Wave propagation analysis in laminated composite plates withtransverse cracks using the wavelet spectral finite element method

Dulip Samaratunga a, Ratneshwar Jha a,n, S. Gopalakrishnan b

a Raspet Flight Research Laboratory, Mississippi State University, 114 Airport Road, Starkville, MS 39759, USAb Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India

a r t i c l e i n f o

Article history:Received 25 November 2013Received in revised form20 May 2014Accepted 27 May 2014Available online 20 June 2014

Keywords:Wave propagationCompositesSpectral finite elementTransverse crackStructural health monitoring

a b s t r a c t

This paper presents a newly developed wavelet spectral finite element (WFSE) model to analyze wavepropagation in anisotropic composite laminate with a transverse surface crack penetrating part-throughthe thickness. The WSFE formulation of the composite laminate, which is based on the first-order sheardeformation theory, produces accurate and computationally efficient results for high frequency wavemotion. Transverse crack is modeled in wavenumber–frequency domain by introducing bendingflexibility of the plate along crack edge. Results for tone burst and impulse excitations show excellentagreement with conventional finite element analysis in Abaquss. Problems with multiple cracks aremodeled 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 cracklocations.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Composite structures are increasingly used in aerospace, auto-motive, and many other industries due to several advantages overtraditional 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 compositesis the damage mechanisms which are not well understood. Mostcommon damage types are delamination between plies, fiber–matrix debonding, fiber breakage, and matrix cracking resultingfrom fatigue, manufacturing defects, foreign object impact, etc.Currently available non-destructive evaluation (NDE) methodssuch as ultrasonic, radiography, and eddy-current are time con-suming, expensive, and may need structural disassembly forinspection. In addition, these techniques are not suitable fordamage detection during operational conditions such as aircraftor space structures in service. Therefore, a structural healthmonitoring (SHM) system capable of performing diagnosticsonline and efficiently is urgently needed. In recent years, Lambwave based SHM techniques have been widely investigated fordamage detection in composite structures [2–10].

SHM is essentially an inverse problem where the measureddata are used to predict the condition of the structure. Therefore,

to be able to differentiate between damage and structural features,prior information is required about the structure in its undamagedstate. This is typically in the form of a baseline data obtained fromthe “healthy state” to use as reference for comparison with the testcase. Alternatively, a mathematical model, such as a conventionalfinite element (FE) simulation, may be used to predict thestructural response for comparison purposes. However, smalldamages such as transverse cracks (resulting from fiber breakagein composite structures) need high frequency Lamb waves fordetection. Lowe et al. [11] observed a sinusoidal variation of thereflection coefficient with the ratio of notch width to A0 modewavelength and determined that the reflection coefficient ismaximized for small notch depths when the ratio stands at 0.5.In another observation, the reflection coefficient is maximized forthe notch depth to A0 mode wavelength ratio of 0.3–0.4. Theseobservations suggest that the diagnostic signal wavelength shouldbe comparable to damage size for SHM. Therefore, high frequencyexcitation signals are generally used in SHM to identify minutedamages present in structures. However, for high frequencyexcitation FE mesh size should be sufficiently refined (usually atleast 20 nodes per wavelength) in order to reach acceptablesolution accuracy which leads to large system size and computa-tional cost. Thus FE simulation becomes prohibitive for SHM,especially onboard diagnostics with limited resources. The spectralfinite element method (SFEM) yields models that are many orderssmaller than FE, making it highly suitable for wave propagationbased SHM. The transverse crack model developed here would beuseful for understanding interactions between Lamb waves and

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Finite Elements in Analysis and Design

http://dx.doi.org/10.1016/j.finel.2014.05.0140168-874X/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: þ1 662 325 3797; fax: þ1 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).

Finite Elements in Analysis and Design 89 (2014) 19–32

cracks, optimizing excitation (interrogation) signal for damagedetection, and inference of damage type (based on damage–waveinteraction features).

The SFEM is essentially a finite element method formulated inthe frequency domain, which uses the exact solution to the waveequations as its interpolating function. The frequency domainformulation of SFEM enables direct relationship between outputand input through the system transfer function (or, frequencyresponse function). In SFEM, nodal displacements are related tonodal tractions through frequency-wave number dependent stiff-ness matrix and mass distribution is captured exactly to derive theaccurate elemental dynamic stiffness matrix. Consequently, in theabsence of any discontinuities, one element is sufficient to model aplate structure of any length. Fast Fourier transform based spectralfinite element (FSFE) method was popularized by Doyle [12], whoformulated FSFE models for isotropic 1-D and 2-D waveguidesincluding elementary rod, Euler Bernoulli beam, and thin plates.Gopalakrishnan et al. [13] extensively investigated FSFE models forbeams and plates-with anisotropic and inhomogeneous materialproperties. The FSFE method is very efficient for wave motionanalysis and it is suitable for solving inverse problems. However,the required assumption of periodicity in time approximationresults in “wraparound” problem for smaller time window, whichtotally distorts the response. In addition, for 2-D problems, FSFEare essentially semi-infinite, that is, they are bounded only in onedirection [12]. Thus, the effect of lateral boundaries cannot becaptured and this can again be attributed to the global basisfunctions of the Fourier series approximation of the spatialdimension. The 2-D wavelet transform based spectral finite ele-ment (WSFE) method presented by Mitra and Gopalakrishnan[14–16] overcomes the “wraparound” problem and can accuratelymodel 2-D plate structures of finite dimensions.

Considerable work has been done for FE modeling of damagesin composite plates [17,18]; however, only a few researchers haveapproached the problem using the spectral finite element method.Palacz et al. [19] presented a spectral element to analyze trans-verse crack in isotropic rod. Gopalakrishnan and associates haveformulated FSFE models for (open and non-propagating) trans-verse cracks and delaminations in composite beams and plates[20,21]. Levy and Rice [22] have shown that the slope disconti-nuity at both sides of a crack due to the bending moments isproportional to the bending compliance of the crack and nominalbending stress. Using the expressions given in [22], Khadem andRezaee [23] obtained slope discontinuity at both sides of thehypothetical boundary along the crack in terms of the character-istics of the crack. Krawczuk et al. [9] used these expressions tomodel a transverse crack using Fourier based spectral element forisotropic plates. In a similar approach, Chakraborty et al. [24]modeled wave propagation in a shear deformable compositelaminate with transverse crack using Fourier based spectral finite

element method. However, FSFE technique suffer from the ‘wrap-around’ problem and cannot model finite structures, as notedearlier.

This study develops WSFE model for a non-propagating andopen transverse crack in shear deformable laminated compositeplates. The developed model has the ability to simulate wavepropagation in 2D finite structures with transverse cracks, withoutthe ‘wraparound’ problem of the FSFE method. In this procedure,the plate with one crack is divided into two sub-elements, one oneither side of the crack. The placement of crack is parallel to one ofthe major axes (X or Y). Following the concept of crack flexibility[22,23], a discontinuity is introduced at the crack interface on theangular displacements about the axis parallel to the crack. Thecrack flexibility depends on crack parameters: length, depth at thecenter, and shape of the crack. A continuity condition is imposed atcrack interface for all other angular and linear displacements, asfollowed by other researchers [9,24]. Composite laminates withmultiple cracks are modeled similarly and the WSFE results arevalidated 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 isintroduced briefly. Next, the transverse crack formulation usingWSFE is presented. The accuracy and efficiency of the WSFEmodeling of crack is demonstrated by comparisons with FSFEand FE simulations. Efficient simulation of multiple cracks usingWSFE is discussed. Concluding remarks are given in Section 5.

2. Wavelet spectral finite element modeling of sheardeformable composite laminates

The 2-D WSFE presented by Gopalakrishnan and Mitra [16]overcomes the “wraparound” problem present in FSFE and canaccurately model plate structures of finite dimensions. This isdue to the use of compactly supported Daubechies scalingfunctions [25] as basis for approximation of the time and spatialdimensions. However, the WSFE plate formulation in [16] is basedon the classical laminated plate theory (CLPT) [26]. The CLPT basedformulations exclude transverse shear deformation and rotaryinertia resulting in significant errors for wave motion analysis athigh frequencies, especially for composite laminates which haverelatively low transverse shear modulus [27,28]. Wave propaga-tion in composite laminates based on the first order sheardeformation theory (FSDT) [26], which accounts for transverseshear and rotary inertia, yields accurate results comparable with3-D elasticity solutions and experiments even at high frequencies[27,28]. For isotropic materials FSDT is known to be exceptionallyaccurate down to wavelengths comparable with the plate thick-ness h, whereas CLPT is of acceptable accuracy only for wave-lengths greater than, say, 20 h [29]. FSDT based 2-D WSFE model

2u

1u1ψ

1φ1w

1v

2ψ

2φ2w

, xxX N

Z

, yyY N

yQ

yxN

xyN

xxM

yyM−

xQ

2v

Fig. 1. (a) Plate element (b) nodal representation with DOFs.

D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–3220

for pristine composite laminates developed by the present authorsis reported in [30,31]. Here, extensive time and frequency domainresults are presented to show the significance of FSDT basedspectral element formulation. The key steps in 2D WSFE elementformulation for shear deformable composite laminates are pre-sented 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 theorigin of the global coordinate system at the mid-plane of theplate and Z axis being normal to the mid-plane as shown inFig. 1(a). Fig. 1(b) shows the corresponding nodal representationwith degrees of freedom (DOFs). Using FSDT, the governing partialdifferential equations (PDEs) for wave propagation have fivedegrees of freedom: u; v; w; ϕ; and; ψ . The terms u(x, y, t) andv(x, y, t) are mid-plane (z¼0) displacements along X and Y axes;w(x, y, t) is transverse displacement in Z direction, and ψ ðx; y; tÞand ϕðx; y; tÞ are the rotational displacements about Y and X axes,respectively. The displacements w, ϕ and ψ do not change alongthe thickness (Z direction). The quantities (Nxx, Nxy, Nyy) are in-plane 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 propa-gation as reported in the authors' previous work [30], but only oneEOM is presented here for conciseness. The first EOM based on theFSDT displacement field is given by [26],

A11∂2u∂x2

þ2A16∂2u∂x∂y

þA66∂2u∂y2

þA16∂2v∂x2

þðA12þA66Þ∂2v∂y∂x

þA26∂2v∂y2

þB11∂2ϕ∂x2

þ2B16∂2ϕ∂x∂y

þB66∂2ϕ∂y2

þB16∂2ψ∂x2

þðB12þB66Þ∂2ψ∂y∂x

þB26∂2ψ∂y2

¼ I0∂2u∂t2

þ I1∂2ϕ∂t2

ð1Þ

where the stiffness constants Aij, Bij, Dij and the inertial coefficientsI0, I1 and I2 are defined as

½Aij;Bij;Dij� ¼ ∑Np

q ¼ 1

Z zqþ 1

zqQ ij½1; z; z2�dz; I0; I1; I2f g ¼

Z h=2

�h=21; z; z2

� �ρ dz

ð2ÞThe term Qij is the stiffness of the qth lamina in laminatecoordinate system, Np is the total number of laminae (plies), ρ isthe mass density, and K is the shear correction factor. Theassociated natural boundary conditions are,

Nnn ¼ n2xNxxþn2

yNyyþ2nxnyNxy;

Nns ¼ �nxnyNxxþnxnyNyyþðn2x �n2

yÞNxy

Qn ¼ QxnxþQyny

Mnn ¼ n2xMxxþn2

yMyyþ2nxnyMxy;

Mns ¼ �nxnyMxxþnxnyMyyþðn2x �n2

y ÞMxy ð3Þ

where n and s denote coordinates normal and tangential to theplate edge, respectively; and nx and ny are unit normal into X andY directions, respectively. The boundary conditions at edgesparallel to Y axis can be derived by setting nx to 1 and ny to zero.Then Nnn and Nns become the specified normal and shear forcesinto X and Y directions, Mnn and Mns are the specified momentsabout Y and X axes, and Qn is the applied shear force in Zdirection. Force and moment resultants are expressed in termsof 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 layerssuch that the axes of material symmetry are parallel to the X–Y

axes of the plate (hence A16¼A26¼A45¼B16¼B26¼D16¼D26¼0) isconsidered 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 tothem, making it very complex to solve. Therefore, Daubechiescompactly supported scaling functions [25] are used to approximatethe time variable which reduces the set of equations to PDEs in x andy only. Compactly supported scaling functions have only a finitenumber of filter coefficients with non-zero values, which enableseasy handling of finite geometries and imposition of boundaryconditions. The use of Daubechies compactly supported wavelets tosolve partial differential wave equations is explained in detail in [14].

Let the time-space variable u(x, y, t) be discretized at n points inthe time window (0, tf) and τ¼0, 1, …, n�1 be the samplingpoints, then t ¼Δtτ where Δt is the time interval between twosampling points. The function u(x, y, t) can be approximated at anarbitrary scale as

uðx; y; tÞ ¼ uðx; y; τÞ ¼∑kukðx; yÞφðτ�kÞ; kAZ ð4Þ

where ukðx; yÞ are the approximation coefficients at a certainspatial dimension (x and y) and φ τð Þ are scaling functionsassociated with Daubechies wavelets. The other translational androtational 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 theseapproximations into Eq. (1). The approximated equation is thenmultiplied with translations of the scaling function (φðτ� jÞ; forj¼ 0;1;…;n�1) and inner product is taken on both sides of theequation. The orthogonal property of Daubechies scaling functionresults in the cancelation of all the terms except when j¼k andyields n simultaneous equations.

A11∂2uj

∂x2

( )þðA66þA12Þ

∂2vj∂y∂x

( )þB11

∂2ϕj

∂x2

( )

þðB12þB66Þ∂2ψ j

∂y∂x

( )þA66

∂2uj

∂y2

( )

þB66∂2ϕj

∂y2

( )¼ I0 Γ1

h i2ujþ I1 Γ1

h i2ϕj ð5Þ

where Γ1 is the first-order connection coefficient matrix obtainedafter using the wavelet extrapolation technique for non-periodicextension. Connection coefficients are the inner product betweenthe scaling functions and its derivatives [32]. The wavelet extra-polation approach of Williams and Amaratunga [33] is used for thetreatment of finite length data before computing the connectioncoefficients. This method uses polynomials to extrapolate thecoefficients lying outside the finite domain and it is particularlysuitable for approximation in time and the ease to impose initialvalues. This extrapolation technique addresses one of the seriousproblems 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 tohandle short waveguides and smaller time windows efficiently.Next, the coupled PDEs are decoupled using eigenvalue analysis ofΓ1 following the procedure in [14]. The final decoupled form of thereduced PDEs given in Eq. (5) can be written as

A11∂2uj

∂x2þðA66þA12Þ

∂2vj∂y ∂x

þB11∂2ϕj

∂x2þðB12þB66Þ

∂2ψ j

∂y ∂xþA66

∂2uj

∂y2þB66

∂2ϕj

∂y2

¼ � I0γ2j uj� I1γ2j ϕj; j¼ 0;1;…;n�1 ð6Þ

where uj is given by uj ¼Φ�1uj, Φ is the eigenvector matrix of Γ1

and iγjði¼ffiffiffiffiffiffiffiffi�1

pÞ are the corresponding eigenvalues. Following

D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32 21

exactly similar steps, the other governing PDEs and the naturalboundary conditions are transformed to PDEs in spatial dimensions(x and y) only. It should be mentioned here that the sampling rateΔt should be less than a certain value to avoid spurious dispersionin the simulation using WSFE. In Mitra and Gopalakrishnan [15], anumerical study has been conducted from which the required Δtcan be determined depending on the order N of the Daubechiesscaling function and frequency content of the load.

2.3. Spatial (Y) approximation

The next step is to further reduce each of the transformed anddecoupled PDEs given by Eq. (6) (and similarly for the othertransformed governing equations and boundary conditions) to aset of coupled ordinary differential equations (ODEs) using Dau-bechies scaling function approximation in one of the spatial (Y)dimension. Similar to time approximation, the time transformedvariable uj is discretized at m points in the spatial window (0, LY),where LY is the length in Y direction. Let ζ¼0, 1, …, m�1 be thesampling points, then y¼ΔYζ where ΔY is the spatial intervalbetween two sampling points.

The function uj can be approximated by scaling function φ ζ� �

at an arbitrary scale as

ujðx; yÞ ¼ ujðx; ζÞ ¼∑luljðxÞφðζ� lÞ; lAZ ð7Þ

where ulj are the approximation coefficients at a certain spatialdimension x. The other four displacements vjðx; yÞ; wj ðx; yÞ;ϕjðx; yÞ; and ψ jðx; yÞ are similarly transformed. Following similarsteps as the time approximation, substituting the above approx-imations in Eq. (6) and taking inner product on both sides with thetranslates of scaling functions φðζ� iÞ, where i¼0, 1, …, m�1 andusing their orthogonality property (which results in the cancela-tion of all the terms except when i¼ l), we get m simultaneousODEs. Eq. (1), which is temporally reduced in Eq. (6), can bespatially reduced as

A11d2uij

dx2þðA66þA12Þ

1ΔY

∑l ¼ iþN�2

l ¼ i�Nþ2

dvljdx

Ω1i� lþðB12þB66Þ

1ΔY

∑l ¼ iþN�2

l ¼ i�Nþ2

dψ lj

dxΩ1

i� l

þB11d2ϕij

dx2þA66

1

ΔY2 ∑l ¼ iþN�2

l ¼ i�Nþ2uljΩ2

i� lþB661

ΔY2 ∑l ¼ iþN�2

l ¼ i�Nþ2ϕljΩ

2i� l ¼ � I0γ2j uij� I1γ2j ϕij

ð8Þ

where N is the order of Daubechies wavelet and Ω1i� l and Ω2

i� lare the connection coefficients for first- and second-order deriva-tives [32].

It can be seen from the ODE given by Eq. (8) that similar to timeapproximation, even here certain coefficients uij near the vicinityof the boundaries (i¼0 and i¼m�1) lie outside the spatialwindow (0, LY) defined by i¼0, 1,…, m�1. Here, after expressingthe unknown coefficients lying outside the finite domain in termsof the inner coefficients considering periodic extension, the ODEsgiven by Eq. (8) can be written as a matrix equation:

A11d2uij

dx2

( )þðA66þA12Þ½Λ1� dvij

dx

� �þB11

d2ϕij

dx2

þðB12þB66Þ½Λ1� dψ ij

dx

� �þA66½Λ1�2 uij

� �þB66½Λ1�2fϕijg ¼ � I0γ2j fuijg� I1γ2j fϕijg ð9Þ

where Λ1 is the first-order connection coefficient matrix obtainedafter periodic extension.

The coupled ODEs given by Eq. (9) are decoupled usingeigenvalue analysis similar to that done in temporal approxima-tion. Let the eigenvalues be iβiði¼

ffiffiffiffiffiffiffiffi�1

pÞ, then the decoupled ODEs

corresponding to Eq. (9) are given by

A11d2 ~uij

dx2� iβiðA66þA12Þ

d ~vij

dxþB11

d2 ~ϕ ij

dx2� iβiðB12þB66Þ

d ~ψ ij

dx

�β2i A66 ~uij�β2

i B66~ϕ ij ¼ � I0γ2j ~uij� I1γ2j

~ϕ ij ð10Þ

where ~uij (and similarly other transformed displacements) aregiven by ~uij ¼Ψ �1uij; Ψ is the eigenvector and iβi are theeigenvalues of connection coefficient matrix Λ1. Exactly the sameprocedure is followed to obtain decoupled form for the otherEOMs as presented in [30]. The natural boundary conditions(Eq. (3)) are also transformed (temporal and spatial approxima-tions) similarly. The final decoupled ODEs are used for 2D WSFEformulation.

2.4. Spectral finite element formulation

The 2D WSFE has five degrees of freedom per node,uij; ~vij; ~wij;

~ϕ ij; and ~ψ ij, as shown in Fig. 1(b). The decoupled ODEspresented above are solved for uij; ~vij; ~wij;

~ϕ ij; ~ψ ij and the finaldisplacements u(x, y, t), v(x, y, t), w(x, y, t), ϕðx; y; tÞ and ψ ðx; y; tÞare obtained using inverse wavelet transform twice for spatial Ydimension and time. For further steps, the subscripts j and i aredropped for simplified notations and all of the following equationsare valid for j¼0, 1,…, n�1 and i¼0, 1,…, m�1 for each j. Let usconsider the displacement vector ~d as follows,

~d ¼

~uðxÞ~vðxÞ~wðxÞ~ϕðxÞ~ψ ðxÞ

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð11Þ

Since the final set of equations (that is, Eq. (10) and the decoupledform of other EOMs) is ordinary equations with constant coeffi-cients, we assume a solution of ~d as

~d ¼ ~d0e� iκx; ~d0 ¼

~u0ðxÞ~v0ðxÞ~w0ðxÞ~ϕ0ðxÞ~ψ 0ðxÞ

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð12Þ

where κ stands for the wavenumber. Substituting Eq. (12) into theEq. (10), the set of equations can be posed in Polynomial EigenvalueProblem (PEP) form [13]. PEP uses the concept of the latent roots andright latent eigenvector of the system matrix (also called the wavematrix) for computing the wavenumber and the amplitude ratiomatrix. The PEP of above Eq. (10) and other EOMs can be expressed as,

fA2κ2þA1κþA0gf ~d0g ¼ 0 ð13ÞThe matrices A2; A1; A0 are given in the Appendix.

The wavenumbers κ are obtained as eigenvalues of the PEPgiven by Eq. (13). Similarly, the vector ~d0 is the eigenvector(representing wave amplitudes) corresponding to each of thewavenumbers. The solution of Eq. (13) gives a 5�10 eigenvectormatrix of the form

½R� ¼ ½fd0g1; fd0g2;…; fd0g10� ð14Þand the solution, ~d can be written as

f ~dg ¼ ½R�½Θ�fag ð15Þwhere Θ

is a diagonal matrix with the diagonal terms

diag½Θ� ¼ fe�k1x; e�k1ðLX � xÞ; e�k2x; e�k2ðLX � xÞ;

e�k3x; e�k3ðLX � xÞ; e�k4x; e�k4ðLX � xÞ; e�k5x; e�k5ðLX �xÞg

D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–3222

Here, fagT ¼ fC1;C2;⋯;C10g are the unknown coefficients which canbe determined as described in [30]. Finally, the transformed nodalforces f ~Feg and transformed nodal displacements f ~ueg are related by

f ~Feg ¼ ½ ~Ke�f ~ueg ð16Þwhere ½ ~Ke� is the exact elemental dynamic stiffness matrix. Thesolution of Eq. (16) and the assembly of the elemental stiffnessmatrices to obtain the global stiffness matrix are similar to the FEmethod. One major difference is that the time integration in FE usesa suitable finite difference scheme; however, the SFEM performsdynamic stiffness generation, assembly, and solution through adouble do-loop over frequency and horizontal wavenumber.Although this procedure is computationally expensive, the problemsize in SFEM is very small which keeps overalllow computational cost. Another major difference is that unlike FEmethod, SFEM deals with only one dynamic stiffness matrix andhence matrix operation and storage require minimumcomputations.

3. Transverse crack modeling with wavelet spectral finiteelement

Although the WSFE method is applicable to general laminates,this study considers symmetric laminates wherein bending-extension coupling does not exist. For symmetric laminates, theequations of motion decouple into two sets of equations governingin-plane and transverse motions. The present model considerstransverse motion of such laminates. A schematic diagram of theplate element with a crack (penetrating part-through the thick-ness and non-propagating) is shown in Fig. 2. The length of theelement in the X direction is Lx and the plate has finite dimensionof Ly in the Y direction. The crack is located at a distance of L1 fromthe left edge of the plate (origin of the coordinate system) and hasa length of 2c in the Y direction.

Two sets of nodal displacements fw1;ϕ1;φ1g and fw2;ϕ2;φ2gare assumed on the left and right hand side of a hypothetical boundaryalong the crack. In the wavenumber–frequency domain, these dis-placements are represented as f ~w1;

~ϕ1; ~φ1g and f ~w2;~ϕ2; ~φ2g and

expressed by

~w1~ϕ1

~ψ 1

8><>:

9>=>;¼ ½R�3X6½Θ1�6X6fC1g6X1;

~w2~ϕ2

~ψ 2

8><>:

9>=>;¼ ½R�3X6½Θ2�6X6fC2g6X1

ð17Þ

where ½R�3�6 is the amplitude ratio matrix, and

½Θ1�6�6 ¼ diag½e� ik1x; e� ik2x; e� ik3x; e� ik1ðL1 � xÞ; e� ik2ðL1 � xÞ; e� ik3ðL1 �xÞ�

½Θ2�6�6 ¼ diag½e� ik1ðL1 þ xÞ; e� ik2ðL1 þ xÞ; e� ik3ðL1 þxÞ; e� ik1ðL�ðL1 þxÞÞ;

e� ik2ðL�ðL1 þ xÞÞ; e� ik3ðL�ðL1 þxÞÞ� ð18Þ

are square matrices containing exponential terms. A total of 12unknown constants of fC1g and fC2g are to be determined using theboundary conditions. These constants can be expressed as a functionof the nodal spectral displacements using the boundary conditions asfollows,

1. At the left edge of the element ðx¼ 0Þ~w1 ¼ ~q1;

~ϕ1 ¼ ~q2; ~ψ 1 ¼ ~q3 ð19Þ

2. At the crack interface (x¼ L1for f ~w1;~ϕ1; ~ψ 1gand x¼ 0 for

f ~w2;~ϕ2; ~ψ 2g)

~w1 ¼ ~w2; ~ψ 1 ¼ ~ψ 2 (Continuity of transverse displacementand slope about X-axis across the crack).~ϕ1� ~ϕ2 ¼ θMxx (Discontinuity of slope about Y-axis across thecrack).Mxx1 ¼Mxx2; Mxy1 ¼Mxy2 (Continuity of bending moments).

Qx1 ¼Qx2 ðContinuity of shear forceÞ: ð20Þ

3. At the right edge of the element (x¼ L�L1 for ~w2).

~w2 ¼ ~q4;~ϕ2 ¼ ~q5; ~ψ 2 ¼ ~q6 ð21Þ

The terms ~q1; :::; ~q6 are nodal spectral displacements at the twoends. The term θ is the non-dimensional bending flexibility of theplate along the crack edge. (The method to calculate θ is given inSection 3.1) The quantities Mxx1;Mxy1 and Qx1 are the bendingmoments and shear force expressed in displacement terms, andMxx2;Mxy2, Qx2 are obtained similarly. These boundary conditionscan be expressed in the matrix form as

½M�fCg ¼M1 0M2 M3

0 M4

264

375fCg ¼

u1

00u2

8>>><>>>:

9>>>=>>>;

ð22Þ

where u1 ¼ f ~q1; ~q2; ~q3gT , u2 ¼ f ~q4; ~q5; ~q6gT , M1;M4Aℂ3X6, M2;M3Aℂ6X6 and C ¼ fC1;C2gT

The shear force and bending moments at the left and rightboundaries of the plate can be written as,

Vx1ð0ÞMxx1ð0ÞMyy1ð0ÞVx2ðLÞMxx2ðLÞMyy2ðLÞ

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

¼

~Q 1~Mxx1~Myy1

~Q 2~Mxx2~Myy2

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

¼P1 00 P2

" #fCg; P1; P2AC3X6 ð23Þ

Sub-matrices Mi; i¼ 1; :::; 4 and P1; P2 can be found in theAppendix. Combining Eqs. (22) and (23),

f ~Q 1; ~Mxx1; ~Myy1; ~Q 2; ~Mxx2; ~Myy2gT ¼ ½ ~K �u1

u2

( )ð24Þ

where ~K is the frequency dependent dynamic stiffness matrix ofthe spectral plate element with transverse (open and non-

XL

YL1L

2c

X

Y

Fig. 2. Plate element with transverse crack.

D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32 23

propagating) crack and is given by

½ ~K � ¼P1 00 P2

" #½N1 N4 �; where N¼ ½N1 N2 N3 N4 � ¼ ½M��1

ð25ÞHere the stiffness matrix ~K is a square matrix with dimensions of6�6. The sub matrices Pj (j¼1, 2) and Ni (i¼1, …, 4) havedimensions 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 beexpressed in dimensionless form [9,22–24] as,

θðyÞ ¼ 6HLy

� �α0bbf ðyÞFðyÞ; y¼ y

Lyð26Þ

where H is the thickness of the plate, FðyÞ is a correction functionand α0

bb is the dimensionless bending compliance coefficient at thecrack center given by

α0bb ¼

1H

� �Z h0

0ξð1:99�2:47ξþ12:97ξ2

�23:117ξ3þ24:80ξ4Þ2dh;ξ¼ hH

ð27Þ

for the range, 0oξo0:7. Here h is the depth of crack at any giveny and h0 is the central crack depth. The crack shape function f ðyÞ isdefined as,

f ðyÞ ¼ e�ðy�y0Þ2ðe2=2c2Þ; c¼ cLy; y0 ¼

y0Ly

ð28Þ

where e is the base of natural logarithm. The correction factor FðyÞis given as,

FðyÞ ¼ 2c=Hþ3ðνþ3Þð1�νÞα0bb½1� f ðyÞ�

2c=Hþ3ðνþ3Þð1�νÞα0bb

ð29Þ

Fig. 3 presents the variation of θ for a few different values ofcrack parameters.

Now, in order to obtain the slope discontinuity (θðyÞMxx inEq. (20)) one would require the knowledge of bending moment atthe crack location which is not known a priori. Therefore an initialassumption is made that the bending moment of the cracked plateis not greatly disturbed from that of the undamaged plate at thecrack location. Using this, a first approximation of the displace-ment and stress field for the cracked plate can be obtained. This

new result can be used again to replace Mxx, and this iteration canbe continued until the convergence is achieved [24]. In this work,no iteration is performed as the objective is to show the qualitativechange induced in the displacement field due to the presence ofa crack.

4. Results and discussions

The formulated 2D WSFE is used to study transverse wavepropagation in several composite laminates (graphite-epoxy, AS4/3501) having single and multiple part-through the thicknesscracks. WSFE results are first compared with Fourier transformbased spectral finite element results. Validations with (conven-tional) FE are performed using Abaqussstandard implicit simula-tions employing nine-node shell elements with reducedintegration (S9R5), which is shear flexible and able to modelmultiple layers [34]. In the numerical examples provided, theproperties of graphite-epoxy, AS4/3501 are taken as follows:E1¼144.48 GPa, E2¼E3¼9.63 GPa, G23¼G13¼G12¼4.128 GPa,ν23¼0.3, ν13¼ν12¼0.02, and ρ¼1389 kg m�3. Time domainresponses are studied using two types of input excitations, namelysinusoidal tone burst and impulse (Fig. 4).

Tone burst is essentially a windowed sinusoidal signal with alimited number of frequencies around the center frequency and itis narrow banded in the frequency domain. This type of input isused when a minimal number of frequencies are desired in theresponse to minimize wave dispersion. On the other hand, impulseexcitation consists of multiple frequencies (due to the broadbandnature in frequency domain) which cause multiple wave frontsbeing propagated through a waveguide. The spatial distribution ofthe input load along the Y-axis (in Fig. 2) is given by

FðYÞ ¼ e�ðY=αÞ2 ð30Þwhere α (as specified for each analysis) is a constant and can bevaried to change Y-distribution of the load. The 2D WSFE isformulated with Daubechies scaling function of order N¼22 fortemporal and N¼4 for spatial approximations. The time samplingrate is Δt¼1–2 ms (as specified for each analysis), while the spatialsampling rate ΔY is varied according to ΔY ¼ LY=ðm�1Þ depend-ing on LY and the number of spatial discretization points m.

4.1. Comparisons with FSFE results

Wave propagation analysis of finite length structures usingconventional spectral finite element analysis based on Fouriertransform (FSFE) requires the structures to be damped or use of athrow-off element to dissipate energy out of the system [12,13].In addition, the time window should be large enough to removethe wraparound problem associated with FSFE. The time windowis dependent on the value of damping and the length of thestructure, and it is required to be higher for lightly damped shortlength structures. WSFE is completely free from these constraintsand the accuracy of solution is independent of such time windowrestrictions or energy dissipation by means of damping or throw-off elements [14]. Fig. 5 shows a composite plate (with transversecrack) considered for wave motion analysis using FSFE and thecurrent WSFE formulation. The plate considered here is semi-infinitely long into X direction and a crack is placed at L1¼0.5 mdistance away from the free end. Relatively large lateral spatialwindow (LY¼5 m) is chosen due to the Fourier series representa-tion of the spatial dimension in FSFE where finite lateral boundaryconditions cannot be imposed. A symmetric ply sequence of [0]10with each ply having a thickness of 1 mm is considered. Timesampling rate is set at Δt¼2 ms with the number of spatial

Fig. 3. Variation of θ with nondimensional Y along the crack (plate dimensionsLx¼Ly¼0.5 m, H¼0.01 m).

D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–3224

discretization points m¼64 (ΔY ¼ 5=64 m) and α¼ 0:1 in Eq. (30).Fig. 6 shows transverse velocities at the tip of the plate due to atone burst input load at the same location (edge AB in Fig. 5). Theincident tone burst appears first in the response and the secondwave packet with relatively smaller amplitude is the reflectionfrom the crack. As mentioned earlier, the FSFE solution is highlydependent upon analysis time window as illustrated in Fig. 6.Three time windows of 1024 μs, 2048 μs, and 4096 μs are used forFSFE solutions, whereas the time window of 1024 μs is sufficientto obtain accurate WSFE results. It is observed that forTw¼1024 μs, FSFE solution is highly distorted and spurious wavecomponents appear. The accuracy of FSFE results graduallyincrease with increasing time windows of 2048 μs (Fig. 6(b)) and

4096 μs (Fig. 6(c)). The FSFE prediction using the largest timewindow (Tw¼4096 μs) is free from spurious distortions andmatches WSFE results very well. Thus, the developed WSFEmethod yields substantial reduction in computational cost as thetime window is directly related to the system size.

4.2. Validation with conventional finite element simulations

The developed 2DWSFE model for transverse crack is validatedwith FE simulations using Abaquss standard implicit directsolver [34]. Fig. 7 shows the graphite-epoxy, AS4/3501, cantileverlaminate used in this numerical example. The dimensions of thelaminate are LX ¼ LY ¼ 0:5 m with a ply sequence of [0]10 whereeach ply has a thickness of 1 mm. Responses are presented for thetwo types of inputs, tone burst and unit impulse (shown in Fig. 4).The tone burst has a central frequency of 20 kHz, while the unitimpulse has duration of 50 μs and occurs between 100 and 150 μs,with a frequency content of 44 kHz. For WSFE analysis, timesampling rate isΔt¼1 ms with the number of spatial discretizationpoints m¼64 (ΔY ¼ 1=64 m). The input load is applied along edgeAB with the load variation given by Eq. (30) with α¼0.05.

The FE simulation in Abaquss exploits symmetry of theproblem about the centerline (parallel to edges AC and BD) andmodels half-plate. The model has 7688, 9-noded shell elements(S9R5) with the total number of nodes equal to 31,250 ensuringthe model captures all the modes at the given excitation fre-quency. The crack is modeled using 62 line spring elements (LS6)

Fig. 4. (a) Unit impulse and (b) 3.5 cycle tone burst input loads with their FFTs.

Fig. 5. Composite plate with infinite length into X direction (width¼LY, andthickness¼0.01 m).

D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32 25

developed for modeling part-through cracks in shells [34]. Thecrack depth is 50% of the laminate thickness (that is, 0.005 m).Fig. 8 shows the comparison of WSFE and FEM results at the tipand middle of the plate with the crack located in the middle ofplate (L1¼0.25 m) and the tone burst input. It is observed that theformulated 2D WSFE element with transverse crack captures thewave scattering due to the crack and the results match very wellwith FE simulations. For the response at the plate tip (Fig. 8(a)),the incident wave (the first wave packet) propagates through theplate and part of this forward propagating wave is reflected due tothe crack. This partial reflection is observed as the second wavepacket with relatively smaller amplitude. The rest of the incident

wave propagates beyond the crack and reflects completely fromthe 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 WSFEaccurately captures the incident wave followed by the fixedboundary reflection.

Fig. 9 shows the response of 2DWSFE with transverse crack hasexcellent match with FE simulation for the impulse input load. Forthe tip response (Fig. 9(a)), the incident impulse occurs between100 and 150 ms, gets partially reflected from the crack around400 ms, and the fixed boundary reflection starts to appear around700 ms. The broad-band response at the center of the plate (Fig. 9(b)) also shows excellent match with FE including the fixedboundary reflection. Computational efficiency is one key advan-tage of WSFE over conventional FEM. For the results presented inFig. 8, computational time for WSFE and FE (Abaquss) simulationswere 26 s and 4102 s, respectively (PC with 8-core, i7 processor).Thus, WSFE resulted in reduction of computational time by morethan 2 orders of magnitude, even though only half-plate wasmodeled 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 toidentify presence and location of cracks through wave scatteringdue to a crack. The cantilever plate and loads used in the previoussection are used for the numerical results given here. Fig. 10 shows

Fig. 6. Transverse velocity response at the tip due to 20 kHz tone burst input load applied at the same location: WSFE time window Tw¼1024 μs for all cases; FSFE timewindows (a) Tw¼1024 μs (b) Tw¼2048 μs (c) Tw¼4096 μs.

C

BD

LYA

LX

L1

Fig. 7. Uniform cantilever plate with a transverse crack (length¼LX, width¼LY, andthickness¼0.01 m).

D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–3226

the transverse velocities at the plate tip (which is also theexcitation point) due to a tone burst input. For the healthy case(Fig. 10(a)), the first wave packet is the incident wave and thesecond larger wave packet is the reflection from the fixedboundary which arrives 582 ms after the excitation. Based on thisarrival time, the group velocity of the flexural wave mode at thisexcitation frequency is calculated to be 1718 m/s. In the damagedplate response (Fig. 10(b)), there is an additional wave packet withsmaller amplitude, which appears soon after the incident wave.Comparing this with the healthy plate response, it can be inferredthat the additional wave packet is reflection from the crack.This reflected wave from the crack appears approximately 290 msafter the incident wave. With the group velocity data available(1718 m/s) the distance to the crack can now be determined to beð1718� 290E�06=2Þ ¼ 0:25 m. Therefore the crack location can bedetermined when time of flight of reflected wave is used alongwith group velocity data. Fig. 11 shows the response of the platefor the same tone burst input with crack placed at 0.15 m and0.35 m away from the tip (excitation point). In these responses too,the arrival time of additional wave packets indicate that theyoriginate from the crack. In addition, when the crack is placed veryclose to the tip (Fig. 11(b)), there are multiple reflections occurringbefore the fixed boundary reflection arrives.

Fig. 12 shows the response for an impulse input load at the tipwith crack placed at 0.15 m, 0.25 m, and 0.35 m from the excita-tion point. A consistent reflection of the input impulse from thecrack is observed for all three locations of the crack. As indicated in

the figure, the presence of crack does not change peak amplitudeor the group velocity at which the pulse is propagated. However,there is an additional reflection from the crack arriving beforereflection from the fixed boundary.

Incident wave

Crack reflection

Fixed boundary reflection

Fig. 8. Transverse velocity using 2D WSFE and FE (Abaquss) for 20 kHz tone burstinput 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, respec-tively, using a PC with 8-core, i7 processor.

Incident wave

Crack reflection

Fixed boundary reflection

Fig. 9. Transverse velocity using 2D WSFE and FE for impulse input applied at tipand 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.

D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32 27

4.4. Multiple cracks analysis

The dynamic stiffness matrix of WSFE can be assembled in thefrequency-wavenumber domain to study the presence of multiplecracks within a laminate. This assembling procedure is similar tothe conventional FE method, except that the frequency domainapproach is used [19,31]. A numerical example is presented here todemonstrate the ability of 2D WSFE method to simulate multiplecracks efficiently. Here a laminate with thickness and ply layupsequence different from above examples is used in order to showthe flexibility of the model to analyze more general situations. An8-ply graphite-epoxy, AS4/3501, laminate with a layup sequenceof [0/90]2S is considered. The laminate with two transverse cracks(Fig. 13(a)) is divided into two sub-laminates along a dashed lineas shown in Figs. 13(b) and (c). The placement of this dashed line issomewhat arbitrary given that it is somewhere in between the twocracks. The in-plane dimensions are LX1¼LX2¼0.5 m, andLY¼1.0 m while the thickness is kept at 0.008 m. The distance tocrack 1 along the X-axis from the origin of its local coordinatesystem (n1) is 0.375 m and the distance to crack 2 from n3 is0.25 m.

The dynamic stiffness matrix of each element (sub-laminate) isrepresented by Eq. (16). During assembly, nodes n2 and n3 coincideand the dynamic stiffness matrix for a laminate with two cracks isobtained. 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 byEq. (30) with α¼ 0:03. Fixed boundary conditions are applied atthe left edge of the laminate (node n1).

Fig. 14 shows the transverse velocity response at the tip oflaminate for three different cases: (a) healthy, (b) crack 1 present,and (c) both crack 1 and crack 2 present. For the healthy plate, theresponses show the excitation wave packet (point n4) and itsreflection from the fixed boundary (point n1), When crack 1 is

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 fromthe tip.

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) 19–3228

introduced, there is an additional reflection arriving from thecrack. When both cracks are present, distinct responses fromthe two cracks are clearly observed (Fig. 14(c)). Fig. 15 shows the

transverse velocity response of each material point on the lami-nate along a straight line connecting nodes n4 and n1. The inputtone burst excited at the tip starts propagating towards the left as

Crack 2 Crack 1

Crack 1

Crack 2

1

2

3

1n

2n

1

2

3

3n

4n

X1LX2L

X

YZ

YL

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.

Fig. 15. Flexural wave propagation in composite laminate with two cracks atvarious time instants: (a) 190 μs, (b) 320 μs, and (c) 540 μs.

D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–32 29

shown in Fig. 15(a). This wave packet interacts with crack 1 result-ing in wave transmission and reflection due to the nature of theboundary created by the crack (Fig. 15(b)). The transmitted wavecontinues propagating towards the left, while the reflected waveheads towards the tip of the plate. The transmitted portion of theforward propagating wave is again scattered by crack 2 (Fig. 15(c)).The propagation direction of the incident wave packet is indicatedby a solid arrow, while the reflected wave packet is indicatedby a dashed arrow with a number to identify the crack causingthe reflection. Following the same procedure, even morecomplex situations with multiple cracks may be simulated usingWSFE, without much increase in computational time. Fig. 16visualizes the entire laminate for the same case (as discussed inFig. 15), showing transverse wave propagation and scatteringdue to the two cracks. Reflections from the lateral boundariesare observed in Fig. 16(c), which is a clear advantage of WSFEover FSFE.

5. Concluding remarks

A 2-D wavelet spectral finite element (WSFE) was presented foraccurate modeling of high frequency wave propagation in aniso-tropic composite laminates with a transverse, non-propagatingcrack. The governing PDEs based on FSDT for laminated compositeplates were used accounting for transverse shear strains androtary inertia to produce accurate results for wave motion at highfrequencies. Daubechies compactly supported, orthonormal scal-ing functions were employed for temporal and spatial approxima-tions of the governing PDEs and the resulting equations weresolved exactly to obtain the shape functions used in the elementformulation. Since the developed element captured the exactinertial distribution, a single element was sufficient to model alaminate of any dimension in the absence of discontinuities.Further, due to the compact nature of Daubechies scaling func-tions, the WSFE was able to model finite waveguides without thesignal ‘wraparound’ problem (associated with Fourier transformbased spectral finite element). A transverse crack having anarbitrary length, depth, and located parallel to any one side ofthe plate was modeled in the wavenumber–frequency domain byintroducing bending flexibility of the plate along the crack edge.The bending flexibility of the plate with crack was presented as afunction of crack parameters, that is, crack length and depth at thecenter. Numerical examples using sinusoidal tone burst andbroadband unit impulse excitations showed excellent agreementwith conventional finite element simulations using shear flexibleelements in Abaquss. WSFE resulted in reduction of computa-tional time by more than 2 orders of magnitude, even though onlyhalf-plate was modeled in Abaquss using symmetry of theproblem. WSFE modeling of transverse crack showed superiorityover the FSFE results which showed signal distortions due to the“wrap around” problem for shorter time windows. To modelmultiple cracks within a laminate, the dynamic stiffness matrixof elements in the frequency-wavenumber domain wereassembled, similar to the conventional finite element method.Time of flight estimations of waves reflected from cracks weresuccessfully used to determine crack locations.

Acknowledgment

The authors gratefully acknowledge partial funding for thisresearch through AFOSR Grant no. FA9550-09-1-0275 (ProgramManagers – Dr. Victor Giurgiutiu and Dr. David Stargel).

Appendix

Matrices for polynomial eigenvalue problem (Eq. (13))

A2 ¼

�A11 0 0 0 00 �A66 0 0 00 0 �KA55 0 00 0 0 �D11 00 0 0 0 �D66

26666664

37777775

A1 ¼

0 �βðA12þA66Þ 0 0 0�βðA12þA66Þ 0 0 0 0

0 0 0 � iKA55 00 0 iKA55 0 �βðD12þD66Þ0 0 0 �βðD12þD66Þ 0

26666664

37777775

Fig. 16. Full field view of flexural wave propagation in composite laminate withtwo cracks: (a) 190 μs, (b) 310 μs, and (c) 520 μs.

D. Samaratunga et al. / Finite Elements in Analysis and Design 89 (2014) 19–3230

Boundary condition matrices (Eq. (22)).

M1 ¼R11 R12 R13 R14e� ik1L1 R15e� ik2L1 R16e� ik3L1

R21 R22 R23 R24e� ik1L1 R25e� ik2L1 R26e� ik3L1

R31 R32 R33 R34e� ik1L1 R35e� ik2L1 R36e� ik3L1

0B@

1CA

M4 ¼R11e� ik1L R12e� ik2L R13e� ik3L R14 R15 R16

R21e� ik1L R22e� ik2L R23e� ik3L R24 R25 R26

R31e� ik1L R32e� ik2L R33e� ik3L R34 R35 R36

0B@

1CA

M2ð1;1 : 3Þ ¼ R1;1:3e� ik1:3L1

M2ð1;4 : 6Þ ¼ R1;4:3

M2ð2;1 : 3Þ ¼ ðR2;1:3þ iθðk1:3D11R2;1:3þβD12R3;1:3ÞÞe� ik1:3L1

M2ð2;4 : 6Þ ¼ ðR2;4:6þ iθð�k1:3D11R2;4:6þβD12R3;4:6ÞÞM2ð3;1 : 3Þ ¼ R3;1:3e� ik1:3L1

M2ð3;4 : 6Þ ¼ R3;4:3

M2ð4;1 : 3Þ ¼ �A55ðik1:3R1;1:3�R2;1:3Þe� ik1:3L1

M2ð4;4 : 6Þ ¼ A55ðik1:3R1;4:6þR2;4:6ÞM2ð5;1 : 3Þ ¼ � iðk1:3D11R2;1:3þβD12R3;1:3Þe� ik1:3L1

M2ð5;4 : 6Þ ¼ iðk1:3D11R2;4:6�βD12R3;4:6ÞM2ð6;1 : 3Þ ¼ � iD66ðk1:3R3;1:3þβR2;1:3Þe� ik1:3L1

M2ð6;4 : 6Þ ¼ iD66ðk1:3R3;4:6�βR2;4:6Þ

M3ð1;1 : 3Þ ¼ �R1;1:3e� ik1:3L1

M3ð1;4 : 6Þ ¼ �R1;4:3e� ik1:3ðL�L1Þ

M3ð2;1 : 3Þ ¼ �R2;1:3e� ik1:3L1

M3ð2;4 : 6Þ ¼ �R2;4:6e� ik1:3ðL�L1Þ

M3ð3;1 : 3Þ ¼ R3;1:3e� ik1:3L1

M3ð3;4 : 6Þ ¼ R3;4:3e� ik1:3ðL� L1Þ

M3ð4;1 : 3Þ ¼ A55ðik1:3R1;1:3�R2;1:3Þe� ik1:3L1

M3ð4;4 : 6Þ ¼ �A55ðik1:3R1;4:6þR2;4:6Þe� ik1:3ðL� L1Þ

M3ð5;1 : 3Þ ¼ iðk1:3D11R2;1:3þβD12R3;1:3Þe� ik1:3L1

M3ð5;4 : 6Þ ¼ ið�k1:3D11R2;4:6þβD12R3;4:6Þe� ik1:3ðL� L1Þ

M3ð6;1 : 3Þ ¼ iD66ðk1:3R3;1:3þβR2;1:3Þe� ik1:3L1

M3ð6;4 : 6Þ ¼ iD66ð�k1:3R3;4:6þβR2;4:6Þe� ik1:3ðL�L1Þ

P1ð1;1 : 3Þ ¼ �A55ð� ik1:3R1;1:3þR2;1:3ÞP1ð1;4 : 6Þ ¼ �A55ðik1:3R1;4:6þR2;4:6Þe� ik1:3L1

P1ð2;1 : 3Þ ¼ iðk1:3D11R2;1:3þβD12R3;1:3ÞP1ð2;4 : 6Þ ¼ ið�k1:3D11R2;4:6þβD12R3;4:6Þe� ik1:3L1

P1ð3;1 : 3Þ ¼ iD66ðk1:3R3;1:3þβR2;1:3ÞP1ð3;4 : 6Þ ¼ iD66ð�k1:3R3;4:6þβR2;4:6Þe� ik1:3L1

P2ð1;1 : 3Þ ¼ A55ð� ik1:3R1;1:3þR2;1:3Þe� ik1:3L

P2ð1;4 : 6Þ ¼ A55ðik1:3R1;4:6þR2;4:6ÞP2ð2;1 : 3Þ ¼ � iðk1:3D11R2;1:3þβD12R3;1:3Þe� ik1:3L

P2ð2;4 : 6Þ ¼ � ið�k1:3D11R2;4:6þβD12R3;4:6ÞP2ð3;1 : 3Þ ¼ � iD66ðk1:3R3;1:3þβR2;1:3Þe� ik1:3L

P2ð3;4 : 6Þ ¼ � iD66ð�k1:3R3;4:6þβR2;4:6Þ

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