VIETNAM NATIONAL UNIVERSITY, HANOI
VIETNAM JAPAN UNIVERSITY
NGO DINH DAT
NONLINEAR DYNAMIC RESPONSE AND
VIBRATION OF SANDWICH PLATES
WITH FG POROUS HOMOGENEOUS
CORE AND NANOTUBE-REINFORCED
COMPOSITE FACE SHEETS
INTEGRATED WITH PIEZOELECTRIC
LAYERS IN THERMAL ENVIROMENTS
MASTER’S THESIS
Ha Noi, 2020
VIETNAM NATIONAL UNIVERSITY, HANOI
VIETNAM JAPAN UNIVERSITY
NGO DINH DAT
NONLINEAR DYNAMIC RESPONSE AND
VIBRATION OF SANDWICH PLATES
WITH FG POROUS HOMOGENEOUS
CORE AND NANOTUBE-REINFORCED
COMPOSITE FACE SHEETS
INTEGRATED WITH PIEZOELECTRIC
LAYERS IN THERMAL ENVIROMENTS
MAJOR: INFRASTRUCTURE ENGINEERING
CODE: 8900201.04QTD
RESEARCH SUPERVISOR:
Prof. Dr. Sci. NGUYEN DINH DUC
Ha Noi, 2020
ACKNOWLEDGEMENT
First of all, I would like to express my deep gratitude to the instructor, Professor
Nguyen Dinh Duc, who devotedly guided, helped, created all favorable conditions
and regularly encouraged me to complete this thesis.
I would like to express my deepest thanks to Professor Kato, Professor Dao Nhu Mai,
Professor Nagayama, Dr. Phan Le Binh and Dr. Nguyen Tien Dung from the
Infrastructure Engineering Program for always caring and helping, supporting and
giving useful advice during the time I study and complete the thesis. In addition, I
feel very happy because of the enthusiastic support from the program assistant Bui
Hoang Tan who assisted in studying at Vietnam Japan University.
In particular, I would like to express my gratitude to Dr. Tran Quoc Quan, Master Vu
Minh Anh for giving me valuable suggestions and advice to help me complete my
thesis during meetings outside the lecture hall. I would like to thank everyone at VJU,
my classmate for creating unforgettable memories. Finally, I would like to thank my
family, my girlfriend Dang Thu Trang, who is always with me at difficult time who
encourage and help me.
I
TABLE OF CONTENTS
ACKNOWLEDGEMENT .......................................................................................... I
LIST OF TABLES .................................................................................................... III
LIST OF FIGURES...................................................................................................IV
LIST OF ABBREVIATIONS .................................................................................... V
ABSTRACT ..............................................................................................................VI
CHAPTER 1 INTRODUCTION ................................................................................ 1
1.1 Background ....................................................................................................... 1
1.2 Research objectives ........................................................................................... 2
1.3 Structure of the thesis ........................................................................................ 2
CHAPTER 2 LITERATURE REVIEW ..................................................................... 4
CHAPTER 3 MODELING & METHODOLOGY ..................................................... 7
3.1 Material properties of sandwich plate ............................................................... 7
3.2 Modeling of sandwich plate ............................................................................ 10
3.3 Methodology ................................................................................................... 11
3.4 Basic Equation ................................................................................................ 11
3.5 Nonlinear vibration analysis ........................................................................... 20
3.5.1 Nonlinear dynamic response ......................................................................... 21
3.5.2 Natural frequencies ........................................................................................ 23
CHAPTER 4 RESULTS AND DISCUSSION ......................................................... 24
4.1 Validation analysis .......................................................................................... 24
4.2 Natural frequencies ......................................................................................... 25
4.3 Nonlinear dynamic response ........................................................................... 27
4.3.1 The influence of geometric parameters ......................................................... 28
4.3.2 The influence of initial imperfection ............................................................. 31
4.3.3 The influence of temperature increment........................................................ 31
4.3.4 The influence of mechanical load.................................................................. 32
4.3.5 The influence of elastic foundation ............................................................... 32
4.3.6 The influence of type of porosity distribution ............................................... 34
CHAPTER 5 CONCLUSIONS................................................................................. 35
5.1 Conclusions ..................................................................................................... 35
APPENDIX ............................................................................................................... 36
LIST OF PUBLICATIONS ...................................................................................... 38
REFERENCES .......................................................................................................... 39
II
LIST OF TABLES
Table
4.1.
Comparison
= a 2 12 (1 − v 2 ) / Ec ( 2h )
( a / b = 1,
Table
of
2
the
with
non-dimensional
that
reported
in
natural
Refs.
frequencies
[1,3,12]
for
2h / a = 0.005) . ............................................................................................ 24
4.2.
Comparison
= 2h 2 (1 + v ) / Ec
with
of
the
that
non-dimensional
reported
in
natural
Refs.
frequencies
[1,32]
for
( a / b = 2, 2h / a = 1/ 20,1/12 ) . ....................................................................................... 25
Table 4.3. The influence of porosity coefficient e0 , ratio width-to-length a / b and
*
volume fraction VCNT
on natural frequencies of the sandwich plate with
b / h = 20, T = 100K , hc / hf = 5, hc / hp = 10, ( m, n ) = (1,1) , ( k1, k2 ) = ( 0,0 ) . .................. 26
Table 4.4. The influence of type of porosity distribution, elastic foundation and
temperature increment T on natural frequencies of the sandwich plate with
a / b = 1, b / h = 20, e0 = 0.2, hc / hf = 5, hc / hp = 10,VCNT = 0.12, ( m, n ) = (1,1) . ............... 27
III
LIST OF FIGURES
Figure 1.1. Application of Advanced material ............................................................ 1
Figure 3.1. Simulation model of the sandwich plate................................................. 10
Figure 4.1. Influence of ratio width-to-length a / b on the nonlinear dynamic
response of the sandwich plate .................................................................................. 28
Figure 4.2. Influence of ratio length-to-thickness b / h on the nonlinear dynamic
response of the sandwich plate .................................................................................. 29
*
Figure 4.3. Influence of volume fraction VCNT on the nonlinear dynamic response of
the sandwich plate ..................................................................................................... 29
Figure 4.4. Influence of porosity coefficient e0 on the nonlinear dynamic response of
the sandwich plate ..................................................................................................... 30
Figure 4.5. Influence of initial imperfection W0 on the nonlinear dynamic response
of the sandwich plate ................................................................................................. 30
Figure 4.6. Influence of temperature increment T on the nonlinear dynamic
response of the sandwich plate .................................................................................. 31
Figure 4.7. Influence of the magnitude Q0 of the external excitation on the nonlinear
dynamic response of the sandwich plate ................................................................... 32
Figure 4.8. Influence of the Winkler foundation k1 on the nonlinear dynamic
response of the sandwich plate .................................................................................. 33
Figure 4.9. Influence of the Pasternak foundation k 2 on the nonlinear dynamic
response of the sandwich plate .................................................................................. 33
Figure 4.10. Influence of the type of porosity distribution on the nonlinear dynamic
response of the sandwich plate .................................................................................. 34
IV
LIST OF ABBREVIATIONS
FG
SWCNTs
CNT
FG-CNTRC
hc
Functional graded
Single-walled carbon nanotubes
Carbon nanotube
Functional graded- carbon nanotube-reinforced composite
GygaPascal
Thickness of FG porous homogeneous core
hf
Thickness of FG-CNTRC face sheet
hp
Thickness of piezoelectric layer
GPa
V
ABSTRACT
Abstract: This thesis analytical solutions for the nonlinear dynamic response and
vibration of sandwich plates with FG porous homogeneous core and nanotubereinforced composite face sheets integrated with piezoelectric layers in thermal
environment. Assuming that the characteristics of the plate depend on temperature
and change consistent with the power functions of the plate thickness. Motion and
compatibility equations are used to base on the Reddy’s higher-order shear
deformation plate theory and consider the influence of initial geometric imperfection
and the thermal stress in the plate. Besides, the Galerkin method and Runge – Kutta
method are used to give clear expressions for nonlinear dynamic response and natural
frequencies of the sandwich plate. The influences of geometrical parameters, type of
porosity distribution, initial imperfection, elastic foundation and temperature
increment on the nonlinear dynamic response and vibration of thick sandwich plate
are demonstrated in detail. The results are reviewed with other authors in possible
cases to check the reliability of the approach used.
Keywords: Nonlinear dynamic response, sandwich plate, FG porous, thermal
environment, the Reddy’s higher order shear deformation theory.
VI
CHAPTER 1 INTRODUCTION
1.1 Background
In all industries, materials are the most important factor to create certain
products and details. Materials determine the design, construction and cost of the
product. Metallic and non-metallic materials are materials commonly used in many
industrial fields. Recently with the development of science and technology has
created a number of new materials such as composite materials, nanocomposite
materials, sandwich materials.
Figure 1.1. Application of Advanced material
In the world, sandwich materials are widely used in many fields of medical,
electronics, energy, aerospace engineering, industry automotive and construction of
civil, … (figure 1.1) Due to the outstanding characteristics of this material like light
weight, heat resistance, energy dissipation reduction and superior vibrational
damping, ... Especially, it is impossible not to mention the porous material. It is
lightweight cellular materials inspired by nature. Wood, bones and sea sponges are
some well-known examples of these types of structures. Foams and other highly
1
porous materials with a cellular structure are known to have many interesting
combinations of physical and mechanical properties, such as high stiffness combined
with very low specific gravity or high gas permeability combined with high thermal
conductivity. Among artificial cell materials, polymer foams are currently the most
important with wide applications in most areas of technology. Less known is that
even metals and alloys can be manufactured in the form of cellular or foam materials,
and these materials have such interesting properties that exciting new applications are
expected in the near future.
1.2 Research objectives
The research objective of this thesis is to research nonlinear dynamic response
and vibration of sandwich plate subjected to thermo-mechanical load combination.
Hence, to solve the problem, this thesis will set out the objectives should be achieved
as below:
Investigations on nonlinear dynamic response and vibration of sandwich plates
subjected to thermo-mechanical load combination. The natural frequency and the
deflection – time curves of sandwich plate structures are determined. In numerical
results, the effects of the geometrical parameters, types of distribution of porosity,
temperature increment, imperfections and elastic foundation on the nonlinear
dynamic response and vibration of the sandwich plate will be studied.
1.3 Structure of the thesis
This thesis provides a detailed explanation of the nonlinear dynamic response and
vibration of sandwich plate structure using analytical method. In order to better
understand the solution method as well as give an appropriate result, the thesis is
presented in the following structure:
➢ Chapter 1: Introduction
Highlights the role and importance of the material, especially the advanced
material for industrial fields. The background and research objective are introduced.
2
➢ Chapter 2: Literature review
Introduction of articles related to research issues. Since then, explains why this
study is necessary.
➢ Chapter 3: Methodology
The material and model properties of the structure are presented. The method used
as well as how to solve the problem are discussed.
➢ Chapter 4: Numerical results and discussion
Check the reliability of the method through comparison with other authors
considered. The results are expressed and discussed as the geometric transformations,
temperature and mechanical load through the deflection amplitude - time curves and
natural frequencies.
➢ Chapter 5: Conclusions
Summarize the results achieved and provide further direction for the study.
3
CHAPTER 2 LITERATURE REVIEW
Nowadays, sandwich structures are widely in many fields of life such as
medical, electronics, energy, aerospace engineering, industry automotive and
construction of civil with advantages such as high rigidity and lightweight Karlsson
et al. [10]. A sandwich structure consists of two thin face sheet and core layer of
low strength but thick brings high bending stiffness. The face sheet is often used
with sheet metal and fiber-reinforced polymers, while the core is usually made of
honeycomb or polymer foam. Recently, the advanced of carbon nanotubes such as
superior strength and stiffness has been of interest to many scientists. Carbon
nanotubes Liew et al. [15] are a potential candidate for sandwich structures with
the replacement of the face sheet with nanocomposite material which is reinforced
with carbon nanotubes that improves the bearing capacity. The sandwich structure
carbon nanotube-reinforced composite face sheets are investigated by a number of
authors. Wang et al. [37], Natarajan et al. [21] studied vibration and bending of
sandwich plates with nanotube-reinforced composite face sheets. By using
Extended High order Sandwich Panel Theory, the bending analysis of sandwich beam
Salami [9] also presented. Di Sciuva et al. [7] investigated additionally buckling
of sandwich plates adopted Refined zigzag theory with Rits method. The vibration
of thermally postbuckled also studied by Shen et al. [28]. The dynamic instability
analysis using shear flexible QUAD-8 serendipity element under periodic load is
present by Sankar et al. [25]. Based on mesh-free method, Moradi-Dastjerdi et al
[19]. studied static analysis of functionally grade nanocomposite sandwich plate
reinforced by defected CNT. Safaei et al. [24] also using mesh-free method to
investigate the influence of loading frequency on dynamic behavior of structure.
Mehar et al. [18] researched thermoelastic nonlinear frequency analysis of CNT
reinforced functional graded sandwich structure. Sobhy et al. [31] studied the
effect of the magnetic field on thermomechanical buckling and vibration of
viscoelastic sandwich nanobeams with CNT reinforced face sheets.
4
On the other hand, a porous core in sandwich structures is capable of
withstanding the transverse normal and shear loads as well as superior energy
dissipation, not only thermal and acoustic insulation but also vibration damping
due to the novel properties of porous materials. Li et al. [14] devoted to
considering the energy-absorption performance of porous materials in sandwich
composites. The paper demonstrate sandwich models can be abilities to prevent
perforation subjected to up 7 km/s projectile hypervelocity impact loading.
Talebitooti et al. [35] investigated the effect of nature of porous material on diffuse
field acoustic transmission of the sandwich aerospace composite doubly curved
shell. Qiao et al. [23] studied the sound insulation of a periodically rib-stiffened
double panel with porous core by using space harmonic series and Biot theory.
Moreover, a graded porosity leads to the continuous variation in material
properties so reducing stress concentration often encountered in conventional
sandwich structures. Chen et al. [5] employed the Ritz method to derive the
nonlinear free vibration of shear deformation of sandwich beam with FG porous
core. Li et al. [13] explored nonlinear vibration and dynamic buckling of sandwich
functionally graded porous plates with reinforced graphene platelet. Chen et al.
[6] analyzed bucking and bending of a novel FG porous plate. The results of
research showed FG porosity were suggested could remove the mismatch stresses
and effective buckling and bending significantly.
In addition, based on the properties of piezoelectric materials, sensors are
made such as ultrasonic transceiver sensors in machines (detecting defects in metal
and concrete), which has made their applications more popular over the past
decade (Liu et al. [16], Tao et al. [36]). Shuyu et al. [30] proposed a study of
vibration properties for piezoelectric sandwich ultrasonic transducers. Masmoudi
et al. [17] investigated mechanical behavior and health monitoring by acoustic
emission of sandwich composite integrated by piezoelectric implant. The acoustic
emission technique can be found damage in materials through transient ultrasonic
detection. Belouettar et al. [4] adopted the Harmonic balance method to study active
5
control of nonlinear vibration of sandwich beam with piezoelectric face sheets. Azrar
et al. [2] studied nonlinear vibration of imperfect sandwich piezoelectric beams. Thus,
the sandwich materials with FG porous core and nanocomposite-reinforced face
sheets and integrated with piezoelectric layers can be considered as new advanced
material. Moradi-Dastjerdi et al. [20] employed the Reddy’s third order and meshfree method to study stability analysis of multifunctional smart sandwich plates with
graphene nanocomposite and porous layers integrated with piezoelectric layers. By
using generalized differential quadrature method, Setoodeh et al. [26] examined
vibrational behavior of doubly curved smart sandwich shells with FG porous core and
FG carbon nanotube-reinforced composite face sheets.
Form the above literature reviews, we can see that although there are few
authors studied smart sandwich material with FG porous core and nanocompositereinforced face sheets integrated with piezoelectric layers, the nonlinear dynamic
response and vibration of sandwich plate with FG nanotube-reinforced composite
face sheets and FG porous homogeneous core is not studied so far. In this study, the
governing equations are used to base on the Reddy’s higher-order shear deformation
plate theory and consider the influence of initial geometric imperfection and the
thermal stress in the plate. Besides, The Galerkin method and Runge – Kutta method
are used for nonlinear dynamic response and vibration of the sandwich plate. The
influences of geometrical parameters, type of porosity distribution, initial
imperfection, elastic foundation and temperature increment on the nonlinear dynamic
of thick sandwich plate are demonstrated in detail. The results are reviewed with other
authors very good agreement to verify of the approach used.
6
CHAPTER 3 MODELING & METHODOLOGY
3.1 Material properties of sandwich plate
In this thesis, sandwich plate with three types of porous core are studied, in
c
which Young’s modulus of core layer E c , mass density of core layer and thermal
c
expansion coefficients of core layer are summed to be, respectively Chen et al. [5]
E c = E1 1 − e0 ( z )
c = 1 1 − em ( z )
(1)
c = 1 1 − em ( z )
where
cos ( z / hc )
( z ) = cos ( z / 2hc + / 4 )
Non-uniform symmetric distribution
Non-uniform asymmetric distribution
(2)
Uniform distribution
in which e0 ( 0 e0 1) shows the coefficient of porosity and
1 12
2
em = 1 − 1 − e 0 , = −
1 − e 0 − + 1
e0 e0
2
(3)
The homogeneous core layer made by titanium alloy, referred to as Ti-6Al4V. Except for Poisson’s ratio, the material properties are supposed to express as a
nonlinear function of temperature Chen et al. [5]
c = 0.29, c = 4429kg / m3
c = 7.5788 (1 + 6.638 10−4 T + 3.147 10−6 T 2 ) 10−6 / K ,
(4)
Ec = 122.56(1 − 4.586 10−4 T ) GPa,
Recently, the FG-CNTRC material is made of Poly (methyl methacrylate),
referred to as PMMA, reinforced by (10,10) SWCNTs. The effective Young’s and
shear modulus of the FG-CNTRC material are determined as Shen [27]
7
E11 = 1VCNT E11CNT + Vm Em ,
2
E22
3
G12
=
VCNT Vm
+
,
CNT
Em
E22
=
VCNT Vm
+
,
G12CNT Gm
(5)
where E11CNT , E11CNT , G12CNT are Young’s and shear modulus of the CNT; Em, Gm are
mechanical properties of the matrix. VCNT and Vm are the volume fractions of the
CNT and the matrix, respectively and i (i = 1,3) are the CNT efficiency parameters.
In this thesis, it is assumed that the volume fractions of the CNTs have linear
variations through the thickness layer as
*
2 ( ( z − hc / 2 ) / h f )VCNT
VCNT ( z ) =
*
2 ( ( −hc / 2 − z ) / h f )VCNT
Top face
Bottom face
(6)
wCNT
,
/ m ) − ( CNT / m ) wCNT
(7)
Vm ( z) = 1 − VCNT ( z),
where
*
VCNT
=
wCNT + ( CNT
in which wCNT is the mass fraction of CNTs, CNT and m are the densities of CNT
and matrix, respectively.
Except Poisson’s ratio, the material properties of the matrix are assumed to
express as a nonlinear function of temperature by Shen [27]
m = 0.34, m = 45 (1 + 0.0005T ) 10−6 / K ,
Em = (3.52 − 0.0034T ) GPa,
(8)
in which T = T0 + T , T0 is the room temperature and T is the increase in
temperature.
8
For the CNT, the material properties of (10,10) SWCNTs which are highly
dependent to temperature. The Poisson’s ratio of SWCNTs is chosen to be constant
12CNT = 0.175. The temperature-dependent material properties for (10, 10) SWCNTs
with 300 T 1000 by Shen et al. [29]
E11CNT = ( 6.18387 − 0.00286T + 4.22867 10 −6 T 2 − 2.2724 10 −9 T 3 ) (TPa)
CNT
E22
= ( 7.75348 − 0.00358T + 5.30057 10 −6 T 2 − 2.84868 10 −9 T 3 ) (TPa)
E11CNT = (1.80126 − 7.7845 10−4 T − 1.1279 10 −6 T 2 + 4.93484 10 −9 T 3 ) (TPa)
11CNT = ( −1.12148 + 0.02289T − 2.88155 10−5 T 2 + 1.13253 10 −8 T 3 ) (10 −6 / K )
22CNT = ( 5.43874 − 9.95498 10 −4 T + 3.13525 10 −7 T 2 − 3.36332 10 −12 T 3 ) (10 −6 / K )
The CNT efficiency parameters i (i = 1,3) used in equation (5) are estimated
by matching Young’s modulus E11 , E22 and the shear modulus G12 of FG-CNTRC
material obtained by the rule of mixtures extended to molecular simulation results
(Kwon et al. [11]).
For three different volume fractions of CNTs, these parameters are as:
*
*
with VCNT
= 0.12 (12%) then 1 = 0.137,2 = 1.022,3 = 0.715 ; with VCNT
= 0.17 (17%)
then 1 = 0.142,2 = 1.626,3 = 1.138 and 1 = 0.141,2 = 1.585,3 = 1.109 corresponds
*
to VCNT
= 0.28 (28%) .
The effective Poisson’s ratio depends weakly on temperature change and
position and is given as
*
CNT
12 = VCNT
v12
+ Vm m ,
(9)
where 12CNT and m are Poisson’s ratio of the CNT and the matrix, respectively.
The thermal expansion coefficients in the longitudinal and transverse directions
of the CNTRCs are expressed by
9
VCNT E11CNT 11CNT + Vm Em m
11 =
,
VCNT E11CNT + Vm Em
22 = (1 +
CNT
12
)VCNT
CNT
22
(10)
+ (1 + m )Vm m − 1211 ,
with 11CNT , 22CNT and m are the thermal expansion coefficients of the CNT and the
matrix, respectively.
3.2 Modeling of sandwich plate
Consider sandwich plate with total thickness h , length a, and width b is
placed in the spatial coordinate system ( x, y, z ) be illustrated in the figure 3.1a. In this
figure can see that sandwich plate is consisted by 5 layers: two layers piezoelectric,
two layers CNTs and one layer FG porous homogeneous core. Besides, in the figure
3.1, z direction is attached in the thickness direction of the sandwich plate while
( x, y ) plane will be attached to the middle face of sandwich plate.
a) 2D model
b) 3D model
Figure 3.1. Simulation model of the sandwich plate
10
3.3 Methodology
In order to obtain the proposed purpose, an analytical method is used. I assume
that the deflection of structures is relatively large, the material is elastic and the
structural damage does not occur. Depending on the form of structures, the problems
are posed in terms of stress and deflection functions. Basic equations will be
established taking into account the influences of geometric nonlinearity and initial
imperfection. Specifically, the Reddy’s higher-order shear deformation plate theory
is used for thick sandwich plate structures. Then these equations are solved by
combination of the Galerkin method and the Runge-Kutta method. I also use popular
software to calculate such as Matlab, Maple, etc. Some numerical results are given
and compared with one of other authors to verify the accuracy of the research.
3.4 Basic Equation
The higher-order shear deformation plate theory is used to set up basic
equations and determine nonlinear dynamic response and vibration of the sandwich
plate. The deformed components of the sandwich plate are at a point away the mid –
plane at the distance z are defined as Reddy [33]
x x0
k x1
k x3
0
2
0
1 3 3 xz xz 2 k xz
y = y + z k y + z k y , = 0 + z k 2 ,
yz
xy xy0
k xy1
k xy3 yz yz
where
u 1 w 2
+
x
2 x
0
x
2
0 v 1 w
=
+
y y 2 y ,
xy0
u + v + w w
y x x y
w
+ x
x
,
= −3c1 w
+ y
y
0
xz
0
yz
11
(11)
x
x
1
3
kx
kx
1
3
y
k
=
,
k
=
−
c
1
y
y
y
1
3
k
k xy
xy
y
x+
x
y y
y
x 2 w
+ 2
x x
w
(12)
+
x
x
k xz2
y 2 w
,
+
, 2 = −3c1 w
k
y y 2
yz
+ y
2
y
y
w
+
+2
y
xy
with u, v, w are displacement components parallel to the coordinates
( x, y, z ) ,
respectively. Also, x , y are respectively the rotations of the transverse normal about
the y and x axes at z = 0 and c1 = 4 / (3h2 ).
The Hooke’s law for the FG porous homogeneous core
xx
Q11C
C
yy
Q12
xy = 0
xz
0
yz
C 0
Q12C
0
0
Q22C
0
0
C
66
0
Q
0
C
44
0
0
Q
0
0
0
( ) − 11T
0 xx C
( yy ) − 22 T
0
C
xy )C
0
(
,
0
( xz )C
Q55C
( yz )C
(13a)
in which
Q11C = Q22C =
EC
1 − ( vC )
, Q12C =
2
vC E C
1 − ( vC )
, Q66C = Q44C = Q55C =
2
EC
2 (1 + vC )
(13b)
The Hooke’s law for FG-CNTRC face sheets
xx
Q11f
f
yy
Q12
xy = 0
xz
0
yz
f 0
f
12
f
22
Q
0
0
Q
0
0
0
Q66f
0
0
0
Q44f
0
0
0
( xx ) f − 11T
0
−
T
( )
22
0 yy f
0
( xy ) f ,
0
( xz ) f
f
Q55
( yz ) f
12
(14a)
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