50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<br>17th
4 - 7 May 2009, Palm Springs, California
AIAA 2009-2342
An Invariant Model for any Composite Plate Theory
and FEM applications: the Generalized Unified
Formulation
Luciano Demasi∗
San Diego State University, San Diego, California 92182
The Generalized Unified Formulation (GUF) presented here is a modern approach for
the Finite Element analysis of sandwich plates and in general multilayered structures. GUF
is a comprehensive formulation which includes practically all possible axiomatic theories.
Any type of theory with any combination of orders of expansion for the different displacement variables can be obtained from the expansion of six 1 × 1 arrays (the kernels or
fundamental nuclei of the Generalized Unified Formulation). Each of the displacement variables is independently treated and different orders of expansions for the different unknowns
can be chosen. Since infinite combinations can be freely chosen for the displacements, GUF
allows the user to write, with a single invariant formulation implemented in a single FEM
code, ∞3 Higher-order Shear Deformation Theories (HSDT), ∞3 Zig-Zag Theories (ZZT)
and ∞3 Layer-Wise Theories (LWT). The six independent fundamental nuclei are formally
invariant with respect to the order used for the expansion or with respect to the type of
theory. GUF is a powerful versatile tool: the user can freely decide the type of theory (e.g.,
LWT or ZZT) and the orders used in the expansions. Thus, a code based on GUF can
be fast (less terms can be used in the expansions of the unknwons) when the case under
investigation is less demanding or when a large number of runs is required (optimization
problems or probabilistic studies) and the software can be very accurate (layerwise models
with high orders used for the unknowns) when the problem is very challenging and the
accuracy is more important than the CPU time.
GUF can be implemented in existing commercial codes and “intelligent” software can be
developed. The codes can, in fact, use any theory or order of expansion to attempt the
convergence in case of complex three-dimensional problems with localized effects without
the actual need of solid elements. This is another advantage because the number of Degrees of Freedoms could be reduced without loosing in computational efficiency and with
the advantage of having the two dimensional mesh typical of plate and shell models. The
Generalized Unified Formulation can be also adopted for mixed variational statements. In
such cases the number of independent kernels would be different but their size would still
be 1 × 1 arrays. Multifield problems such as thermoelastic applications and multilayered
plates embedding piezo-layers can also be analyzed as well.
This paper assesses bending of sandwich structures. Analytical and elasticity solution are
compared. The effect of the Zig-Zag form of the displacement is discussed.
Nomenclature
ux
uy
uz
u
bx , u
by
u
bz
σ
bzx , σ
bzy , σ
bzz
σ
bxx , σ
byy , σ
bxy
Displacement in the x direction
Displacement in the y direction
Displacement in the z direction
Dimensionless in-plane displacements
Dimensionless out-of-plane displacement
Dimensionless out-of-plane stresses
Dimensionless in-plane stresses
∗ Assistant
Professor, Department of Aerospace Engineering & Engineering Mechanics. Member AIAA.
Email: ldemasi@mail.sdsu.edu
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Copyright © 2009 by Luciano Demasi. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
x, y
z
zbotk
ztopk
m, n
εxx , εyy , γxy
εzz , γxz , γyz
σxx , σyy , σxy
σzz , σxz , σyz
x
Ft , yFt , zFt
x
Fb , yFb , zFb
x
Fl , yFm , zFn
α
β
Nux
Nuy
Nuz
ζk
kα β
Kux uuxx ux
k αu βu
Kux uyx y
kα β
Kux uuzx uz
k αu βu
Kuy uyy y
k αu βu
Kuy uzy z
kα β
Kuz uuzz uz
t αux βux
D ux ux
t αuy βuy
D uy uy
t αuz βuz
D uz uz
b αux βux
Dux ux
b αuy βuy
Duy uy
b αuz βuz
Duz uz
εp
σp
εn
σn
In-plane coordinates
Out-of-plane coordinate
z coordinate of the bottom surface of layer k
z coordinate of the top surface of layer k
Wave numbers
In-plane strains
Out-of-plane strains
In-plane stresses
Out-of-plane stresses
Known functions used in the expansion along the thickness
Known functions used in the expansion along the thickness
Known functions used in the expansion along the thickness
Thickness primary master index
Thickness secondary master index
Order of expansion used for the displacement ux
Order of expansion used for the displacement uy
Order of expansion used for the displacement uz
Non-dimensional coordinate (see equation 39)
Invariant Kernel (1 × 1 array)
Invariant Kernel (1 × 1 array)
Invariant Kernel (1 × 1 array)
Invariant Kernel (1 × 1 array)
Invariant Kernel (1 × 1 array)
Invariant Kernel (1 × 1 array)
Invariant pressure Kernel (1 × 1 array)
Invariant pressure Kernel (1 × 1 array)
Invariant pressure Kernel (1 × 1 array)
Invariant pressure Kernel (1 × 1 array)
Invariant pressure Kernel (1 × 1 array)
Invariant pressure Kernel (1 × 1 array)
Vector containing the in-plane strains
Vector containing the in-plane stresses
Vector containing the out-of-plane strains
Vector containing the out-of-plane stresses
Subscript
k
ux
uy
uz
G
H
Superscript
Referred to layer k
Referred to displacement ux
Referred to displacement uy
Referred to displacement uz
Quantity calculated by using the geometric relations (see equation 11)
Quantity calculated by using Hooke’s law (see equation 62)
T
k
Transpose
Referred to layer k
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I.
A.
Introduction
Background and Motivation
INITE Element Method1−3 is a powerful tool widely used in the industry. With the growing computational capabilities of the modern computers increasingly complex problems can be solved.
Most of the aerospace structures can be modeled with shell and plate models. Therefore, the importance of
finite element applications of such structures is great. Different problems have been solved in the past: free
vibration of plates,4−9 analysis of sandwich structures,10−13 thermoelastic problems,14 piezoelectric plates15
and multifield loadings.16
A very large amount of literature of theory of plates is available and it will not be discussed here.
With focus on axiomatic methods, each existing approach presents a range of applicability and when the
hypotheses used to formulate the theory are no longer satisfied the approach has to be replaced with another
one usually named by the authors as “refined theory” or “improved theory”. In the framework of the mechanical case the Classical Plate Theory (CPT), also known as Kirchoff theory,17 has the advantage of being
simple and reliable for thin plates. However, if there is strong anisotropy of the mechanical properties, or if
the composite plate is relatively thick, other advanced models such as First-order Shear Deformation Theory
(FSDT) are required.18−20 Higher-order Shear Deformation Theories (HSDT) have also been used,5−7 giving
the possibility to increase the accuracy of numerical evaluations for moderately thick plates. But even these
theories are not sufficient if local effects are important or accuracy in the calculation of transverse stresses
is sought. Therefore, more advanced plate theories have been developed to include zig-zag effects.21−32
F
When a new problem with unknown behavior is analyzed, the user may attack it with the most simple formulation or with the most advanced and refined models or with an intermediate solution which is a compromise
between the accuracy and the CPU time needs. This operation is usually costly. For example, the different
capabilities required for this approach are not available. In that case in-house software is often created. But
this in-house software obviously does not contain all possible models between the least, and usually fast,
accurate one and the most advanced, and usually time consuming, one. Even if different approaches are
implemented, each one requires a different formulation of the FE stiffness matrix and the creation of an
independent software. A major problem may arise in that case. The implemented capability may be not
sufficient to solve the challenging case the user is facing or the CPU time is unacceptable and a less accurate
theory would be sufficient. These aspects are also problem-dependent and even if the solution of a previous
case was satisfactory, the same software may be no longer adequate for a new task. Therefore, an extra cost
must be taken into account to face a new situation.
The problem of adequately modeling the behavior of the real structure is then very challenging. For example,
if two-dimensional multilayered structures cases are considered, it may be necessary to abandon the so called
Equivalent Single Layer Models (ESLM) and adopt Layerwise Models (LWM),8,33−40 in which the variables
are layer-dependent. If the anisotropy in the thickness direction is very strong, the accuracy can be preserved
by increasing the number of layers used in the mathematical model or/and by increasing the orders of the
axiomatic expansion along the thickness for the different variables. If this approach is chosen, the use of solid
elements can be avoided and the advantage of having a two-dimensional mesh can be preserved. Versatility
is also required when a commercial or in-house software is considered. Unfortunately this is not usually the
case. An optimization problem or a probabilistic analysis (Monte Carlo) needs to be very fast whereas a
detailed study of the interlaminar stresses requires an advanced model.
What is proposed in this paper meets all the above mentioned requirements and is a powerful tool for
the modelization of most structures in the aerospace field.
B.
What are the contributions of this work
An invariant model, named Generalized Unified Formulation (GUF) is introduced. It can be considered a
generalization of the Unified Formulation41 (named here as Carrera’s Unified Formulation). With the present
GUF all the needs earlier mentioned are satisfied. In particular, the Generalized Unified Formulation allows
to treat each displacement variable a independently from the others and this could lead the derivation of new
FEM numerical techniques. The orders of each variable are freely chosen by the user and can be different
than the orders used for the thickness expansion of the other variables. This property means that practically
a If
a mixed variational statement is used or if the case includes multifield quantities other variables may be modeled.
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all “classical” Equivalent Single Layer Models can be addressed by the Generalized Unified Formulation.
For example, an advanced Higher-order Shear Deformation Theory in which the in-plane displacements
are expanded along the thickness by using a cubic polynomial and in which the out-of-plane displacement is
parabolic can be represented with GUF as well as the theory in which the orders for the in-plane displacements
are parabolic and the out-of-plane displacement is linear. Any combination is allowed. But what is important
is that all the possible combinations that can be generated (which are ∞3 since three displacement variables
ux , uy and uz are here considered) are obtained from the expansion of six kernels which are invariant with
respect to the order adopted for the displacements variables. That is, all the Higher-order Shear Deformation
Theories with any combination of orders can be generated from the same six invariant kernels. The kernels
of the Generalized Unified Formulation are 1 × 1 matrices. If other variational statements different than
the Principle of Virtual Displacements are used, the number of independent kernels and their mathematical
form will be different but the size will always be 1 × 1.
It is possible to demonstrate that when the so called Zig-Zag form of the displacements is included by
adopting Murakami’s Zig-Zag Function,21 the same invariant kernels can describe the ∞3 Zig-Zag theories
that could be generated by changing the orders of the expansions of the displacements. But these features are
not limited to the Equivalent Single Layer Models. All possible Layerwise Theories with any combination of
orders can be obtained, again, from the same six invariant 1 × 1 kernels. The GUF clearly allows to have all
these infinite models in the same software. From the least accurate to the most advanced one. The freedom
to change the orders of expansion in a layerwise model allows to reach a quasi-3D solution without the actual
need of solid elements and so with the preservation of the advantages of having a two-dimensional mesh.
This is very useful when local effects are important or when the analyzed structure may not be considered
as a “thin structure” and so when the classical two-dimensional formulation fails.
Another advantage of the GUF is in the possibility of a numerical “experiment” or a “sensitivity” analysis
of the problem under investigation. For example, it is possible to explore if a particular configuration of
the constraints and loads is affected more by the order of expansion of one variable with respect to the
orders used for the other variables. This feature is particularly appealing in the mixed or multifield cases in
which the order of expansion of some unknown quanties may be determinant to have the correct numerical
simulation. But this is not all. The Generalized Unified Formulation can be used to “adapt” the software to
the optimization or probabilistic simulation. The user can in fact study with a small number of runs the best
combination of type of theory (e.g. Equivalent Single Layer Theory with Zig-Zag Function vs a Higher-order
Shear Deformation Theory) and orders that minimize the CPU time and maximize the accuracy within the
requested level. Then the user can start the optimization or probabilistic study knowing that the software
gives the best performances because it is “tailored” to the case under investigation. The Generalized Unified
Formulation could also be used to design automatic and “intelligent” FEM codes: the codes could understand
(with a designed algorithm) where the accuracy is required the most and so they could “try” any of the types
of theories and any of the ∞3 combinations allowed by GUF and improve the accuracy of the prediction. As
stated earlier this capability is not limited to the pure mechanical case presented in this paper and it can be
easily extended to the multifield problems and functionally graded materials.
A final but not less important advantage of the Generalized Unified Formulation is in the educational
possibilities for young engineers. The engineers could learn a single formulation (GUF) and apply all the
powerful features of GUF to a very vast class of problems without the need of learning new formulations of
commercial softwares. With the Generalized Unified Formulation the “classical” axiomatic approaches and
the distinction between theories are just an exercise and the user can move from one approach to another
one without difficulties. The Generalized Unified Formulation can be then considered a modern view of
multilayered plates and shells with possible applications in high fidelity codes in the different disciplines such
as aeroelasticity.
II.
Classification of the theories
The main feature of the Generalized Unified Formulation is that the descriptions of Layerwise Theories,
Higher-order Shear Deformation Theories and Zig-Zag Theories of any combination of orders do not show any
formal differences and can all be obtained from six invariant kernels. So, with just one theoretical model an
infinite number of different approaches can be considered. For example, in the case of moderately thick plates
a higher order theory could be sufficient but for thick plates layerwise models may be required. With GUF
the two approaches are formally identical because the kernels are invariant with respect to the type of theory.
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In the present work the concepts of type of theory and class of theories are introduced. The following
types of displacement-based theories are discussed. The first type is named as Advanced Higher-order Shear
Deformation Theories (AHSDT). These theories are Equivalent Single Layer models because the displacement field is unique and independent of the number of layers. The effects of the transverse normal strain
εzz are retained.
The second type of theories is named as Advanced Higher-order Shear Deformation Theories with Zig-Zag
effects included (AHSDTZ). These theories are Equivalent Single Layer models and the so called Zig-Zag
form of the displacements is taken into account by using Murakami’s Zig-Zag Function (MZZF). The effects
of the transverse normal strain εzz are included. The third type of theories is named Advanced LayerWise
Theories (ALWT). These theories are the most accurate ones because all the displacements have a layerwise
description. The effects of the transverse normal strain εzz are included as well. These models are necessary
when local effects need to be described. The price is of course (in FEM applications) in higher computational time. An infinite number of theories which have a particular logic in the selection of the used orders of
expansion is defined as class of theories. For example, the infinite layerwise theories which have the displacements ux , uy and uz expanded along the thickness with a polynomial of order N are a class of theories. The
infinite theories which have the in-plane displacements ux and uy expanded along the thickness with order
N , the out of plane displacement expanded along the thickness with order N −1 are another class of theories.
III.
The Generalized Unified Formulation for Multilayered Composite Plates
Both layerwise and Equivalent Single Layer models are axiomatic approaches. That is, the unknowns are
expanded along the thickness by using a chosen series of functions.
When the Principal of Virtual Displacements is used, the unknowns are the displacements ux , uy and uz .
When other variational statements are used the unknowns may also be all or some of the stresses and other
quantities as well (multifield case).
The Generalized Unified Formulation is introduced here considering a generic layer k of a multilayered plate
structure. This is the most general approach and the Equivalent Single Layer theories, which consider the
displacement unknowns to be layer-independent, can be derived from this formulation with some simple
formal techniques as will be demonstrated in this paper. Consider a theory denoted as Theory I, in which
Figure 1. Multilayered plate: notations and definitions.
the displacement in x direction ukx has four Degrees of Freedom. Here by Degrees of Freedom it is intended
the number of unknown quantities that are used to expand a variable. In the case under examination four
Degrees of Freedom for the displacement ukx means that four unknowns are considered. Each unknown
multiplies a known function of the thickness coordinate z. Where the origin of the coordinate z is measured
is not important. However, from a practical point of view it is convenient to assume that the middle plane
of the plate is also the plane with z = 0. This assumption does not imply that there is a symmetry with
respect to the plane z = 0. The formulation is general.
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For layer k the following relation holds: zbotk ≤ z ≤ ztopk . zbotk is the global coordinate z of the bottom
surface of layer k and ztopk is the global coordinate z of the top surface of layer k (see Figure 1). hk =
ztopk − zbotk is the thickness of layer k and h is the thickness of the plate.
In the case of Theory I, ukx is expressed as follows:
unknown#1
known
unknown#2
known
z }| { z }| { z }| { z }| {
= f1k (z) · ukx1 (x, y) + f2k (z) · ukx2 (x, y)
ukx (x, y, z)
+
f3k (z) · ukx3 (x, y) + f4k (z) · ukx4 (x, y)
| {z } | {z } | {z } | {z }
known
known
unknown#3
zbotk ≤ z ≤ ztopk
(1)
unknown#4
The functions f1k (z), f2k (z), f3k (z) and f4k (z) are known functions (axiomatic approach). These functions
could be, for example, a series of trigonometric functions of the thickness coordinate z. Polynomials (or even
better orthogonal polynomials) could be selected. In the most general case each layer has different functions.
For example, f1k (z) 6= f1k+1 (z). The next formal step is to modify the notation.
The following functions are defined:
x k
Ft
(z) = f1k (z)
x k
F2
(z) = f2k (z)
x k
F3
(z) = f3k (z)
x k
Fb
(z) = f4k (z)
(2)
The logic behind these definitions is the following. The first function f1k (z) is defined as xFtk . Notice the
superscript x. It was added to clarify that the displacement in x direction, ukx , is under investigation. The
subscript t identifies the quantities at the “top” of the plate and, therefore, will be useful in the assembling
of the stiffness matrices in the thickness direction. This aspect will be discussed later.
The last function f4k (z) is defined as xFbk . Notice again the superscript x. The subscript b means “bottom”
and, again, its utility will be clear when the matrices are assembled.
The intermediate functions f2k (z) and f3k (z) are defined simply as xF2k and xF3k . To be consistent with the
definitions of equation 2, the following unknown quantities are defined:
ukxt (x, y) = ukx1 (x, y)
ukxb (x, y) = ukx4 (x, y)
(3)
Using the definitions reported in equations 2 and 3, equation 1 can be rewritten as
known
ukx
(x, y, z)
unknown#1
known
unknown#2
z }| { z }| { z }| { z }| {
= xFtk (z) · ukxt (x, y) + xF2k (z) · ukx2 (x, y)
+
x k
F3 (z) · ukx3 (x, y) + xFbk (z) · ukxb (x, y)
| {z
} | {z } | {z } | {z }
known
unknown#3
known
zbotk ≤ z ≤ ztopk
(4)
unknown#4
It is supposed that each function of z is a polynomial. The order of the expansion is then 3 and indicated as
Nukx . Each layer has in general a different order. Thus, in general Nukx 6= Nuk+1
. If the functions of z are not
x
polynomials (for example, this is the case if trigonometric functions are used) then Nukx is just a parameter
related to the number of terms or Degrees of Freedom used to describe the displacement ukx in the thickness
direction. This concept will be clear later. The expression representing the displacement ukx (see equation 4)
can be put in a compact form typical of the Generalized Unified Formulation presented here. In particular
it is possible to write:
ukx (x, y, z) = xFαkux (z) · ukxαux (x, y)
αux = t, l, b;
l = 2, ..., Nukx
(5)
where, in the example, Nukx = 3. The thickness primary master index α has the subscript ux . This subscript
from now on will be called slave index. It is introduced to show that the displacement ux is considered.
Figure 2 explains these definitions. Consider another example. Suppose that the displacement ukx of a
particular theory is expressed with 3 Degrees of Freedom. In that case it is possible to write:
known
ukx (x, y, z)
unknown#1
known
unknown#2
known
unknown#3
z }| { z }| { z }| { z }| { z }| { z }| {
= f1k (z) · ukx1 (x, y) + f2k (z) · ukx2 (x, y) + f3k (z) · ukx3 (x, y)
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(6)
Figure 2. Generalized Unified Formulation. Master and slave indices.
By adopting the definitions earlier used for the case of 4 Degrees of Freedom it is possible to rewrite equation
6 in the following equivalent form:
known
ukx (x, y, z)
unknown#1
known
unknown#2
known
unknown#3
z }| { z }| { z }| { z }| { z }| { z }| {
= xFtk (z) · ukxt (x, y) + F2k (z) · ukx2 (x, y) + xFbk (z) · ukxb (x, y)
(7)
which can be put again in the form shown in equation 5 with Nukx = 2. In general Nukx is DOFukx − 1, where
DOFukx is the number of Degrees of Freedom (at layer level) used for the displacement ukx . In the case of
Zig-Zag theories it is possible to demonstrate that Nukx = DOFukx − 2 because one Degree of Freedom is used
for the Zig-Zag function.
The minimum number of Degrees of Freedom is chosen to be 2. This is a choice used to facilitate the
assembling in the thickness direction. In fact, the “top” and “bottom” terms will be always present. In the
case in which DOFukx = 2 the Generalized Unified Formulation is simply
ukx (x, y, z) = xFαkux (z) · ukxαux (x, y)
αux = t, b
(8)
In this particular case the “l ” term of equation 5 is not present.
An infinite number of theories can be included in equation 5. It is in fact sufficient to change the value of
Nukx . It should be observed that formally there is no difference between two distinct theories (obtained by
changing Nukx ). It is deduced that ∞1 theories can be represented by equation 5.
The other displacements uky and ukz can be treated in a similar fashion. The Generalized Unified Formulation
for all the displacements is the following:
ukx = xFt ukxt + xFl ukxl + xFb ukxb = xFαux ukxαux
αux = t, l, b;
uky = yFt ukyt + yFm ukym + yFb ukyb = yFαuy ukyαuy
αuy = t, m, b; m = 2, ..., Nuy
ukz = zFt ukzt + zFn ukzn + zFb ukzb = zFαuz ukzαuz
αuz = t, n, b;
l = 2, ..., Nux
(9)
n = 2, ..., Nuz
In equation 9, for simplicity it is assumed that the type of functions is the same for each layer and that
the same number of terms is used for each layer. This assumption will make it possible to adopt the same
Generalized Unified Formulation for all types of theories, and layerwise and equivalent single layer theories
will not show formal differences. This concept means, for example, that if displacement uy is approximated
with five terms in a particular layer k then it will be approximated with five terms in all layers of the
multilayered structure.
Each displacement variable can be expanded in ∞1 combinations. In fact, it is sufficient to change the
number of terms used for each variable. Since there are three variables (the displacements ux , uy and uz ), it
is concluded that equation 9 includes ∞3 different theories. For now the quantities are defined in a layerwise
sense but it will be shown that the same concept is valid for the Equivalent Single Layer cases too.
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A.
Governing Equations
A multilayered structure composed of Nl layers is considered. The Principle of Virtual Displacements for
the case of two pressures applied at the top and bottom of each layer k is:
ztopk
Z
Z
£
δε
kT
k
pG σ pH
+ δε
kT
k
nG σ nH
¤
dzdxdy = δLke =
Ωk zbotk
Z
¡
¢
¡
¢
δukx x, y, z = ztopk Pxk t x, y, z = ztopk dxdy+
+
Ωk
Z
¡
¢
¡
¢
δuky x, y, z = ztopk Pyk t x, y, z = ztopk dxdy+
+
Ωk
Z
¡
¡
¢
¢
δukz x, y, z = ztopk Pzk t x, y, z = ztopk dxdy+
+
Ωk
(10)
Z
δukx (x, y, z = zbotk ) Pxk b (x, y, z = zbotk ) dxdy+
+
Ωk
Z
δuky (x, y, z = zbotk ) Pyk b (x, y, z = zbotk ) dxdy+
+
Ωk
Z
δukz (x, y, z = zbotk ) Pzk b (x, y, z = zbotk ) dxdy+
+
Ωk
Z
ztopk
Z
+
¤
£ k k
k
k
+ δukz σ nz
dzds
δun σ nn + δuks σ ns
z
Γk
σ botk
where Pxk t , Pyk t and Pzk t are the top pressures (applied at z = ztopk ), Pxk b , Pyk b and Pzk b are the bottom
k
k
k
, σ ns
and σ nz
are the specified normal and tangential components
pressures (applied at z = zbotk ), σ nn
k
k
k
measured per unit area, un , us and uz are the normal and tangential displacements on the edge Γkσ in which
the stresses are specified. It is assumed that Ωk = Ω.
All the stresses and strains are retained. No restricting hypotheses are formulated. The relation between
the stresses and strains is Hooke’s law (see Appendix A). The geometric relations relate the strains to the
derivative of the displacements. Using the formalism of the Generalized Unified Formulation it is possible
to write:
∂ukx
εkxxG =
= x Fαux ukxαux ,x
∂x
∂uky
εkyyG =
= y Fαuy ukyαuy
,y
∂y
k
γxyG
=
k
γxzG
k
γyzG
∂uky
∂ukx
+
= x Fαux ukxαux ,y + y Fαuy ukyαuy
,x
∂y
∂x
∂ukx
∂ukz
+
= z Fαuz ukzαuz ,x + x Fαux ,z ukxαux
=
∂x
∂z
∂uky
∂ukz
=
+
= z Fαuz ukzαuz ,y + y Fαuy ,z ukyαuy
∂y
∂z
(11)
∂ukz
= z Fαuz ,z ukzαuz
∂z
Where the symbol “,” indicates the derivative. For example z Fαuz ,z indicates the derivative of z Fαuz with
k
respect to z. The explicit form for the function δε kT
pG σ pH under the sign of integral (see equation 10) is:
εkzzG =
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δε
kT
k
pG σ pH
=
k k
k k
k k
e11
e12
e16
δεkxxG C
εxxG + δεkxxG C
εyyG + δεkxxG C
γxyG
+
k k
k k
k k
e13
e12
e22
δεkxxG C
εzzG + δεkyyG C
εxxG + δεkyyG C
εyyG
+
k k
e26
δεkyyG C
γxyG
+
k
k
e k εk + δγ k C
ek k
ek k
δγxyG
C
26 yyG
xyG 66 γxyG + δγxyG C36 εzzG
+
k k
e23
δεkyyG C
εzzG
+
(12)
k
k k
e16
δγxyG
C
εxxG
Using the geometric relations (equation 11):
δε
kT
k
pG σ pH
The function δε
kT
k
nG σ nH
δε
=
k x
k x
e12
e11
C
Fαux x Fβux δukxαux ,x ukxβux ,x + C
Fαux y Fβuy δukxαux ,x ukyβuy
,y
+
k
y
k
k
e k x Fα x Fβ δuk
ek x
C
16
xαux ,x uxβux ,y + C16 Fαux Fβuy δuxαux ,x uyβuy
ux
ux
,x
+
k x
k y
e13
e12
C
Fαux z Fβuz ,z δukxαux ,x ukzβuz + C
Fαuy x Fβux δukyαuy ukxβux ,x
+
k y
e22
C
Fαuy y Fβuy δukyαuy ukyβuy
+
k
e k y Fα y Fβ δuk
C
26
yαuy uyβuy
uy
uy
+
k x
k y
e16
e16
C
Fαux x Fβux δukxαux ,y ukxβux ,x + C
Fαuy x Fβux δukyαuy ukxβux ,x
+
k x
e26
C
Fαux y Fβuy δukxαux ,y ukyβuy
+
k x
k y
e66
e66
C
Fαux x Fβux δukxαux ,y ukxβux ,y + C
Fαuy x Fβux δukyαuy ukxβux ,y
+
k x
e66
C
Fαux y Fβuy δukxαux ,y ukyβuy
+
k x
k y
e36
e36
C
Fαux z Fβuz ,z δukxαux ,y ukzβuz + C
Fαuy z Fβuz ,z δukyαuy ukzβuz
,y
,y
,y
,y
k y
e26
+C
Fαuy x Fβux δukyαuy ukxβux ,y
,x
k
e k y Fα z Fβ
+C
ukzβuz
23
uy
uz ,z δuyαuy
,y
,y
(13)
,x
,y
k y
e26
+C
Fαuy y Fβuy δukyαuy ukyβuy
,x
,y
,x
,x
k y
e66
+C
Fαuy y Fβuy δukyαuy ukyβuy
,x
,x
,x
can be obtained by using a similar procedure:
kT
k
nG σ nH
=
k k
k k
k
k k
k
k
e45
e45
e55
C
γxzG
C
γyzG + δγyzG
C
γxzG + δγxzG
δγxzG
+
k k
k k
k k
k
e23
e13
e44
εyyG
εxxG + δεkzzG C
C
γyzG + δεkzz C
δγyzG
+
k k
e36
γxyG
δεkzzG C
+
(14)
k k
e33
εzzG
δεkzzG C
Using the geometric relations (equation 11):
δε
kT
k
nG σ nH
k z
k x
e55
e55
= C
Fαuz z Fβuz δukzαuz ,x ukzβuz ,x + C
Fαux ,z z Fβuz δukxαux ukzβuz ,x
k z
k x
e55
e55
+ C
Fαuz x Fβux ,z δukzαuz ,x ukxβux + C
Fαux ,z x Fβux ,z δukxαux ukxβux
k z
k x
e45
e45
+ C
Fαuz z Fβuz δukzαuz ,x ukzβuz ,y + C
Fαux ,z z Fβuz δukxαux ukzβuz ,y
k z
k x
e45
e45
+ C
Fαuz y Fβuy ,z δukzαuz ,x ukyβuy + C
Fαux ,z y Fβuy ,z δukxαux ukyβuy
k z
k y
e45
e45
+ C
Fαuz z Fβuz δukzαuz ,y ukzβuz ,x + C
Fαuy ,z z Fβuz δukyαuy ukzβuz ,x
(15)
k z
k y
e45
e45
+ C
Fαuz x Fβux ,z δukzαuz ,y ukxβux + C
Fαuy ,z x Fβux ,z δukyαuy ukxβux
k z
k y
e44
e44
+ C
Fαuz z Fβuz δukzαuz ,y ukzβuz ,y + C
Fαuy ,z z Fβuz δukyαuy ukzβuz ,y
k z
k y
e44
e44
+ C
Fαuz y Fβuy ,z δukzαuz ,y ukyβuy + C
Fαuy ,z y Fβuy ,z δukyαuy ukyβuy
k z
k z
e13
e23
+ C
Fαuz ,z x Fβux δukzαuz ukxβux ,x + C
Fαuz ,z y Fβuy δukzαuz ukyβuy
,y
k z
k z
e36
e36
+ C
Fαuz ,z x Fβux δukzαuz ukxβux ,y + C
Fαuz ,z y Fβuy δukzαuz ukyβuy
,x
z
k
k
e k z Fα
+ C
33
uz ,z Fβuz ,z δuzαuz uzβuz
Equations 13 and 15 (see also equation 10) are integrated over the volume of layer k. The integrals along
the thickness can be immediately performed. The notation used for that integrals is shown in Figure
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American Institute of Aeronautics and Astronautics
Figure 3. GUF. Examples of some definitions related to the integrals along the thickness.
3. An integration by parts is performed (details omitted for brevity). Now the focus is on the external
virtual work. Consider the applied pressures. They are written in a slightly different manner in order to
use ¡the power ¢of the¡ present Generalized
Unified Formulation.
Consider
in particular the pressure term
¢
¡
¢
δukx x, y, ztopk Pxk t x, y, ztopk (see equation 10). Pxk t x, y, ztopk is the force per unit of area in direction
x (subscript x) and
¡ applied ¢at the top surface (z = ztopk ) of the layer k (the superscript t means “top”).
The pressure Pxk t x, y, ztopk is thought as a function along the thickness (this is in order to use GUF) as
follows:
¡
¢
¡
¢
kt
Pxk t x, y, ztopk = x Fαux z = ztopk · Pxα
(x, y)
(16)
ux
Notice that the same number of terms used for the expansion in the thickness direction of the displacement
ukx has been considered. This is a natural choice, considering the fact that the pressure in x direction works
with the displacement in x direction.
Now the following notation is introduced:
¡
¢
x t
(17)
Fαux =
˙ x Fαux z = ztopk
The superscript t means that the functions x Fαux are all calculated at the z which corresponds to the top
surface of layer k (i.e., z = ztopk ) where the pressure is supposed to be applied. Using this definition, the
pressure can be written by adopting the formalism already used for the displacements:
¡
¢
kt
kt
Pxk t x, y, ztopk = x Fαt ux · Pxα
(x, y) = x Fαt ux Pxα
(18)
ux
ux
Observing that
¡
¢
¡
¢
δukx x, y, ztopk = x Fαux z = ztopk · δukxαux (x, y) = x Fαt ux δukxαux
(19)
¡
¢
the contribution of the pressure Pxk t x, y, ztopk to the external virtual work can be written as (the secondary
master index β needs to be used):
¡
¢
¡
¢
kt
δukx x, y, ztopk Pxk t x, y, ztopk = δukxαux x Fαt ux x Fβtux Pxβ
(20)
ux
or
where
¡
¢
¡
¢
kα β
kt
δukx x, y, ztopk Pxk t x, y, ztopk = δukxαux t Dux uuxx ux Pxβ
ux
t
kα
Dux uuxx
βux
= x Fαt ux x Fβtux
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American Institute of Aeronautics and Astronautics
(21)
(22)
The other applied pressures are similarly treated. Superscript b is used to indicate the bottom pressure; for
example, x Fαbux =
˙ x Fαux (z = zbotk )). The virtual external work can then be rewritten as
Z
δLke
Z
=+
t
δuxαux
kα β
kt
Dux uuxx ux Pxβ
dxdy
ux
k αu βuy
δuyαuy t Duy uyy
+
Ωk
kt
Pyβ
dxdy
uy
Ωk
Z
Z
kα
δuzαuz t Duz uuzz
+
βuz
kt
Pzβ
dxdy +
uz
kα
βux
kb
dxdy
Pxβ
ux
kα
βuz
kb
dxdy
Pzβ
uz
δuxαux b Dux uuxx
Ωk
Ωk
Z
+
b
δuyαuy
k α u βu
kb
Duy uyy y Pyβ
dxdy
uy
Z
δuzαuz b Duz uuzz
+
(23)
Ωk
Ωk
ztopk
Z
Z
+
£ k k
¤
k
k
δun σ nn + δuks σ ns
+ δukz σ nz
dz ds
z
Γk
σ botk
kα
β
kt
The pressure terms (for example Pxβ
) are inputs of the problem. Terms of the type t Dux uuxx ux are defined
ux
as pressure fundamental kernels and are 1 × 1 matrices.
The pressure has been treated in a particular form because it is a natural choice for the case of multilayered
structures analyzed with GUF.
The term in equation 23, which contains the prescribed stresses, is elaborated to obtain an expression that
contains the displacements ukx , uky and ukz . This elaboration is required since, in general, different orders
of expansions are used for different variables. After a few elaborations it can be demonstrated that the
contribution of layer k to the expression of virtual external work is:
Z
Z
k α u βu
kα β
kt
kt
δLke = + δukxαux t Dux uuxx ux Pxβ
dxdy
+
δukyαuy t Duy uyy y Pyβ
dxdy
ux
uy
Ωk
Ωk
Z
Z
kα
δukzαuz t Duz uuzz
+
βuz
kt
Pzβ
dxdy +
uz
Ωk
βux
kb
Pxβ
dxdy
ux
kα
βuz
kb
Pzβ
dxdy
uz
Ωk
Z
+
kα
δukxαux b Dux uuxx
k αu βuy
δukyαuy b Duy uyy
Ωk
Z
kα
+Zux uuxx
βux
Z
kb
Pyβ
dxdy +
uy
δukzαuz b Duz uuzz
Ωk
k αu βuy
k
x
Γk
σ
Z
kα
+Zuz uuzz
Z
δukxαux txβux ds + Zuy uyy
βuz
(24)
k
δukyαuy tyβuy ds
y
Γk
σ
k
δukzαuz tzβuz ds
z
Γk
σ
k
It is now assumed that where txαux is assigned the corresponding quantity ukxαux is not assigned. Similarly,
k
k
where tyαuy is assigned ukyαuy is not assigned and where tzαuz is assigned ukzαuz is not assigned. When the
displacements are assigned, the virtual variations of the displacements are zero. For example, consider the
displacement ukxαux , which is assigned (see above) only on the boundary portion Γk − Γkσ x . Therefore, in
this portion of the boundary δukxαux = 0. Similar considerations can be made for the other displacements
and, thus, the following relations can be written:
δukxαux = 0
on
Γk − Γkσ x
δukyαuy = 0
on
Γk − Γkσ y
δukzαuz = 0
on Γk − Γkσ z
Considering these last relations and with the help of Figure 3 the governing equations are:
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American Institute of Aeronautics and Astronautics
(25)
δukxαux : −
k αu βu
β
k αux βuz,z
k αu βu
−
Z16 uxx uy y ukyβuy
−
Z26 uxx uy y ukyβuy
−
k αux βuz
Z36 ux uz ,z
+
Z45 ux ,zuy
−
Dux uxux
δukyαuy : −
δukzαuz :
kα
Z11 uuxx uuxx ukxβux ,x,x − Z12 uxx uy y ukyβuy
ktα
βux
− Z66 uuxx uuxx ukxβux ,y,y − Z66 uxx uy y ukyβuy
kα
+
k αu βu
β
k αux βuz
Z55 ux ,zuz ukzβuz ,x
k αux
βuz
ukyβuy + Z45 ux ,zuz
kbα
kt
Pxβ
− Dux uxux
ux
k αu βu
βux
−
Z26 uyy uy y ukyβuy
−
k αuy βuz
Z36 uy uz ,z
+
Z44 uy ,zuz
−
Duy uy y
−
Z55 uuzz uuz z ukzβuz ,x,x − Z55 uz ux
ukzβuz ,y
−
Z45 uz uy
−
Z44 uuzz uuz z ukzβuz ,y,y − Z44 uz uy
+
Z23 uz ,zuy
+
k αuz βuz
Z33 uz ,zuz ,z
kα
k αuz
k αu βu
ukzβuz ,y − Z16 uyy uxx ukxβux ,x,x
− Z66 uyy uxx ukxβux ,y,x − Z66 uyy uy y ukyβuy
k α u βu
+
k αu βu
k αuy βuz
Z45 uy ,zuz ukzβuz ,x
k αu y
βuy,z
ukzβuz ,y + Z44 uy ,zuy
k b αu βuy
kt
Pyβ
− Duy uy y
uy
ukyβuy
kα
,x
k αu z
ukzβuz
(27)
kα
β
ukxβux ,x − Z45 uuzz uuz z ukzβuz ,y,x
β
k αuz βuy,z
,y
βux
+ Z36 uz ,zux
−
,x,x
ukxβux
ukyβuy
k αuz βux,z
ukxβux ,y
k αu z
βux
ukxβux ,x
k αuz
βuy
ukyβuy
− Z45 uuzz uuz z ukzβuz ,x,y − Z45 uz ux
β
ukyβuy
+
k αuy βux
Z45 uy ,zux ,z
kb
Pyβ
=0
uy
k αuz βux,z
βuy
k αu βu
− Z26 uyy uxx ukxβux ,y,y
,y,y
,y,x
β
k αuz βuy,z
(26)
− Z23 uy uz
ukzβuz ,x
,x,y
ukxβux
,x,y
k αu βu
kα
k αux βux
Z55 ux ,zux ,z
kb
Pxβ
=0
ux
k αuy βuz,z
k αu βu
Z26 uyy uy y ukyβuy
k t αu βuy
+
k αu βu
−
βuz
β
,y,y
Z12 uyy uxx ukxβux ,x,y − Z22 uyy uy y ukyβuy
k αuy
kα
ukzβuz ,x − Z16 uuxx uuxx ukxβux ,x,y
− Z13 ux uz
ukzβuz ,y
βuy,z
β
,x,x
k αu βu
k αux
kα
− Z16 uuxx uuxx ukxβux ,y,x
,y,x
ukyβuy
,y
+ Z13 uz ,zux
ukxβux ,y + Z36 uz ,zuy
ktα β
kt
Duz uzuz uz Pzβ
uz
−
kbα β
kb
Duz uzuz uz Pzβ
uz
,x
=0
(28)
Boundary conditions omitted for brevity. The governing equations here obtained are formally independent
on the actual orders used to expand the displacements. This feature will lead to the invariant kernels from
which all possible theories are generated.
B.
Navier-Type Solution
To demonstrate how the invariant model is built, a Navier-type solution is considered. The invariant models
are built under some hypothesis which can be easily removed when the Finite Element Method is used. The
FEM procedure is not much different from a formal point of view and will be presented in future works.
In the Navier-type solution only lamination schemes with angles 0 or 90 are used. Thus, from equation 63
e16 = C
e26 = C
e36 = C
e45 = 0. Suppose also that the reference plane of
of Appendix A it is deduced that C
the plate is a rectangle with lenght a in the x direction and b in the y direction. The external loads and
displacements are assumed to have a sinusoidal distribution:
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American Institute of Aeronautics and Astronautics
kt
Pxβ
= x Pβkutx Camπx Sbnπy
ux
kb
Pxβ
= x Pβkubx Camπx Sbnπy
ux
kt
Pyβ
= y Pβkuty Samπx Cbnπy
uy
kb
Pyβ
= y Pβkuby Samπx Cbnπy
uy
kt
Pzβ
uz
=
z
Pβkutz Samπx Sbnπy
kb
Pzβ
uz
=
z
(29)
Pβkubz Samπx Sbnπy
ukxαux = x Uαkux Camπx Sbnπy
ukyαuy = y Uαkuy Samπx Cbnπy
(30)
ukzαuz = z Uαkuz Samπx Sbnπy
where the following definitions have been used:
mπx
Camπx = cos
a
mπx
mπx
= sin
Sa
a
nπy
b
nπy
= cos
b
Sbnπy = sin
Cbnπy
(31)
kt
Where m and n are the wave numbers. The symbol Pxβ
which indicates the pressure (along x) at the
ux
top surface of layer k should not be confused with the amplitude x Pβkutx associated to the trigonometric
expansion (see equation 29). Similar considerations are valid for the other pressures applied at the top and
bottom surfaces of layer k. The Navier-type assumptions used for the displacements and applied pressures
solve exactly the problem of simply supported plate.35 Substituting equations 29 and 30 into relations 26,
27 and 28 the governing equations become:
α
αu βuy y
β
δuxαux : +Kuxuxux ux x Uβux + Kux xuy
αu βux x
δuyαuy : +Kuy yux
α
αu βuy y
Uβux + Kuy yuy
αu βuy y
β
α
β
Uβuy + Kuxuxuz uz z Uβuz = x Rαux
αu βuz z
Uβuy + Kuy yuz
α
β
Pβkutx + Dux uxux
βux
x
k b αu βuy
y
k bα
z
δuzαuz : +Kuzuzux ux x Uβux + Kuz zuy
Uβuz = y Rαuy
(32)
Uβuy + Kuzuzuz uz z Uβuz = z Rαuz
where the loads have been defined as follows:
k tα
βux
x
k t αu βuy
y
k tα
z
x
k
Rα
= Dux uxux
ux
y
k
Rα
= Duy uy y
uy
z
k
Rα
= Duz uzuz
uz
βuz
k bα
Pβkuty + Duy uy y
Pβkutz + Duz uzuz
βuz
Pβkubx
Pβkuby
(33)
Pβkubz
The 9 fundamental nuclei or kernels of the Generalized Unified Formulation (see equation 32) are defined as
follows:
k αux βux
2 2
2 2
kα β
kα β
kα β
Kux uuxx ux = +Z11 uuxx uuxx ma2π + Z66 uuxx uuxx nb2π + Z55 ux ,zux ,z
k αu βuy
Kux uyx
kα
Kux uuzx
βuz
k αu βu
2
k αu βu
x
y mnπ
= +Z12 uxx uy y mnπ
ab + Z66 ux uy
ab
k αux βuz,z mπ
a
k αux
βuz mπ
a
+ Z55 ux ,zuz
= −Z13 ux uz
k αu βux
y
x mnπ
= +Z12 uyy uxx mnπ
ab + Z66 uy ux
ab
k αu βuy
= +Z22 uyy uy y nb2π + Z66 uyy uy y ma2π + Z44 uy ,zuy
Kuy uxy
Kuy uyy
k αu βuz
Kuy uzy
kα
Kuz uuxz
βux
k αu βuy
Kuz uyz
kα
Kuz uuzz
βuz
k αu βu
2
k αu βu
2
2
2
k αuy βuz,z nπ
b
= −Z23 uy uz
k αuz βux,z mπ
a
= +Z55 uz ux
k αuz βuy,z nπ
b
= +Z44 uz uy
kα
β
2
2
k αu βu
k αu βu
k αuy
2
2
2
k αu y
βuy,z
k αuz
βuz,z
βuz nπ
b
+ Z44 uy ,zuz
k αuz
βux mπ
a
− Z13 uz ,zux
k αuz
βuy nπ
b
− Z23 uz ,zuy
kα
β
2
2
= +Z55 uuzz uuz z ma2π + Z44 uuzz uuz z nb2π + Z33 uz ,zuz
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American Institute of Aeronautics and Astronautics
(34)
But only six kernels are required to generate any theory. This can be seen from the following derivations.
At structural level (after the assembling process) the governing equations can be written as:


Kux ux x U + Kux uy y U + Kux uz z U = Dtux ux x Pt + Dbux ux x Pb = x P



(35)
Kuy ux x U + Kuy uy y U + Kuy uz z U = Dtuy uy y Pt + Dbuy uy y Pb = y P




Kuz ux x U + Kuz uy y U + Kuz uz z U = Dtuz uz z Pt + Dbuz uz z Pb = z P
In a matrix form, the system becomes:

Kux ux Kux uy



Kuy uy


Symm
Kux uz
 
 
 
Kuy uz  · 
 
 
Kuz uz
x
U


 

U =
 
 
z
U
y
x
P



P 

z
P
y
(36)
kα
It is deduced (see equation 36) that with the following six invariant fundamental kernels Kux uuxx
kα β
Kux uuzx uz ,
k αu βu
Kuy uyy y ,
k αu βu
Kuy uzy z
βux
k αu βuy
, Kux uyx
,
kα β
Kuz uuzz uz
and
can generate all types of theories with any order of expansion
in the thickness direction. Each variable can be represented with a different order with respect the other
variables. This property is also valid in the FEM representation.
The main properties of the Generalized Unified Formulation are described in Figure 4.
IV.
∞3 Advanced Layerwise Theories Generated by Using the Generalized
Unified Formulation
Equivalent Single Layer theories give a sufficiently accurate description of the global laminate response.
However, these theories are not adequate for determining the stress fields at ply level. Layerwise theories
assume separate displacement field expansions within each layer. The accuracy is then greater but the price is
in the increased computational cost. Many layerwise plate models have been proposed in the past by applying
Classical Plate Theory or Higher-order Shear Theories at each layer. Generalizations of these approaches
were also given, and the displacements variables were expressed in terms of Lagrange polynomials. Many
papers are devoted on the subject of layerwise theories.8,9,33−38 The conceptual differences between the
displacement fields in layerwise and Equivalent Single Layer Theories are depicted in Figure 5. Layerwise
models are computationally more expensive than the less accurate Equivalent Single Layer models. Therefore,
layerwise models can be used in regions of the structure in which an accurate description is required,40
whereas Equivalent Single Layer models are employed in other parts of the structure. The Generalized
Unified Formulation form for the displacements has been derived at layer level in equation 9. LWT are
going to be developed and so equation 9 applies for this case. The functions of the thickness coordinate (see
equation 9) are introduced in a general form. For example, xFt is a function of z and can be a polynomial,
trigonometric, exponential or another function chosen a priori. To have the assembling process along the
thickness direction immediate and intuitive and indicated for the case of multilayered structures a convenient
expansion along the thickness is introduced.
The displacements must be continuous functions along the thickness to ensure the compatibility between
two adjacent layers. Therefore, it is convenient for the axiomatic expansions along the thickness to have the
following properties. The first property is that for z = zbotk (the bottom surface of layer k) all the functions
along the thickness are zero except the one which multiplies the term corresponding to the bottom (subscript
b). For example, in the case of the displacement ukx , the functions calculated at the bottom of layer k should
give the following values:
x
Ft (z = zbotk ) = 0
x
Fl (z = zbotk ) = 0
x
Fb (z = zbotk ) = 1
(37)
If the previous conditions are satisfied then ukxb is not just a term in the thickness expansion of the variable
ukx but assumes the meaning of the value that the displacement ukx takes when the bottom surface of layer k
is considered (i.e., z = zbotk ). This now explains why the subscript “b” is introduced in the notation. Similar
concept is applied for the top surface of the layer. A good set of function satisfies the following relations:
¡
¢
¡
¢
¡
¢
x
x
x
(38)
Ft z = ztopk = 1
Fl z = ztopk = 0
Fb z = ztopk = 0
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Figure 4. Generalized Unified Formulation for displacement-based theories.
Figure 5. Layerwise theories vs equivalent single layer theories in a three layered structure.
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and so ukxt is not just a term in the thickness expansion of the variable ukx but assumes the meaning of
the value that the displacement ukx takes when the top surface of layer k is considered (i.e., z = ztopk ).
This now explains why the subscript “t” is introduced. The second property is that numerical stability (no
ill conditioning when the orders are increased) is important. Therefore, orthogonal polynomials should be
preferred.
A good set of functions which satisfy the above mentioned properties should be selected. It is possible to
demonstrate that all the previous properties are satisfied if particular combination of Legendre polynomials
is used. Legendre polynomials are defined in the interval [−1, 1]. Thus, a transformation is necessary:
ζk =
ztopk
ztopk + zbotk
2
z−
− zbotk
ztopk − zbotk
− 1 ≤ ζk ≤ +1
(39)
where ζk is a non-dimensional coordinate. The following formula is also valid:
ztopk = zbotk + hk
The transformation is then
ζk =
(40)
ztopk + zbotk
2
z−
hk
ztopk − zbotk
(41)
The Legendre polynomial of order zero is P0 (ζk ) = 1. The Legendre polynomial of order one is P1 (ζk ) = ζk .
The higher order polynomials can be obtained by using Bonnet’s recursion:42
Pn+1 (ζk ) =
(2n + 1) ζk Pn (ζk ) − nPn−1 (ζk )
n+1
(42)
Bonnet’s formula is a convenient method to calculate the Legendre polynomials in a practical code based on
the Generalized Unified Formulation.
The same type of functions for all displacements are used. This is not necessary with the Generalized Unified
Formulation but it is more practical. The following combination of Legendre functions is used:
x
Ft = yFt = zFt =
P0 +P1
,
2
x
Fb = yFb = zFb =
P0 −P1
2
x
Fl = Pl − Pl−2 , l = 2, 3, ..., Nux
y
Fm = Pm − Pm−2 , m = 2, 3, ..., Nuy
(43)
z
Fn = Pn − Pn−2 , n = 2, 3, ..., Nuz
in which Pj = Pj (ζk ) is the Legendre polynomial of j-order. The chosen functions have the following
properties:

x
 +1,
Ft , yFt , zFt = 1, xFb , yFb , zFb = 0, xFl , yFm , zFn = 0
ζk =
(44)
 −1,
x
Ft , yFt , zFt = 0, xFb , yFb , zFb = 1, xFl , yFm , zFn = 0
Thus, the properties earlier mentioned are all satisfied and this set of functions is a good choice to build
the advanced layerwise theories. It is then possible to create any class of theories by changing the orders of
displacements. Suppose, for example, that a theory has the following data: Nux = 3, Nuy = 2, Nuz = 4.
The corresponding theory is indicated as LD324 . The first letter “L” means “Layerwise” theory, the second
letter “D” means that a “Displacement-based” theory is formulated (i.e. the variational statement is the
Principle of Virtual Displacements) The subscripts are the orders of the Legendre polynomials used for the
displacements. In general, the acronym is then built as follows: LDNux Nuy Nuz .
The logic used to build the acronyms and one example are shown in Figure 6.
A.
Expansion of the Six 1 × 1 Invariant Kernels: Matrices at Layer Level
In this paper the expansion used in the different variables does not change and each layer is treated in the
same way. Thus, for example, Nukx = Nuk+1
= Nux . The expansion of the kernels is the most important
x
part of the generation of one of the possible ∞3 layerwise theories. This operation is done at layer level. To
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Figure 6. Acronyms used to define the Advanced Layerwise Theories using the Generalized Unified Formulation.
explain how this operation is performed, consider the case of theory LD326 , in which the number of Degrees
of Freedom, at layer level, is the following:
k
[DOF ]ux = Nux + 1 = 3 + 1 = 4
k
[DOF ]uy = Nuy + 1 = 2 + 1 = 3
k
(45)
[DOF ]uz = Nuz + 1 = 6 + 1 = 7
From the number of Degrees of Freedom it is possible to calculate the size of the layer matrices. For example,
kα β
k
k
when matrix Kux uuzx uz is expanded then the final size at layer level will be [DOF ]ux × [DOF ]uz . In the
kα
example relative to theory LD326 , matrix Kux uuzx
and obtained as explained in Figure 7.
βuz
at layer level (indicated as Kkux uz ) is a 4 × 7 matrix
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Figure 7. Generalized Unified Formulation: example of expansion from a kernel to a layer matrix. Case of
kα
β
theory LD326 . From Kux uuzx uz to Kkux uz .
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B.
Assembling in the Thickness Direction: from Layer to Multilayer Level
The assembling must consider the compatibility of the displacements between two adjacent layers. Figure
8 shows how the assembling of a typical matrix is performed. The pressure matrices are obtained from the
pressure kernels using the same method shown in Figures 7 and 8. The use of combinations of Legendre
Figure 8. Generalized Unified Formulation: example of assembling from layer matrices to multilayer matrix.
(k+1)
Case of theory LD326 . From Kkux uz and Kux uz to Kux uz .
polynomials ensures the continuity of the functions, which are used to expand the displacements. Figure 9
shows this concept for the case of theory LD324 . The pressure amplitudes at multilayer level are input of the
problem. Some input examples are shown in Figure 10. The layer pressure matrices (for example t Dkux ux )
are obtained using a method similar to the one adopted for the stiffness matrices. They are generated from
the pressure fundamental kernels and assembled. In the practice, all the pressure matrices are set to be with
only zeros except the top and bottom layers in which the pressure is actually applied. Once the matrices are
all assembled, equation 36 is obtained and the displacement amplitudes can be found. In the case of FEM
applications the unknown will be the nodal displacements and the concepts are the same.
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Figure 9. Case of theory LD324 . Multilayer unknown displacement and out-of-plane stresses for the case in
which the number of layers is two.
V.
∞3 Advanced Higher-order Shear Deformation Theories Generated by
Using the Generalized Unified Formulation
The derivation is started by considering a particular theory in which the in-plane displacements are
expanded along the thickness by using a cubic polynomial and the out-of-plane displacement uz is parabolic.
In this case it is possible to write the displacements as follows:


ux = ux0 + zφ1ux + z 2 φ2ux + z 3 φ3ux



Theory I :
(46)
uy = uy0 + zφ1uy + z 2 φ2uy + z 3 φ3uy




uz = uz0 + zφ1uz + z 2 φ2uz
For each displacement component the concepts of the Generalized Unified Formulation can be applied. For
example, the displacement ux is written as
ux = ux0 + zφ1ux + z 2 φ2ux + z 3 φ3ux = xFt uxt + xF2 ux2 + xF3 ux3 + xFb uxb
where
x
Ft = 1;
uxt = ux0 ;
x
F2 = z;
x
F3 = z 2 ;
ux2 = φ1ux ;
Fb = z 3
(47)
x
ux3 = φ2ux ;
(48)
uxb = φ3ux
The GUF for the displacement ux is:
ux = xFαux uxαux
αux = t, l, b;
l = 2, ..., Nux
(49)
where, in the example, Nux = 3. Notice that the superscript “x” in xFαux is used to clearly enhance that
the displacement ux (displacement in the x direction) is being considered.
For the displacement uy the formal procedure produces similar results (notice that now the slave index is
uy ):
uy = yFαuy uyαuy αuy = t, m, b;
m = 2, ..., Nuy
(50)
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Figure 10. Case of theory LD324 . Example of pressure amplitudes and inputs at multilayer level for the case
in which the number of layers is two.
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The displacement uz is only parabolic (three terms are used in the expansion), but the representation is
formally the same:
n = 2, ..., Nuz
uz = zFαuz uzαuz αuz = t, n, b;
(51)
The superscript k is not present, whereas in the layerwise case it was. In fact, in Equivalent Single Layer
Models the displacement fields have a description at plate level and not at layerwise level. The Generalized
Unified Formulation for this case (equations 49, 50 and 51) is formally equivalent to the writing of equation
9. This similarity suggests that it is possible to use the layerwise Generalized Unified Formulation for the
Equivalent Single Layer case too. That is, equation 9 can be used for the Equivalent Single Layer case. The
fact that the displacement field does not have a layerwise description (see, for example, Theory I explicitly
written in equation 46) is taken into account when the assembling in the thickness direction of the layer
matrices is considered. For the Advanced Higher-order Shear Deformation Theories it is then possible to use
equation 9. However, now the functions of the thickness coordinates are going to be different. The functions
of the thickness coordinate are assumed to be of the type 1 z z 2 z 3 .... This choice is made for “consistency”
with the usual approach used in the literature for Equivalent Single Layer Models. The actual functions are
reported below:
x
Ft = 1 yFt = 1 zFt = 1
x
F2 = z
y
x
F3 = z 2
F2 = z
z
F2 = z
y
F3 = z 2
z
F3 = z 2
...
Fl = z l−1
x
(52)
y
Fm = z m−1
z
Fn = z n−1
Fb = z Nuy
z
...
x
Fb = z Nux
y
Fb = z Nuz
The Advanced Higher-order Shear Deformation Theories (AHSDT) are indicated with acronyms shown in
Figure 11. As for the layerwise case it is possible to create a class of theories by changing the order used for
Figure 11. Acronyms used in the case of Advanced Higher-order Shear Deformation Theories (AHSDT).
the displacements. Suppose, for example, that a theory has the following data: Nux = 3, Nuy = 2, Nuz = 4.
The corresponding AHSDT theory is indicated as ED324 . The first letter “E” means Equivalent Single
Layer theory, the second letter “D” means that a displacement-based formulation is used. The subscripts
are the orders of the polynomials used for the displacements. In general, the acronym is then built as follows:
EDNux Nuy Nuz .
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A.
Expansion of the Six 1 × 1 Invariant Kernels: Matrices at Layer Level
Besides the different form of the functions used to expand the displacements, AHSDT do not present formal
differences with respect to the layerwise cases. For example, the generation of layer matrix Kkux uz from
kα β
the kernel Kux uuzx uz in the cases of theories LD326 and ED326 (which have the same number of Degrees of
Freedom at layer level) is formally the same. Thus, Figure 7 applies for the case of theory ED326 as well.
Of course, the actual integrals along the thickness are different because different functions have been used
(compare equations 52 and 43 and see the definitions of Figure 3).
B.
Assembling in the Thickness Direction: from Layer to Multilayer Level
The displacement fields are treated as equivalent single layers quantities, and this makes the assembling
procedure different with respect to the layerwise case. The continuity of the displacements in the thickness
direction must be imposed. So, it is not difficult to show that the assembling is performed as shown in
Figure 12). The pressure matrices are obtained using similar procedure and the details are omitted for
Figure 12. Generalized Unified Formulation: example of assembling from layer matrices to multilayer matrix.
(k+1)
Case of theory ED324 . From Kkux uy and Kux uy to Kux uy .
brevity. About the pressure amplitudes, inputs of the problem (see Figure 13), the difference between
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this ESL case and the corresponding Layerwise case (Figure 10) should be noted. The final equations at
Figure 13. Case of theory ED324 . Example of pressure amplitudes and inputs at multilayer level for the case
in two-layered case.
multilayer level are again given by relation 36. This equivalence is another advantage of the Generalized
Unified Formulation.
VI.
∞3 Advanced Higher-order Shear Deformation Theories with Zig-Zag
Effects Generated by Using the Generalized Unified Formulation
The Higher-order Shear Deformation Theories may be not sufficiently accurate in some challenging cases.
One option would be to abandon the Equivalent Single Layer Description and use Layerwise Theories.
However, this may be computationally too expensive. Is there anything more accurate than HSDT with
a similar number of Degrees of Freedoms? The answer is yes, zig-zag theories.30 The concept behind zigzag theories and zig-zag form of the displacements is the following. The equilibrium between two adjacent
layers implies that the out-of-plane stresses are equal at the interface. These stresses can be thought as a
combination of strains multiplied by some coefficients that depend on the material of each layer (Hooke’s
law). In general, two layers have different mechanical properties and, therefore, different strains are required
to obtain equilibrium. The strains are related to the derivatives of the displacements (geometric relations).
Thus, different strains imply different slopes of the displacements. This fact leads to the zig-zag form of the
displacements (see Figure 14).
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A very large amount of literature has been devoted to the formulation of axiomatic zig-zag theories that
take into account these requirements. Three different categories30 of zig-zag theories can be created. The
first category is Lekhnitskii Multilayered Theory (LMT). The second is Ambartsumian Multilayered Theory
(AMT) and the third is Reissner Multilayered Theory (RMT). LMT was introduced for the particular case
Figure 14. Zig-Zag form of the displacements and Murakami’s Zig-Zag Function.
of cantilevered multilayered beam22 and almost ignored in subsequent works with a few exceptions.27,28 A
summary of the main facts of LMT is presented in reference [30]. Ambartsumian work23−26 was an extension
of Reissner-Mindlin theory.18,19 RMT is based on Reissner’s Mixed Variational Theorem (RMVT).43,44
The present paper takes into account the Zig-Zag effects by adopting of Murakami’s Zig-Zag Function21 (see
Figure 14). MZZF has the advantage of being simple and reproducing the discontinuity of the first derivative
of the displacements in the thickness direction. It will be demonstrated that the usage of MZZF is more
effective than increasing the orders of the expansions of the variables along the thickness.
The derivation is started from the Advanced Higher-order Shear Deformation Theory expressed by equation
46. Theory I (see equation 46) is then “improved” as follows (Theory II is the resulting theory):

Extra term for Zig−Zag effect

}|
{
z



k
2
3


(−1)
ζ
u
u
=
u
+
zφ
+
z
φ
+
z
φ
+
k xZ
x0
1u x
2ux
3u x

 x




Extra term for Zig−Zag effect

z
}|
{
Theory II :
(53)
k
2
3

u
=
u
+
zφ
+
z
φ
+
z
φ
+
(−1)
ζ
u
y
y0
1uy
2uy
3uy
k yZ







Extra term for Zig−Zag effect


z
}|
{



k
2
uz = uz0 + zφ1uz + z φ2uz + (−1) ζk uzZ
The quantity ζk is defined in equation 39. In equation 53, valid for a theory with Zig-Zag form of the
displacements included, the following can be observed.
k
In Murakami’s Zig-Zag functions the term (−1) is present. k is the integer representing the ID of a generic
layer, k = 1 is for the bottom layer and k = Nl is for the top layer (Nl is the number of layers). The
k
term (−1) enforces the discontinuity of the first derivative (thickness direction) of the displacement. For
example, in layer k the derivative with respect to z of the Zig-Zag term relative to the component ux is
h
i
k
d (−1) ζk uxZ
dζk
2
k
k
(54)
= (−1) uxZ
= (−1) uxZ
dz
dz
ztopk − zbotk
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k
As can be seen the term (−1) strongly affects the sign of the derivative. The displacements still have an
Equivalent Single Layer description. In fact, the terms uxZ , uyZ and uzZ are independent on the actual layer
and defined for the whole plate. The zig-zag form of the displacements is taken into account a priori by
adding only three Degrees of Freedom (uxZ , uyZ and uzZ respectively). This is a general property and does
not depend on the orders used for the expansion of the different variables. That is, a generic theory can
take into account the zig-zag form of the displacements by adding only three extra Degrees of Freedoms, as
was done in equations 46 and 53 for the case of Theory I. For each displacement component the concepts of
the Generalized Unified Formulation is still valid. However, in this case equations 49, 50 and 51 have to be
slightly modified because they have to contain an extra term which comes from the Zig-Zag Function. This
is explained if for example displacement ux is considered:
k
ux = ux0 + zφ1ux + z 2 φ2ux + z 3 φ3ux + (−1) ζk uxZ
= xFt uxt + xF2 ux2 + xF3 ux3 + xF4 ux4 + xFb uxb
where
x
x
Ft = 1;
x
F3 = z 2 ;
F2 = z;
uxt = ux0 ;
ux2 = φ1ux ;
x
F4 = z 3 ;
ux3 = φ2ux ;
(55)
k
x
Fb = (−1) ζk
ux4 = φ3ux ;
(56)
uxb = uxZ
The GUF for the displacement ux is:
ux = xFαux uxαux
αux = t, l, b;
l = 2, ..., Nux + 1
(57)
where, in the example, Nux = 3. For the displacement uy the formal procedure produces similar results:
uy = yFαuy uyαuy
αuy = t, m, b;
m = 2, ..., Nuy + 1
(58)
The displacement uz is only parabolic, but the representation is formally the same:
uz = zFαuz uzαuz
αuz = t, n, b;
n = 2, ..., Nuz + 1
(59)
For the case of Zig-Zag theories, the Generalized Unified Formulation (equations 57, 58 and 59) is formally
equivalent to a layerwise case in which the same number of terms is considered. Thus (see equation 9),
where
ukx = xFαux ukxαux
αux = t, l, b;
uky = yFαuy ukyαuy
αuy = t, m, b; m = 2, ..., Nuy + 1
ukz = zFαuz ukzαuz
αuz = t, n, b; n = 2, ..., Nuz + 1
x
y
x
y
Ft = 1
F2 = z
x
F3 = z 2
l = 2, ..., Nux + 1
(60)
z
Ft = 1
Ft = 1
z
F2 = z
F2 = z
y
F3 = z 2
z
F3 = z 2
...
(61)
x
Fl = z l−1
y
Fm = z m−1
z
Fn = z n−1
...
x
k
Fb = (−1) ζk
k
y
Fb = (−1) ζk
z
k
Fb = (−1) ζk
The Advanced Zig-Zag Theories (AZZT) are obtained by adding the zig-zag form of the displacements to
the Advanced Higher Order Shear Deformation Theories presented above. This is also evident if equations
52 and 61 are compared. How the acronym are built is explained in Figure 15. It is possible to create a class
of theories by changing the order used for the displacements. Suppose, for example, that a theory has the
following data: Nux = 3, Nuy = 2, Nuz = 4. The corresponding AZZT theory is indicated as EDZ324 . The
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Figure 15. Advanced Zig-Zag Theoris: acronyms used.
first letter “E” means “Equivalent Single Layer” theory, the second letter “D” means displacement-based
theory, the third letter “Z” means that the zig-zag form of the displacements is enforced a priori. The
subscripts are the orders of the polynomials used for the displacements. In general the acronym is then
built as follows: EDZNux Nuy Nuz . The formal generation of the stiffness matrices from the kernels of the
Generalized Unified Formulation is done as in the case of AHSDT. The difference is only in the sizes of
the matrices and of course in the actual values of the integrals along the thickness. For example, consider
theories ED324 and EDZ324 . They both have the same orders. However, in the Zig-Zag theory there is an
extra degree of freedom for each displacement variable. This means that the layer matrix Kkux uy has size
4 × 3 in the case of theory ED324 whereas the size is 5 × 4 in the case of theory EDZ324 . The assembling
at multilayered level follows the same procedure outlined in the case of AHSDT. The final system that has
to be solved is again equation 36.
Appendix A. Coefficients of Hooke’s Law















σxx


 

σyy 
 
 
 
σxy  
 
=

σxz 
 
 

σyz 
 
 
σzz
e11
C
e12
C
e16
C
0
0
e12
C
e22
C
e26
C
0
0
e16
C
e26
C
e66
C
0
0
0
0
0
e55
C
e45
C
0
0
0
e45
C
e44
C
e13
C
e23
C
e36
C
0
0
e13
C



e23 
C



e36 
C




0



0


e
C33
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εxx


εyy 



γxy 


γxz 


γyz 


εzz
(62)
where
e11 = C 4 C 11 + 2S 2 C 2 C 12 + S 4 C 22 + 4S 2 C 2 C 66
C
e12 = S 2 C 2 C 11 + C 4 C 12 + S 4 C 12 + S 2 C 2 C 22 − 4S 2 C 2 C 66
C
e16 = SC 3 C 11 + S 3 CC 12 − SC 3 C 12 − S 3 CC 22 − 2SC 3 C 66 + 2S 3 CC 66
C
e13 = C 2 C 13 + S 2 C 23
C
e22 = S 4 C 11 + 2S 2 C 2 C 12 + C 4 C 22 + 4S 2 C 2 C 66
C
e26 = S 3 CC 11 + SC 3 C 12 − S 3 CC 12 − SC 3 C 22 + 2SC 3 C 66 − 2S 3 CC 66
C
e23 = S 2 C 13 + C 2 C 23
C
e66 = S 2 C 2 C 11 − 2S 2 C 2 C 12 + S 2 C 2 C 22 + C 66 C 4 − 2S 2 C 2 C 66 + C 66 S 4
C
e36 = SCC 13 − SCC 23
C
(63)
e55 = C 2 C 55 + S 2 C 44
C
e45 = SCC 55 − SCC 44
C
e44 = S 2 C 55 + C 2 C 44
C
e33 = C 33
C
The definitions C = cos ϑ and S = sin ϑ were used, where ϑ is the rotation angle between the material
e16 = C
e26 = C
e36 = C
e45 = 0. The coefficients of
coordinates and problem coordinates.35 If ϑ = 0/90 then C
Hooke’s law in material coordinates are:
C 11 =
C 13 =
1−υ23 υ32
13 υ31
E11 ; C 12 = υ21 +υ∆23 υ31 E11 ; C 22 = 1−υ∆
E22
∆
υ21 υ32 +υ31
υ32 +υ12 υ31
1−υ12 υ21
E11 ; C 23 =
E22 ; C 33 =
E33
∆
∆
∆
C 44 = G23 ; C 55 = G13 ; C 66 = G12
(64)
∆ = 1 − υ23 υ32 − υ12 υ21 − υ13 υ31 − 2υ21 υ32 υ13
υ32 =
E33
E22 υ23 ;
υ21 =
E22
E11 υ12 ;
υ31 =
E33
E11 υ13
The independent parameters used in the definition of the material properties are nine: υ12 , υ13 , υ23 , G12 ,
G13 , G23 , E11 , E22 and E33 . In the case of isotropic materials only two parameters are needed: the Poisson’s
ratio and the elastic modulus.
VII.
Results
The multilayered structure is a sandwich plate (see Figure 16) made of two skins and a core [hlower skin =
Elower skin
= 5/4. The plate is simply supported and
h/10; hupper skin = 2h/10; hcore = (7/10)h]. It is also E
upper skin
the load is a sinusoidal pressure applied at the top surface of the plate (m = n = 1). Different cases are
proposed here:
• Face-to-Core Stiffness Ratio = F CSR =
Elower skin
Ecore
= 101 ; a/h = 4, 100
• Face-to-Core Stiffness Ratio = F CSR =
Elower skin
Ecore
= 105 ; a/h = 4, 100
As far as Poisson’s ratio is concerned, the following values are used: υlower skin = υupper skin = υcore = υ =
0.34. In all cases b = 3a. In this test case there is no symmetry with respect the plane z = 0. The following
non-dimensional quantities are introduced:
Ecore
zP t h a
h
core
u
by = uy z Et corea 3 ; u
bz = uz z100E
;
3;
a 4
P h( h )
P t h( h
( )
)
σ
bzy = zP tzya ; σ
bzz = σzPzzt ;
σ
bzx = zPσtzxa ; σ
(h)
(h)
σ
σ
σ
bxx = z σt xxa 2 ; σ
byy = z t yya 2 ; σ
bxy = z t xya 2 ;
P (h)
P (h)
P (h)
u
bx = ux
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American Institute of Aeronautics and Astronautics
(65)
Figure 16. Test Case 2. Geometry of the plate sandwich structure.
All the results have been compared with the solution obtained by solving the “exact” problem.45 The exact
value is indicated with the terminology “elasticity” and is the reference value corresponding to the solution
of the differential equations that govern the problem. The details of this elasticity solution are here omitted
for brevity.45
Tables 1 and 2 compare a large number of ALWT, AHSDT and AHSDTZ. The orders of the expansions
along the thickness is changed. FCSR is dramatically changed to clearly show the limitations of some of
the possible approaches. From Tables 1 and 2 it is deduced that when FCSR is very high most of the the
Advanced Higher-order Shear Deformation Theories fail to predict the correct displacements or stresses. For
example, when F CSR = 105 even the AHSDT with order 7 of all the displacements (ED777 ) has a large
error. The Zig Zag theories present a better compromise (see Figure 17) but only the Layerwise Theories
can capture this complex behavior well. The Generalized Unified Formulation allows to compare different
types of theories and different types of expansions without any effort. It can be then considered a natural
tool to study the numerical performances of multilayered structures in general (see for example reference
[46]). The “sensitivity” with respect to the type of theory and/or orders of expansion is an impractical task
with the classical approaches but this does not present problems when the Generalized Unified Formulation
is used.
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American Institute of Aeronautics and Astronautics
a/h
4
Elasticity
LD111
LD115
LD225
LD555
ED111
ED115
ED225
ED335
ED444
ED445
ED555
ED777
EDZ111
EDZ115
EDZ225
EDZ335
EDZ444
EDZ445
EDZ555
EDZ777
3.01123
2.98058
3.00603
3.00983
3.01123
1.58218
2.00048
2.03295
2.73658
2.79960
2.80707
2.84978
2.86875
2.34412
2.68809
2.97597
2.98229
2.97886
2.98242
2.98737
2.99670
Elasticity
LD111
LD115
LD225
LD555
ED111
ED115
ED225
ED335
ED444
ED445
ED555
ED777
EDZ111
EDZ115
EDZ225
EDZ335
EDZ444
EDZ445
EDZ555
EDZ777
1.31593 · 10−02
9.79008 · 10−03
1.31222 · 10−02
1.31473 · 10−02
1.31593 · 10−02
1.79831 · 10−04
2.49683 · 10−04
4.09818 · 10−04
7.83554 · 10−04
1.16851 · 10−03
1.31972 · 10−03
4.29224 · 10−03
1.08119 · 10−02
8.36735 · 10−04
7.93264 · 10−03
9.52709 · 10−03
1.02216 · 10−02
1.26288 · 10−02
1.26702 · 10−02
1.30409 · 10−02
1.31363 · 10−02
100
F CSR = 101
Err.%
1.51021
(−1.02)
1.47242
(−0.17)
1.51020
(−0.05)
1.51021
(0.00)
1.51021
(−47.5)
1.10845
(−33.6)
1.50862
(−32.5)
1.50864
(−9.12)
1.50979
(−7.03)
1.50989
(−6.78)
1.50989
(−5.36)
1.50996
(−4.73)
1.50999
(−22.2)
1.15866
(−10.7)
1.50975
(−1.17)
1.51015
(−0.96)
1.51017
(−1.07)
1.51017
(−0.96)
1.51017
(−0.79)
1.51018
(−0.48)
1.51019
F CSR = 105
Err.% 2.08948 · 10−03
(−25.6) 1.96509 · 10−03
(−0.28) 2.08948 · 10−03
(−0.09) 2.08948 · 10−03
(0.00)
2.08949 · 10−03
(−98.6) 1.19941 · 10−04
(−98.1) 1.63241 · 10−04
(−96.9) 1.63247 · 10−04
(−94.0) 1.64006 · 10−04
(−91.1) 1.64835 · 10−04
(−90.0) 1.64835 · 10−04
(−67.4) 1.73120 · 10−04
(−17.8) 2.96304 · 10−04
(−93.6) 1.63329 · 10−04
(−39.7) 1.64323 · 10−04
(−27.6) 2.24726 · 10−04
(−22.3) 2.70584 · 10−04
(−4.03) 1.16305 · 10−03
(−3.72) 1.16345 · 10−03
(−0.90) 1.78411 · 10−03
(−0.17) 2.02060 · 10−03
Err.%
(−2.50)
(0.00)
(0.00)
(0.00)
(−26.6)
(−0.11)
(−0.10)
(−0.03)
(−0.02)
(−0.02)
(−0.02)
(−0.01)
(−23.3)
(−0.03)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
DOF
12
24
30
48
6
10
12
14
15
16
18
24
9
13
15
17
18
19
21
27
Err.%
(−5.95)
(0.00)
(0.00)
(0.00)
(−94.3)
(−92.2)
(−92.2)
(−92.2)
(−92.1)
(−92.1)
(−91.7)
(−85.8)
(−92.2)
(−92.1)
(−89.2)
(−87.0)
(−44.3)
(−44.3)
(−14.6)
(−3.30)
12
24
30
48
6
10
12
14
15
16
18
24
9
13
15
17
18
19
21
27
Table 1.
Comparison of various theories to evaluate the transverse displacements amplitude (center plate
upper skin
3
core
deflection) b
uz = uz z100E
h, x = a/2, y = b/2.
= 10
a 4 in z = zbottom
P t h( h
)
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American Institute of Aeronautics and Astronautics
a/h
4
Elasticity
LD111
LD115
LD225
LD555
ED111
ED115
ED225
ED335
ED444
ED445
ED555
ED777
EDZ111
EDZ115
EDZ225
EDZ335
EDZ444
EDZ445
EDZ555
EDZ777
0.32168
0.31730
0.32157
0.32142
0.32168
0.33178
0.32509
0.32503
0.32502
0.33240
0.32976
0.32884
0.32707
0.34184
0.32854
0.32856
0.32803
0.32913
0.32783
0.32755
0.32530
Elasticity
LD111
LD115
LD225
LD555
ED111
ED115
ED225
ED335
ED444
ED445
ED555
ED777
EDZ111
EDZ115
EDZ225
EDZ335
EDZ444
EDZ445
EDZ555
EDZ777
5.40842 · 10−04
1.05700 · 10−04
3.25506 · 10−04
5.40033 · 10−04
5.40842 · 10−04
0.33242
0.33277
0.33166
0.32018
0.30529
0.30566
0.21639
3.96907 · 10−02
0.30971
8.48754 · 10−02
5.88764 · 10−02
4.74947 · 10−02
6.84336 · 10−03
6.84955 · 10−03
1.87520 · 10−03
8.02443 · 10−04
Err.
100
F CSR = 101
Err.% 0.33176
(−1.36) 0.32345
(−0.03) 0.33177
(−0.08) 0.33176
(0.00)
0.33176
(+3.14) 0.33178
(+1.06) 0.33177
(+1.04) 0.33177
(+1.04) 0.33177
(+3.33) 0.33178
(+2.51) 0.33178
(+2.23) 0.33178
(+1.68) 0.33177
(+6.27) 0.34497
(+2.13) 0.33178
(+2.14) 0.33178
(+1.97) 0.33178
(+2.32) 0.33178
(+1.91) 0.33178
(+1.82) 0.33177
(+1.12) 0.33177
F CSR = 105
Err.% 0.27797
(−80.5) 0.26143
(−39.8) 0.27797
(−0.15) 0.27797
(0.00)
0.27797
(> 100) 0.33242
(> 100) 0.33243
(> 100) 0.33243
(> 100) 0.33241
(> 100) 0.33238
(> 100) 0.33238
(> 100) 0.33214
(> 100) 0.32865
(> 100) 0.33077
(> 100) 0.33240
(> 100) 0.33069
(> 100) 0.32939
(> 100) 0.30392
(> 100) 0.30405
(> 100) 0.28655
(+48.4) 0.27994
Err.
Err.%
(−2.50)
(0.00)
(0.00)
(0.00)
(+0.01)
(0.00)
(0.00)
(0.00)
(+0.01)
(+0.01)
(+0.01)
(0.00)
(+3.98)
(+0.01)
(+0.01)
(+0.01)
(+0.01)
(+0.01)
(0.00)
(+0.00)
DOF
12
24
30
48
6
10
12
14
15
16
18
24
9
13
15
17
18
19
21
27
Err.%
(−5.95)
(0.00)
(0.00)
(0.00)
(+19.6)
(+19.6)
(+19.6)
(+19.6)
(+16.6)
(+16.6)
(+19.5)
(+18.2)
(+19.0)
(+19.6)
(+19.0)
(+18.5)
(+9.34)
(+9.38)
(+3.09)
(+0.71)
12
24
30
48
6
10
12
14
15
16
18
24
9
13
15
17
18
19
21
27
upper skin
in z = zbottom
=
( )
3
h,
x
=
0,
y
=
b/2.
The
indefinite
equilibrium
equations
have
been
integrated
along
the
thickness.
10
Table 2.
Comparison of various theories to evaluate the transverse shear stress b
σzx =
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American Institute of Aeronautics and Astronautics
σzx
zP t a
h
Figure 17. Dimensionless in-plane displacement b
ux .
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American Institute of Aeronautics and Astronautics
VIII.
Conslusions
An invariant methodology, the Generalized Unified Formulation, to include practically any type of axiomatic theory has been presented. ∞3 Higher-order Shear Deformation Theories, ∞3 Zig-Zag Theories and
∞3 Layer-Wise Theories, with any combination of orders of expansion for the displacement variables can be
derived from six invariant 1 × 1 matrices. This property gives the possibility to include in a single software
and FEM formulation classical and advanced models. It is then possible to find the best formulation, among
the infinite possible choices, for the problem under investigation. The applicability of the Generalized Unified
Formulation is not confined to the displacement-based case. Mixed variational statements can in fact be used
and multifield loadings could be included as well. In any case all the theories are derived from 1 × 1 invariant
matrices (the so called kernels of the Generalized Unified Formulation). The number of independent kernels
depend on the variational statement that is used. In the case of displacement-based theories only six kernels
are required, whereas in the case of mixed variational statements this number is different.
The possibility to select the type of theory among the infinite possibilities without actually change the software makes the Generalized Unified Formulation particularly indicated for the solution of optimization and
probabilistic problems and for the quasi-3D analysis of structures when localized effects are important. The
Generalized Unified Formulation can also have an important educational feature because it allows the user
to adopt practically all the existing approaches and to compare with an infinite number of more advanced
models. The engineer has only to learn a single invariant formulation and software and this can be applied
to study a very large variety of new cases. When the CPU time is more important then a low order theory
can be selected and when the accuracy is the determinant factor the order can be increased or the theory
can be changed for a more advanced analysis.
The Generalized Unified Formulation is also a powerful tool to evaluate the sensitivity with respect the order
of expansion or the type of theory (e.g., Higher-Order Shear Deformation Theories with or without Zig-Zag
effects included in the model). This property makes possible to tailor the software to the peculiarity of a new
problem with unknown requirements in terms of axiomatic theories necessary to study it. An “intelligent”
FEM software can be designed using the Generalized Unified Formulation: the software can change the
orders or the theory without the actual need of a new code or theoretical development.
The paper presented an application of these concepts to a sandwich structure. It is demonstrated that
when the Face to Core Stiffness Ratio is very high the classical approaches fail and a layerwise formulation
is required. This finding is immediate with the Generalized Unified Formulation because all the possible
combinations of orders are allowed and also different types of theories can be experimented as well. It is concluded that the Generalized Unified Formulation is a modern approach for the theoretical and computational
analysis of multilayered structures.
Acknowledgements
The author acknowledges the support by San Diego State University (University Grant Program).
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American Institute of Aeronautics and Astronautics
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