modeling of lamb waves in composites using new third-order plate theories
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Modeling of Lamb waves in composites using new third-order plate theories
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8/10/2019 Modeling of Lamb waves in composites using new third-order plate theories
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Smart Materials and Structures
Smart Mater. Struct. 23 (2014) 045017 (14pp) doi:10.1088/0964-1726/23/4/045017
Modeling of Lamb waves in composites
using new third-order plate theoriesJinling Zhao, Hongli Ji and Jinhao Qiu
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of
Aeronautics and Astronautics, Nanjing 210016, Peoples Republic of China
E-mail:[email protected]
Received 2 July 2013, revised 22 January 2014
Accepted for publication 27 January 2014
Published 5 March 2014
AbstractThe effect of shear deformation and rotatory inertia should be taken into account in modeling
Lamb waves in composite laminates. However, the first and second-order shear deformation
plate theories which do not satisfy the stress free boundary condition are uneconomic
approximations, because one has to develop a complicated scheme to compute the shear
correction factors. The stress free boundary condition requires an in-plane displacement field
expanded at least as cubic functions of the thickness coordinate. Hence, in this paper, the
dispersive curves of Lamb waves in laminates are calculated according to two new third-order
shear deformation plate theories, considering the stress free boundary condition. The lower
anti-symmetric Lamb mode results of the new theories are closest to the exact solutions from
3D elasticity theory, when compared to several existing plate theories.
Keywords: Lamb waves, composites, third-order plate theory, 3D elasticity theory
(Some figures may appear in colour only in the online journal)
1. Introduction
Composite structures are increasingly used in many engi-
neering fields such as marine, aerospace, automotive, civil
and other applications. This is because of their high perfor-
mance and reliability due to high strength-to-weight and high
stiffness-to-weight ratios, excellent fatigue strength and, most
importantly, design flexibility for the desired applications. The
increasing use of laminated composite has led to extensive
research activities in the fields ofin situstructural health mon-
itoring [15]and other applications such as micro electrome-
chanical systems [6]. For active diagnosis that utilizes transient
Lamb waves in damage detection, the complex characteristics
of Lamb waves in composite laminates need to be thoroughly
studied [7,8].
Numerical research focusing on laminate damage such
as delamination, matrix cracking etc always requires the use
of layer-wise theories[9] or 3D finite element models [10].
While highly accurate in predicting local effects, these models
are computationally expensive, especially for modeling an
entire laminated structure. There are two common theoreti-cal approaches to investigate global Lamb wave propagation
characteristics in composite laminates: one is the 3D elas-
ticity theory calculating exact solutions, and the other is the
equivalent single layer theory (ESLT), depicting approximate
solutions. As for the exact solutions, Nayfeh [11] gave the
dispersion characteristics of Lamb waves in laminates. Zhao
et al[12] verified the dispersion results by the finite element
method and studied the directivity characteristics of Lamb
waves numerically in laminates. Although the exact solutionscan provide accurate results, the complexity of 3D elasticity
theory depends heavily on the stacking sequence of the plies
(along the plane of symmetry or not) and the number of plies.
The ESLTs show great efficiency over the 3D elasticity theory,
especially for laminates of complicated stacking sequence and
large number of plies. Also, the approximate plate theories
are more efficient for solving large-scale problems, such as
the reconstruction of the unknown stiffness coefficients in
composites based on Lamb wave phase velocities [13]. Such
optimization problems based on genetic algorithms, which
have large populations and large numbers of generations, can
be very time consuming using 3D elasticity theory.
To improve computational efficiency, many researchershave applied various ESLTs to estimate Lamb wave properties
0964-1726/14/045017+14$33.00 1 c2014 IOP Publishing Ltd Printed in the UK
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z y
x
z
x
N
N-1
N/2+1
N/2
2
1
(a) (b)
Figure 1. Geometry schematic of a composite laminate: (a) global coordinate system; (b) coordinate of each layer in a symmetric laminate.
in composites. In ESLTs, the material properties of the con-
stituent layers are combined to form a hypothetical single layer
whose properties are equivalent to the through-the-thickness
integrated sum of its constituents. The classical plate theory
(CPT) based on the Kirchhoff hypothesis has been generally
recognized to be accurate only if the wavelength is about ten
times the laminate thickness[14]. Composite laminates often
exhibit significant transverse shear deformation due to theirlow transverse shear stiffness. CPT which neglects the inter-
laminar shear deformation is not valid over the high frequency
range where wavelength is near the laminate thickness. The
Mindlin plate theory[15] which includes transverse shear and
rotary inertia effects provides accurate prediction of the lowest
anti-symmetric wave mode. Since constant transverse shear
deformation is assumed in the first-order shear deformation
theories (FSDTs) [16, 17] (including the Mindlin plate theory),
the higher modes of wave propagation cannot be described
accurately using the FSDTs. More accurate theories such
as higher-order shear deformation theories (HSDTs) assume
quadratic [18], cubic [19]and higher variations of displace-
ment through the entire thickness of the laminate.
Since both the FSDTs and some HSDTs neglect stress
free boundary conditions on the top and bottom surfaces of
the panel, a complicated scheme was developed to calculate
the shear correction factors [18]. The correction factors are
not unique for different laminations. They vary when the
laminate properties (the stacking sequence of the plies, the
number of plies and the ply properties, etc) change. The usual
procedure for determining the correction factors is to match
specific cut-off frequencies from the approximate theories to
the ones obtained from the exact theory. So the real CPU time
of the conventional ESLTs should include the time one needs
to calculate the exact solutions. To avoid computing thecomplex shear correction factors, Reddy [9] adjusted the
displacement field considering the vanishing of transverse
shear stresses on the top and bottom of a general laminate.
But when Lamb waves propagate in panels, the stress free
condition on the panel surfaces refers not only to the vanishing
of transverse shear stresses but also to the vanishing of normal
stress [20].
Inspired by Reddy, the authors deduced two new third-
order plate theories considering the transverse shear defor-
mation and stress free boundary conditions to model Lamb
waves efficiently in composite laminates. The results of the
new plate theories are compared with those of the FSDTs
and the HSDTs in [1518], in aspects of computing time andaccuracy for predicting Lamb wave dispersion properties.
2. Displacement fields of existing plate theories
Lamb wave modes can be generally classified into symmetric
and anti-symmetric modes according to the symmetrical char-
acteristics of the displacement distribution. Separately, u, v
and w are the displacement components in directions x,y
andz as shown in figure1(a). As for anti-symmetric modes,
displacement components u and v are anti-symmetric aboutthezaxis, thus the odd-order terms with respect tozinuand v
describe the anti-symmetric modes. In addition, the even-order
terms with respect toz in w also depict anti-symmetric modes
becausew is symmetric about the z axis for anti-symmetric
modes. Similarly, the even-order terms with respect to z inu
and v, together with the odd-order terms with respect tozin w,
describe symmetric modes. All of the under-mentioned ESLTs
follow these instructions.
2.1. FSDT
Taking theeffectsof rotatory inertia andsheardeformation into
account, Mindlin[15] assumed the displacement field which
is defined in a coordinate system shown in figure1(a) as
u=zx(x,y, t)v=zy(x,y, t)w= w0(x,y, t).
(1)
The displacement field in (1) predicts three anti-symmetric
modes and cannot be utilized to calculate any symmetric mode
according to the explanation mentioned above.
Hu and Liu [16] extended Mindlins displacement field
to construct a pseudo-spectral plate element of 5 degrees of
freedom (DOFs)
u= u0(x,y, t)+zx(x,y, t)v= v0(x,y, t)+zy(x,y, t)w= w0(x,y, t)
(2)
where u0 and v0 represent the displacement components of
the plate mid-plane. The displacement field in (2) depicts
three anti-symmetric modes and two symmetric modes. For
symmetric composite laminates, the characteristic matrix of
ESLTs can be decoupled into two sub-matrices to solve
anti-symmetric modes and symmetric modes separately. The
three anti-symmetric modes predicted by (2)are exactly the
same as the results obtained by (1). Meanwhile, there is
no term in w describing symmetric modes, which meansthat the calculated two symmetric modes share zero-value
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displacement componentw. Therefore the displacement field
in(2) may not be appropriate for predicting symmetric modes,
which will be proved later.
Zak [17]introduced the full form of FSDT displacement
field to study wave propagation in isotropic structures. There
exist six independent variations in (3) representing three anti-
symmetric modes and three symmetric modes, respectively.
u= u0(x,y, t)+zx(x,y, t)v= v0(x,y, t)+zy(x,y, t)w= w0(x,y, t)+zz(x,y, t).
(3)
The three variations depicting anti-symmetric modes are
all the same in (1)(3). Therefore, the phase velocities of
anti-symmetric Lamb wave modes obtained by the 3 3
characteristic matrices in (1)(3) are all equivalent.
2.2. HSDT
The displacement components are near linear along the
laminate thickness coordinate for lower-order Lamb wavemodes [21]. However, the displacement curves of higher-order
modes become complex and are not simply linear along the
plate thickness coordinate. More accurate theories assume
quadratic, cubic or higher variations of displacements through
the laminate thickness. Theories higher than third order are
not used because the accuracy gained is so little that the effort
required to solve the equations is not worthwhile[22].
Whitney [18] assumed a quadratic displacement field
through the laminate thickness for studying extensional motion
of laminate composites,
u= u0(x,y, t)+z
x(x,y, t)+z2
x(x,y, t)
v= v0(x,y, t)+zy(x,y, t)+z2y(x,y, t)
w= w0(x,y, t)+zz(x,y, t).(4)
The displacement field of (4) depicts three anti-symmetric
modes and five symmetric modes. Whitneys theory is ex-
panded from the FSDTs for estimating symmetric Lamb wave
modes more accurately, while the anti-symmetric results cal-
culated by (4)are exactly the same as those obtained by the
FSDTs.
Wang and Yuan[19] used a third-order displacement field
to describe as many as six anti-symmetric Lamb wave modes
and five symmetric Lamb wave modes,
u= u0(x,y, t)+zx (x,y, t)+z2x (x,y, t)
+z 3x(x,y, t)
v= v0(x,y, t)+zy(x,y, t)+z2y(x,y, t)
+z 3y(x,y, t)
w= w0(x,y, t)+zz(x,y, t)+z2z(x,y, t).
(5)
The two HSDTs of (4) and (5) can predict more Lamb modes
than the FSDTs mentioned in (1)(3). However, since the
stress free boundary condition is neglected in all of the five
theories above, shear correction factors have to be introduced
to adjust the strain components in relation to z . The accuracy
of these HSDTs and FSDTs will be strongly dependent upon
the accuracy of estimation for the shear correction factorski . The usual procedure for determining ki in a dynamic
problem is to match specific cut-off frequencies from the
approximate theories to the cut-off frequencies obtained from
exact theory [18]. For the general case of a laminate, this
procedure becomes very complex as the values ofki depend
on the stacking sequence of the plies and the number of
plies, as well as the ply properties. In particular, with an
increasing number of plies it becomes very tedious to calculatethe solutions from exact theories. Since the plate theories are
approximations, it is self-defeating to develop an elaborate
scheme for calculating shear correction factors.
In order to avoid computing complex shear correction
factors, Reddy[9] modified the third-order theory considering
the transverse shear stresses on the top and bottom surfaces as
zero yzx z
Nt
=
yzxz
1b
=
00
(6)
where the subscript Nt means the top surface of layer N,
and 1b means the bottom surface of layer 1, as shown infigure1(b). The displacement field of Reddys theory is
u= u0+zx z2
1
2
z
x
z 3
C1
w0
x+x
+
1
3
z
x
v= v0+zyz2
1
2
z
y
z 3
C1
w0
y+y
+
1
3
z
y
w= w0+zz+z
2z
(7)
where C1 = 4/3h2, and h is the laminate thickness. As
opposed to the 11 independent variables in (5), the number
of independent variables is only seven. The displacement
field in(7) can depict quadratic variation of transverse shear
strains (and hence stresses) and vanishing of transverse shear
stresses on the top and bottom of a general laminate. However,
displacement functions in (7) are not appropriate for Lamb
wave modeling, since the normal stresszz is not zero on the
panel surfaces.
3. A new third-order plate theory
In order to avoid computing complex shear correction fac-tors when studying Lamb wave propagation characteristics
in plates, researchers should take the following stress free
boundary condition into consideration:zzyzx z
Nt
=
zzyzx z
1b
=
000
. (8)
To avoid the tedious work on estimating stress correction
factors, the authors developed two new third-order plate
theories based on Reddys idea of considering the boundary
condition first. The stress free boundary condition expanded
in (8) is taken into account for modeling Lamb waves incomposite laminates efficiently.
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3.1. Displacement field for Lamb waves
Consider a laminate of constant thickness h composed of
anisotropic laminas perfectly bonded together. The origin of
a global Cartesian coordinate system is located at the middle
xyplane with thezaxis being normalto the mid-plane, so two
outer surfaces of the laminate are atz= h/2. A packet of the
transient Lamb waves propagates in the composite laminatein an arbitrary direction , which is defined relative to the
x axis. Each lamina with an arbitrary orientation in the global
coordinate system can be considered as a monoclinic material
having x y as a plane of symmetry, thereby the stressstrain
relations of a single lamina can be expressed in the following
matrix form:
123456
=
C11 C12 C13 0 0 C16C12 C22 C23 0 0 C26C13 C23 C33 0 0 C36
0 0 0 C44 C45 00 0 0 C45 C55 0
C16
C26
C36
0 0 C66
123456
(9)
where subscript 1 denotes x, 2 denotes y, 3 denotes
z, 4 denotes yz , 5 denotes x z and 6 denotes x y.
When the global coordinate system (x,y,z) does not coincide
with the principal material coordinate system (x ,y,z), the
6 6 stiffness matrixC in the (x,y,z) system can be obtained
from the lamina stiffness matrix C in the (x ,y ,z) system by
multiplying the transforming matrix.
We begin with the displacement field in (5). The vari-
ables x , x , y, y, z and z will be determined with the
boundary condition in (8)that the stresses,zz = 3,yz = 4and x z = 5, vanish on the top and bottom surfaces of the
laminate panels. For orthotropic laminates,4= 0 and5= 0are equivalent to the corresponding strains (4 and 5) being
zero on the surfaces. However for the condition 3 = 0 on
the surfaces, the strain components (1, 2, 3 and 6) are
coupled with the stiffness coefficients (C13,C23,C33andC36)
for the two laminas on the top and bottom surfaces according
to the constitutive equation in (9). For symmetric laid-up
laminates, the boundary condition can be expressed in the
form of displacement parameters as
(v,z+ w,y)z=h/2= 0
(u,z+ w,x)z=h/2= 0
[Q13u,x + Q 23v,y+ Q33w,z+Q 36(u,y+ v,x)]z=h/2= 0
(10)
whereQ13,Q 23,Q33andQ36respectively denote the stiffness
coefficientsC13,C23,C33 andC36 of the two laminas on the
top and the bottom surfaces. Assume the solution forms of
Lamb waves asu0, x , x , x , v0, y, y, y,w0, z, z
=
U0, x ,x ,Xx , V0,y, y,Xy, W0, z,z
expi(kxx+ kyy t) (11)
where is the angular frequency, and wavevector k=
[kx , ky ]T points to the direction of wave propagation () in
the x y plane. The Symbolic Math Toolbox of Matlab is
utilized to solve the six equations in (10). The six variablesx , x , y, y, z and z are expressed by the left five
independent variables, then the displacement field in (5) is
modified as
u= u0+zx +z2(4Q13u0,x x + 4Q36u0,x y
+4 Q36v0,x x+ 4Q23v0,x y)/a1
+z 3(96Q33x+ 12h2 Q13x,x x
+16h2 Q36x,x y+ 4h2 Q23x,yy
+8h2 Q36y,x x + 8h2 Q23y,x y
96 Q33w0,x )/a2v= v0+zy+z
2(4Q13u0,x y+ 4Q36u0,yy
+4 Q36v0,x y+ 4Q23v0,yy )/a1
+z 3(96Q33y+ 12h2 Q23y,yy
+16h2 Q36y,x y+ 4h2 Q13y,x x
+8h2 Q36x,yy + 8h2 Q13x,x y
96 Q33w0,y)/a2w= w0z(8Q13u0,x + 8Q36u0,y+ 8Q36v0,x
+8 Q23v0,y)/a1+z2(4Q13w0,x x
+8 Q36w0,x y+ 4Q23w0,yy
8 Q13x,x 8Q36x,y
8 Q23y,y 8Q36y,x)/a3
(12)
where
a1= Q 13h2k2x+ 2Q36h
2kx ky
+ Q 23h2k2y+ 8Q33
a2= 3Q13h4k2x + 6Q36h
4kx ky
+3 Q23h4k2y+ 72Q33h
2
a3= Q 13h2k2x+ 2Q36h2kx ky
+ Q 23h2k2y+ 24Q33.
(13)
The displacement field described in (12) is in connection
with the stiffness coefficients of the two laminas on the
top and the bottom surfaces and with the wavevector. The
five independent variables (x , y,w0, u0,v0) denote three
anti-symmetric modes and two symmetric modes respectively.
Computation of stress correction factors is avoided because
the displacement function is deduced with consideration of
the stress free boundary condition.
3.2. Equations of motion
With the linear straindisplacement relations, the equations of
motion of the higher-order theory can be derived using the
principle of virtual displacement of Hamiltons principle [23]
T0
(U T+ W)dt= 0 (14)
where U, T and Ware virtual strain energy, virtual kinetic
energy and virtual work done by applied forces, respectively.
For Lamb wave modeling, the condition that the stresses are
free on the surfaces is considered, thus the virtual workW is
zero. With substitution of stress and strain components into(14), the final integral equation for plate elasticity is given as
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Smart Mater. Struct. 23 (2014) 045017 J Zhaoet al T0
0
h/2h/2
(11+ 22+ 33+ 44
+55+ 66)dz dAdt
T0
0
h/2h/2
(uu+ vv
+ w w) dz dAdt= 0. (15)
Plate inertias and stress resultants per unit length are definedin the following:
I1,...,7=N
n=1
zn+1zn
n (1,z,z2,z3,z4,z5,z6)dz
(Ni ,Mi ,Pi ,Si ) =N
n=1
zn+1zn
i (1,z,z2,z3)dz
(i= 1, 2, 3, 4, 5, 6)
(16)
where the index n refers to the layer number in a laminate.
Using fundamental lemma of calculus of variation, the equa-
tions of motion can be derived, as shown in appendixA.
In order to express the equations of motion in appendixA
with displacement field parameters, the constitutive equation
of a laminate with arbitrary lay-up can be utilized,N
M
P
S
=
[A] [B] [D] [F][D] [E] [F]
[F] [H]Sym [J]
0
1
2
3
. (17)
In (17), the stress and moment resultant vectors are defined as
N =N1 N2 N3 N4 N5 N6
TM =
M1 M2 M3 M4 M5 M6T
P =
P1 P2 P3 P4 P5 P6T
S =
S1 S2 S3 S4 S5 S6T
.
(18)
The elements of stiffnessmatrices[A], [B], [D], [E], [F], [H]
and[J]are
(Ai j ,Bi j ,Di j ,Ei j ,Fi j ,Hi j ,Ji j )
=
Nn=1
zn+1zn
Ci j (1,z,z2,z3,z4,z5,z6)dz
(i, j= 1, 2, 3, 4, 5, 6). (19)
The strain vectors represent
i =
i1
i2
i3
i4
i5 i6
T(i= 0, 1, 2, 3) (20)
whereij (i = 0, 1, 2, 3)are defined as the strain coefficients
j = 0j +z
1j+z
22j+z33j
(j= 1, 2, 3, 4, 5, 6) (21)
and the strain components j in (21) can be obtained according
to the displacement field of(12).
Substituting (11) and (17) into the equations of motion
in appendixAyields a generalized eigen-value problem. Five
real positive eigen-values related to three anti-symmetric andtwo symmetric Lamb modes can be obtained from the 5 5
Figure 2. Comparison of dispersion curves for anti-symmetricmodes among 3D elasticity theory, the HSDT by Whitney and thenew HSDT.
Table 1. Stiffness coefficients of composite lamina (unit: GPa).
C11 C12 C13 C22 C23 C33 C44 C55 C66
155.43 3.72 3.72 16.34 4.96 16.34 3.37 7.48 7.48
order characteristic matrix. The determinant of the matrix is
a function of the phase velocity (cp), the frequency (f) and
the propagation direction (). That leads to a characteristic
equation, solution of which gives the cpfcurves (dispersion
curves) in a rectangular coordinate system for a given angle
or the cp
curves in a polar coordinate system for a given
frequency f.
4. Numerical studies and results
The equations described in the previous sections were im-
plemented using Matlab, because it can seamlessly combine
symbolic and numeric computation. The material in this study
is epoxy (35%)/graphite (65%) composite as shown in table1,
with a density of 1700 kg m3. A laminate[+456/ 456]Sis
used and the thickness of each lamina is 0.125 mm. Numerical
results consist of phase velocity dispersion curves in rectangu-
lar coordinate systems when phase velocity travels along the
direction of 30. The results of the new HSDT are compared
with those obtained by several existing ESLTs and the exact
solutions based on 3D elasticity theory.
4.1. Accuracy comparison
The dispersion results are classified by the number of Lamb
modes in figures 2, 3, 5, 6 and 7. Figure 2 displays the
dispersion results for three anti-symmetric Lamb modes of
different theories. The exact solutions of 3D elasticity theory
are drawn in red solid lines in all of the following figures. The
HSDT-Whitney method has a displacement field expressed
in (4), describing three anti-symmetric modes exactly the same
as the FSDTs do. So the anti-symmetric mode results of the
FSDTs are not redundant in figure 2. The shear correctioncoefficientski of HSDT-Whitney are chosen as in [16], but
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Figure 3. Comparison of dispersion curves for anti-symmetricmodes among 3D elasticity theory, the HSDT by Reddy and the new
HSDT.
Figure 4. Difference of the 3D elasticity results for A 0 mode in lowfrequency range among the HSDT by Whitney, the HSDT by Reddyand the new HSDT.
the cut-off frequency of the SH1 mode of HSDT-Whitney
does not match the exact solution in figure2,which indicates
that the laminate parameters do have influence in determining
ki . The dispersion curves for anti-symmetric modes of the new
HSDT proposed in this paper are closer to the exact solutions
compared to HSDT-Whitney based on shear correction.
Besides, unlike the HSDT-Whitney method, there exists
mode flip between A1 and SH1 modes in the new HSDT
and 3D elasticity theory.
Stresses in connection toz are all free on the surfaces of
the laminate in the new HSDT, while HSDT-Reddy, based
on the displacement field in (7) proposed by Reddy, half
satisfies the stress free boundary condition. The comparison
of these two methods is shown is figure3where there are four
anti-symmetric modes of HSDT-Reddy but only three of the
new HSDT. The phase velocities are almost identical between
the two theories except for A0mode in the low frequency range
and A1 mode in the high frequency range. As the propagationcharacteristics of A0 mode in the low frequency range are more
Figure 5. Comparison of dispersion curves for symmetric modesamong 3D elasticity theory, the FSDT by Hu and the new HSDT.
Figure 6. Comparison of dispersion curves for symmetric modesamong 3D elasticity theory, the HSDT by Reddy and the FSDT byZak.
essential to experimental researchers, the differences between
the ESLT results and the exact solutions in the low frequency
range are drawn in figure4. It is obvious that the new HSDT
gives the best estimation of A0 mode in the low frequency
range compared with other existing plate theories.As mentioned above, the displacement field in (2) may
not be appropriate for predicting symmetric modes, because
there is no term describing symmetric modes in displacement
component w. This statement is proved in figure 5 where
the FSDT-Hu dashed lines are dispersion curves based on
the displacement field in (2). The two symmetric modes
of FSDT-Hu share the same initial phase velocity in low
frequency as those exact solutions. However, FSDT-Hu fails
to describe the steep descent of S0 mode. The results do not
show any dispersive characteristics as the phase velocities of
the two symmetric modes are constant in the frequency range.
Displacement fields utilized by Reddy in (7) and Zak in(3) can both predict three symmetric modes, as displayed in
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Figure 7. Comparison of dispersion curves for symmetric modesbetween 3D elasticity theory and the HSDT by Whitney.
figure6. Whitney modeled extensional modes with displace-
ment field in (4), and the five dispersion curves of HSDT-
Whitney in figure 7 show more accurate steep descents of
symmetric modes. This implies that the increasing number of
independent variables in each displacement component is the
key to improve the accuracy of modeling Lamb waves using
the ESLTs.
The new HSDT proposed in (12) can depict two symmet-
ric modes. Unlike FSDT-Hu in (2), there is one term in w
describing symmetric modes in the new HSDT. However, this
term has no independent variable. As shown in figure 5, the
new HSDT is not appropriate for modeling symmetric Lambmodes.
4.2. A new HSDT of six DOFs
As mentioned above, the new HSDT of five DOFs is not
appropriate for modeling symmetric Lamb modes due to
the absence of independent variables describing symmetric
modes in displacement component w. Thus, in order to depict
symmetric modes more precisely, an independent variable zis introduced in w of a new six-DOF HSDT. Considering
the stress free boundary condition, the authors deduced the
displacement functions of the new six-DOF HSDT as
u= u0+zx +z2(4Q13u0.x x + 4Q36v0,x x
8 Q33z,x+ 4Q36u0.xy + 4Q23v0,xy )/a3
+z3(96Q33x+ 12h2 Q13x,x x
+16h2 Q36x,x y+ 4h2 Q23x,yy
+8h2 Q36y,x x + 8h2 Q23y,x y
96 Q33w0,x )/a2
v= v0+zy+z2(4Q13u0,x y+ 4Q36u0,yy
8 Q33z,y+ 4Q36v0,x y+ 4Q23v0,yy )/a3
+z3(96Q33y+ 12h2 Q23y,yy
Figure 8. Comparison of symmetric Lamb mode dispersion curvesfrom different ESLTs and the 3D elasticity theory.
+16h2 Q36y,x y+ 4h2 Q13y,x x
+8h2 Q36x,yy + 8h2 Q13x,x y
96 Q33w0,y)/a2
w= w0+zz+z2(4Q13w0,x x+ 8Q36w0,x y
+4 Q23w0,yy 8Q13x,x 8Q36x,y
8 Q23y,y 8Q36y,x )/a3
+z 3(32Q33z+ 4Q13h2z,x x
+4 Q23h2z,yy 32Q13u0,x
32 Q36u0,y 32Q23v0,y 32Q36v0,x
+8 Q36h2z,x y)/(a3h
2) (22)
where the independent variables x , y, w0 describe anti-
symmetric Lamb modes, the other three independent variables
u0, v0, z describe symmetric Lamb modes, and a1, a2, a3are
defined as in (13). In appendix B, the authors obtained the
equations of motion of the new six-DOF HSDT using the
principle of virtual displacement of Hamiltons principle.
Phase velocities of anti-symmetric Lamb modes obtained
by the new five-DOF HSDT are the same as those calculated
by the new six-DOF HSDT, because the terms depicting
anti-symmetric Lamb modes in (12) and (22) are all the
same. Emphasis is laid on the estimation accuracy for lower
symmetric modes, and the dispersion curves of the first three
symmetric modes from different theories are drawn in figure 8.
Whitneys HSDT predicts symmetric modes most accu-
rately because of theexistence of fiveindependentvariables for
estimating symmetric modes. However, one has to develop a
complex scheme to estimate the shear correction coefficients.
The authors deduced the new HSDT of five DOFs in (12)
considering the stress free boundary condition. But this HSDT
performs poorly in calculating symmetric modes. Unlike the
new HSDT of five DOFs, the new six-DOF HSDT can predict
better the steep descent of S0 mode, which proves that theincreasing number of independent variables is essential to
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Table 2. The relative computing time of the ESLTs and the 3D elasticity theory.
Theory Mindlin Hu Zak Whitney ReddyNew HSDT(5 DOFs)
New HSDT(6 DOFs)
3D elasticitytheory
Matrixorder
3 5 6 8 7 5 6
RelativeCPU time 1.0+1659
.5
1.7+1659
.5
2.0+1659
.5
2.7+1659
.5
7.0 139.7 214.4 1659
.5
improve accuracy. Reddys HSDT is half-satisfying the stress
free boundary condition. In comparison to Reddys theory,
the new six-DOF HSDT takes the full boundary condition
into consideration and gives better estimation. The FSDT by
Zak has a linear displacement field with respect to thickness
coordinate z. Each of the three displacement components,
u,v ,w, i n (22) has two more terms than those of Zaks
displacement field. Although the two terms do not include
any independent variables, the dispersion curves of the new
six-DOF HSDT are still much closer to the exact solutions
compared with those of Zaks theory.
4.3. Computation time comparison
3D elasticity theory has advantages over all ESLTs: (a) 3D
elasticity theory predicts exact solutions because there is no
assumption during the modeling process; (b) 3D elasticity
theory canpredict an infinite number of Lamb modes as thefre-
quency increases, while the number of Lamb modes modeled
by the ESLTs is equivalent to the number of the independent
variables in displacement fields. However, 3D elasticity theory
is very time consumingespecially for laminatesof complicatedstacking sequences or large numbers of plies. This encourages
researchers studying other approximation approaches to solve
large projects efficiently. In order to compare the computation
efficiency between ESLTs and 3D elasticity theory, the relative
computing time (with respect to FSDT by Mindlin) of the
existing plate theories and the 3D elasticity theory are listed
in table2.The CPU times of the conventional ESLTs include
the time one needs to calculate the correction factors. All the
results are solved by the ergodic searching method with the
same searching range and the same step interval.
Since the solution methods are the same for all these
theories, the computing time of plate theories depends on the
order of the characteristic matrix and on the complexity of
each matrix element. The order of the characteristic matrix is
determined by the independent variables in the displacement
field. The statistics in table 2 indicate that, for the C0 continuity
displacement functions by Mindlin, Hu, Zak and Whitney in
equations (1)(4), the computing time is linear to the matrix
order. The 11 terms of the displacement field in (7) can be
separated to 7 terms of C0 continuity and 4 terms of C1
continuity. The terms of C1 continuity make the matrix element
more complex, which leads to the computing time by Reddys
theory of 7 times that by Mindlins. The two new HSDTs
satisfying the vanishing of stresses on the laminate surfaces
have displacement fields containing several C2 continuitycomponents. The complicated C2 continuity matrix elements
make the two new HSDTs the most time consuming among all
the existing ESLTs. Even so, the new HSDTs still run much
faster than the 3D elasticity theory.
5. Conclusions
When modeling Lamb waves in plate-like structures, 3D
elasticity theory which calculates the exact solutions is time
consuming. As approximate theories to the 3D elasticitytheory, conventional ESLTs neglect the stress free boundary
condition. Thus tedious work on estimating stress correction
factors by comparing the approximate results to the exact ones
cannot be avoided.
In order to analyze the propagation properties of Lamb
waves efficiently and accurately, the authors introduced a new
HSDT of five DOFs considering the stress free boundary
condition and thus avoided calculating stress correction fac-
tors. Several existing ESLTs are discussed and the dispersion
results of Lamb waves based on these theories are compared
with the exact solutions carried out by 3D elasticity theory.
Although the new five-DOF HSDT is more time consumingthan the conventional ESLTs because of the C2 continuous
displacement components, it still runs almost ten times faster
than the 3D elasticity theory. In addition, the new HSDT could
be the prime choice among all the existing ESLTs to predict
lower anti-symmetric Lamb modes accurately, according to
the dispersion curves compared to the exact solutions.
However, the new HSDT is not appropriate for modeling
symmetric Lamb modes due to the absence of efficient terms
describing symmetric modes in displacement component w.
Another new HSDT of six DOFs with an independent variable
describing symmetric modes in w was developed, and the
dispersion curves of symmetric modes by the six-DOF HSDTare more accurate than those of the five-DOF HSDT.
Acknowledgments
This work is funded by the National Natural Science Foun-
dation of China (no. 51375228) and the Doctoral Program
Foundation of Institutions of Higher Education of China
(no. 20113218110026). This work is also supported by
the Independent Team Project of the State Key Laboratory
(no. 0513G01), the Graduate Education Innovation Project
of Jiangsu Province (no. CXLX13 135) and the Priority
Academic Program Development of Jiangsu Higher EducationInstitutions (PAPD).
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Appendix A. Equations of motion for the new HSDT
u0:
(8Q36 P6,x yy 8Q13 P6,x xy )/a1
+(4Q13 P1,x x x 4Q36 P1,x x y)/a1
+(4Q36 P2,yy y 4Q13 P2,x yy )/a1
+(8Q13N3,x+ 8Q36N3,y)/a1N1,x N6,y
=[I1u02 +I2x
2 +I32(4Q13u0,x x
+4 Q36u0,x y+ 4Q36v0,x x+ 4Q23v0,x y)/a1
+ I42(96Q33x + 12h
2 Q13x,x x
+16h2 Q36x,x y+ 4h2 Q23x,yy + 8h
2 Q36y,x x
+8h2 Q23y,x y 96Q33w0,x )/a2] (4Q13k2
x
+4 Q36kx ky) [I3u02 +I4x
2 +I5
2(4Q13u0,x x+ 4Q36u0,x y+ 4Q36v0,x x
+4 Q23v0,x y)/a1+I62(96Q33x
+12h2 Q13x,x x+ 16h2 Q36x,x y+ 4h
2 Q23x,yy
+8h2 Q36y,x x+ 8h2 Q23y,x y
96 Q33w0,x)/a2]/a1 (4Q36k2
y+ 4Q13kx ky)
[I3v02 +I4y
2 +I52(4Q13u0,x y
+4 Q36u0,yy + 4Q36v0,x y+ 4Q23v0,yy )/a1
+ I62(96Q33y+ 12h
2 Q23y,yy
+16h2 Q36y,x y+ 4h2 Q13y,x x+ 8h
2 Q36x,yy
+8h2 Q13x,x y 96Q33w0,y)/a2]/a1
(i8Q13kx + i8Q36ky) [I2w02 +I3
2
(8Q13u0,x+ 8Q36u0,y+ 8Q36v0,x
+8 Q23v0,y)/a1I42(4Q13w0,x x+ 8Q36w0,xy
+4 Q23w0,yy 8Q13x,x 8Q36x,y 8Q23y,y
8 Q36y,x )/a3]/a1
x :
(20Q13h2 S6,x x y 24Q36h
2 S6,x yy
4 Q23h2 S6,yy y+ 96Q33 S6,y)/a2
+(12Q13h2 S1,x x x 16Q36h
2 S1,x x y
4 Q23h2 S1,x yy + 96Q33 S1,x)/a2
+(8Q36h2 S2,yy y 8Q13h
2 S2,x yy )/a2
+3(4Q13h2 P5,x x+ 8Q36h
2 P5,x y+ 4Q23h2 P5,yy
+96 Q33)/a2+ 2(8Q13M3,x + 8Q36M3,y)/a3
M1,x M6,y+ N5= [I2u02 +I3x
2
+ I42(4Q13u0,x x+ 4Q36u0,x y
+4 Q36v0,x x+ 4Q23v0,x y)/a1+I52(96Q33x
+12h2 Q13x,x x+ 16h2 Q36x,x y+ 4h
2 Q23x,yy
+8h2 Q36y,x x+ 8h2 Q23y,x y 96Q33w0,x)/a2]
(96Q33+ 12Q13h2k2x+ 4Q23h
2k2y
+16 Q36h2
kx ky) [I4u02
+I5x2
+ I62(4Q13u0,x x+ 4Q36u0,x y
+4 Q36v0,x x+ 4Q23v0,x y)/a1+I72(96Q33x
+12h2 Q13x,x x+ 16h2 Q36x,x y+ 4h
2 Q23x,yy
+8h2 Q36y,x x+ 8h2 Q23y,x y
96 Q33w0,x)/a2]/a2 (8Q36h2k2y
+8 Q13h2kx ky) [I4v0
2 +I5y2
+ I62(4Q13u0,x y+ 4Q36u0,yy + 4Q36v0,x y
+4 Q23v0,yy )/a1+I72(96Q33y
+12h2 Q23y,yy + 16h2 Q36y,xy + 4h
2 Q13y,x x
+8h2 Q36x,yy + 8h2 Q13x,x y
96 Q33w0,y)/a2]/a2 (i8Q13kx 8Q36ky)
[I3w02 +I4
2(8Q13u0,x+ 8Q36u0,y
+8 Q36v0,x + 8Q23v0,y)/a1I52(4Q13w0,x x
+8 Q36w0,x y+ 4Q23w0,yy 8Q13x,x
8 Q36x,y 8Q23y,y 8Q36y,x )/a3]/a3
v0:
(8Q23 P6,x yy 8Q36 P6,x x y)/a1
+(4Q36 P1,x x x 4Q23 P1,x x y)/a1
+(4Q23 P2,yy y 4Q36 P2,x yy )/a1
+(8Q36N3,x+ 8Q23N3,y)/a1N2,y N6,x
= (4Q36k2
x + 4Q23kx ky) [I3u02 +I4x
2
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+ I52(4Q13u0,x x+ 4Q36u0,x y+ 4Q36v0,x x
+4 Q23v0,x y)/a1+I62(96Q33x
+12h2 Q13x,x x+ 16h2 Q36x,x y
+4h2 Q23x,yy + 8h2 Q36y,x x + 8h
2 Q23y,x y
96 Q33w0,x)/a2]/a1+ [I1v02 +I2y
2
+ I32(4Q13u0,x y+ 4Q36u0,yy
+4 Q36v0,x y+ 4Q23v0,yy )/a1+I42(96Q33y
+12h2 Q23y,yy + 16h2 Q36y,x y+ 4h
2 Q13y,x x
+8h2 Q36x,yy + 8h2 Q13x,x y
96 Q33w0,y)/a2] (4Q23k2y+ 4Q36kx ky)
[I3v02
+I4y2
+I52(4Q13u0,x y
+4 Q36u0,yy + 4Q36v0,x y+ 4Q23v0,yy )/a1
+ I62(96Q33y+ 12h
2 Q23y,yy
+16h2 Q36y,x y+ 4h2 Q13y,x x+ 8h
2 Q36x,yy
+8h2 Q13x,x y 96Q33w0,y)/a2]/a1
(i8Q36kx + i8Q23ky) [I2w02
+ I32(8Q13u0,x+ 8Q36u0,y+ 8Q36v0,x
+8 Q23v0,y)/a1I42(4Q13w0,x x+ 8Q36w0,xy
+4 Q23w0,yy 8Q13x,x 8Q36x,y
8 Q23y,y 8Q36y,x )/a3]/a1
y :
(20Q23h2 S6,x yy 24Q36h
2 S6,x x y
4 Q13h2 S6,x x x+ 96Q33 S6,x)/a2
+(4Q13h2 S2,x x y 16Q36h
2 S2,x yy
12 Q23h2 S2,yy y+ 96Q33 S2,y)/a2
+(8Q36h2 S1,x x x 8Q23h
2 S1,x x y)/a2
+3(4Q23h2 P4,yy + 8Q36h
2 P4,x y+ 4Q13h2 P4,x x
+96 Q33)/a2+ 2(8Q23M3,y+ 8Q36M3,x)/a3
M2,yM6,x + N4= (8Q36h2k2x
+8 Q23h2kx ky) [I4u0
2 +I5x2
+ I62(4Q13u0,x x+ 4Q36u0,x y
+4 Q36v0,x x+ 4Q23v0,x y)/a1+I72
(96Q33x + 12h2 Q13x,x x+ 16h
2 Q36x,x y
+4h2 Q23x,yy + 8h2 Q36y,x x + 8h
2 Q23y,x y
96 Q33w0,x)/a2]/a2+ [I2v02 +I3y
2
+ I42(4Q13u0,x y+ 4Q36u0,yy + 4Q36v0,x y
+4 Q23v0,yy )/a1+I52(96Q33y
+12h2 Q23y,yy + 16h2 Q36y,xy + 4h
2 Q13y,x x
+8h2 Q36x,yy + 8h2 Q13x,x y 96Q33w0,y)/a2]
(96Q33+ 4Q13h2k2x + 12Q23h
2k2y
+16 Q36h2kx ky) [I4v0
2 +I5y2
+ I62(4Q13u0,x y+ 4Q36u0,yy + 4Q36v0,x y
+4 Q23v0,yy )/a1+I72(96Q33y
+12h2 Q23y,yy + 16h2 Q36y,xy + 4h
2 Q13y,x x
+8h2 Q36x,yy + 8h2 Q13x,x y
96 Q33w0,y)/a2]/a2 (i8Q23ky 8Q36kx)
[I3w02 +I4
2(8Q13u0,x
+8 Q36u0,y+ 8Q36v0,x+ 8Q23v0,y)/a1
I52(4Q13w0,x x+ 8Q36w0,x y+ 4Q23w0,yy
8 Q13x,x 8Q36x,y
8 Q23y,y 8Q36y,x )/a3]/a3
w0:
(4Q13 P5,x x x 8Q36 P5,x x y 4Q23 P5,x yy )/a3
+288 Q33 P5,x/a2+ (4Q13 P4,x x y 8Q36 P4,x yy
4 Q23 P4,yy y)/a3+ 288Q33 P4,y/a2
96 Q33 S1,x x/a2 96Q33 S2,yy/a2
192 Q33 S6,x y/a2N5,x N4,y+ (8Q13M3,x x
+16 Q36M3,x y+ 8Q23M3,yy )/a3= i96Q33kx
[I4u02 +I5x
2 +I62(4Q13u0,x x
+4 Q36u0,x y+ 4Q36v0,x x+ 4Q23v0,x y)/a1
+ I72(96Q33x+ 12h
2 Q13x,x x
+16h2 Q36x,xy + 4h2 Q23x,yy + 8h
2 Q36y,x x
+8h2 Q23y,x y 96Q33w0,x )/a2]/a2
+i96 Q33ky [I4v02 +I5y
2
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+ I62(4Q13u0,x y+ 4Q36u0,yy + 4Q36v0,x y
+4 Q23v0,yy )/a1+I72(96Q33y
+12h2 Q23y,yy + 16h2 Q36y,x y
+4h2 Q13y,x x+ 8h2 Q36x,yy + 8h
2 Q13x,x y
96 Q33w0,y)/a2]/a2 [I1w02
+ I22(8Q13u0,x+ 8Q36u0,y+ 8Q36v0,x
+8 Q23v0,y)/a1I32(4Q130,x x+ 8Q360,x y
+4 Q230,yy 8Q13x,x 8Q36x,y
8 Q23y,y 8Q36y,x )/a3]
+(4Q13k2
x + 4Q23k2y+ 8Q36kx ky)
[I3w02
+I42
(8Q13u0,x + 8Q36u0,y
+8 Q36v0,x+ 8Q23v0,y)/a1I52(4Q13w0,x x
+8 Q36w0,x y+ 4Q23w0,yy 8Q13x,x
8 Q36x,y 8Q23y,y 8Q36y,x )/a3]/a3
wherea1, a2 and a3 are the same as those defined in (13).
Appendix B. Equations of motion for the newsix-DOF HSDT
u0:
(32Q13 S5,x x 32Q36 S5,y)/(a3h2)
+(32Q13 S4,x y 32Q36 S4,y)/(a3h2)
+(8Q36 P6,xy y 8Q13 P6,x y)/a3+ (96Q13 P3,x
+96 Q36 P3,y)//(a3h2)+ (4Q13 P1,x x x
4 Q36 P1,x x y)/a3+ (4Q36 P2,yy y
4 Q13 P2,xy y)/a3+ (8Q36M4,yy + 8Q13M4,x y)/a3
+(8Q13M5,x x + 8Q36M5,x y)/a3N1,x N6,y
=I12u0+I2
2x+ I32(4Q13u0,x x+ 4Q36v0,x x
8 Q33z,x + 4Q36u0,x y+ 4Q23v0,xy )/a3
+ I42(96Q33x + 12Q13h
2x,x x+ 4Q23h2x,yy
+8 Q36h2y,x x 96Q33w0,x+ 16Q36h
2x,x y
+8 Q23h2y,x y)/a2 (4Q13k
2x + 4Q36kx ky)
[I32u0+I4
2x + I52(4Q13u0,x x+ 4Q36v0,x x
8 Q33z,x + 4Q36u0,x y+ 4Q23v0,xy )/a3
+ I62(96Q33x+ 12Q13h
2x,x x + 4Q23h2x,yy
+8 Q36h2y,x x 96Q33w0,x + 16Q36h
2x,x y
+8 Q23h2y,x y)/a2]/a3 (4Q36k
2y+ 4Q13kx ky)
[I32v0+I4
2y+I52(4Q36u0,yy + 4Q23v0,yy
8 Q33z,y+ 4Q13u0,x y+ 4Q36v0,x y)/a3
+ I62(96Q33y+ 8Q36h
2x,yy + 4Q13h2y,x x
+12 Q23h2y,yy 96Q33w0,y+ 16Q36h
2y,x y
+8 Q13h2x,x y)/a2]/a3+ (i32Q13kx+ i32Q36ky)
[I42w0+I5
2z+I62(4Q13w0,x x + 4Q23w0,yy
8 Q13x,x 8Q36x,y 8Q23y,y 8Q36y,x
+8 Q36w0,x y)/a3+I7
2
(32Q33z+ 4Q13h
2
z,x x
+4 Q23h2z,yy 32Q13u0,x 32Q36u0,y
32 Q23v0,y 32Q36v0,x
+8 Q36h2z,x y)/(a3h
2)]/(a3h2)
x :
(20Q13h2 S6,x x y 24Q36h
2 S6,x yy
4 Q23h2 S6,yy y+ 96Q33 S6,y)/a2+ (12Q13h
2
S1,x x x 16Q36h2 S1,x x y 4Q23h2 S1,x yy
+96 Q33 S1,x)/a2+ (8Q36h2 S2,yy y
8 Q13h2 S2,x yy )/a2+ (12Q13h
2 P5,x x+ 16Q36h2
P5,x y+ 4Q23h2 P5,yy 96Q33h
2 P5)/(a2/3)
+(8Q13 P5,x x 8Q36h2 P5,x y)/a3+ (24Q36h
2
P4,yy + 24Q13h2 P4,x y)/a2+ (8Q13 P4,x y
8 Q36h2 P4,yy )/a3+ (16Q13M3,x+ 16Q36M3,y)/
a3M1,x M6,y+N5= I22u0+I3
2x
+ I42(4Q13u0,x x+ 4Q36v0,x x 8Q33z,x
+4 Q36u0,x y+ 4Q23v0,x y)/a3+I52(96Q33x
+12 Q13h2x,x x+ 4Q23h
2x,yy + 8Q36h2y,x x
96 Q33w0,x+ 16Q36h2x,x y+ 8Q23h
2y,x y)/a2
(96Q33+ 12Q13h2k2x+ 4Q23h
2k2y
+16 Q36h2kx ky) [I4
2u0+I52x
+ I62(4Q13u0,x x+ 4Q36v0,x x 8Q33z,x
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+4 Q36u0,x y+ 4Q23v0,x y)/a3+I72(96Q33x
+12 Q13h2x,x x+ 4Q23h
2x,yy + 8Q36h2y,x x
96 Q33w0,x+ 16Q36h2x,x y+ 8Q23h
2y,xy )/a2]/
a2 (8Q36h2k2y+ 8Q13h
2kx ky) [I42v0
+ I52y+ I6
2(4Q36u0,yy + 4Q23v0,yy
8 Q33z,y+ 4Q13u0,xy + 4Q36v0,xy )/a3
+ I72(96Q33y+ 8Q36h
2x,yy + 4Q13h2y,x x
+12 Q23h2y,yy 96Q33w0,y+ 16Q36h
2y,x y
+8 Q13h2x,x y)/a2]/a2+ (i8Q13kx + i8Q36ky)
[I32w0+I4
2z+I52(4Q13w0,x x
+4 Q23w0,yy 8Q13x,x 8Q36x,y 8Q23y,y
8 Q36y,x + 8Q36w0,x y)/a3+I62(32Q33z
+4 Q13h2z,x x+ 4Q23h
2z,yy 32Q13u0,x
32 Q36u0,y 32Q23v0,y 32Q36v0,x
+8 Q36h2z,x y)/(a3h
2)]/a3
v0:
(32Q23 S5,x y 32Q36 S5,x x)/(a3h2)
+(32Q23 S4,yy 32Q36 S4,x y)/(a3h2)
+(8Q23 P6,xy y 8Q36 P6,x x y)/a3
+(96Q23 P3,y+ 96Q36 P3,x)/(a3h2)
+(4Q36 P1,x x x 4Q23 P1,x x y)/a3
+(4Q23 P2,yy y 4Q36 P2,x yy )/a3
+(8Q23M4,yy + 8Q36M4,x y)/a3
+(8Q36M5,x x + 8Q23M5,x y)/a3N2,y N6,x
= (4Q36k2
x+ 4Q23kx ky) [I32u0+I4
2x
+ I52(4Q13u0,x x+ 4Q36v0,x x 8Q33z,x
+4 Q36u0,x y+ 4Q23v0,x y)/a3+I62(96Q33x
+12 Q13h2x,x x+ 4Q23h
2x,yy + 8Q36h2y,x x
96 Q33w0,x+ 16Q36h2x,x y
+8 Q23h2y,x y)/a2]/a3+I1
2v0+I22y
+ I32(4Q36u0,yy + 4Q23v0,yy 8Q33z,y
+4 Q13u0,x y+ 4Q36v0,x y)/a3+I42
(96Q33y+ 8Q36h2x,yy + 4Q13h
2y,x x
+12 Q23h2y,yy 96Q33w0,y+ 16Q36h
2y,x y
+8 Q13h2x,x y)/a2 (4Q23k
2y+ 4Q36kx ky)
[I32v0+I4
2y+I52(4Q36u0,yy
+4 Q23v0,yy 8Q33z,y+ 4Q13u0,x y
+4 Q36v0,x y)/a3+I62(96Q33y
+8 Q36h2x,yy + 4Q13h
2y,x x + 12Q23h2y,yy
96 Q33w0,y+ 16Q36h2y,x y+ 8Q13h
2
x,x y)/a2]/a3+ (i32Q23ky+ i32Q36kx )
[I42w0+I5
2z+I62(4Q13w0,x x
+4 Q23w0,yy 8Q13x,x 8Q36x,y
8 Q23y,y 8Q36y,x + 8Q36w0,x y)/a3
+ I72(32Q33z+ 4Q13h
2z,x x+ 4Q23h2
z,yy 32Q13u0,x 32Q36u0,y 32Q23v0,y
32 Q36v0,x + 8Q36h2z,x y)/(a3h
2)]/(a3h2)
y :
(20Q23h2 S6,x yy 24Q36h
2 S6,x x y
4 Q13h2 S6,x x x+ 96Q33 S6,x )/a2+ (12Q23
h2 S2,yy y 16Q36h2 S2,x yy 4Q13h
2 S2,x x y
+96 Q33 S2,y)/a2+ (8Q36h2 S1,x x x
8 Q23h2 S1,x x y)/a2+ (12Q23h
2 P4,yy
+16 Q36h2 P4,x y+ 4Q13h
2 P4,x x 96Q33h2
P4)/(a2/3)+ (8Q23 P4,yy 8Q36h2
P4,x y)/a3+ (24Q36h2 P5,x x+ 24Q23h
2
P5,x y)/a2+ (8Q23 P5,x y 8Q36h2 P5,x x)/a3
+(16Q23M3,y+ 16Q36M3,x )/a3M6,x
M2,y+N4= (8Q36h2k2x+ 8Q23h
2kx ky)
[I42u0+I5
2x + I62(4Q13u0,x x
+4 Q36v0,x x 8Q33z,x+ 4Q36u0,x y
+4 Q23v0,x y)/a3+I72(96Q33x
+12 Q13h2x,x x+ 4Q23h
2x,yy + 8Q36h2y,x x
96 Q33w0,x+ 16Q36h2x,x y+ 8Q23h
2
12
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Smart Mater. Struct. 23 (2014) 045017 J Zhaoet al
y,x y)/a2]/a2+I22v0+I3
2y
+ I42(4Q36u0,yy + 4Q23v0,yy 8Q33z,y
+4 Q13u0,x y+ 4Q36v0,x y)/a3+I52
(96Q33y+ 8Q36h2x,yy + 4Q13h
2y,x x
+12 Q23h2y,yy 96Q33w0,y+ 16Q36h
2y,x y
+8 Q13h2x,x y)/a2 (96Q33+ 4Q13h
2k2x
+12 Q23h2k2y+ 16Q36h
2kx ky) [I42v0
+ I52y+ I6
2(4Q36u0,yy + 4Q23v0,yy
8 Q33z,y+ 4Q13u0,xy + 4Q36v0,xy )/a3
+ I72(96Q33y+ 8Q36h
2x,yy + 4Q13h2y,x x
+12 Q23h2
y,yy 96Q33w0,y+ 16Q36h2
y,x y
+8 Q13h2x,x y)/a2]/a2+ (i8Q23ky+ i8Q36kx)
[I32w0+I4
2z+I52(4Q13w0,x x
+4 Q23w0,yy 8Q13x,x 8Q36x,y 8Q23y,y
8 Q36y,x + 8Q36w0,x y)/a3+I62(32Q33z
+4 Q13h2z,x x+ 4Q23h
2z,yy 32Q13u0,x
32 Q36u0,y 32Q23v0,y 32Q36v0,x
+8 Q36h2z,x y)/(a3h2)]/a3
w0:
(4Q13 P5,x x x 8Q36 P5,x x y 4Q23 P5,x yy )/a3
+288Q33 P5,x/a2+ (4Q13 P4,x x y 8Q36
P4,x yy 4Q23 P4,yy y)/a3+ 288Q33 P4,y/a2
96 Q33 S1,x x/a2 96Q33 S2,yy/a2 192Q33
S6,x y/a2+ (4Q13M3,x x + 8Q36M3,x y
+4 Q23M3,yy )N5,x N4,y= i96Q33kx
[I42u0+I5
2x + I62(4Q13u0,x x
+4 Q36v0,x x 8Q33z,x+ 4Q36u0,x y
+4 Q23v0,x y)/a3+I72(96Q33x
+12 Q13h2x,x x+ 4Q23h
2x,yy + 8Q36h2y,x x
96 Q33w0,x+ 16Q36h2x,x y+ 8Q23h
2
y,x y)/a2]/a2+ i96Q33ky [I42v0
+ I52y+ I6
2(4Q36u0,yy + 4Q23v0,yy
8 Q33z,y+ 4Q13u0,x y+ 4Q36v0,x y)/a3
+ I72(96Q33y+ 8Q36h
2x,yy
+4 Q13h2y,x x+ 12Q23h
2y,yy 96Q33w0,y
+16 Q36h2y,xy + 8Q13h
2x,x y)/a2]/a2
+ I12w0+I2
2z+ I32(4Q13w0,x x
+4 Q23w0,yy 8Q13x,x 8Q36x,y 8Q23y,y
8 Q36y,x + 8Q36w0,x y)/a3+I42
(32Q33z+ 4Q13h2z,x x+ 4Q23h
2z,yy
32 Q13u0,x 32Q36u0,y 32Q23v0,y
32 Q36v0,x + 8Q36h2z,x y)/(a3h
2)
(4Q13k
2
x + 4Q23k
2
y+ 8Q36kx ky)
[I32w0+I4
2z+I52(4Q13w0,x x
+4 Q23w0,yy 8Q13x,x 8Q36x,y 8Q23y,y
8 Q36y,x + 8Q36w0,x y)/a3+I62(32Q33z
+4 Q13h2z,x x+ 4Q23h
2z,yy 32Q13u0,x
32 Q36u0,y 32Q23v0,y 32Q36v0,x
+8 Q36h2z,x y)/(a3h
2)]/a3
z :
(4Q13h2 S5,x x x 8Q36h
2 S5,x x y 4Q23h2 S5,x yy
+32 Q33 S5,x)/(a3h2)+ (4Q13h
2 S4,x x y 8Q36h2
S4,x yy 4Q23h2 S4,yy y+ 32Q33 S4,y)/(a3h
2)
8 Q33 P1,x x/a3+ (4Q13h2 P3,x x+ 8Q36h
2 P3,x y
+4 Q23h2 P3,yy 32Q33 P3)/(a3h
2/3) 8Q33
P2,yy/a3 16Q33 P6,x y/a3+ 16Q33M5,x/a3
+16 Q33M4,y/a3M5,x M4,y+N3
=i8Q33kx [I32u0+I4
2x+ I52(4Q13u0,x x
+4 Q36v0,x x 8Q33z,x+ 4Q36u0,x y
+4 Q23v0,x y)/a3+I62(96Q33x
+12 Q13h2x,x x+ 4Q23h
2x,yy + 8Q36h2y,x x
96 Q33w0,x+ 16Q36h2x,x y+ 8Q23h
2
y,x y)/a2]/a3+ i8Q33ky [I32v0+I4
2y
+ I52(4Q36u0,yy + 4Q23v0,yy 8Q33z,y
13
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8/10/2019 Modeling of Lamb waves in composites using new third-order plate theories
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Smart Mater. Struct. 23 (2014) 045017 J Zhaoet al
+4 Q13u0,x y+ 4Q36v0,x y)/a3+I62
(96Q33y+ 8Q36h2x,yy + 4Q13h
2y,x x
+12 Q23h2y,yy 96Q33w0,y+ 16Q36h
2y,x y
+8 Q13h2x,x y)/a2]/a3+I2
2w0+I32z
+ I42(4Q13w0,x x+ 4Q23w0,yy 8Q13x,x
8 Q36x,y 8Q23y,y 8Q36y,x
+8 Q36w0,x y)/a3+I52(32Q33z
+4 Q13h2z,x x+ 4Q23h
2z,yy 32Q13u0,x
32 Q36u0,y 32Q23v0,y 32Q36v0,x
+8 Q36h2z,x y)/(a3h
2) (32Q33+ 4Q13h2k2y
+8 Q36h2
kx ky) [I42
w0+I52
z+I62
(4Q13w0,x x+ 4Q23w0,yy 8Q13x,x
8 Q36x,y 8Q23y,y 8Q36y,x
+8 Q36w0,x y)/a3+I72(32Q33z
+4 Q13h2z,x x+ 4Q23h
2z,yy 32Q13u0,x
32 Q36u0,y 32Q23v0,y 32Q36v0,x
+8 Q36h2z,x y)/(a3h
2)]/(a3h2)
wherea1, a2 and a3 are the same as those defined in (13).
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