y

!ð23Þ

The thickness derivative of the transverse normal stress is nowevaluated at the bottom surface of layer k + 1:

@rðkþ1Þ bzz

@z¼ � @rðkþ1Þ b

zx

@xþ @r

ðkþ1Þ bzy

@y

!ð24Þ

The equilibrium condition implies that the transverse shear stressesmust be a continuous function at the interface between any twoconsecutive layers. This is also true for any order of the in-planederivatives (a discussion similar to the one presented in Eq. (6)can be made for this case too). This means that Eq. (24) can be writ-ten as

@rðkþ1Þ bzz

@z¼ � @rk t

zx

@xþ@rk t

zy

@y

!ð25Þ


The right hand side of Eq. (25) is identical to the right hand side ofEq. (23). This means that the left hand side of both equations mustbe the same. In other words, it has been demonstrated that@rðkþ1Þ b

zz@z ¼ @rk t

zz@z even if the materials and/or fiber orientations of layer

k and k + 1 are different.

5. Zig-Zag form of the transverse shear stresses rzx and rzy

To discuss the properties of the transverse shear stresses it isconvenient to write Hooke’s law in its mixed form (note that themixed form is perfectly equivalent to the classical form; moredetails can be found for example in Ref. [23]) denoted here asMixed Form of Hooke’s Law (MFHL):

rxx

ryy

rxy

czx

czy

ezz

266666666664

377777777775¼

C11 C12 C16 0 0 C13

C12 C22 C26 0 0 C23

C16 C26 C66 0 0 C36

0 0 0 C55 C45 0

0 0 0 C45 C44 0

�C13 �C23 �C36 0 0 C33

266666666664

377777777775

exx

eyy

cxy

rzx

rzy

rzz

266666666664

377777777775ð26Þ

In this demonstration the mixed form is preferred to the ClassicalForm of Hooke’s Law (see Eq. (1)) because in the writing of thein-plane stresses there is no formal dependence on ezz ¼ @uz

@z whichis not a continuous function in the thickness direction (Zig-Zag formof displacement uz).

Starting from MFHL (Eq. (26)) and using the geometric relationsit is possible to write the derivative of rxx with respect to x and thederivative of the stress rxy with respect to y and evaluate bothderivatives at the top surface of layer k as follows:

@rk txx

@x¼ Ck

11@2uk t

x

@x2 þ Ck12

@2uk ty

@x@yþ Ck

16@2uk t

x

@x@yþ@2uk t

y

@x2

!þ Ck

13@rk t

zz

@x

ð27Þ

@rk txy

@y¼ Ck

16@2uk t

x

@x@yþ Ck

26

@2uk ty

@y2 þ Ck66

@2uk tx

@y2 þ@2uk t

y

@x@y

!þ Ck

36@rk t

zz

@y

ð28Þ

Similarly, observing that all the in-plane derivatives of any order ofthe displacements and transverse stresses are continuous functions

at the interface (so, for example it is@2uðkþ1Þ b

y

@y2 ¼ @2uk ty

@y2 ), it is possible to

write the derivative with respect to x of the stress rxx and the deriv-ative with respect to y of the stress rxy (see Eq. (26)) and evaluateboth derivatives at the bottom surface of layer k + 1 as follows:

@rðkþ1Þ bxx

@x¼ Cðkþ1Þ

11@2uk t

x

@x2 þ Cðkþ1Þ12

@2uk ty

@x@yþ Cðkþ1Þ

16@2uk t

x

@x@yþ@2uk t

y

@x2

!

þ Cðkþ1Þ13

@rk tzz

@xð29Þ

@rðkþ1Þ bxy

@y¼ Cðkþ1Þ

16@2uk t

x

@x@yþ Cðkþ1Þ

26

@2uk ty

@y2 þ Cðkþ1Þ66

@2uk tx

@y2 þ@2uk t

y

@x@y

!

þ Cðkþ1Þ36

@rk tzz

@yð30Þ

The first equilibrium equation written at the top surface of layer kcan be written as

@rk tzx

@z¼ � @rk t

xx

@xþ@rk t

xy

@y

!ð31Þ

Substituting Eqs. (27) and (28) into Eq. (31) it is possible to write:




@rk tzx

@z¼ �Ck

11@2uk t

x

@x2 � Ck12

@2uk ty

@x@y� Ck

16@2uk t

x

@x@yþ@2uk t

y

@x2

!

� Ck13@rk t

zz

@x� Ck

16@2uk t

x

@x@y� Ck

26

@2uk ty

@y2

� Ck66

@2uk tx

@y2 þ@2uk t

y

@x@y

!� Ck

36@rk t

zz

@yð32Þ

Similar derivations can be written for the derivative with respect toz of the transverse shear stress rzx evaluated at the bottom surfaceof layer k + 1. Using Eqs. (29) and (30) and the first equilibriumequation:

@rðkþ1Þ bzx

@z¼ �Cðkþ1Þ

11@2uk t

x

@x2 � Cðkþ1Þ12

@2uk ty

@x@y� Cðkþ1Þ

16@2uk t

x

@x@yþ@2uk t

y

@x2

!

� Cðkþ1Þ13

@rk tzz

@x� Cðkþ1Þ

16@2uk t

x

@x@y� Cðkþ1Þ

26

@2uk ty

@y2

� Cðkþ1Þ66

@2uk tx

@y2 þ@2uk t

y

@x@y

!� Cðkþ1Þ

36@rk t

zz

@yð33Þ

Subtracting Eq. (33) from Eq. (32):

@rk tzx

@z� @r

ðkþ1Þ bzx

@z¼ Cðkþ1Þ

11 � Ck11

� � @2uk tx

@x2 þ Cðkþ1Þ12 � Ck

12

� � @2uk ty

@x@y

þ Cðkþ1Þ16 � Ck

16

� � @2uk tx

@x@yþ@2uk t

y

@x2

!


13

� � @rk tzz

@x


16

� � @2uk tx

@x@y


26

� � @2uk ty

@y2


66

� � @2uk tx

@y2 þ@2uk t

y

@x@y

!


36

� � @rk tzz

@yð34Þ

From which it is deduced that @rk tzx

@z – @rðkþ1Þ bzx@z unless the two layers

present the same Hooke’s coefficients. Similar derivations can bewritten for the transverse shear stress rzy (the details are omittedfor brevity). Summarizing the main properties of a multilayeredcomposite structures are the following:

� All the displacements ux, uy, and uz must be continuous func-tions in the thickness direction (compatibility conditions).� All the transverse stresses rzx, rzy, and rzz must be continuous

functions in the thickness direction (equilibrium conditions).� All the in-plane stresses rxx, rxy, and ryy are in general discon-

tinuous functions at the interface between layers.� All the displacements ux, uy, and uz present a Zig-Zag form (first

thickness derivative of the displacements is not a continuousfunction at the interfaces between two consecutive layers).The Zig-Zag form of the in-plane displacements can be directlydemonstrated from the equilibrium conditions of the transverseshear stresses. The Zig-Zag form of the transverse displacementuz can be directly derived from the equilibrium condition of thetransverse normal stress rzz.� The transverse shear stresses rzx and rzy present a Zig-Zag form

(first thickness derivative of the displacements is not a continu-ous function at the interfaces between adjacent layers). This canbe shown starting from the first 2 equilibrium equations andHooke’s equation written in its mixed form.


� The transverse normal stress rzz does not present any Zig-Zagpattern along the thickness even if the materials and/or fiberorientations adopted for the different layers are different. Thisproperty can be easily deduced from the third equilibriumequation and can be shown to be a direct consequence of theequilibrium condition for the transverse shear stresses rzx andrzy.

6. The case of thin plates: Zig-Zag form of the transverse shearstresses rzx and rzy and non-Zig Zag form of the displacements

For thin plates the Classical Plate Theory is adequate. Actually, ithas been analytically demonstrated [20] that for a simply sup-ported rectangular isotropic plate the elasticity solution leads toKirchhoff plate formulation when the thickness of the plateapproaches zero. The assumptions adopted in the Classical PlateTheory imply that the transverse normal strain ezz and transverseshear strains czx and czy are zero. The condition ezz = 0 applied atthe bottom surface of layer (k + 1) and at the top surface at layerk implies:

eðkþ1Þ bzz ¼ 0) @uðkþ1Þ b

z@z ¼ 0

ek tzz ¼ 0) @uk t

z@z ¼ 0

8<: ) 0 ¼ @uk tz

@z¼ @uðkþ1Þ b

z

@zð35Þ

thus, no Zig-Zag form of the displacement uz is present. Similarly,the condition czx = 0 implies:

cðkþ1Þ bzx ¼ 0) @uðkþ1Þ b

z@x þ @uðkþ1Þ b

x@z ¼ 0

ck tzx ¼ 0) @uk t

z@x þ

@uk tx@z ¼ 0

8<: ð36Þ

However, the compatibility of the displacement uz implies (see also

Eq. (6)) @uðkþ1Þ bz@x ¼ @uk t

z@x . Thus, Eq. (36) can be rewritten as

cðkþ1Þ bzx ¼ 0) @uk t

z@x þ

@uðkþ1Þ bx@z ¼ 0

ck tzx ¼ 0) @uk t

z@x þ

@uk tx@z ¼ 0

8<: ) @uðkþ1Þ bx

@z¼ @uk t

x

@zð37Þ

Thus, it can then be deduced that no Zig-Zag form of the displace-ment ux is present. Similar conclusion can be obtained for the dis-placement uy when the condition czy = 0 is analyzed (the detailsare omitted for brevity).

As far as the transverse shear stress rzx is considered, it shouldbe noted that Eq. (34) is not modified in the case of thin plates andso it can be deduced that rzx presents a Zig-Zag form even for verythin plates when Hooke’s coefficients of two consecutive adjacentlayers are different. Using the same procedure that led to Eq. (34) asimilar conclusion can also be reached for rzy.

For thin plates the uz presents no Zig-Zag form: @uðkþ1Þ bz@z ¼ @uk t

z@z .

Nevertheless, as can be directly deduced by a comparison of Eqs.(22) and (20), the in-plane stresses are discontinuous functions ifHooke’s coefficients of two adjacent layers are different.

7. Extension of the Generalized Unified Formulation to includepartially Zig-Zag theories

A virtually infinite number of AHSDTs can be derived from aremoval of the assumptions used to define the displacement fieldof Classical Plate Theory. Two examples are reported in Eqs. (38)and (39):

AHSDT I :

ux ¼ ux0ðx; yÞ þ z/1uxðx; yÞ þ z2/2ux

ðx; yÞuy ¼ uy0

ðx; yÞ þ z/1uyðx; yÞ þ z2/2uy

ðx; yÞuz ¼ uz0ðx; yÞ þ z/1uz

ðx; yÞ

8>><>>: ð38Þ




AHSDT II :

ux ¼ ux0 ðx; yÞ þ z/1uxðx; yÞ þ z2/2ux

ðx; yÞ þ z3/3uxðx; yÞ

uy ¼ uy0ðx; yÞ þ z/1uy

ðx; yÞ þ z2/2uyðx; yÞ þ z3/3uy

ðx; yÞuz ¼ uz0 ðx; yÞ þ z/1uz

ðx; yÞ

8>><>>:ð39Þ

In AHSDT I the in-plane displacements are expanded with a para-bolic Taylor polynomial in the thickness direction. A linear expan-sion is used for the transverse displacement. In AHSDT II a linearexpansion is kept for the uz displacement whereas a cubic expan-sion is adopted for the in-plane displacements. To show how theGeneralized Unified Formulation can be effectively used to repre-sent any Advanced Higher Order Shear Deformation Theory, it issufficient to discuss how the displacement in the x direction isrewritten in the framework of GUF’s notation. We can start fromthe particular case reported in Eq. (39):

ux ¼ ux0 þ z/1uxþ z2/2ux

þ z3/3uxð40Þ

With a simple formal manipulation of Eq. (40) it is possible to writethe displacement ux as

ux ¼ xFtuxt þ xF2ux2 þ xF3ux3 þ xFbuxbð41Þ

where the following definitions have been used:

xFt ¼ 1; xF2 ¼ z; xF3 ¼ z2; xFb ¼ z3

uxt ¼ ux0 ; ux2 ¼ /1ux; ux3 ¼ /2ux

; uxb¼ /3ux

ð42Þ

Eq. (41) can be put in a more compact form:

ux ¼X

aux¼t;l;b

xFauxuxaux

l ¼ 2; . . . ;Nux ð43Þ

or, by using the summation convention (if a subscript appears twicein an given term, that subscript will take the values t;2; . . . ;Nux ; b(or equivalently t, l, b) and the resulting terms are summed):

ux ¼ xFauxuxaux

aux ¼ t; l; b; l ¼ 2; . . . ;Nux ð44Þ

Similar formal operations can be performed on the other primaryunknowns (displacements uy, and uz). This means that anyAdvanced Higher Order Shear Deformation Theory can berepresented as

Generic AHSDT

GUF Notation:

ux ¼ xFauxuxaux

aux ¼ t; l; b; l ¼ 2; . . . ;Nux

uy ¼ yFauyuyauy

auy ¼ t;m; b; m ¼ 2; . . . ;Nuy

uz ¼ zFauzuzauz

auz ¼ t;n; b; n ¼ 2; . . . ;Nuz

8>><>>:ð45Þ

Nux ;Nuy , and Nuz are the orders used in the expansions of the dis-placements ux, uy, and uz respectively. Eq. (45) is the GeneralizedUnified Formulation for any AHSDT. It has been shown in previouspapers that this form is also valid for LW theories [24] with anycombination of orders as well as Zig-Zag theories. All the theoriescan be generated from 6 independent and invariant 1 � 1 arraysdenoted as kernels [28] of the Generalized Unified Formulation.

The Generalized Unified Formulation is a multi-fidelity and amulti-theory architecture which can generate a large amount ofindependent theories. It was used in the past to address cases inwhich all the variables had a LW or ESL or Zig-Zag description.

The following question will be answered in this work:

� Is it possible to attempt a further generalization and have everysingle displacement variable described differently than theremaining variables?

In the past every variable was independently treated as far asthe order of expansion was concerned. Now, with this generaliza-tion it is possible (in addition of the existing features of GUF) for


example to have a Zig-Zag description for a variable and just a‘‘regular’’ ESL description of the other variables. Another possibilitycould be to have 2 variables enhanced with a Zig-Zag functionwhereas the remaining variable has a standard ESL description.This is different than what was done in the past for Zig-Zag theo-ries when all the displacement variables presented a Zig-Zagdescription. To explain how a Zig-Zag form for the displacementis a priori enforced, focus is on the displacement component ux.It is possible to enhance its ESL description by adopting MZZFwhich enforces a priori the discontinuity (at the interface betweenlayers) of the derivative of ux with respect to z. To explain the con-cept, suppose that the expansion along the thickness is done with aparabolic order:

uxðx; y; zÞ ¼ ux0ðx; yÞ þ z/1uxðx; yÞ þ z2/2ux

ðx; yÞ ð46Þ

It is possible to augment this axiomatic model with the addition ofan extra degree of freedom uxZ as follows (note that MZZF is repre-sented by the term ð�1ÞkfkuxZ ðx; yÞ):

uxðx; y; zÞ ¼ ux0ðx; yÞ þ z/1uxðx; yÞ þ z2/2ux

ðx; yÞ

þ ð�1ÞkfkuxZ ðx; yÞ ð47Þ

k is the identity of each layer. For example, in a three-layered struc-ture the bottom layer will have k = 1, the mid-layer will have k = 2,and the top layer will have k = 3. fk is a dimensionless thicknesscoordinate defined at layer level. It is related with the physical zcoordinate (measured from the mid-plane of the plate) by the fol-lowing relation:

fk ¼2

ztopk� zbotk

z�ztopk

þ zbotk

ztopk� zbotk

ð48Þ

where zbotkindicates the thickness coordinate of the bottom surface

of layer k and ztopkindicates the thickness coordinate of the top sur-

face of layer k. Note that �1 6 fk 6 +1. From Eq. (46) (which is astandard ESL representation of ux without the additional Zig-Zagfunction) it can be immediately inferred that

@ux

@z

� �z¼ztopk

� @uk tx

@z¼ /1ux

ðx; yÞ þ 2ztopk/2uxðx; yÞ ð49Þ

From Eq. (46) the slope of the same displacement evaluated at thebottom surface of layer k + 1 can also be obtained:

@ux

@z

� �z¼zbotðkþ1Þ

� @uðkþ1Þ bx

@z¼ /1ux

ðx; yÞ þ 2zbotðkþ1Þ/2uxðx; yÞ ð50Þ

However, by definition of interface the z coordinate at the interfacehas a unique value: zbotðkþ1Þ ¼ ztopk

. This implies that Eq. (50) can berewritten as

@uðkþ1Þ bx

@z¼ /1ux

ðx; yÞ þ 2ztopk/2uxðx; yÞ ð51Þ

which presents the same right hand side of Eq. (49). Thus, it isdeduced that Eq. (46) which reports the standard ESL expansionfor ux implies no discontinuity of the slope of displacement ux inthe thickness direction (no Zig-Zag form is then present and thiscontradicts a physical requirement as it was earlier demonstrated).On the other hand, from Eq. (47) (which contains the additional de-gree of freedom thanks to the presence of MZZF) it is possible to cal-culate the slopes in the thickness direction of the displacement ux:

@uðkþ1Þ bx

@z¼ /1ux

þ 2zbotðkþ1Þ/2uxþ ð�1Þðkþ1Þ @fðkþ1Þ

@zuxZ

@uk tx

@z¼ /1ux

þ 2ztopk/2uxþ ð�1Þk @fk

@zuxZ

ð52Þ

or (see Eq. (48) for the definition of fk)



Fig. 1. Partially Zig-Zag Advanced Higher Order Shear Deformation Theories (PZZAHSDT) and Generalized Unified Formulation (GUF): definition of the acronyms.


@uðkþ1Þ bx

@z¼ /1ux

þ 2zbotðkþ1Þ/2uxþ ð�1Þðkþ1Þ 2

ztopðkþ1Þ � zbotðkþ1Þ

uxZ

@uk tx

@z¼ /1ux

þ 2ztopk/2uxþ ð�1Þk 2

ztopk� zbotk

uxZ

ð53Þ

Fig. 2. PZZAHSDT and GUF: examples of po


These relations can be further elaborated by introducing hðkþ1Þ ¼ztopðkþ1Þ � zbotðkþ1Þ (thickness of layer (k + 1)) and hk ¼ ztopk

� zbotk

(thickness of layer k) and using the fact that zbotðkþ1Þ ¼ ztopk(this is

true by definition of interface):

ssible theories that can be generated.




@uðkþ1Þ bx

@z¼ /1ux

þ 2ztopk/2uxþ ð�1Þðkþ1Þ 2

hðkþ1ÞuxZ

@uk tx

@z¼ /1ux

þ 2ztopk/2uxþ ð�1Þk 2

hkuxZ

ð54Þ

since the exponent k is an integer it is always (�1)(k+1) = � (�1)k

thus, it is possible to deduce from Eq. (54):

@uðkþ1Þ bx

@z¼ @uk t

x

@z� 2 �1ð ÞkuxZ

1hkþ 1

hðkþ1Þ

� �) @uðkþ1Þ b

x

@z–@uk t

x

@zð55Þ

which demonstrates that MZZF is an effective methodology to a pri-ori enforce the interlaminar discontinuity of the first derivative ofthe displacement in the thickness direction. With the GeneralizedUnified Formulation it is now possible to have Partially Zig-ZagAdvanced Higher Order Shear Deformation Theories (PZZAHSDT):some variables are described in ESL sense and other are enhancedwith the inclusion of MZZF. A virtually infinite number of theorieswith any combination of orders can be obtained with GUF. Any the-ory that can be generated is indicated with an acronym described inFigs. 1 and 2. It should be noted that PZZAHSDT coincide with theZig-Zag theories introduced in other previous works (see Refs.[26,28]) only when all displacements variables are described in aZig-Zag sense. This implies for example that ZZZPVD213 � EDZ213

and ZEZPVD213 – EDZ213. EEEPVD213 is an example of acronymdescribing a generic AHSDT (with no Zig-Zag term; see also Ref.[28]). It should be noted that EEEPVD213 � ED213.

8. Results

As it has been shown in this work, the Zig-Zag form for thetransverse shear stresses is always present (even for very thinplates) when the materials and/or fiber orientations of two adja-cent layers are different. However, the Zig-Zag form of the dis-placements appears when the composite plate presents differentHooke’s coefficients along the thickness (usual case) and the platecan be considered a relatively thick or thick structure (practicallythis is the case when a/h < 20). Thus, enforcing the Zig-Zag formof the displacements a priori will give great beneficial especially

Fig. 3. Test case: thick sandwich structur


when the plate is moderately thick or thick. For that reason, theanalyzed test case of this work is a thick sandwich structure pre-sented in Fig. 3. The accuracy of the different types of displace-ment-based Partially Zig-Zag Advanced Higher Order ShearDeformation theories will be assessed. The results have been gen-erated with the following logic. First a fourth-order AHSDT is com-pared against the elasticity solution [21]. Then, several types ofPZZAHSDT are introduced by enhancing one or more displacementvariables of the baseline theory with the Zig-Zag functions earlierdescribed. The study is reported below.

8.1. Enhancement of AHSDT with an additional Zig-Zag term used onlyfor one displacement variable

The question that is being answered here is the following:

� If only one displacement variable is enhanced with a Zig-Zagterm what is the best option that provides the maximumaccuracy?

In other words, the goal is to decide whether it is convenient toenhance ux, uy or uz. For example, a ‘‘baseline’’ fourth-order AHSDT(EEEPVD444) can be enhanced by the addition of one degree of free-dom and generate three different types of PZZAHSDT:

� Type # 1: Zig-Zag form enforced a priori only for displacementux

� Type # 2: Zig-Zag form enforced a priori only for displacementuy

� Type# 3: Zig-Zag form enforced a priori only for displacement uz

GUF can generate any type of of Partially Zig-Zag theories. More-over, the baseline theory can be general. For example, if the baselinetheory is EEEPVD234 then the model represented by ZEEPVD234 is aType # 1 of PZZAHSDT. Similarly, theory EZEPVD234 is a Type # 2 ofPZZAHSDT. To assess the accuracy of these enhancements of AHSDTthe following models are investigated in this work: ZEEPVD444 (Type# 1), EZEPVD444 ( Type # 2), and EEZPVD444 (Type # 3). These theoriespresent the same number of degrees of freedom and, therefore,

e. Geometry and material properties.



(a) (b)

Fig. 4. Thick sandwich structure: dimensionless displacement ux ¼ ux½E22 �Layer 2

z Pt h ahð Þ

3 .

(a) (b)

Fig. 5. Thick sandwich structure: dimensionless displacement uy ¼ uy½E22 �Layer 2

zPt h ahð Þ

3 .

(a) (b)

Fig. 6. Thick sandwich structure: dimensionless displacement uz ¼ uz100½E22 �Layer 2

z Pt h ahð Þ

4 .

(a) (b)

Fig. 7. Thick sandwich structure: dimensionless in-plane stress rxy ¼ rxy

z Pt ahð Þ

2 . Classical Form of Hooke’s Law has been used to calculate this stress.


Please cite this article in press as: Demasi L. Partially Zig-Zag Advanced Higher Order Shear Deformation Theories Based on the Generalized Unified For-mulation. Compos Struct (2011), doi:10.1016/j.compstruct.2011.07.022


(a) (b)

Fig. 8. Thick sandwich structure: dimensionless in-plane stress rxx ¼ rxx

z Pt ahð Þ

2. Classical Form of Hooke’s Law has been used to calculate this stress.

(b)(a)

Fig. 9. Thick sandwich structure: dimensionless in-plane stress ryy ¼ ryy

z Pt ahð Þ

2 . Classical Form of Hooke’s Law has been used to calculate this stress.

(a) (b)

Fig. 10. Thick sandwich structure: dimensionless transverse shear stress rzx ¼ rzxzPt a

hð Þ. The integration of the equilibrium equations has been used to calculate this stress.

(a) (b)

Fig. 11. Thick sandwich structure: dimensionless transverse shear stress rzy ¼ rzyzPt a

hð Þ. The integration of the equilibrium equations has been used to calculate this stress.


Please cite this article in press as: Demasi L. Partially Zig-Zag Advanced Higher Order Shear Deformation Theories Based on the Generalized Unified For-mulation. Compos Struct (2011), doi:10.1016/j.compstruct.2011.07.022


(a) (b)

Fig. 12. Thick sandwich structure: dimensionless transverse normal stress rzz ¼ rzzz Pt . The integration of the equilibrium equations has been used to calculate this stress.


require the same amount of computational resources. All of themhave one extra degree of freedom with respect to the baselinetheory EEEPVD444. The accuracy of these three different theories inwhich only one variable is enhanced can be assessed from Figs.4–12a. In particular, from Fig. 4a it is possible to see that theenhancement consisting of the Zig-Zag function applied only tothe primary variable ux (theory ZEEPVD444) greatly improves the pre-diction of the dimensionless displacement ux in the x direction.However, the very same theory presents much less accurate predic-tion of the other in-plane dimensionless displacement uy (seeFig. 5a) when compared with the ‘‘baseline’’ theory EEEPVD444. Thisis especially evident in the bottom layer of the sandwich structure.Symmetrically, if the Zig-Zag enhancement is applied only on theprimary variable uy (theory EZEPVD444) the prediction for uy isgreatly improved (see Fig. 5a) whereas the prediction of ux is dete-riorated with respect to the baseline theory EEEPVD444 (see Fig. 4a).The theories ZEEPVD444 and EZEPVD444 improve the accuracy for thedimensionless displacement uz and present identical performancesas far as uz is concerned (see Fig. 6a). This is an obvious consequenceof the symmetry properties of the case under investigation. The testcase has been designed with symmetry properties to clearly iden-tify and separate the effects on the different discretizations forthe different variables due to the Zig-Zag term. For the in-planestress rxy both theories ZEEPVD444 and EZEPVD444 present the sametype of performance (see Fig. 7a). For the dimensionless in-planestresses rxx and ryy both these theories present an improvementof the accuracy (Figs. 8 and 9a) but in some points identified forexample in the mid-planes of the skins the baseline theory is aslightly better choice. For the dimensionless transverse shear stressrzx (see Fig. 10a) theory ZEEPVD444 presents a better overall approx-imation when compared to theory EZEPVD444. However the maxi-mum value of rzx which occurs in the top layer is betterapproximated by theory EZEPVD444. A symmetric argument is validfor the dimensionless transverse shear stress rzy (Fig. 11a). Boththeories present an improvement (with respect to the baseline the-ory EEEPVD444) when the dimensionless transverse normal stress rzz

(see Fig. 12a) is considered.

8.2. Enhancement of AHSDT with additional Zig-Zag terms used onlyfor two displacement variables

The question that is being answered here is the following:

� If only two displacement variables are enhanced with Zig-Zagterms what is the best option that provides the maximumaccuracy?

In other words, the goal is to decide which combination of twodisplacement variables should be enhanced with the Zig-Zag term.For example, a ‘‘baseline’’ fourth order Higher Order Shear


Deformation Theory (EEEPVD444) can be enhanced by the additionof two degrees of freedom and generate three different types ofPZZAHSDT:

� Type # 4: Zig-Zag form enforced a priori only for displacementsux and uy (theory ZZEPVD444)� Type # 5: Zig-Zag form enforced a priori only for displacements

ux and uz (theory ZEZPVD444)� Type # 6: Zig-Zag form enforced a priori only for displacements

uy and uz (theory EZZPVD444)

Other baseline theories (not discussed in this paper for brevitybut they are discussed in Ref. [28] with different acronym defini-tions) are possible. Thus, for example, theory ZZEPVD234 is a Type# 4 PZZAHSDT. Theory ZEZPVD221 is a Type # 5 PZZAHSDT. To assessthe accuracy of these enhancements of AHSDT the following theo-ries are investigated: ZZEPVD444 (Type # 4), ZEZPVD444 (Type # 5),and EZZPVD444 (Type # 6). These theories present the same numberof degrees of freedom and, therefore, require the same amount ofcomputational resources. All of them have two extra degrees offreedom with respect to the baseline theory EEEPVD444. Moreover,these theories present one extra degree of freedom when com-pared with theories ZEEPVD444, EZEPVD444, and EEZPVD444 previouslydiscussed. The accuracy of Types # 4, # 5, and # 6 can be assessedfrom Figs. 4–12b. In particular, from Fig. 4b it is possible to see thatif the Zig-Zag enhancement includes the displacement ux then theaccuracy is greatly improved with respect to the baseline referencefourth-order AHSDT. Thus, theories ZZEPVD444 and ZEZPVD444 are thepreferable choice with respect to the baseline theory or with re-spect to theory EZZPVD444 (which does not present the Zig-Zagenhancement for the displacement ux). From Fig. 5b it is possibleto deduce that theories which include the Zig-Zag term for the uy

displacement are the preferable choice. For the out-of-plane dimensionless displacement (Fig. 6b) there is a hugeimprovement of the results if both the in-plane displacementsare enhanced with the Zig-Zag term (theory ZZEPVD444). If onlyone in-plane displacement is enhanced in conjunction with theenhancement of the out-of-plane displacement then the accuracyis not much improved when compared to the baseline (see boththeories ZEZPVD444 and EZZPVD444 in Fig. 6b). Similar observationscan be made for the in-plane stresses (Figs. 7–9b) and the trans-verse stresses (Figs. 10–12b).

8.3. Enhancement of AHSDT with additional Zig-Zag terms used for alldisplacement variables (fully Zig-Zag theories)

In this case the Zig-Zag term is adopted for all displacementvariables (theory ZZZPVD444). From Figs. 4–12b it can be deducedthat the accuracy is greatly improved. This theory presents threeadditional degrees of freedom compared to the baseline theory




EEE PVD444 and has the highest computational cost when comparedto all the other theories analyzed for this test case.

9. Conclusions

This paper demonstrates that in the most general case a multi-layered plate presents a Zig-Zag form of all displacements andstresses. The continuity of transverse shear stresses along thethickness also implies that the transverse normal stress cannotpresent a Zig-Zag pattern: the derivative of rzz with respect tothe thickness must be continuous even if the material properties/fiber orientations of two consecutive adjacent layers are different.

A further generalization of the Generalized Unified Formulation(GUF) has also been presented in this work. Every variable can nowbe treated with a different type of theory and/or order of expan-sion. For example, with this generalization is now possible withGUF to generate a generic theory which presents an ESL descriptionwith cubic expansion in the thickness direction for a user-selecteddisplacement unknown and and a parabolic Zig-Zag description foranother user-selected displacement variable. This is an extremelyversatile multi-fidelity approach which gives the user the possibil-ity to tailor the computational cost and performance to the actualcase under investigation. Among the infinite possible choices avail-able with GUF this paper presented the investigation of a sandwichstructure with different material properties for the three layers ofthe structure. A baseline fourth-order AHSDT EEEPVD444 (the trans-verse strain effects are retained) has been compared against theelasticity solution developed in a previous work for this class ofproblems. This baseline theory has been augmented with Zig-Zagfunctions with the enhancement of only one displacement compo-nent which can be either ux (Type # 1 of PZZAHSDT) or uy (Type # 2of PZZAHSDT) or uz (Type # 3 of PZZAHSDT). This work also pre-sented other types of PZZAHSDT in which two displacement vari-ables are enhanced with the Zig-Zag function (Types # 4, # 5,and # 6 respectively). Finally, all the results have been comparedwith the fully Zig-Zag Theory in which all displacement variablesare enhanced with the Zig-Zag function. The following conclusionscan be made:

� The Zig-Zag form of the in-plane displacements is a conse-quence of the equilibrium of the transverse shear stresses rzx

and rzy. The Zig-Zag form of the in-plane displacements disap-pears in the case of thin plates.� The Zig-Zag form for the displacement uz is a consequence of

the equilibrium condition for the transverse normal stress rzz.The Zig-Zag form of uz disappears in the case of thin plates.� The shear stresses rzx and rzy present in general a Zig-Zag form.

The in-plane stresses are in general discontinuous functions atthe interfaces. For thin multilayered structures the Zig-Zag formof rzx and rzy is still present. Moreover, the in-plane stresses arediscontinuous functions even for thin multilayered plates.� For thick plates: enhancing only an in-plane displacement vari-

able (for example only ux or uy) with a Zig-Zag function in gen-eral improves the results of the starting baseline theory.� For thick plates: enhancing only the out-of-plane displacement

variable uz with a Zig-Zag function presents a very marginalimprovement of the results in comparison with the startingbaseline theory. Moreover, the enhancement requires an addi-tional degree of freedom which increases the computationalcost. The enhancement of uz only is then not an efficient useof the computational resources.� For thick plates: enhancing at the same time an in-plane dis-

placement variable (for example only ux or uy) and uz withZig-Zag functions presents similar numerical accuracy of


enhancing only a single in-plane variable. Thus, if the computa-tional cost is the major constraint it is better to enhance only anin-plane displacement variable.� For thick plates: enhancing both the in-plane displacements ux

and uy presents an excellent improvement of the results com-pared with the starting baseline theory. Moreover, the accuracyis practically coincident with the fully Zig-Zag approach (butwith a smaller computational cost).

Further generalizations of the Generalized Unified Formulationwill include the following features:

� Partially Layerwise Advanced Higher Order Shear DeformationTheories (PLAHSDT): some a priori selected variables have anESL description with a user-selected order of expansion andother variables have a LW description with another user-selected order of expansion.� Partially Layerwise Advanced Zig-Zag Theories (PLAZZT): some

a priori selected variables have an ESL description with a Zig-Zag enhancement and with a user-selected order of expansionand other variables have a LW description with another user-selected order of expansion.� Partially Layerwise Advanced Zig-Zag and Higher Order Shear

Deformation Theories (PLAZZHSDT): one displacement variablehas an ESL description, one displacement variable has an Equiv-alent Single Layer description with Zig-Zag enhancement viaMZZF and another displacement has a LW description. All vari-ables may have different orders of expansion along thethickness.� Extension to the mixed cases and in particular use of Reissner’s

Mixed Variational Theorem (RMVT) where the primary vari-ables are the three displacements and the three transversestresses. For the mixed case based on the RMVT there are manyoptions that can be selected and with potentially very differentnumerical properties:– A displacement variable has an ESL or a Zig-Zag-enhanced

ESL or a LW description.– A transverse stress variable has an ESL or a Zig-

Zag-enhanced ESL or a LW description.

Furthermore, the orders of expansion can be separately decided bythe user. A quite large number of combinations is then possiblewith enormous versatility given a computational cost constraint.The accuracy of the results and possible numerical problems/advantages will be assessed in upcoming papers.

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