Finite Elements in Analysis and Design 89 (2014) 19–32
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Finite Elements in Analysis and Design
journal homepage: www.elsevier.com/locate/finel
Wave propagation analysis in laminated composite plates with
transverse cracks using the wavelet spectral finite element method
Dulip Samaratunga a, Ratneshwar Jha a,n, S. Gopalakrishnan b
a
b
Raspet Flight Research Laboratory, Mississippi State University, 114 Airport Road, Starkville, MS 39759, USA
Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India
art ic l e i nf o
a b s t r a c t
Article history:
Received 25 November 2013
Received in revised form
20 May 2014
Accepted 27 May 2014
Available online 20 June 2014
This paper presents a newly developed wavelet spectral finite element (WFSE) model to analyze wave
propagation in anisotropic composite laminate with a transverse surface crack penetrating part-through
the thickness. The WSFE formulation of the composite laminate, which is based on the first-order shear
deformation theory, produces accurate and computationally efficient results for high frequency wave
motion. Transverse crack is modeled in wavenumber–frequency domain by introducing bending
flexibility of the plate along crack edge. Results for tone burst and impulse excitations show excellent
agreement with conventional finite element analysis in Abaquss. Problems with multiple cracks are
modeled by assembling a number of spectral elements with cracks in frequency-wavenumber domain.
Results show partial reflection of the excited wave due to crack at time instances consistent with crack
locations.
& 2014 Elsevier B.V. All rights reserved.
Keywords:
Wave propagation
Composites
Spectral finite element
Transverse crack
Structural health monitoring
1. Introduction
Composite structures are increasingly used in aerospace, automotive, and many other industries due to several advantages over
traditional metals such as superior specific strength and stiffness,
lower weight, corrosion resistance, and improved fatigue life [1].
One of the main concerns that restrict wider usage of composites
is the damage mechanisms which are not well understood. Most
common damage types are delamination between plies, fiber–
matrix debonding, fiber breakage, and matrix cracking resulting
from fatigue, manufacturing defects, foreign object impact, etc.
Currently available non-destructive evaluation (NDE) methods
such as ultrasonic, radiography, and eddy-current are time consuming, expensive, and may need structural disassembly for
inspection. In addition, these techniques are not suitable for
damage detection during operational conditions such as aircraft
or space structures in service. Therefore, a structural health
monitoring (SHM) system capable of performing diagnostics
online and efficiently is urgently needed. In recent years, Lamb
wave based SHM techniques have been widely investigated for
damage detection in composite structures [2–10].
SHM is essentially an inverse problem where the measured
data are used to predict the condition of the structure. Therefore,
n
Corresponding author. Tel.: þ 1 662 325 3797; fax: þ 1 662 325 3864.
E-mail addresses: dulip@raspet.msstate.edu (D. Samaratunga),
jha@raspet.msstate.edu (R. Jha),
krishnan@aerospace.iisc.ernet.in (S. Gopalakrishnan).
http://dx.doi.org/10.1016/j.finel.2014.05.014
0168-874X/& 2014 Elsevier B.V. All rights reserved.
to be able to differentiate between damage and structural features,
prior information is required about the structure in its undamaged
state. This is typically in the form of a baseline data obtained from
the “healthy state” to use as reference for comparison with the test
case. Alternatively, a mathematical model, such as a conventional
finite element (FE) simulation, may be used to predict the
structural response for comparison purposes. However, small
damages such as transverse cracks (resulting from fiber breakage
in composite structures) need high frequency Lamb waves for
detection. Lowe et al. [11] observed a sinusoidal variation of the
reflection coefficient with the ratio of notch width to A0 mode
wavelength and determined that the reflection coefficient is
maximized for small notch depths when the ratio stands at 0.5.
In another observation, the reflection coefficient is maximized for
the notch depth to A0 mode wavelength ratio of 0.3–0.4. These
observations suggest that the diagnostic signal wavelength should
be comparable to damage size for SHM. Therefore, high frequency
excitation signals are generally used in SHM to identify minute
damages present in structures. However, for high frequency
excitation FE mesh size should be sufficiently refined (usually at
least 20 nodes per wavelength) in order to reach acceptable
solution accuracy which leads to large system size and computational cost. Thus FE simulation becomes prohibitive for SHM,
especially onboard diagnostics with limited resources. The spectral
finite element method (SFEM) yields models that are many orders
smaller than FE, making it highly suitable for wave propagation
based SHM. The transverse crack model developed here would be
useful for understanding interactions between Lamb waves and
20
D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32
cracks, optimizing excitation (interrogation) signal for damage
detection, and inference of damage type (based on damage–wave
interaction features).
The SFEM is essentially a finite element method formulated in
the frequency domain, which uses the exact solution to the wave
equations as its interpolating function. The frequency domain
formulation of SFEM enables direct relationship between output
and input through the system transfer function (or, frequency
response function). In SFEM, nodal displacements are related to
nodal tractions through frequency-wave number dependent stiffness matrix and mass distribution is captured exactly to derive the
accurate elemental dynamic stiffness matrix. Consequently, in the
absence of any discontinuities, one element is sufficient to model a
plate structure of any length. Fast Fourier transform based spectral
finite element (FSFE) method was popularized by Doyle [12], who
formulated FSFE models for isotropic 1-D and 2-D waveguides
including elementary rod, Euler Bernoulli beam, and thin plates.
Gopalakrishnan et al. [13] extensively investigated FSFE models for
beams and plates-with anisotropic and inhomogeneous material
properties. The FSFE method is very efficient for wave motion
analysis and it is suitable for solving inverse problems. However,
the required assumption of periodicity in time approximation
results in “wraparound” problem for smaller time window, which
totally distorts the response. In addition, for 2-D problems, FSFE
are essentially semi-infinite, that is, they are bounded only in one
direction [12]. Thus, the effect of lateral boundaries cannot be
captured and this can again be attributed to the global basis
functions of the Fourier series approximation of the spatial
dimension. The 2-D wavelet transform based spectral finite element (WSFE) method presented by Mitra and Gopalakrishnan
[14–16] overcomes the “wraparound” problem and can accurately
model 2-D plate structures of finite dimensions.
Considerable work has been done for FE modeling of damages
in composite plates [17,18]; however, only a few researchers have
approached the problem using the spectral finite element method.
Palacz et al. [19] presented a spectral element to analyze transverse crack in isotropic rod. Gopalakrishnan and associates have
formulated FSFE models for (open and non-propagating) transverse cracks and delaminations in composite beams and plates
[20,21]. Levy and Rice [22] have shown that the slope discontinuity at both sides of a crack due to the bending moments is
proportional to the bending compliance of the crack and nominal
bending stress. Using the expressions given in [22], Khadem and
Rezaee [23] obtained slope discontinuity at both sides of the
hypothetical boundary along the crack in terms of the characteristics of the crack. Krawczuk et al. [9] used these expressions to
model a transverse crack using Fourier based spectral element for
isotropic plates. In a similar approach, Chakraborty et al. [24]
modeled wave propagation in a shear deformable composite
laminate with transverse crack using Fourier based spectral finite
element method. However, FSFE technique suffer from the ‘wraparound’ problem and cannot model finite structures, as noted
earlier.
This study develops WSFE model for a non-propagating and
open transverse crack in shear deformable laminated composite
plates. The developed model has the ability to simulate wave
propagation in 2D finite structures with transverse cracks, without
the ‘wraparound’ problem of the FSFE method. In this procedure,
the plate with one crack is divided into two sub-elements, one on
either side of the crack. The placement of crack is parallel to one of
the major axes (X or Y). Following the concept of crack flexibility
[22,23], a discontinuity is introduced at the crack interface on the
angular displacements about the axis parallel to the crack. The
crack flexibility depends on crack parameters: length, depth at the
center, and shape of the crack. A continuity condition is imposed at
crack interface for all other angular and linear displacements, as
followed by other researchers [9,24]. Composite laminates with
multiple cracks are modeled similarly and the WSFE results are
validated with conventional FE analysis using Abaquss.
The rest of paper is organized as follows. In the next section,
the WSFE modeling of shear deformable composite laminates is
introduced briefly. Next, the transverse crack formulation using
WSFE is presented. The accuracy and efficiency of the WSFE
modeling of crack is demonstrated by comparisons with FSFE
and FE simulations. Efficient simulation of multiple cracks using
WSFE is discussed. Concluding remarks are given in Section 5.
2. Wavelet spectral finite element modeling of shear
deformable composite laminates
The 2-D WSFE presented by Gopalakrishnan and Mitra [16]
overcomes the “wraparound” problem present in FSFE and can
accurately model plate structures of finite dimensions. This is
due to the use of compactly supported Daubechies scaling
functions [25] as basis for approximation of the time and spatial
dimensions. However, the WSFE plate formulation in [16] is based
on the classical laminated plate theory (CLPT) [26]. The CLPT based
formulations exclude transverse shear deformation and rotary
inertia resulting in significant errors for wave motion analysis at
high frequencies, especially for composite laminates which have
relatively low transverse shear modulus [27,28]. Wave propagation in composite laminates based on the first order shear
deformation theory (FSDT) [26], which accounts for transverse
shear and rotary inertia, yields accurate results comparable with
3-D elasticity solutions and experiments even at high frequencies
[27,28]. For isotropic materials FSDT is known to be exceptionally
accurate down to wavelengths comparable with the plate thickness h, whereas CLPT is of acceptable accuracy only for wavelengths greater than, say, 20 h [29]. FSDT based 2-D WSFE model
Z
Qy
w1
M xx
N yx Y , N yy
Qx
φ1
v1
ψ1
u1
N xy
− M yy
X , N xx
Fig. 1. (a) Plate element (b) nodal representation with DOFs.
w2
φ2
v
ψ22
u2
D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32
for pristine composite laminates developed by the present authors
is reported in [30,31]. Here, extensive time and frequency domain
results are presented to show the significance of FSDT based
spectral element formulation. The key steps in 2D WSFE element
formulation for shear deformable composite laminates are presented below and the reader is referred to [30] for further details.
2.1. Governing differential equations for wave propagation
Consider a laminated composite plate of thickness h with the
origin of the global coordinate system at the mid-plane of the
plate and Z axis being normal to the mid-plane as shown in
Fig. 1(a). Fig. 1(b) shows the corresponding nodal representation
with degrees of freedom (DOFs). Using FSDT, the governing partial
differential equations (PDEs) for wave propagation have five
degrees of freedom: u; v; w; ϕ; and; ψ . The terms u(x, y, t) and
v(x, y, t) are mid-plane (z ¼0) displacements along X and Y axes;
w(x, y, t) is transverse displacement in Z direction, and ψ ðx; y; tÞ
and ϕðx; y; tÞ are the rotational displacements about Y and X axes,
respectively. The displacements w, ϕ and ψ do not change along
the thickness (Z direction). The quantities (Nxx, Nxy, Nyy) are inplane force resultants, (Mxx, Mxy, Myy) are moment resultants, and
(Qx, Qy) denote the transverse force resultants.
There are five governing equations of motion for wave propagation as reported in the authors' previous work [30], but only one
EOM is presented here for conciseness. The first EOM based on the
FSDT displacement field is given by [26],
∂2 u
∂2 u
∂2 u
∂2 v
þ A66 2 þ A16 2
þ 2A16
∂x∂y
∂x2
∂y
∂x
∂2 v
∂2 v
∂2 ϕ
þA26 2 þ B11 2
þ ðA12 þ A66 Þ
∂y∂x
∂y
∂x
A11
þ 2B16
þ B26
∂2 ϕ
∂2 ϕ
∂2 ψ
∂2 ψ
þ B66 2 þ B16 2 þ ðB12 þ B66 Þ
∂x∂y
∂y∂x
∂y
∂x
∂2 ψ
∂2 u
∂2 ϕ
¼ I0 2 þ I1 2
2
∂y
∂t
∂t
ð1Þ
where the stiffness constants Aij, Bij, Dij and the inertial coefficients
I0, I1 and I2 are defined as
Z h=2
N p Z zq þ 1
½Aij ; Bij ; Dij ¼ ∑
Q ij ½1; z; z2 dz; fI 0 ; I 1 ; I2 g ¼
1; z; z2 ρ dz
q¼1
zq
h=2
ð2Þ
The term Q ij is the stiffness of the qth lamina in laminate
coordinate system, Np is the total number of laminae (plies), ρ is
the mass density, and K is the shear correction factor. The
associated natural boundary conditions are,
N nn ¼ n2x Nxx þ n2y N yy þ 2nx ny N xy ;
N ns ¼ nx ny N xx þ nx ny N yy þ ðn2x n2y ÞN xy
Q n ¼ Q x nx þ Q y ny
M nn ¼ n2x M xx þ n2y M yy þ 2nx ny M xy ;
M ns ¼ nx ny M xx þ nx ny M yy þ ðn2x n2y ÞM xy
ð3Þ
where n and s denote coordinates normal and tangential to the
plate edge, respectively; and nx and ny are unit normal into X and
Y directions, respectively. The boundary conditions at edges
parallel to Y axis can be derived by setting nx to 1 and ny to zero.
Then N nn and N ns become the specified normal and shear forces
into X and Y directions, M nn and M ns are the specified moments
about Y and X axes, and Q n is the applied shear force in Z
direction. Force and moment resultants are expressed in terms
of the stiffness constants Aij, Bij, Dij and displacements [26].
Without loss of generality in all essential aspects of the problem,
a laminate composed of an arbitrary number of orthotropic layers
such that the axes of material symmetry are parallel to the X–Y
21
axes of the plate (hence A16 ¼ A26 ¼A45 ¼B16 ¼B26 ¼D16 ¼ D26 ¼0) is
considered for further analysis.
2.2. Temporal approximation
The governing PDEs (Eq. (1)) and boundary conditions (Eq. (3))
have three variables (x, y, and t), and derivatives with respect to
them, making it very complex to solve. Therefore, Daubechies
compactly supported scaling functions [25] are used to approximate
the time variable which reduces the set of equations to PDEs in x and
y only. Compactly supported scaling functions have only a finite
number of filter coefficients with non-zero values, which enables
easy handling of finite geometries and imposition of boundary
conditions. The use of Daubechies compactly supported wavelets to
solve partial differential wave equations is explained in detail in [14].
Let the time-space variable u(x, y, t) be discretized at n points in
the time window (0, tf) and τ ¼ 0, 1, …, n 1 be the sampling
points, then t ¼ Δt τ where Δt is the time interval between two
sampling points. The function u(x, y, t) can be approximated at an
arbitrary scale as
uðx; y; tÞ ¼ uðx; y; τÞ ¼ ∑ uk ðx; yÞφðτ kÞ;
kAZ
ð4Þ
k
where uk ðx; yÞ are the approximation coefficients at a certain
spatial dimension (x and y) and φðτÞ are scaling functions
associated with Daubechies wavelets. The other translational and
rotational displacements v(x, y, t), w(x, y, t), ϕðx; y; tÞ and ψ(x, y, t)
are transformed similarly. The next step is to substitute these
approximations into Eq. (1). The approximated equation is then
multiplied with translations of the scaling function (φðτ jÞ; for
j ¼ 0; 1; …; n 1) and inner product is taken on both sides of the
equation. The orthogonal property of Daubechies scaling function
results in the cancelation of all the terms except when j¼k and
yields n simultaneous equations.
(
)
(
)
(
)
∂2 ϕj
∂ 2 uj
∂ 2 vj
A11
þ
A
Þ
þ
ðA
þB
66
12
11
∂y∂x
∂x2
∂x2
(
)
(
)
∂2 ψ j
∂ 2 uj
þ A66
þ ðB12 þB66 Þ
∂y∂x
∂y2
(
)
2
h i2
h i2
∂ ϕj
1
1
ð5Þ
¼ I 0 Γ uj þ I 1 Γ ϕj
þ B66
∂y2
where Γ is the first-order connection coefficient matrix obtained
after using the wavelet extrapolation technique for non-periodic
extension. Connection coefficients are the inner product between
the scaling functions and its derivatives [32]. The wavelet extrapolation approach of Williams and Amaratunga [33] is used for the
treatment of finite length data before computing the connection
coefficients. This method uses polynomials to extrapolate the
coefficients lying outside the finite domain and it is particularly
suitable for approximation in time and the ease to impose initial
values. This extrapolation technique addresses one of the serious
problems with Fourier based SFE method, namely, the ‘wraparound’
problem caused by treating the boundaries as periodic extensions.
By solving the ‘wraparound’ problem, WSFE method is able to
handle short waveguides and smaller time windows efficiently.
Next, the coupled PDEs are decoupled using eigenvalue analysis of
Γ1 following the procedure in [14]. The final decoupled form of the
reduced PDEs given in Eq. (5) can be written as
1
A11
^
^
∂2 ϕ
∂2 ψ^ j
∂2 ϕ
∂2 u^ j
∂2 v^ j
∂2 u^ j
j
j
þ ðA66 þ A12 Þ
þ B11 2 þ ðB12 þ B66 Þ
þ A66 2 þ B66 2
∂y ∂x
∂y ∂x
∂x2
∂x
∂y
∂y
^ ;
¼ I 0 γ 2j u^ j I 1 γ 2j ϕ
j
j ¼ 0; 1; …; n 1
1
ð6Þ
1
^
given
where u^ j is p
ffiffiffiffiffiffiffiffi by uj ¼ Φ uj , Φ is the eigenvector matrix of Γ
and iγ j ði ¼ 1Þ are the corresponding eigenvalues. Following
22
D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32
exactly similar steps, the other governing PDEs and the natural
boundary conditions are transformed to PDEs in spatial dimensions
(x and y) only. It should be mentioned here that the sampling rate
Δt should be less than a certain value to avoid spurious dispersion
in the simulation using WSFE. In Mitra and Gopalakrishnan [15], a
numerical study has been conducted from which the required Δt
can be determined depending on the order N of the Daubechies
scaling function and frequency content of the load.
2.3. Spatial (Y) approximation
The next step is to further reduce each of the transformed and
decoupled PDEs given by Eq. (6) (and similarly for the other
transformed governing equations and boundary conditions) to a
set of coupled ordinary differential equations (ODEs) using Daubechies scaling function approximation in one of the spatial (Y)
dimension. Similar to time approximation, the time transformed
variable u^ j is discretized at m points in the spatial window (0, LY),
where LY is the length in Y direction. Let ζ ¼0, 1, …, m 1 be the
sampling points, then y ¼ ΔY ζ where ΔY is the spatial interval
between two sampling points.
The function u^ j can be approximated by scaling function φ ζ
at an arbitrary scale as
u^ j ðx; yÞ ¼ u^ j ðx; ζ Þ ¼ ∑ u^ lj ðxÞφðζ lÞ;
lAZ
ð7Þ
l
where u^ lj are the approximation coefficients at a certain spatial
^ j ðx; yÞ;
dimension x. The other four displacements v^ j ðx; yÞ; w
ϕ^ j ðx; yÞ; and ψ^ j ðx; yÞ are similarly transformed. Following similar
steps as the time approximation, substituting the above approximations in Eq. (6) and taking inner product on both sides with the
translates of scaling functions φðζ iÞ, where i ¼0, 1, …, m 1 and
using their orthogonality property (which results in the cancelation of all the terms except when i¼ l), we get m simultaneous
ODEs. Eq. (1), which is temporally reduced in Eq. (6), can be
spatially reduced as
d u^ ij
1 l ¼ i þ N 2 dv^ lj 1
1 l ¼ i þ N 2 dψ^ lj 1
þ ðA66 þ A12 Þ
Ω þ ðB12 þ B66 Þ
Ω
∑
∑
dx2
ΔY l ¼ i N þ 2 dx i l
ΔY l ¼ i N þ 2 dx i l
2
A11
þ B11
2^
d ϕ
ij
dx2
þ A66
1
l ¼ iþN 2
∑
ΔY 2 l ¼ i N þ 2
2
u^ lj Ωi l þ B66
1
l ¼ iþN 2
∑
ΔY 2 l ¼ i N þ 2
ϕ^ lj Ω2i l ¼ I0 γ 2j u^ ij I1 γ 2j ϕ^ ij
ð8Þ
and Ωi l
where N is the order of Daubechies wavelet and Ω
are the connection coefficients for first- and second-order derivatives [32].
It can be seen from the ODE given by Eq. (8) that similar to time
approximation, even here certain coefficients u^ ij near the vicinity
of the boundaries (i ¼0 and i¼m 1) lie outside the spatial
window (0, LY) defined by i¼0, 1,…, m 1. Here, after expressing
the unknown coefficients lying outside the finite domain in terms
of the inner coefficients considering periodic extension, the ODEs
given by Eq. (8) can be written as a matrix equation:
( 2 )
2 ^
d ϕ
d u^ ij
dv^ ij
ij
1
A11
þ
B
þ
A
Þ½
Λ
þ
ðA
66
12
11
dx
dx2
dx2
dψ^ ij
1
1 þ A66 ½Λ 2 u^ ij
þ ðB12 þ B66 Þ½Λ dx
1
il
1
^ g ¼ I γ 2 fu^ g I γ 2 fϕ
^ g
þ B66 ½Λ 2 fϕ
0 j
1 j
ij
ij
ij
2
ð9Þ
where Λ is the first-order connection coefficient matrix obtained
after periodic extension.
The coupled ODEs given by Eq. (9) are decoupled using
eigenvalue analysis similar to thatpffiffiffiffiffiffiffiffi
done in temporal approximation. Let the eigenvalues be iβ i ði ¼ 1Þ, then the decoupled ODEs
1
corresponding to Eq. (9) are given by
2 ~
2
d ϕ
dψ~ ij
d u~ ij
dv~ ij
ij
þ B11
iβi ðA66 þ A12 Þ
iβi ðB12 þ B66 Þ
2
dx
dx
dx
dx2
2
2
~ ¼ I γ 2 u~ I γ 2 ϕ
~
βi A66 u~ ij β i B66 ϕ
0 j ij
1 j
ij
ij
A11
ð10Þ
where u~ ij (and similarly other transformed displacements) are
1
given by u~ ij ¼ Ψ u^ ij ; Ψ is the eigenvector and iβi are the
1
eigenvalues of connection coefficient matrix Λ . Exactly the same
procedure is followed to obtain decoupled form for the other
EOMs as presented in [30]. The natural boundary conditions
(Eq. (3)) are also transformed (temporal and spatial approximations) similarly. The final decoupled ODEs are used for 2D WSFE
formulation.
2.4. Spectral finite element formulation
The 2D WSFE has five degrees of freedom per node,
~ ; and ψ~ , as shown in Fig. 1(b). The decoupled ODEs
~ ij ; ϕ
u^ ij ; v~ ij ; w
ij
ij
~ ; ψ~ and the final
~ ij ; ϕ
presented above are solved for u^ ij ; v~ ij ; w
ij
ij
displacements u(x, y, t), v(x, y, t), w(x, y, t), ϕðx; y; tÞ and ψ ðx; y; tÞ
are obtained using inverse wavelet transform twice for spatial Y
dimension and time. For further steps, the subscripts j and i are
dropped for simplified notations and all of the following equations
are valid for j¼ 0, 1,…, n 1 and i¼0, 1,…, m 1 for each j. Let us
consider the displacement vector d~ as follows,
9
8~
uðxÞ >
>
>
>
>
>~
>
>
>
vðxÞ >
>
>
=
<
~
ð11Þ
d~ ¼ wðxÞ
>
>
>
~ ðxÞ >
>
>ϕ
>
>
>
>
>
>
: ~
ψ ðxÞ ;
Since the final set of equations (that is, Eq. (10) and the decoupled
form of other EOMs) is ordinary equations with constant coefficients, we assume a solution of d~ as
9
8~
u 0 ðxÞ >
>
>
>
>
>
>
~
>
>
v ðxÞ >
>
>
=
< 0
~ 0 ðxÞ
ð12Þ
d~ ¼ d~ 0 e iκx ; d~ 0 ¼ w
>
>
>
~ ðxÞ >
>
>ϕ
>
>
0
>
>
>
>
: ~
ψ 0 ðxÞ ;
where κ stands for the wavenumber. Substituting Eq. (12) into the
Eq. (10), the set of equations can be posed in Polynomial Eigenvalue
Problem (PEP) form [13]. PEP uses the concept of the latent roots and
right latent eigenvector of the system matrix (also called the wave
matrix) for computing the wavenumber and the amplitude ratio
matrix. The PEP of above Eq. (10) and other EOMs can be expressed as,
fA2 κ 2 þA1 κ þ A0 gfd~ 0 g ¼ 0
ð13Þ
The matrices A2 ; A1 ; A0 are given in the Appendix.
The wavenumbers κ are obtained as eigenvalues of the PEP
given by Eq. (13). Similarly, the vector d~ 0 is the eigenvector
(representing wave amplitudes) corresponding to each of the
wavenumbers. The solution of Eq. (13) gives a 5 10 eigenvector
matrix of the form
½R ¼ ½fd0 g1 ; fd0 g2 ; …; fd0 g10 ð14Þ
and the solution, d~ can be written as
~ ¼ ½R½Θfag
fdg
where Θ is a diagonal matrix with the diagonal terms
diag½Θ ¼ fe k1 x ; e k1 ðLX xÞ ; e k2 x ; e k2 ðLX xÞ ;
e k3 x ; e k3 ðLX xÞ ; e k4 x ; e k4 ðLX xÞ ; e k5 x ; e k5 ðLX xÞ g
ð15Þ
D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32
23
Here, fagT ¼ fC 1 ; C 2 ; ⋯; C 10 g are the unknown coefficients which can
be determined as described in [30]. Finally, the transformed nodal
e
e
forces fF~ g and transformed nodal displacements fu~ g are related by
where ½R36 is the amplitude ratio matrix, and
fF~ g ¼ ½K~ fu~ g
½Θ2 66 ¼ diag½e ik1 ðL1 þ xÞ ; e ik2 ðL1 þ xÞ ; e ik3 ðL1 þ xÞ ; e ik1 ðL ðL1 þ xÞÞ ;
e
e
e
ð16Þ
where ½K~ is the exact elemental dynamic stiffness matrix. The
solution of Eq. (16) and the assembly of the elemental stiffness
matrices to obtain the global stiffness matrix are similar to the FE
method. One major difference is that the time integration in FE uses
a suitable finite difference scheme; however, the SFEM performs
dynamic stiffness generation, assembly, and solution through a
double do-loop over frequency and horizontal wavenumber.
Although this procedure is computationally expensive, the problem
size
in
SFEM
is
very
small
which
keeps
overall
low computational cost. Another major difference is that unlike FE
method, SFEM deals with only one dynamic stiffness matrix and
hence matrix operation and storage require minimum
computations.
½Θ1 66 ¼ diag½e ik1 x ; e ik2 x ; e ik3 x ; e ik1 ðL1 xÞ ; e ik2 ðL1 xÞ ; e ik3 ðL1 xÞ e
3. Transverse crack modeling with wavelet spectral finite
element
Although the WSFE method is applicable to general laminates,
this study considers symmetric laminates wherein bendingextension coupling does not exist. For symmetric laminates, the
equations of motion decouple into two sets of equations governing
in-plane and transverse motions. The present model considers
transverse motion of such laminates. A schematic diagram of the
plate element with a crack (penetrating part-through the thickness and non-propagating) is shown in Fig. 2. The length of the
element in the X direction is Lx and the plate has finite dimension
of Ly in the Y direction. The crack is located at a distance of L1 from
the left edge of the plate (origin of the coordinate system) and has
a length of 2c in the Y direction.
Two sets of nodal displacements fw1 ; ϕ1 ; φ1 g and fw2 ; ϕ2 ; φ2 g
are assumed on the left and right hand side of a hypothetical boundary
along the crack. In the wavenumber–frequency domain, these dis~ ;φ
~ ;φ
~ 1; ϕ
~ 2; ϕ
~ 1 g and fw
~ 2 g and
placements are represented as fw
1
2
expressed by
9
9
8
8
~
~
w
w
>
>
=
=
< 1>
< 2>
ϕ~ 1 ¼ ½R3X6 ½Θ1 6X6 fC 1 g6X1 ;
ϕ~ 2 ¼ ½R3X6 ½Θ2 6X6 fC 2 g6X1
>
>
>
;
: ψ~ ;
: ψ~ >
1
2
e ik2 ðL ðL1 þ xÞÞ ; e ik3 ðL ðL1 þ xÞÞ ð18Þ
are square matrices containing exponential terms. A total of 12
unknown constants of fC 1 g and fC 2 g are to be determined using the
boundary conditions. These constants can be expressed as a function
of the nodal spectral displacements using the boundary conditions as
follows,
1. At the left edge of the element ðx ¼ 0Þ
ϕ~ 1 ¼ q~ 2 ;
~ 1 ¼ q~ 1 ;
w
ψ~ 1 ¼ q~ 3
ð19Þ
~ ; ψ~ gand x ¼ 0 for
~ 1; ϕ
2. At the crack interface (x ¼ L1 for fw
1
1
~
~ 2 ; ϕ 2 ; ψ~ 2 g)
fw
~ 1 ¼w
~ 2 ; ψ~ 1 ¼ ψ~ 2 (Continuity of transverse displacement
w
and slope about X-axis across the crack).
ϕ~ 1 ϕ~ 2 ¼ θM xx (Discontinuity of slope about Y-axis across the
crack).
M xx1 ¼ M xx2 ; M xy1 ¼ M xy2 (Continuity of bending moments).
Q x1 ¼ Q x2 ðContinuity of shear forceÞ:
ð20Þ
~ 2 ).
3. At the right edge of the element (x ¼ L L1 for w
ϕ~ 2 ¼ q~ 5 ;
~ 2 ¼ q~ 4 ;
w
ψ~ 2 ¼ q~ 6
ð21Þ
The terms q~ 1 ; :::; q~ 6 are nodal spectral displacements at the two
ends. The term θ is the non-dimensional bending flexibility of the
plate along the crack edge. (The method to calculate θ is given in
Section 3.1) The quantities M xx1 ; M xy1 and Q x1 are the bending
moments and shear force expressed in displacement terms, and
M xx2 ; M xy2 , Q x2 are obtained similarly. These boundary conditions
can be expressed in the matrix form as
2
M1
6
½MfCg ¼ 4 M 2
0
0
3
M3 7
5fCg ¼
M4
ð17Þ
8 9
u1 >
>
>
>
>
=
<0 >
ð22Þ
0 >
>
>
>
>
;
: >
u2
where u1 ¼ fq~ 1 ; q~ 2 ; q~ 3 gT , u2 ¼ fq~ 4 ; q~ 5 ; q~ 6 gT , M 1 ; M 4 A ℂ3X6 , M 2 ; M 3 A
ℂ6X6 and C ¼ fC 1 ; C 2 gT
The shear force and bending moments at the left and right
boundaries of the plate can be written as,
Y
9
8
V x1 ð0Þ >
>
>
>
>
>
>
>
>
M xx1 ð0Þ >
>
>
>
>
>
>
>
=
< M yy1 ð0Þ >
V x2 ðLÞ >
>
>
>
>
>
>
>
>
M xx2 ðLÞ >
>
>
>
>
>
>
>
;
: M yy2 ðLÞ >
2c
LY
X
L1
LX
Fig. 2. Plate element with transverse crack.
¼
9
8 ~
Q1 >
>
>
>
>
> ~
>
>
>
M xx1 >
>
>
>
>
>
>
>
~ yy1 >
=
<M
>
>
Q~ 2 >
>
>
>
>
>
>
~ xx2 >
>
>
M
>
>
>
>
>
>
;
:M
~
"
¼
P1
0
0
P2
#
fCg;
P 1 ; P 2 A C3X6
ð23Þ
yy2
Sub-matrices M i ; i ¼ 1; :::; 4 and P 1 ; P 2 can be found in the
Appendix. Combining Eqs. (22) and (23),
( )
u1
T
~
~
~
~
~
~
~
fQ 1 ; M xx1 ; M yy1 ; Q 2 ; M xx2 ; M yy2 g ¼ ½K ð24Þ
u2
where K~ is the frequency dependent dynamic stiffness matrix of
the spectral plate element with transverse (open and non-
24
D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32
propagating) crack and is given by
"
#
P1 0
½ N 1 N 4 ; where N ¼ ½ N 1
½K~ ¼
0
P2
N2
N3
N 4 ¼ ½M 1
ð25Þ
Here the stiffness matrix K~ is a square matrix with dimensions of
6 6. The sub matrices Pj (j¼1, 2) and Ni (i¼1, …, 4) have
dimensions of 6 12 and 12 3 consecutively.
3.1. Bending flexibility due to a surface crack
The bending flexibility of the plate with a crack can be
expressed in dimensionless form [9,22–24] as,
6H 0
y
θðyÞ ¼
αbb f ðyÞFðyÞ; y ¼
ð26Þ
Ly
Ly
where H is the thickness of the plate, FðyÞ is a correction function
and α0bb is the dimensionless bending compliance coefficient at the
crack center given by
Z h0
1
α0bb ¼
ξð1:99 2:47ξ þ 12:97ξ2
H
0
h
3
4
ð27Þ
23:117ξ þ 24:80ξ Þ2 dh; ξ ¼
H
for the range, 0 o ξ o 0:7. Here h is the depth of crack at any given
y and h0 is the central crack depth. The crack shape function f ðyÞ is
defined as,
2
f ðyÞ ¼ e ðy y0 Þ
2
ðe2 =2c Þ
;
c¼
c
;
Ly
y0 ¼
y0
Ly
ð28Þ
where e is the base of natural logarithm. The correction factor FðyÞ
is given as,
FðyÞ ¼
2c=H þ 3ðν þ 3Þð1 νÞα0bb ½1 f ðyÞ
2c=H þ 3ðν þ 3Þð1 νÞα0bb
new result can be used again to replace Mxx, and this iteration can
be continued until the convergence is achieved [24]. In this work,
no iteration is performed as the objective is to show the qualitative
change induced in the displacement field due to the presence of
a crack.
4. Results and discussions
The formulated 2D WSFE is used to study transverse wave
propagation in several composite laminates (graphite-epoxy, AS4/
3501) having single and multiple part-through the thickness
cracks. WSFE results are first compared with Fourier transform
based spectral finite element results. Validations with (conventional) FE are performed using Abaqussstandard implicit simulations employing nine-node shell elements with reduced
integration (S9R5), which is shear flexible and able to model
multiple layers [34]. In the numerical examples provided, the
properties of graphite-epoxy, AS4/3501 are taken as follows:
E1 ¼144.48 GPa,
E2 ¼E3 ¼ 9.63 GPa,
G23 ¼G13 ¼G12 ¼ 4.128 GPa,
ν23 ¼0.3, ν13 ¼ ν12 ¼0.02, and ρ ¼1389 kg m 3. Time domain
responses are studied using two types of input excitations, namely
sinusoidal tone burst and impulse (Fig. 4).
Tone burst is essentially a windowed sinusoidal signal with a
limited number of frequencies around the center frequency and it
is narrow banded in the frequency domain. This type of input is
used when a minimal number of frequencies are desired in the
response to minimize wave dispersion. On the other hand, impulse
excitation consists of multiple frequencies (due to the broadband
nature in frequency domain) which cause multiple wave fronts
being propagated through a waveguide. The spatial distribution of
the input load along the Y-axis (in Fig. 2) is given by
FðYÞ ¼ e ðY=αÞ
2
ð29Þ
Fig. 3 presents the variation of θ for a few different values of
crack parameters.
Now, in order to obtain the slope discontinuity (θðyÞM xx in
Eq. (20)) one would require the knowledge of bending moment at
the crack location which is not known a priori. Therefore an initial
assumption is made that the bending moment of the cracked plate
is not greatly disturbed from that of the undamaged plate at the
crack location. Using this, a first approximation of the displacement and stress field for the cracked plate can be obtained. This
Fig. 3. Variation of θ with nondimensional Y along the crack (plate dimensions
Lx ¼ Ly ¼ 0.5 m, H¼ 0.01 m).
ð30Þ
where α (as specified for each analysis) is a constant and can be
varied to change Y-distribution of the load. The 2D WSFE is
formulated with Daubechies scaling function of order N ¼ 22 for
temporal and N ¼4 for spatial approximations. The time sampling
rate is Δt ¼1–2 ms (as specified for each analysis), while the spatial
sampling rate ΔY is varied according to ΔY ¼ LY =ðm 1Þ depending on LY and the number of spatial discretization points m.
4.1. Comparisons with FSFE results
Wave propagation analysis of finite length structures using
conventional spectral finite element analysis based on Fourier
transform (FSFE) requires the structures to be damped or use of a
throw-off element to dissipate energy out of the system [12,13].
In addition, the time window should be large enough to remove
the wraparound problem associated with FSFE. The time window
is dependent on the value of damping and the length of the
structure, and it is required to be higher for lightly damped short
length structures. WSFE is completely free from these constraints
and the accuracy of solution is independent of such time window
restrictions or energy dissipation by means of damping or throwoff elements [14]. Fig. 5 shows a composite plate (with transverse
crack) considered for wave motion analysis using FSFE and the
current WSFE formulation. The plate considered here is semiinfinitely long into X direction and a crack is placed at L1 ¼0.5 m
distance away from the free end. Relatively large lateral spatial
window (LY ¼ 5 m) is chosen due to the Fourier series representation of the spatial dimension in FSFE where finite lateral boundary
conditions cannot be imposed. A symmetric ply sequence of [0]10
with each ply having a thickness of 1 mm is considered. Time
sampling rate is set at Δt ¼2 ms with the number of spatial
D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32
25
Fig. 4. (a) Unit impulse and (b) 3.5 cycle tone burst input loads with their FFTs.
4096 μs (Fig. 6(c)). The FSFE prediction using the largest time
window (Tw ¼4096 μs) is free from spurious distortions and
matches WSFE results very well. Thus, the developed WSFE
method yields substantial reduction in computational cost as the
time window is directly related to the system size.
4.2. Validation with conventional finite element simulations
Fig. 5. Composite plate with infinite length into X direction (width ¼LY, and
thickness¼0.01 m).
discretization points m ¼64 (ΔY ¼ 5=64 m) and α ¼ 0:1 in Eq. (30).
Fig. 6 shows transverse velocities at the tip of the plate due to a
tone burst input load at the same location (edge AB in Fig. 5). The
incident tone burst appears first in the response and the second
wave packet with relatively smaller amplitude is the reflection
from the crack. As mentioned earlier, the FSFE solution is highly
dependent upon analysis time window as illustrated in Fig. 6.
Three time windows of 1024 μs, 2048 μs, and 4096 μs are used for
FSFE solutions, whereas the time window of 1024 μs is sufficient
to obtain accurate WSFE results. It is observed that for
Tw ¼1024 μs, FSFE solution is highly distorted and spurious wave
components appear. The accuracy of FSFE results gradually
increase with increasing time windows of 2048 μs (Fig. 6(b)) and
The developed 2D WSFE model for transverse crack is validated
with FE simulations using Abaquss standard implicit direct
solver [34]. Fig. 7 shows the graphite-epoxy, AS4/3501, cantilever
laminate used in this numerical example. The dimensions of the
laminate are LX ¼ LY ¼ 0:5 m with a ply sequence of [0]10 where
each ply has a thickness of 1 mm. Responses are presented for the
two types of inputs, tone burst and unit impulse (shown in Fig. 4).
The tone burst has a central frequency of 20 kHz, while the unit
impulse has duration of 50 μs and occurs between 100 and 150 μs,
with a frequency content of 44 kHz. For WSFE analysis, time
sampling rate is Δt ¼ 1 ms with the number of spatial discretization
points m ¼64 (ΔY ¼ 1=64 m). The input load is applied along edge
AB with the load variation given by Eq. (30) with α ¼0.05.
The FE simulation in Abaquss exploits symmetry of the
problem about the centerline (parallel to edges AC and BD) and
models half-plate. The model has 7688, 9-noded shell elements
(S9R5) with the total number of nodes equal to 31,250 ensuring
the model captures all the modes at the given excitation frequency. The crack is modeled using 62 line spring elements (LS6)
26
D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32
Fig. 6. Transverse velocity response at the tip due to 20 kHz tone burst input load applied at the same location: WSFE time window Tw ¼1024 μs for all cases; FSFE time
windows (a) Tw ¼ 1024 μs (b) Tw ¼ 2048 μs (c) Tw ¼4096 μs.
D
B
L1
A
C
LY
LX
Fig. 7. Uniform cantilever plate with a transverse crack (length ¼ LX, width ¼ LY, and
thickness¼0.01 m).
developed for modeling part-through cracks in shells [34]. The
crack depth is 50% of the laminate thickness (that is, 0.005 m).
Fig. 8 shows the comparison of WSFE and FEM results at the tip
and middle of the plate with the crack located in the middle of
plate (L1 ¼0.25 m) and the tone burst input. It is observed that the
formulated 2D WSFE element with transverse crack captures the
wave scattering due to the crack and the results match very well
with FE simulations. For the response at the plate tip (Fig. 8(a)),
the incident wave (the first wave packet) propagates through the
plate and part of this forward propagating wave is reflected due to
the crack. This partial reflection is observed as the second wave
packet with relatively smaller amplitude. The rest of the incident
wave propagates beyond the crack and reflects completely from
the fixed boundary (CD in Fig. 7), which is the third wave packet.
The response at the center of the plate (Fig. 8(b)) shows that WSFE
accurately captures the incident wave followed by the fixed
boundary reflection.
Fig. 9 shows the response of 2D WSFE with transverse crack has
excellent match with FE simulation for the impulse input load. For
the tip response (Fig. 9(a)), the incident impulse occurs between
100 and 150 ms, gets partially reflected from the crack around
400 ms, and the fixed boundary reflection starts to appear around
700 ms. The broad-band response at the center of the plate (Fig. 9
(b)) also shows excellent match with FE including the fixed
boundary reflection. Computational efficiency is one key advantage of WSFE over conventional FEM. For the results presented in
Fig. 8, computational time for WSFE and FE (Abaquss) simulations
were 26 s and 4102 s, respectively (PC with 8-core, i7 processor).
Thus, WSFE resulted in reduction of computational time by more
than 2 orders of magnitude, even though only half-plate was
modeled in Abaquss using symmetry of the problem.
4.3. Identification of presence and location of crack using WSFE
The WSFE formulation presented in this paper may be used to
identify presence and location of cracks through wave scattering
due to a crack. The cantilever plate and loads used in the previous
section are used for the numerical results given here. Fig. 10 shows
D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32
27
Incident wave
Crack reflection
Incident wave
Fixed boundary reflection
Fig. 8. Transverse velocity using 2D WSFE and FE (Abaquss) for 20 kHz tone burst
input applied at tip and response sensed at (a) tip (b) middle of the plate.
Computational time for WSFE and FE simulations were 26 s and 4102 s, respectively, using a PC with 8-core, i7 processor.
the transverse velocities at the plate tip (which is also the
excitation point) due to a tone burst input. For the healthy case
(Fig. 10(a)), the first wave packet is the incident wave and the
second larger wave packet is the reflection from the fixed
boundary which arrives 582 ms after the excitation. Based on this
arrival time, the group velocity of the flexural wave mode at this
excitation frequency is calculated to be 1718 m/s. In the damaged
plate response (Fig. 10(b)), there is an additional wave packet with
smaller amplitude, which appears soon after the incident wave.
Comparing this with the healthy plate response, it can be inferred
that the additional wave packet is reflection from the crack.
This reflected wave from the crack appears approximately 290 ms
after the incident wave. With the group velocity data available
(1718 m/s) the distance to the crack can now be determined to be
ð1718 290E 06=2Þ ¼ 0:25 m. Therefore the crack location can be
determined when time of flight of reflected wave is used along
with group velocity data. Fig. 11 shows the response of the plate
for the same tone burst input with crack placed at 0.15 m and
0.35 m away from the tip (excitation point). In these responses too,
the arrival time of additional wave packets indicate that they
originate from the crack. In addition, when the crack is placed very
close to the tip (Fig. 11(b)), there are multiple reflections occurring
before the fixed boundary reflection arrives.
Fig. 12 shows the response for an impulse input load at the tip
with crack placed at 0.15 m, 0.25 m, and 0.35 m from the excitation point. A consistent reflection of the input impulse from the
crack is observed for all three locations of the crack. As indicated in
Crack reflection
Fixed boundary reflection
Fig. 9. Transverse velocity using 2D WSFE and FE for impulse input applied at tip
and response sensed at (a) tip and (b) middle of the plate.
Fig. 10. Response at the tip of 0.5 0.5 m2 [0]10 laminate for a tone burst input:
(a) healthy plate, and (b) crack at 0.25 m from the tip.
the figure, the presence of crack does not change peak amplitude
or the group velocity at which the pulse is propagated. However,
there is an additional reflection from the crack arriving before
reflection from the fixed boundary.
28
D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32
4.4. Multiple cracks analysis
Fig. 11. Response at the tip of 0.5 0.5 m [0]10 laminate for a tone burst input:
(a) healthy plate, and (b) crack at 0.15 m from the tip, and (c) crack at 0.35 m from
the tip.
The dynamic stiffness matrix of WSFE can be assembled in the
frequency-wavenumber domain to study the presence of multiple
cracks within a laminate. This assembling procedure is similar to
the conventional FE method, except that the frequency domain
approach is used [19,31]. A numerical example is presented here to
demonstrate the ability of 2D WSFE method to simulate multiple
cracks efficiently. Here a laminate with thickness and ply layup
sequence different from above examples is used in order to show
the flexibility of the model to analyze more general situations. An
8-ply graphite-epoxy, AS4/3501, laminate with a layup sequence
of [0/90]2S is considered. The laminate with two transverse cracks
(Fig. 13(a)) is divided into two sub-laminates along a dashed line
as shown in Figs. 13(b) and (c). The placement of this dashed line is
somewhat arbitrary given that it is somewhere in between the two
cracks. The in-plane dimensions are LX1 ¼LX2 ¼0.5 m, and
LY ¼1.0 m while the thickness is kept at 0.008 m. The distance to
crack 1 along the X-axis from the origin of its local coordinate
system (n1) is 0.375 m and the distance to crack 2 from n3 is
0.25 m.
The dynamic stiffness matrix of each element (sub-laminate) is
represented by Eq. (16). During assembly, nodes n2 and n3 coincide
and the dynamic stiffness matrix for a laminate with two cracks is
obtained. A tone burst load is applied at the tip of the laminate
(at node n4 in Fig. 13(c)) which has a spatial distribution given by
Eq. (30) with α ¼ 0:03. Fixed boundary conditions are applied at
the left edge of the laminate (node n1).
Fig. 14 shows the transverse velocity response at the tip of
laminate for three different cases: (a) healthy, (b) crack 1 present,
and (c) both crack 1 and crack 2 present. For the healthy plate, the
responses show the excitation wave packet (point n4) and its
reflection from the fixed boundary (point n1), When crack 1 is
Fig. 12. Response at the tip of 0.5 0.5 m [0]10 laminate for an impulse input: (a) healthy plate, (b) crack at 0.15 m, (c) crack at 0.25 m, and (d) crack at 0.35 m from the tip.
D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32
29
Z
Y
Crack 1
X
Crack 2
LY
L X1
3
L X2
3
n1
n3
2
2
Crack 2
1
1
n2
Crack 1
n4
Fig. 13. (a) Composite laminate with two cracks (b) element 1 and (c) element 2 with nodes.
Fig. 14. Transverse velocity measured at the tip of laminate when (a) healthy
(b) only crack 1 present, and (c) both crack 1 and crack 2 present.
introduced, there is an additional reflection arriving from the
crack. When both cracks are present, distinct responses from
the two cracks are clearly observed (Fig. 14(c)). Fig. 15 shows the
Fig. 15. Flexural wave propagation in composite laminate with two cracks at
various time instants: (a) 190 μs, (b) 320 μs, and (c) 540 μs.
transverse velocity response of each material point on the laminate along a straight line connecting nodes n4 and n1. The input
tone burst excited at the tip starts propagating towards the left as
30
D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32
5. Concluding remarks
A 2-D wavelet spectral finite element (WSFE) was presented for
accurate modeling of high frequency wave propagation in anisotropic composite laminates with a transverse, non-propagating
crack. The governing PDEs based on FSDT for laminated composite
plates were used accounting for transverse shear strains and
rotary inertia to produce accurate results for wave motion at high
frequencies. Daubechies compactly supported, orthonormal scaling functions were employed for temporal and spatial approximations of the governing PDEs and the resulting equations were
solved exactly to obtain the shape functions used in the element
formulation. Since the developed element captured the exact
inertial distribution, a single element was sufficient to model a
laminate of any dimension in the absence of discontinuities.
Further, due to the compact nature of Daubechies scaling functions, the WSFE was able to model finite waveguides without the
signal ‘wraparound’ problem (associated with Fourier transform
based spectral finite element). A transverse crack having an
arbitrary length, depth, and located parallel to any one side of
the plate was modeled in the wavenumber–frequency domain by
introducing bending flexibility of the plate along the crack edge.
The bending flexibility of the plate with crack was presented as a
function of crack parameters, that is, crack length and depth at the
center. Numerical examples using sinusoidal tone burst and
broadband unit impulse excitations showed excellent agreement
with conventional finite element simulations using shear flexible
elements in Abaquss. WSFE resulted in reduction of computational time by more than 2 orders of magnitude, even though only
half-plate was modeled in Abaquss using symmetry of the
problem. WSFE modeling of transverse crack showed superiority
over the FSFE results which showed signal distortions due to the
“wrap around” problem for shorter time windows. To model
multiple cracks within a laminate, the dynamic stiffness matrix
of elements in the frequency-wavenumber domain were
assembled, similar to the conventional finite element method.
Time of flight estimations of waves reflected from cracks were
successfully used to determine crack locations.
Acknowledgment
Fig. 16. Full field view of flexural wave propagation in composite laminate with
two cracks: (a) 190 μs, (b) 310 μs, and (c) 520 μs.
shown in Fig. 15(a). This wave packet interacts with crack 1 resulting in wave transmission and reflection due to the nature of the
boundary created by the crack (Fig. 15(b)). The transmitted wave
continues propagating towards the left, while the reflected wave
heads towards the tip of the plate. The transmitted portion of the
forward propagating wave is again scattered by crack 2 (Fig. 15(c)).
The propagation direction of the incident wave packet is indicated
by a solid arrow, while the reflected wave packet is indicated
by a dashed arrow with a number to identify the crack causing
the reflection. Following the same procedure, even more
complex situations with multiple cracks may be simulated using
WSFE, without much increase in computational time. Fig. 16
visualizes the entire laminate for the same case (as discussed in
Fig. 15), showing transverse wave propagation and scattering
due to the two cracks. Reflections from the lateral boundaries
are observed in Fig. 16(c), which is a clear advantage of WSFE
over FSFE.
The authors gratefully acknowledge partial funding for this
research through AFOSR Grant no. FA9550-09-1-0275 (Program
Managers – Dr. Victor Giurgiutiu and Dr. David Stargel).
Appendix
Matrices for polynomial eigenvalue problem (Eq. (13))
2
3
0
0
0
A11 0
60
7
0
0
A66 0
6
7
6
7
7
0
0
0
0
KA
A2 ¼ 6
55
6
7
60
7
0
0
D11 0
4
5
0
2
0
0
6 β ðA þ A Þ
12
66
6
6
0
A1 ¼ 6
6
6
0
4
0
0
D66
0
βðA12 þ A66 Þ
0
0
0
0
0
0
0
iKA55
0
0
iKA55
0
0
βðD12 þ D66 Þ
0
0
3
7
7
7
7
7
βðD12 þ D66 Þ 7
5
0
0
D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32
2
6
6
6
6
A0 ¼ 6
6
6
6
4
β A66 þ γ 2 I 0
0
0
0
0
0
β A22 þ γ 2 I 0
0
0
0
0
0
β KA44 þ γ 2 I 0
0
iβKA44
0
0
0
β D66 KA55 þ γ 2 I 2
0
0
0
iβ KA44
0
β D22 KA44 þ γ 2 I 2
2
2
2
Boundary condition matrices (Eq. (22)).
0
R11 R12 R13 R14 e ik1 L1 R15 e ik2 L1
B
M 1 ¼ @ R21 R22 R23 R24 e ik1 L1 R25 e ik2 L1
R31
0
R32
R11 e ik1 L
B R e ik1 L
M 4 ¼ @ 21
R31 e ik1 L
R33
R34 e ik1 L1
2
R16 e
ik3 L1
C
R26 e ik3 L1 A
ik3 L1
R36 e
R35 e ik2 L1
R12 e ik2 L
R22 e ik2 L
R13 e ik3 L
R23 e ik3 L
R14
R24
R15
R25
1
R16
R26 C
A
R32 e ik2 L
R33 e ik3 L
R34
R35
R36
M 2 ð1; 1 : 3Þ ¼ R1;1:3 e ik1:3 L1
M 2 ð1; 4 : 6Þ ¼ R1;4:3
M 2 ð2; 1 : 3Þ ¼ ðR2;1:3 þ iθðk1:3 D11 R2;1:3 þ βD12 R3;1:3 ÞÞe ik1:3 L1
M 2 ð2; 4 : 6Þ ¼ ðR2;4:6 þ iθð k1:3 D11 R2;4:6 þ β D12 R3;4:6 ÞÞ
M 2 ð3; 1 : 3Þ ¼ R3;1:3 e ik1:3 L1
M 2 ð3; 4 : 6Þ ¼ R3;4:3
M 2 ð4; 1 : 3Þ ¼ A55 ðik1:3 R1;1:3 R2;1:3 Þe ik1:3 L1
M 2 ð4; 4 : 6Þ ¼ A55 ðik1:3 R1;4:6 þ R2;4:6 Þ
M 2 ð5; 1 : 3Þ ¼ iðk1:3 D11 R2;1:3 þ βD12 R3;1:3 Þe ik1:3 L1
M 2 ð5; 4 : 6Þ ¼ iðk1:3 D11 R2;4:6 β D12 R3;4:6 Þ
M 2 ð6; 1 : 3Þ ¼ iD66 ðk1:3 R3;1:3 þ βR2;1:3 Þe ik1:3 L1
M 2 ð6; 4 : 6Þ ¼ iD66 ðk1:3 R3;4:6 β R2;4:6 Þ
M 3 ð1; 1 : 3Þ ¼ R1;1:3 e ik1:3 L1
M 3 ð1; 4 : 6Þ ¼ R1;4:3 e ik1:3 ðL L1 Þ
M 3 ð2; 1 : 3Þ ¼ R2;1:3 e ik1:3 L1
M 3 ð2; 4 : 6Þ ¼ R2;4:6 e ik1:3 ðL L1 Þ
M 3 ð3; 1 : 3Þ ¼ R3;1:3 e ik1:3 L1
M 3 ð3; 4 : 6Þ ¼ R3;4:3 e ik1:3 ðL L1 Þ
M 3 ð4; 1 : 3Þ ¼ A55 ðik1:3 R1;1:3 R2;1:3 Þe ik1:3 L1
M 3 ð4; 4 : 6Þ ¼ A55 ðik1:3 R1;4:6 þ R2;4:6 Þe ik1:3 ðL L1 Þ
M 3 ð5; 1 : 3Þ ¼ iðk1:3 D11 R2;1:3 þ β D12 R3;1:3 Þe ik1:3 L1
M 3 ð5; 4 : 6Þ ¼ ið k1:3 D11 R2;4:6 þ βD12 R3;4:6 Þe ik1:3 ðL L1 Þ
M 3 ð6; 1 : 3Þ ¼ iD66 ðk1:3 R3;1:3 þ β R2;1:3 Þe ik1:3 L1
M 3 ð6; 4 : 6Þ ¼ iD66 ð k1:3 R3;4:6 þ βR2;4:6 Þe ik1:3 ðL L1 Þ
P 1 ð1; 1 : 3Þ ¼ A55 ð ik1:3 R1;1:3 þ R2;1:3 Þ
P 1 ð1; 4 : 6Þ ¼ A55 ðik1:3 R1;4:6 þ R2;4:6 Þe ik1:3 L1
P 1 ð2; 1 : 3Þ ¼ iðk1:3 D11 R2;1:3 þ β D12 R3;1:3 Þ
P 1 ð2; 4 : 6Þ ¼ ið k1:3 D11 R2;4:6 þ βD12 R3;4:6 Þe ik1:3 L1
P 1 ð3; 1 : 3Þ ¼ iD66 ðk1:3 R3;1:3 þ β R2;1:3 Þ
P 1 ð3; 4 : 6Þ ¼ iD66 ð k1:3 R3;4:6 þ βR2;4:6 Þe ik1:3 L1
P 2 ð1; 1 : 3Þ ¼ A55 ð ik1:3 R1;1:3 þ R2;1:3 Þe ik1:3 L
P 2 ð1; 4 : 6Þ ¼ A55 ðik1:3 R1;4:6 þ R2;4:6 Þ
P 2 ð2; 1 : 3Þ ¼ iðk1:3 D11 R2;1:3 þ βD12 R3;1:3 Þe ik1:3 L
P 2 ð2; 4 : 6Þ ¼ ið k1:3 D11 R2;4:6 þ βD12 R3;4:6 Þ
P 2 ð3; 1 : 3Þ ¼ iD66 ðk1:3 R3;1:3 þ βR2;1:3 Þe ik1:3 L
P 2 ð3; 4 : 6Þ ¼ iD66 ð k1:3 R3;4:6 þ βR2;4:6 Þ
1
2
31
3
7
7
7
7
7
7
7
7
5
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