Lamb wave propagation in composite laminates using a higher-order
plate theory
Lei Wang and F. G. Yuan*
Department of Mechanical and Aerospace Engineering, Campus box 7921,
North Carolina State University, Raleigh, NC 27695-7921, USA
ABSTRACT
A new consistent higher-order plate theory is developed for composites with the aim of accurately and efficiently
modeling multiple higher-order Lamb waves over a higher frequency range. The dispersion relations based on this
theory that can be analytically determined comprise five symmetric and six anti-symmetric wave modes. Computational
procedures for phase and group velocities are discussed. Meanwhile, characteristic wave curves including velocity,
slowness, and wave curves are introduced to investigate the dispersive and anisotropic behavior of Lamb wave
propagation in composites. From numerical results of Lamb waves in both lamina and symmetric laminate, it shows that
the higher-order plate theory not only gives good agreement with three-dimensional (3-D) elasticity theory over a wide
high frequency range, but also provides a more robust method than 3-D elasticity theory. This study demonstrates a
feasibility of using the proposed theory for realizing near real-time Structural Health Monitoring for composites at a
higher frequency range.
Keywords: Lamb waves; SH waves; Higher-order plate theory; Dispersion relation; Phase/Group velocity; Slowness;
Composite; Structural Health Monitoring (SHM)
1. INTRODUCTION
The increasing use of advanced composites in structural components, such as aerospace structures, marine vehicles,
automotive parts, and many other applications has led to extensive research activities in the fields of NDE (NonDestructive Evaluation)1, 2 and in-situ SHM3. For the active diagnosis that utilizes transient Lamb waves in damage
detection, the characteristics of Lamb waves in composite laminates are needed to be thoroughly studied. Because of
inherent anisotropy and heterogeneity of multi-layered laminates, the wave behavior becomes much more complex than
in isotropic plates. Besides angular dependence, the most significant consequence of elastic anisotropy is the loss of pure
wave modes for general propagation directions. Therefore the distinction between wave mode types in composites is
somewhat artificial since three distinct types of wave modes, i.e., symmetric (extensional, S), anti-symmetric (flexural,
A) and symmetric/anti-symmetric shear horizontal (SH) waves, observed in isotropic plates are generally coupled. In
practice, symmetric laminates are commonly used in the design of composite structures. The Lamb waves for symmetric
laminates can be separated into symmetric and anti-symmetric modes. For symmetric modes, one type is designated as
quasi-extensional (qSn) (n = 0, 1, 2, ⋅⋅⋅), where the dominant component of the polarization vector is along the
propagation direction, and the other type is quasi-horizontal shear (qSH2n), where the polarization vector is mainly
parallel to the plane of the plate. Similarly for the anti-symmetric types of wave modes, the quasi-flexural (qAn) and
quasi-horizontal shear (qSH2n-1) are generated. For ease of notation, in the paper the prefix quasi is omitted unless stated
otherwise.
There are two common theoretical approaches in investigating Lamb wave propagation in composite laminates: one
is exact solutions by 3-D elasticity theory; the other is approximate solutions by laminated plate theories. As for the
exact solutions, Nayfeh and Chimenti4 gave the dispersion relations of Lamb waves in a lamina. Later, Nayfeh5 extended
the formulation to Lamb waves in laminates. Recently, Wang and Yuan6 established dispersion relations and
characteristic wave curves for both phase and group velocities of Lamb waves in general laminates with arbitrary layup,
and verified the results by experiments. Although the exact solutions can provide accurate results, the computational
*
Corresponding author. E-mail: yuan@ncsu.edu; phone 1 919 515 5947; fax 1 919 515 5934;
http://www.mae.ncsu.edu/research/SSML/index.htm
effort is highly intensive because the transcendental equations have to be numerically solved and matrix expansion
incurs more analytical difficulty.
To make solutions tractable and improve computational efficiency, many researchers have applied various laminated
plate theories to approximate the exact solutions of Lamb waves in composites. 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 of the
laminate thickness7. It is noted that CPT contains two non-dispersive symmetric and one dispersive anti-symmetric
modes. Due to relatively low transverse shear modulus in composite laminates, the transverse shear deformation effects
neglected by CPT are not valid and make CPT results in error over the high frequency range where wavelength is not
much greater than the laminate thickness8. The Mindlin plate theory which includes transverse shear deformation and
rotary inertia provides very accurate prediction even for very short wavelengths in the lowest anti-symmetric wave
mode. It is also well known that the Mindlin type of plate theory, even though very good for the lowest anti-symmetric
mode, cannot describe accurately the higher-mode of wave propagation9, 10.
A higher-order plate theory can represent the kinematics better and yield more accurate inter-laminar stress
distributions at the cost of computational effort in elastostatic problems. In principle, it is possible to expand the
displacement fields in terms of the thickness coordinate up to any degree. Due to the algebraic complexity and
computational cost involved with higher-order plate theory in return for marginal gain in accuracy, theory higher than
third-order has not been attempted11. Higher-order shear deformation theories have been proposed by several
researchers12.However most of these studies have not paid enough attention to group velocity of Lamb waves in
laminates. It has been analytically proved that the group velocity in lossless plates is identical to the energy velocity13.
Accordingly most of Lamb-wave-based damage detection or imaging techniques evaluate arrival time (or time-of-flight)
of scattered waves from damage. Knowing the group velocity, the location of damage can be determined14. Besides
group velocity dispersion, another important issue is the angular dependency of group velocity (or wave curve) in
composite because wave curve gives the locus of wave front, at unit time, by the disturbance emitted at time zero by a
point elastic point source15, 16. Therefore group velocities
of Lamb waves are vital in SHM and NDE for
composites.
The main objective of this paper is to develop a new
consistent higher-order plate theory not only improving
computational efficiency but also more accurately
θ
approximating Lamb waves in composites. It models
symmetric waves in second-order displacement
expansion and anti-symmetric waves in third-order
expansion, which extends Mindlin plate theory (or firstorder plate theory) to a third-order plate theory; therefore
the proposed theory can approximate five symmetric and
Fig. 1: Force and moment resultants on a composite laminate
six anti-symmetric of Lamb wave modes. Meanwhile,
computational procedures to analytically obtain phase and group velocity dispersions are carefully discussed.
Furthermore, the characteristic wave curves are introduced to analyze the anisotropy of Lamb wave propagation. Then
numerical results based on the plate theory are compared with the exact solutions from 3-D elasticity in the cases of
Lamb waves in both lamina and laminate. Finally, some conclusions and guidelines for the practice of NDE and SHM
are drawn from the theory and experiments.
2. A NEW HIGHER-ORDER PLATE THEORY
In this section, a new consistent higher-order plate theory for Lamb waves in composites is developed by appending
up to third-order terms to the displacement expansion about the mid-plane in a manner similar to that of Kane and
Mindlin17 for homogeneous isotropic plates. The governing equations reveal that general unsymmetric laminates display
a coupling phenomenon among all the wave modes. In most applications symmetric laminates are commonly used, so
the decoupled dispersion relations of symmetric and anti-symmetric Lamb waves will be investigated.
Consider a laminate of constant thickness h composed of anisotropic laminas perfectly bonded together. As shown in
Fig. 1, the origin of a Cartesian coordinate system is located at the middle x-y plane with z axis being normal to the midplane, so two outer surfaces the laminate are at z = ±h/2. A packet of transient Lamb waves propagates in the composite
laminate in an arbitrary direction θ, which is defined counterclockwise relative to x axis.
Each lamina with an arbitrary orientation in the global coordinate system (x, y, z) can be considered as a monoclinic
material having x-y as a plane of mirror symmetry, thereby the stress-strain relations take the following matrix form:
0 C16 ⎤ ⎧ ε x ⎫
⎧σ x ⎫ ⎡C11 C12 C13 0
⎪ ⎪ ⎢
⎥⎪ ⎪
⎪σ y ⎪ ⎢C12 C22 C23 0
0 C26 ⎥ ⎪ ε y ⎪
⎪ ⎪ ⎢
⎥⎪ ⎪
(1)
⎪σ z ⎪ ⎢C13 C23 C33 0
⎥ ⎪εz ⎪
0
C
36
⎪ ⎪ ⎢
⎪ ⎪
⎥
=
⎨ ⎬
⎨ ⎬
0
0 C44 C45 0 ⎥ ⎪γ yz ⎪
⎪τ yz ⎪ ⎢ 0
⎥⎪ ⎪
⎪ ⎪ ⎢
0
0 C45 C55 0 ⎥ ⎪γ xz ⎪
⎪τ xz ⎪ ⎢ 0
⎥⎪ ⎪
⎪ ⎪ ⎢
⎪⎩τ xy ⎪⎭ ⎢⎣C16 C26 C36 0
0 C66 ⎥⎦ ⎪⎩γ xy ⎪⎭
When the global coordinate system (x, y, z) does not coincide with the principal material coordinate system (x’, y’, z)
but rotates with respect to z axis, the stiffness matrix Cij (i, j = 1, 2, ⋅⋅⋅, 6) in (x, y, z) system can be obtained from the
lamina stiffness matrix C ij′ in (x’, y’, z) system by using a transformation matrix18. The material stiffness matrix C ij′
can be calculated from lamina properties Ek, νkl, and Gkl (k, l = 1, 2, 3)19.
The linear engineering strain-displacement relations are
ε x = u , x , ε y = v, y , ε z = w, z , γ yz = v, z + w, y , γ xz = u , z + w, x ,
(2)
γ xy = u , y + v, x
where subscript comma denotes partial differential, u, v, and w are the displacements in x, y, and z directions,
respectively.
The displacement field in the x, y, and z directions can be described by
(3a)
u ( x, y, z , t ) = u 0 ( x, y, t ) + zψ x ( x, y, t ) + z 2φ x ( x, y, t ) + z 3 χ x ( x, y, t )
2
3
(3b)
v( x, y, z, t ) = v0 ( x, y, t ) + zψ y ( x, y, t ) + z φ y ( x, y, t ) + z χ y ( x, y, t )
(3c)
w( x, y , z , t ) = w0 ( x, y , t ) + zψ z ( x, y, t ) + z 2φ z ( x, y, t )
where u0, v0, and w0 represent the displacements of every point of the mid-plane, ψx and ψy physically denote the
rotations of section x = constant and y = constant respectively. It is worth noting that the odd-order terms with respect to
z in u and v together with even-order terms in w with respect to z describe anti-symmetric wave modes; the other terms
in Eq. (3) depict symmetric wave modes. Furthermore, it may be predicted that the accuracy of flexural motion will be
higher than that of extensional motion because the displacement field of flexural motion is one-order higher than that of
extensional motion. The strain-displacement relations in conjunction with Eq. (3) yield the following relations:
ε x = u0, x + zψ x , x + z 2φ x , x + z 3 χ x, x , ε y = v0, y + zψ y , y + z 2φ y , y + z 3 χ y , y , ε z = ψ z + 2 zφ z
(4)
γ xy = u0, y + v0, x + z (ψ x , y + ψ y , x ) + z 2 (φ x , y + φ y , x ) + z 3 ( χ x , y + χ y , x )
2
2
γ xz = ψ x + w0, x + z (ψ z , x + 2φ x ) + z (φ z , x + 3χ x ) , γ yz = ψ y + w0, y + z (ψ z , y + 2φ y ) + z (φ z , y + 3χ y )
Additionally, the discrepancies between the actual displacement field and that of the higher-order plate theory need
to be corrected by make the following substitutions related to the thickness strains
κ 1 (ψ x + w0 , x ) for ψ x + w0, x , κ 2 (ψ y + w0, y ) for ψ y + w0, y , κ 3ψ z for ψ z ,
κ 4 (ψ z , x + 2φ x ) for (ψ z , x + 2φ x ) ,
κ 5 (ψ z , y + 2φ y ) for (ψ z , y + 2φ y ) ,
(5)
κ 6φ z for κ 6φ z , κ 7 (φ z , x + 3χ x ) for φ z , x + 3χ x , and κ 8 (φ z , y + 3χ y ) for φ z , y + 3χ y
where κi (i = 1,2,…, 8) are the correction factors similar to those introduced by Mindlin and Medick20 for homogeneous
isotropic plates. By matching the cut-off frequencies of A1, S1, SH1, and SH2 modes from 3-D elasticity theory, the
correction factors are chosen as κ 1 = κ 2 = κ 7 = κ 8 = π 90 − 2 1605 , κ 3 = π 12 , and κ 4 = κ 5 = κ 6 = π 15 .
Stress and moment resultants per unit length are defined as in the following:
( N x , N y , N z , N xy , Qx , Q y ) = ∫
h/2
−h / 2
( M x , M y , M z , M xy , R x , R y ) = ∫
( S x , S y , S xy , Px , Py ) = ∫
h/2
−h / 2
(σ x , σ y , σ z , τ xy , τ xz , τ yz ) dz
h/2
−h / 2
(σ x , σ y , σ z , τ xy , τ xz , τ yz ) zdz
(σ x , σ y , τ xy , τ xz , τ yz ) z 2 dz
(6a)
(6b)
(6c)
(Tx , T y , Txy ) = ∫
h/2
−h / 2
(6d)
(σ x , σ y , τ xy ) z 3 dz
The constitutive equations for a linear elastic laminate can be derived from the strain energy in the 3-D elasticity
theory with the corrections made by Eq. (5). The constitutive equations of the laminate with arbitrary lay-up can be
obtained by rearranging Eq. (6) in matrix form:
)
)
⎧ N ⎫ ⎡[ A] [ B]
[ D] [ F ] ⎤ ⎧ε (0) ⎫
⎪ ⎪ ⎢
)
) ⎥⎪ ⎪
⎪⎪M ⎪⎪ ⎢[ B] [ D] [ F ] [ H ]⎥ ⎪⎪ε (1) ⎪⎪
(7)
⎥⎨ ⎬
⎨ ⎬=⎢ ) T
) T
[ H ] [ J ] ⎥ ⎪ε (2) ⎪
⎪ S ⎪ ⎢[ D ] [ F ]
⎥⎪ ⎪
⎪ ⎪ ⎢ )
)
⎪⎩ T ⎪⎭ ⎢⎣[ F ]T [ H ]T [ J ] [ K ] ⎥⎦ ⎪⎩ε (3) ⎪⎭
(0)
⎧Q ⎫ ⎡[ A] [ B] [ D] ⎤ ⎧γ ⎫
⎥ ⎪⎪ ⎪⎪
⎪⎪ ⎪⎪ ⎢
(8)
(1)
⎨ R ⎬ = ⎢[ B ] [ D ] [ F ] ⎥ ⎨ γ ⎬
⎥⎪ ⎪
⎪ ⎪ ⎢
⎪⎩ P ⎪⎭ ⎢⎣[ D] [ F ] [ H ]⎥⎦ ⎪⎩γ (2) ⎪⎭
where the stress and moment resultant vectors are defined by
N = {N x , N y , N z κ 3 , N xy }T , M = {M x , M y , M z (2κ 6 ) , M xy }T
S = {S x , S y , S xy }T ,
Q = {Qx κ 1 , Q y κ 2 }T
T = {Tx , Ty , Txy }T ,
R = {Rx κ 4 , R y κ 5 }T ,
P = {Px κ 7 , Py κ 8 }T
ε
(0)
= {u0, x , v0, y , κ 3ψ z , u 0, y + v0, x } ,
ε
(2)
= {φ x , x , φ y , y , φ x , y + φ y , x } ,
T
T
ε
ε
(3)
(9)
= {ψ x , x , ψ y , y , 2κ 6φ z , ψ x , y + ψ y , x }
T
(1)
= {χ x , x , χ y , y , χ x , y + χ y , x }T ,
γ (0) = {κ 1 (ψ x + w0, x ), κ 2 (ψ y + w0, y )}T ,
γ (1) = {κ 4 (ψ z , x + 2φ x ), κ 5 (ψ z , y + 2φ y )}T
γ (2) = {κ 7 (φ z , x + 3χ x ), κ 8 (φ z , y + 3χ y )}T
the parenthesis head in Eq. (7) indicates non-square matrices, and all submatrices in Eqs. (7) and (8) are given by
⎡ β11 β12 β13 β16 ⎤
⎡υ11 υ12 υ16 ⎤
⎢
⎥
⎢
⎥
⎢ β12 β 22 β 23 β 26 ⎥
; ) ⎢υ12 υ 22 υ 26 ⎥
;
⎥, ( β = A, B, D) [υ ] = ⎢
⎥, (υ = D, F , H )
[β ] = ⎢
⎢ β13 β 23 β 33 β 36 ⎥
⎢υ13 υ 23 υ 36 ⎥
⎢
⎢
⎥
⎥
⎢⎣ β16 β 26 β 36 β 66 ⎥⎦
⎢⎣υ16 υ 26 υ 66 ⎥⎦
⎡ς 11 ς 12 ς 16 ⎤
ζ 45 ⎤
⎢
⎥
⎡ζ
;
[ζ ] = ⎢ 55
[ς ] = ⎢ς 12 ς 22 ς 26 ⎥, (ς = H , J , K )
⎥, (ζ = A, B, D, F , H )
⎣ζ 45 ζ 44 ⎦
⎢
⎥
⎢⎣ς 16 ς 26 ς 66 ⎥⎦
in which ( Aij , Bij , Dij , Fij , H ij , J ij , K ij ) =
∫
h/2
−h / 2
(10)
Cij (1, z, z 2 , z 3 , z 4 , z 5 , z 6 ) dz . The coupling between anti-symmetric
and symmetric modes in Eqs. (7) and (8) is through the stiffness matrices Bij, Fij, and Jij.
With the linear strain-displacement relations, the equations of motion of the higher-order theory can be derived using
the principle of virtual displacement or Hamilton’s principle21
t2
(11)
0=
(δU + δV − δK )dt
∫
t1
where δU is the virtual strain energy, δV virtual work done by applied force, and δK the virtual kinetic energy. A set of
equations of motion is
(12a)
N x , x + N xy , y + q x = I 0u&&0 + I1ψ&&x + I 2φ&&x + I 3 χ&&x
(12b)
N + N + q = I v&& + I ψ&& + I φ&& + I χ&&
xy , x
y, y
Rx , x + R y , y
y
0 0
1
y
2
y
&&0 + I 2ψ&&z + I 3φ&&z
− N z + m = I1 w
3
y
(12c)
S x , x + S xy , y − 2 Rx + n x = I 2u&&0 + I 3ψ&&x + I 4φ&&x + I 5 χ&&x
S + S − 2 R + n = I v&& + I ψ&& + I φ&& + I χ&&
(12d)
&&0 + I1ψ&&z + I 2φ&&z
Qx , x + Q y , y + q = I 0 w
(12f)
M x , x + M xy , y − Qx + m x = I1u&&0 + I 2ψ&&x + I 3φ&&x + I 4 χ&&x
M + M − Q + m = I v&& + I ψ&& + I φ&& + I χ&&
(12g)
xy , x
y, y
xy , x
y
y, y
y
y
2 0
y
3
1 0
y
2
4
y
y
3
5
y
(12e)
y
4
y
&&0 + I 3ψ&&z + I 4φ&&z
Px , x + Py , y − M z + n = I 2 w
Tx , x + Txy , y − 3Px + rx = I 3u&&0 + I 4ψ&&x + I 5φ&&x + I 6 χ&&x
T + T − 3P + r = I v&& + I ψ&& + I φ&& + I χ&&
xy , x
where I j =
∫
h/ 2
− h/ 2
y, y
y
y
3 0
4
y
5
y
6
y
(12h)
(12i)
(12j)
(12k)
ρ z j dz, ( j = 0, 1, 2, K, 6) and the surface loads qα, nα, mα, rα, q, n, and m are expressed by
⎧⎪ 1 ⎫⎪
⎧⎪qα ⎫⎪
⎧⎪ 1 ⎫⎪ ⎧⎪q ⎫⎪
,
,
⎨ ⎬ = [τ α z (h / 2) − τ α z ( − h / 2)] ⎨ 2 ⎬ ⎨ ⎬ = [σ z ( h / 2) − σ z (− h / 2)] ⎨ 2 ⎬
⎪⎩h 4⎪⎭
⎪⎩h 4⎪⎭ ⎪⎩n ⎪⎭
⎪⎩nα ⎪⎭
⎧⎪mα ⎫⎪
⎧⎪ h 2 ⎫⎪
,
⎨ ⎬ = [τ α z (h / 2) + τ α z (− h / 2)] ⎨ 3 ⎬ m = [σ z (h / 2) + σ z (−h / 2)] h 2
⎪⎩h 8⎪⎭
⎪⎩ rα ⎪⎭
here α = x, y. Assuming the solution form as
{u 0 , v0 , w0 } = {U 0 , V0 , W0 } exp i[(k x x + k y y ) − ω t ]
(13a)
(13b)
(14a)
{φ x , φ y , φ z } = {Φ x , Φ y , Φ z } exp i[( k x x + k y y ) − ω t ]
(14b)
{ψ x , ψ y , ψ z } = {Ψ x , Ψ y , Ψ z } exp i[( k x x + k y y ) − ω t ]
(14c)
{χ x , χ y } = { Χ x , Χ y } exp i[(k x x + k y y ) − ω t ]
(14d)
where ω is the angular frequency, wave vector k = [kx, ky] points to the direction of wave propagation θ in x-y plane,
and its magnitude is k = k = k x2 + k y2 .
T
Substituting Eqs. (7), (8), and (14) into Eq. (12) in the absence of surface loads yields the displacement equations of
motion in generalized eigenvalue problem form:
(15)
(L − ω 2 I)ξ = 0
T
where ξ = {U 0 V0 Ψ z Φx Φ y W0 Ψ x Ψ y Φz Χ x Χ y } , variables on the left and right sides of the dashed line
are associated with symmetric motion and anti-symmetric motion, respectively; L and I are 11×11 matrices.
For symmetric laminates, Bij = Fij = Jij = 0. Eq. (15) can be decoupled into symmetric wave modes governed by:
(16)
(L s − ω 2 I s )ξ s = 0
T
where subscript s indicates symmetric modes, ξ s = {U 0 V0 Ψ z Φx Φ y } , Ls and Is are 5×5 matrices.
and anti-symmetric wave modes governed by:
(17)
( L a − ω 2 I a )ξ a = 0
T
where subscript a indicates anti-symmetric modes, ξ a = {W0 Ψ x Ψ y Φz Χ x Χ y } , La and Ia are 6×6 matrices.
Eqs. (15), (16), and (17) are generalized eigenvalue problems, where I, Is, and Ia are real symmetric matrices;
furthermore L, Ls, and La are positive definite Hermitian matrices. For a given wave normal k, eleven real positive
eigenvalues ω2, or eleven coupled Lamb wave modes, can be obtained from Eq. (15). For a symmetric laminate, these
eleven modes are decoupled into symmetric and anti-symmetric modes; in other words, five real positive eigenvlaues ω2
or five symmetric wave modes can be computed from Eq. (16), and similarly six real positive eigenvlaues ω2 or six antisymmetric modes can be obtained from Eq. (17). Vanishing determinant of Eq. (15), (16), or (17), and solving the
resulting polynomial gives the dispersion relations. When the value of k varies in a fixed direction φ, each pair of k and
the individual solution of ω represents the dispersion relation of a single Lamb wave mode in the composite laminate.
3. ANALYTICAL FORMS OF DISPERSIONS AND CHARACTERISTIC WAVE CURVES
In contrast to exact solutions from 3-D elasticity, it is feasible to obtain the approximate solutions in analytical forms
because only polynomial equations are required to be solved without the need of solving transcendental equations. From
Eq. (15), the dispersion relation between ω and k can be symbolically represented by an implicit functional form
G (ω , k ) = 0 , or G (ω,k ,θ ) = L − ω 2 I = 0 , where | | is a determinant operator. Meanwhile, this relation can be explicitly
solved in the form of real roots of ω = W (k ) , or ω = W (k ,θ ) . There are an infinite number of possible solutions, in
general, such solutions correspond to different wave modes. For plane waves, the phase velocity vector is defined as
c p = (ω / k )(k / k ) = (ω / k 2 )k and thus its magnitude is c p = ω k . A curve generated by all choices of k from the
origin for cp at a given frequency is called velocity curve. The radius vectors of velocity curves in the direction of a given
k represent the admissible phase velocity dispersion of different wave modes.
Similarly a slowness (or inverse velocity) curve can be introduced by defining a slowness vector s = k/ω. The
slowness curve can be simply formed from the velocity curve by reciprocal. The slowness vector has the same direction
as the phase velocity vector. Thus the inverse of the phase velocities can be measured from the origin to the slowness
curves. Phase velocity is numerically equal to the distance traveled in unit time; while slowness is numerically equal to
the time required to travel unit distance. For the bulk (non-dispersive) waves, it is convenient to use the slowness curve
since this curve is independent of ω – rather than the velocity curve. It is worth of noting that in isotropic materials the
phase velocity depends only on the magnitude of the wave vector k; while in anisotropic materials the phase velocity
depends on the wave vector k, both magnitude and direction in which the wave propagates.
The group velocity, which can be measured by tracking envelopes of a wave packet, is defined by
c g = grad k W = ∂W / ∂k . In a polar coordinate system, grad k W has a radial component ∂W / ∂k in the direction of
k and an angular component ∂W / k∂θ perpendicular to k. Using coordinate transformation, the group velocity in a
Cartesian coordinate can be attained as
⎧ ∂W ⎫
⎧⎪c gx ⎫⎪ ⎡cos θ − sin θ ⎤ ⎪
⎪
(18)
⎥ ⎨ ∂k ⎬
⎨ ⎬=⎢
W
∂
⎪⎩c gy ⎪⎭ ⎢⎣ sin θ cos θ ⎥⎦ ⎪
⎪
⎩ k∂θ ⎭
where the subscripts x and y represent the components in x and y axes, respectively.
The magnitude of group velocity cg and the angle θg from the x axis are given by
2
2
and θ = tan −1 c gy
(19)
c g = c gx
+ c gy
g
c gx
The skew angle
ϑ
or steering angle22 is defined as
ϑ = θg −θ
(20)
In isotropic plates since ω is only function of k, i.e., ∂W / ∂θ = 0 , the direction of group velocity coincides with the
direction of wave vector, (i.e., ϑ = 0 ). The magnitude of the group velocity is c g = dW / dk . However for Lamb waves
in composites, ∂W / ∂θ does not vanish in general; thus the direction of group velocity is not parallel to k, i.e., the
skew angle ϑ may not be zero6.
The locus of group (ray) velocity vector along all choices of cg from the origin at a given frequency is referred to as
wave curve (or wave front curve). It is worth of noting that the radius vector joining the origin (or source point) to a
point on a wave curve represents the distance traveled by the elastic disturbance in unit time. The wave curve therefore
gives the locus of wave front, at a unit time, by the disturbance emitted by a point source acting through the origin at
time t = 0. Thus wave curves are of great importance in damage detection of SHM or NDE.
These three characteristic curves were initially used to analyze bulk waves in 3-D infinite domain15, 18. Wang and
Yuan6 have recently adopted them to investigate the propagation of Lamb waves in composites. Note that some valid
conclusions on characteristic curves of bulk waves are not applicable for Lamb waves in composites. For instance, in 3D domain the polar reciprocal of the slowness curve is the wave curve (i.e., s ⋅ c g = 1 ) and c p = c g cosϑ 15. However,
these two relations break down for Lamb waves in composites because of the dispersive behavior 6, 22.
The computation of dispersion relations should start with calculating the relation of ω = W (k ,θ ) by fixing θ and
solving for ω values at each varying k. Then according to the definition, phase velocity dispersion can be analytically
obtained from the relation of (ω ∼ k). For the dispersion of group velocity, the procedure however is not as direct as that
of phase velocity because ∂W / ∂k and ∂W / ∂θ are required in Eq. (18). The two derivatives ∂W / ∂k and ∂W / ∂θ
are obtained by explicitly differentiating the determinant of matrix (L − ω 2 I ) :
∂ L − ω 2I ∂ k
∂G (ω,k ,θ ) ∂ k
∂W
=−
=−
∂k
∂G (ω,k ,θ ) ∂ω
∂ L − ω 2 I ∂ω
(21a)
∂ L − ω 2I ∂θ
∂G (ω,k ,θ ) ∂ θ
∂W
=−
=−
∂θ
∂G (ω,k ,θ ) ∂ω
∂ L − ω 2 I ∂ω
(21b)
where L can be replaced with Ls or La for mode decoupling in symmetric laminates. The closed-forms of ∂W / ∂k and
∂W / ∂θ are possibly expanded out, however they are too lengthy to be expressed here; in practice, analytical forms of
Eq. (21) can be conveniently evaluated by symbolic computation software such as Matlab® or Maple®. Then, plugging
Eq. (21) into Eq. (18) yields the analytical forms of group velocity dispersions.
To compute characteristic curves, one should start with calculating the slowness curve by fixing ω and solving for k
values at each varying θ. Then the velocity and slowness curves can be analytically calculated from their own
definitions. For the wave curves, the two derivatives ∂W / ∂k and ∂W / ∂θ are similarly evaluated from Eq. (21), and
then the final analytical forms of wave curves would be yielded by plugging Eq. (21) into Eq. (18).
4. NUMERICAL RESULTS
The formulation described in the previous sections has been implemented by Matlab®, because it can seamlessly fuse
symbolic and numeric computation. The composite material used in this study is AS4/3502 graphite/epoxy shown in
Table1, and the dimensions and stacking sequences are listed in Table 2. Numerical examples demonstrate dispersion
curves and characteristic curves in the lamina and laminate, and compare the approximate solutions from the proposed
theory with the exact solutions based on 3-D elasticity6. Note that all figures in this section are shown in both
dimensionless coordinate system (located at the bottom and left sides) and dimensional coordinate system (located at the
top and right sides). Dimensionless frequency ωh/cT and dimensionless velocity cp/cT together with cg/cT are employed to
normalize the physical wave frequency and velocity respectively; additionally cT defined as G12 ρ is the transverse
(or, in-plane shear) wave velocity in the lamina. Moreover the exact and approximate solutions are marked by solid and
dashed lines, respectively.
Table 1 Material properties of AS4/3502 composite lamina
Table 2 Properties of AS4/3502 laminates
E1
(GPa)
E2
(GPa)
E3
(GPa)
G12
(GPa)
G13
(GPa)
G23
(GPa)
ν12
ν13
ν23
ρ
(kg/m3)
127.6
11.3
11.3
5.97
5.97
3.75
0.3
0.3
0.34
1578
I
Stacking sequence
Dimension
[+456/-456]s
46×43×3 mm3
In the figures of dispersion curves, five S modes are obtained from the higher-order plate theory including three
symmetric Lamb wave modes (S0, S1, and S2) and two symmetric SH modes (SH0 and SH2). Meanwhile six A modes
exist in the higher-order plate theory containing four anti-symmetric Lamb modes (A0, A1, A2, and A3) and two antisymmetric SH wave modes (SH1 and SH3). In order to clearly compare the dispersion curves from the two theories, only
the first five S and six A modes from 3-D elasticity theory are shown.
Fig. 2 displays the compared results for dispersion curves of Lamb waves in the lamina. In Figs. 2(a) and (b), the first
four S modes and the beginning portion of SH2 mode from the higher-order plate theory have good agreement with those
obtained from 3-D elasticity, but exact solutions have more complicated behavior than the higher-order plate theory
when ωh/cT exceeds 10. As shown in Fig. 2(c) and (d), the approximate solutions of A0 and SH1 match exact solutions
well. For A1 mode, the approximate solution matches the exact solution very well over the beginning portion up to
frequencyωh/cT = 7, and then it approaches the exact solution of A2 mode. Similar phenomenon on approximate solution
of SH3 mode can be observed. Therefore in the real SHM employing A modes as diagnostic waves, wave behavior of A0
and SH1 modes and even the portion of A1 mode prior to frequency ωh/cT = 7, can be conveniently modeled without the
need of using more complicated exact solutions because the proposed plate theory provides sufficient accuracy and realtime computation.
T
p
Dimensionless Velocity c /c
2
0
3
2
3
1
p
2
c (km/s)
1
cp (km/s)
Dimensionless Velocity cp/cT
The
dispersion
f⋅h (MHz⋅mm)
f⋅h (MHz⋅mm)
curves of Lamb waves
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
20
20
10
10
3−D elasticity theory
3−D elasticity theory
in the laminate [+456/18
Higher−order plate theory
Higher−order plate theory 18
9
9
456]s are shown in Fig.
16
16
8
8
3. The approximate
14
14
A
7
7
S
S
SH
A
SH
solutions of S0 and SH0
12
12
6
6
A
10
10
modes shown in Fig.
5
5
8
8
4
4
SH
3(a)
give
good
S
6
6
3
3
agreement with exact
4
4
2
2
solutions except for the
SH
A
2
2
1
1
steep descent of S0
0
0
0
0
0
2
4
6
8
10
12
14
16
0
2
4
6
8
10
12
mode. However in Fig.
Dimensionless Frequency ωh/cT
Dimensionless Frequency ωh/cT
3(b) the approximate
(a) cp of symmetric modes
(c) cp of anti-symmetric modes
solutions of S0 mode
have worse agreement
f⋅h (MHz⋅mm)
f⋅h (MHz⋅mm)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
with exact solutions and
11
9
3−D elasticity theory
3−D elasticity theory
10
the discrepancy of S0
8
5
Higher−order plate theory
Higher−order
plate
theory
SH
4
9
mode at the steep
7
S
S
8
S
4
descent
becomes
A
6
3
A
7
A
bigger. Furthermore, it
5
6
3
is interesting that there
5
4
2
SH
4
is a “mode flip” of
2
3
3
SH
A
exact solutions of SH0
2
1
2
1
and S0 modes at ωh/cT =
1
SH
1
4.5 in Fig. 3(b). This
0
0
0
0
0
2
4
6
8
10
12
14
16
0
2
4
6
8
10
12
phenomenon
also
Dimensionless Frequency ωh/c
Dimensionless Frequency ωh/c
T
T
occurs in the higher(b) cg of symmetric modes
(d) cg of anti-symmetric modes
order plate theory but in
Fig. 2: Comparison between 3-D elasticity and higher-order plate theory for dispersion in the lamina
a discontinuous manner.
with θ = 45°
From Figs. 3(c) and (d)
the two solutions of A0 mode match quiet well over the whole frequency range; moreover the two solutions of SH1 show
good agreement in phase velocity over whole frequency range and in group velocity over frequency less than ωh/cT = 4.
Furthermore, it can be found the higher-order plate theory provides considerable accuracy on both phase and group
velocities of A1 until ωh/cT = 4. In comparison with Fig. 2, the bigger discrepancies between approximate and exact
solutions of high modes such as SH2, S2, A3, and SH3 at their cut-off frequencies may result from the fact that the
accuracy of correction factors κi (i = 1,2,…, 8) calculated from homogeneous isotropic plates becomes lower in the
laminate than in the lamina.
Fig. 4 shows the anti-symmetric dispersion curves in the same laminate [+456/-456]s by using Mindlin plate theory.
Compared Fig. 3(c) and (d) with Fig. 4, it can be seen that (1) in high frequency range the group velocity of A0 from
Mindlin plate theory is not so accurate as from the higher-order plate theory; (2) Mindlin plate theory does not give
correct cut-off frequencies of SH1 and A1 modes; (3) Mindlin plate theory can not approximate the steep descent of A1
mode in the frequency range of ωh/cT = (4~8). Therefore the higher-order plate theory can provide higher accuracy than
Mindlin plate theory.
Figs. 5 and 6 visualize the characteristic curves including velocity, slowness, and wave curves of Lamb waves
propagating in composites at a given frequency. It is interesting that all curves have polar symmetry with respect to
origin; in other words every point on a curve of a given Lamb wave mode still remains located on the same curve after
rotated 180º. The reason is that composite lamina comprises paralleled fibers. Moreover, all characteristic wave curves
are certainly functions of frequency.
From characteristic curves in the lamina as shown in Fig. 5, it can be seen that the higher-order plate theory can
match 3-D elasticity theory except for the wave curves of S0 modes. The remarkable angular dependence of Lamb wave
propagation can be also observed in Fig. 6; velocity and slowness reach their maximum and minimum respectively in
1
0
0
2
0
1
2
3
0
1
cg (km/s)
Dimensionless Velocity cg/cT
1
cg (km/s)
Dimensionless Velocity cg/cT
3
2
0
2
0
3
2
cp (km/s)
2
Dimensionless Velocity cp/cT
1
cp (km/s)
Dimensionless Velocity cp/cT
fiber directions of
f (MHz)
f (MHz)
the lamina (0º or 180º)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
20
20
10
10
3−D elasticity theory
3−D elasticity theory
because
of
higher
Higher−order plate theory 18
Higher−order plate theory 18
9
9
stiffness
in
those
16
16
8
8
SH
directions.
Similar
14
14
7
7
A
S
result is also found in A
12
12
6
6
SH
S
10
10
modes of Lamb waves
5
5
8
8
4
4
in the laminate [+456/S
A
A
6
6
3
3
456]s as shown in Fig. 6.
SH
4
4
2
2
SH
Since the fibers in the
A
2
2
1
1
outer
lamina
are
0
0
0
0
0
2
4
6
8
10
12
0
2
4
6
8
10
12
orientated in 45º and
Dimensionless Frequency ωh/c
Dimensionless Frequency ωh/c
T
T
225º directions, the
(a) cp of symmetric modes
(c) cp of anti-symmetric modes
bending stiffness of the
laminate reaches its
f (MHz)
f (MHz)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
maximum in those
9
9
3−D elasticity theory
3−D elasticity theory
directions. The higher8
8
Higher−order
plate
theory
Higher−order
plate
theory
4
4
order plate theory can
7
7
S
provide
acceptable
S
6
6
SH
3
3
accurate for A modes,
A
5
5
SH
but it can not make
4
4
2
2
SH
good agreement with
3
3
the exact solutions of S0
A
2
2
1
1
modes in the first and
1
1
SH
A
S
A
third
quadrants
as
0
0
0
0
0
2
4
6
8
10
12
0
2
4
6
8
10
12
shown in Fig. 6(a)-(c).
Dimensionless Frequency ωh/c
Dimensionless Frequency ωh/c
T
T
The reason is that the
(b) cg of symmetric modes
(d) cg of anti-symmetric modes
wave frequency ωh/cT =
4 is right at the steep
Fig. 3: Comparison between 3-D elasticity and higher-order plate theory in the laminate [+456/-456]s
descent of S0 mode as
with θ = 30°
shown in Fig. 4(a) and
(c), and according to previous discussion, it is known that the accuracy of the higher-order plate theory becomes lower at
that portion.
From Fig. 5(c), (g),
f (MHz)
f (MHz)
Fig. 6 c), and (g), an
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
9
20
10
interesting
3−D elasticity theory
3−D elasticity theory
18
8
Mindlin plate theory
Mindlin plate theory
9
4
phenomenon of energy
16
7
8
focusing, well-known
14
7
6
3
for bulk waves in
12
6
5
anisotropic
solids,
A
10
5
4
2
occurs for the SH wave
8
4
A
3
modes
as
well.
6
3
SH
A
2
1
4
2
Slowness
curves
A
1
SH
2
1
indeed have many
0
0
0
0
inflexion points in a
0
2
4
6
8
10
12
0
2
4
6
8
10
12
Dimensionless Frequency ωh/cT
Dimensionless Frequency ωh/cT
90º sector. It implies
(b) cg of anti-symmetric modes
(a) cp of anti-symmetric modes
that the same group
velocity direction may
Fig. 4: Comparison between 3-D elasticity and Mindlin plate theory for dispersion curves of anticorrespond to several
symmetric Lamb waves in the laminate [+456/-456]s with θ = 30°
phase
front
15
directions . These particular shapes are responsible for the cusps of the associated group velocity or wave curve. It is
seen that the higher-order plate theory is able to model the cuspidal phenomena of Lamb waves in composites. It should
be note the wave frequency ωh/cT = 4 is considered as high frequency, and both CPT and Mindlin plate theory is not
capable of modeling Lamb waves because the frequency already exceeds the cut-off frequency of A1 and SH1.
1
3
1
0
2
3
c (km/s)
1
g
2
0
Dimensionless Velocity cg/cT
0
cg (km/s)
Dimensionless Velocity cg/cT
0
0
1
2
1
1
0
3
cg (km/s)
Dimensionless Velocity cg/cT
cp (km/s)
Dimensionless Velocity cp/cT
1
1
0
1
ο
ο
90
90
120ο
60ο
120ο
60ο
5 (c /c )
p
ο
p
ο
3.75
150
8 (c /c )
T
ο
30
2.5
ο
30
4
1.25
2
SH0
0
ο
180
A0
S0
210ο
ο
0
ο
0
ο
180
330ο
A1
SH1
ο
0
210ο
ο
240
330ο
ο
300
ο
240
ο
300
ο
270
270
(a) Velocity curves of symmetric modes
(d) Velocity curves of anti-symmetric modes
90ο
90ο
ο
ο
120
ο
60
ο
120
60
1.2 (S⋅cT)
ο
150
1.6 (S⋅cT)
ο
0.9
ο
30
150
0
A0
0.4
S0
A1
SH0
210ο
ο
0
ο
SH1
0
ο
180
330ο
ο
0
210ο
ο
240
330ο
ο
300
ο
240
270ο
300
270ο
(b) Slowness curves of symmetric modes
(e) Slowness curves of anti-symmetric modes
ο
90ο
90
ο
ο
120
ο
60
ο
120
60
4 (cg/cT)
ο
3.2 (cg/cT)
ο
3
150
ο
30
0.8
SH
0
S
0ο
0
210ο
330ο
ο
ο
240
30
1.6
1
0
ο
2.4
150
2
180ο
30
0.8
0.3
ο
ο
1.2
0.6
180
T
6
150
300
270ο
(c) Wave curves of symmetric modes
0
180ο
SH
1
A
A
0
0ο
1
210ο
330ο
ο
ο
240
300
270ο
(f) Wave curves of anti-symmetric modes
Fig. 5: Comparison between 3-D elasticity and higher-order plate theory for characteristic wave curves of Lamb
waves in the lamina at ωh/cT = 4
90ο
90ο
ο
ο
120
ο
60
ο
120
60
2.8 (c /c )
p
ο
p
ο
2.1
150
6 (c /c )
T
ο
30
1.4
0.7
3
0
0
0
180ο
A1
1.5
A
SH
T
30ο
4.5
150
0ο
0
180ο
0ο
SH1
S
0
ο
ο
210
ο
330
240ο
ο
210
300ο
330
240ο
300ο
270ο
270ο
(a) Velocity curves of symmetric modes
(d) Velocity curves of anti-symmetric modes
ο
ο
90
90
120ο
60ο
120ο
60ο
1.2 (S⋅cT)
ο
150
1.6 (S⋅cT)
ο
0.9
ο
30
150
0.6
SH
0
ο
0
S0
210ο
ο
A1
ο
180
330ο
SH1
0
ο
0
210ο
ο
240
A0
0.4
0
ο
330ο
ο
300
ο
240
300
270ο
270ο
(b) Slowness curves of symmetric modes
(e) Slowness curves of anti-symmetric modes
ο
90ο
90
120ο
ο
60ο
ο
120
60
3.6 (c /c )
2 (cg/cT)
ο
1.5
150
g
ο
ο
2.7
150
30
1
SH1
0
ο
ο
0
ο
ο
1.8
ο
180
ο
0
A0
S0
330ο
ο
330
240ο
300
300ο
ο
270ο
(c) Wave curves of symmetric modes
ο
210
ο
240
T
30
0
SH0
210ο
A1
0.9
0.5
180
30
0.8
0.3
180
ο
1.2
270
(f) Wave curves of anti-symmetric modes
Fig. 6: Comparison between 3-D elasticity and higher-order plate theory for characteristic wave curves of Lamb
waves in the laminate [+456/-456]s at ωh/cT = 4
5. CONCLUSIONS
A new higher-order plate theory has been developed to more accurately approximate five S and six A modes of Lamb
waves in composite laminates with less computational effort. According to the authors’ experience, the higher-plate
theory runs almost fifty times faster than 3-D elasticity theory. The computational procedures to analytically obtain
phase and group velocity dispersions are discussed. Furthermore, the characteristic wave curves are introduced to
analyze the anisotropy of Lamb wave propagation. Then numerical results from the higher-order plate theory are
compared with the exact solutions for the cases of Lamb waves in both lamina and laminate. From these numerical
examples, it is seen that the higher-order can provide considerable accuracy for A modes over a wide frequency range up
to the peak-frequency of A1 mode. Moreover the higher-order plate theory gives good approximation for S modes in
relative low frequency. The reason for lower accuracy of S modes may be found from Eq. (3) that the displacement
expansion of S modes is one-order lower less than that of A modes. From the comprised results, it shows that the
proposed theory effectively approximates dispersive behaviors and wave curves of Lamb waves in laminates, and A0
mode Lamb wave would be more suited for the practice of SHM and NDE. Future study will apply the higher-order
theory to achieve real-time SHM for composites via Lamb-wave-based damage detection or imaging techniques.
ACKNOWLEDGEMENT
This research is supported by the Sensors and Sensor Networks Program from the National Science Foundation (Grant
No. CMS-0329878). The authors would like to thank NASA Langley Research Center providing two composites.
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