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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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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Smart Mater. Struct. 23 (2014) 045017 J Zhaoet al

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

2


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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.

3


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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

5


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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

6


8/15


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

7


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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).

8


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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

9


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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

10


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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

11


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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


14/15


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


15/15


+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).

References

[1] Staszewski W, Boller C and Tomlinson G R 2004Health

Monitoring of Aerospace Structures: Smart Sensor

Technologies and Signal Processing(New York: Wiley)

[2] Boller C, Chang F-K and Fujino Y 2009Encyclopedia of

Structural Health Monitoring(New York: Wiley)

[3] Boller C 2000 Next generation structural health monitoring

and its integration into aircraft designInt. J. Syst. Sci.

31133349

[4] Giurgiutiu V 2007Structural Health Monitoring With

Piezoelectric Wafer Active Sensors(Amsterdam: Elsevier)[5] Chang F-K and Chang K-Y 1987 A progressive damage model

for laminated composites containing stress concentrations

J. Compos. Mater. 21 83455

[6] Lin C-M, Yen T-T, Lai Y-J, Felmetsger V V, Hopcroft M A,

Kuypers J H and Pisano A P 2010 Temperature-

compensated aluminum nitride Lamb wave resonatorsIEEE

Trans. Ultrason. Ferroelectr. Freq. Control 57 52432

[7] Su Z, Ye L and Lu Y 2006 Guided Lamb waves for

identification of damage in composite structures: a review

J. Sound Vib. 295 75380

[8] Kessler S S, Spearing S M and Soutis C 2002 Damagedetection in composite materials using Lamb wave methods

Smart Mater. Struct. 11 269

[9] Robbins D H and Reddy J N 1993 Modelling of thick

composites using a layerwise laminate theory Int. J. Numer.

Methods Eng. 36 65577

[10] Moser F, Jacobs L J and Qu J 1999 Modeling elastic wave

propagation in waveguides with the finite element method

NDT & E Int. 32 22534

[11] Nayfeh A H 1995Wave Propagation in Layered Anisotropic

Media With Application to Composites (Amsterdam:

Elsevier) pp 11733

[12] Zhao J-L, Qiu J-H, Ji H-L and Hu N 2013 Four vectors of

Lamb waves in composites: semianalysis and numerical

simulationJ. Intell. Mater. Syst. Struct. 24 198594

[13] Zhao J-L, Qiu J-H and Ji H-L 2013 Reconstruction of stiffness

coefficients in orthotropic plates based on Lamb waves

phase velocities using genetic algorithmProc. 1st Int. Symp.

on Smart Layered Materials and Structures for Energy

Savingpp 2930

[14] Ashton J E and Whitney J M 1970Theory of Laminated Plates

(Stanford, CA: Technomic)

[15] Mindlin R D 1951 Influence of rotary inertia and shear on

flexural motions of isotropic elastic platesJ. Appl.

Mech.Trans. ASME18 318

[16] Liu Y, Hu N, Yan C, Peng X and Yan B 2009 Construction of

a Mindlin pseudospectral plate element and evaluating

efficiency of the elementFinite Elem. Anal. Des. 45 53846

[17] Zak A 2009 A novel formulation of a spectral plate element

for wave propagation in isotropic structures Finite Elem.Anal. Des. 45 6508

[18] Whitney J M and Sun C T 1973 A higher order theory for

extensional motion of laminated compositesJ. Sound Vib.

308597

[19] Wang L and Yuan F-G 2007 Lamb wave propagation in

composite laminates using a higher-order plate theoryProc.

SPIE653165310I

[20] Zhang H, Ta D and Liu Z 2008Lamb Waves in Anisotropic

Composite Laminate(Beijing: Science Press) pp 378

(in Chinese)

[21] Rose J L 2004Ultrasonic Waves in Solid Media (Cambridge:

Cambridge University Press)

[22] Robbins D H and Reddy J N 1994 Structural theories and

computational models for composite laminatesAppl. Mech.Rev. 47 14770

[23] Washizu K 1982Variational Methods in Elasticity and

Plasticity3rd edn (Oxford: Pergamon)

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http://dx.doi.org/10.1080/00207720050197730http://dx.doi.org/10.1080/00207720050197730http://dx.doi.org/10.1177/002199838702100904http://dx.doi.org/10.1177/002199838702100904http://dx.doi.org/10.1109/TUFFC.2010.1443http://dx.doi.org/10.1109/TUFFC.2010.1443http://dx.doi.org/10.1016/j.jsv.2006.01.020http://dx.doi.org/10.1016/j.jsv.2006.01.020http://dx.doi.org/10.1088/0964-1726/11/2/310http://dx.doi.org/10.1088/0964-1726/11/2/310http://dx.doi.org/10.1002/nme.1620360407http://dx.doi.org/10.1002/nme.1620360407http://dx.doi.org/10.1016/S0963-8695(98)00045-0http://dx.doi.org/10.1016/S0963-8695(98)00045-0http://dx.doi.org/10.1177/1045389X13488250http://dx.doi.org/10.1177/1045389X13488250http://dx.doi.org/10.1016/j.finel.2009.03.004http://dx.doi.org/10.1016/j.finel.2009.03.004http://dx.doi.org/10.1016/j.finel.2009.05.002http://dx.doi.org/10.1016/j.finel.2009.05.002http://dx.doi.org/10.1016/S0022-460X(73)80052-5http://dx.doi.org/10.1016/S0022-460X(73)80052-5http://dx.doi.org/10.1117/12.716499http://dx.doi.org/10.1117/12.716499http://dx.doi.org/10.1115/1.3111076http://dx.doi.org/10.1115/1.3111076http://dx.doi.org/10.1115/1.3111076http://dx.doi.org/10.1115/1.3111076http://dx.doi.org/10.1117/12.716499http://dx.doi.org/10.1117/12.716499http://dx.doi.org/10.1016/S0022-460X(73)80052-5http://dx.doi.org/10.1016/S0022-460X(73)80052-5http://dx.doi.org/10.1016/j.finel.2009.05.002http://dx.doi.org/10.1016/j.finel.2009.05.002http://dx.doi.org/10.1016/j.finel.2009.03.004http://dx.doi.org/10.1016/j.finel.2009.03.004http://dx.doi.org/10.1177/1045389X13488250http://dx.doi.org/10.1177/1045389X13488250http://dx.doi.org/10.1016/S0963-8695(98)00045-0http://dx.doi.org/10.1016/S0963-8695(98)00045-0http://dx.doi.org/10.1002/nme.1620360407http://dx.doi.org/10.1002/nme.1620360407http://dx.doi.org/10.1088/0964-1726/11/2/310http://dx.doi.org/10.1088/0964-1726/11/2/310http://dx.doi.org/10.1016/j.jsv.2006.01.020http://dx.doi.org/10.1016/j.jsv.2006.01.020http://dx.doi.org/10.1109/TUFFC.2010.1443http://dx.doi.org/10.1109/TUFFC.2010.1443http://dx.doi.org/10.1177/002199838702100904http://dx.doi.org/10.1177/002199838702100904http://dx.doi.org/10.1080/00207720050197730http://dx.doi.org/10.1080/00207720050197730

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Modeling of Lamb waves in composites using new third-order plate theories

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